TPTP Problem File: SLH0254^1.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_Calc2/0018_EPathHintikka/prob_00639_026803__13293458_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1782 ( 502 unt; 503 typ;   0 def)
%            Number of atoms       : 4004 ( 978 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 12764 ( 393   ~;  62   |; 322   &;10090   @)
%                                         (   0 <=>;1897  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :   97 (  96 usr)
%            Number of type conns  : 1968 (1968   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  410 ( 407 usr;  30 con; 0-4 aty)
%            Number of variables   : 3987 ( 229   ^;3530   !; 228   ?;3987   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 08:45:51.383
%------------------------------------------------------------------------------
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thf(ty_n_t__Stream__Ostream_It__SeCaV__Ofm_J,type,
    stream_fm: $tType ).

thf(ty_n_t__Stream__Ostream_It__Nat__Onat_J,type,
    stream_nat: $tType ).

thf(ty_n_t__List__Olist_It__Prover__Orule_J,type,
    list_rule: $tType ).

thf(ty_n_t__FSet__Ofset_It__Prover__Orule_J,type,
    fset_rule: $tType ).

thf(ty_n_t__Set__Oset_It__Prover__Orule_J,type,
    set_rule: $tType ).

thf(ty_n_t__List__Olist_It__SeCaV__Otm_J,type,
    list_tm: $tType ).

thf(ty_n_t__List__Olist_It__SeCaV__Ofm_J,type,
    list_fm: $tType ).

thf(ty_n_t__FSet__Ofset_It__SeCaV__Otm_J,type,
    fset_tm: $tType ).

thf(ty_n_t__FSet__Ofset_It__SeCaV__Ofm_J,type,
    fset_fm: $tType ).

thf(ty_n_t__FSet__Ofset_It__Nat__Onat_J,type,
    fset_nat: $tType ).

thf(ty_n_t__Set__Oset_It__SeCaV__Otm_J,type,
    set_tm: $tType ).

thf(ty_n_t__Set__Oset_It__SeCaV__Ofm_J,type,
    set_fm: $tType ).

thf(ty_n_t__Prover__Orule,type,
    rule: $tType ).

thf(ty_n_t__SeCaV__Otm,type,
    tm: $tType ).

thf(ty_n_t__SeCaV__Ofm,type,
    fm: $tType ).

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

% Explicit typings (407)
thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__List__Olist_It__SeCaV__Ofm_J_001t__Prover__Orule,type,
    abstra1396063136455076235m_rule: ( list_fm > rule > fset_rule > $o ) > stream_list_fm > set_rule > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Prover__Orule_001t__List__Olist_It__SeCaV__Ofm_J,type,
    abstra3805608168546402149ist_fm: ( rule > list_fm > fset_list_fm > $o ) > stream_rule > set_list_fm > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra3967921542344661089ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > set_Pr5202636777678657877ist_fm > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Prover__Orule_001t__Prover__Orule,type,
    abstra2971612018481630886e_rule: ( rule > rule > fset_rule > $o ) > stream_rule > set_rule > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Prover__Orule_001t__SeCaV__Ofm,type,
    abstra3960359434150717791ule_fm: ( rule > fm > fset_fm > $o ) > stream_rule > set_fm > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Prover__Orule_001t__SeCaV__Otm,type,
    abstra3960359434151636177ule_tm: ( rule > tm > fset_tm > $o ) > stream_rule > set_tm > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__SeCaV__Ofm_001t__Prover__Orule,type,
    abstra6710624115968083717m_rule: ( fm > rule > fset_rule > $o ) > stream_fm > set_rule > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
    abstra1085421490916510144_fm_fm: ( fm > fm > fset_fm > $o ) > stream_fm > set_fm > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
    abstra1085421490917428530_fm_tm: ( fm > tm > fset_tm > $o ) > stream_fm > set_tm > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__SeCaV__Otm_001t__Prover__Orule,type,
    abstra6151302760690159735m_rule: ( tm > rule > fset_rule > $o ) > stream_tm > set_rule > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
    abstra5645505808712133326_tm_fm: ( tm > fm > fset_fm > $o ) > stream_tm > set_fm > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
    abstra5645505808713051712_tm_tm: ( tm > tm > fset_tm > $o ) > stream_tm > set_tm > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem__axioms_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra9069655090277518462ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > set_Pr5202636777678657877ist_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__List__Olist_It__SeCaV__Ofm_J_001t__Prover__Orule,type,
    abstra6744069043672362530m_rule: ( list_fm > rule > fset_rule > $o ) > stream_list_fm > set_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
    abstra6353961794954871768ist_fm: ( produc340336539035504054m_rule > list_fm > fset_list_fm > $o ) > stream2709947120125613254m_rule > set_list_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra406286406349174356ist_fm: ( produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream2709947120125613254m_rule > set_Pr5202636777678657877ist_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Prover__Orule,type,
    abstra7792261008698710771e_rule: ( produc340336539035504054m_rule > rule > fset_rule > $o ) > stream2709947120125613254m_rule > set_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__SeCaV__Ofm,type,
    abstra6746019703889232722ule_fm: ( produc340336539035504054m_rule > fm > fset_fm > $o ) > stream2709947120125613254m_rule > set_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__SeCaV__Otm,type,
    abstra6746019703890151108ule_tm: ( produc340336539035504054m_rule > tm > fset_tm > $o ) > stream2709947120125613254m_rule > set_tm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Prover__Orule_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    abstra8554342375581516207m_rule: ( rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o ) > stream_rule > set_Ab3010718383674909480m_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Prover__Orule_001t__List__Olist_It__SeCaV__Ofm_J,type,
    abstra9153614075763688444ist_fm: ( rule > list_fm > fset_list_fm > $o ) > stream_rule > set_list_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra5221733350967904376ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > set_Pr5202636777678657877ist_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Prover__Orule_001t__Prover__Orule,type,
    abstra6805424310982398927e_rule: ( rule > rule > fset_rule > $o ) > stream_rule > set_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Prover__Orule_001t__SeCaV__Ofm,type,
    abstra4909020524819817846ule_fm: ( rule > fm > fset_fm > $o ) > stream_rule > set_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Prover__Orule_001t__SeCaV__Otm,type,
    abstra4909020524820736232ule_tm: ( rule > tm > fset_tm > $o ) > stream_rule > set_tm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__SeCaV__Ofm_001t__Prover__Orule,type,
    abstra7659285206637183772m_rule: ( fm > rule > fset_rule > $o ) > stream_fm > set_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
    abstra966159493038840937_fm_fm: ( fm > fm > fset_fm > $o ) > stream_fm > set_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
    abstra966159493039759323_fm_tm: ( fm > tm > fset_tm > $o ) > stream_fm > set_tm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__SeCaV__Otm_001t__Prover__Orule,type,
    abstra7099963851359259790m_rule: ( tm > rule > fset_rule > $o ) > stream_tm > set_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
    abstra5526243810834464119_tm_fm: ( tm > fm > fset_fm > $o ) > stream_tm > set_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
    abstra5526243810835382505_tm_tm: ( tm > tm > fset_tm > $o ) > stream_tm > set_tm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__List__Olist_It__SeCaV__Ofm_J_001t__Prover__Orule,type,
    abstra6915028855912523229m_rule: ( list_fm > rule > fset_rule > $o ) > stream_list_fm > rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
    abstra3212481807891478163ist_fm: ( produc340336539035504054m_rule > list_fm > fset_list_fm > $o ) > stream2709947120125613254m_rule > list_fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra7932678668315420303ist_fm: ( produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream2709947120125613254m_rule > produc6018962875968178549ist_fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Prover__Orule,type,
    abstra898225168387205880e_rule: ( produc340336539035504054m_rule > rule > fset_rule > $o ) > stream2709947120125613254m_rule > rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__SeCaV__Ofm,type,
    abstra5959417719433056141ule_fm: ( produc340336539035504054m_rule > fm > fset_fm > $o ) > stream2709947120125613254m_rule > fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__SeCaV__Otm,type,
    abstra5959417719433974527ule_tm: ( produc340336539035504054m_rule > tm > fset_tm > $o ) > stream2709947120125613254m_rule > tm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Prover__Orule_001t__List__Olist_It__SeCaV__Ofm_J,type,
    abstra101201851149073335ist_fm: ( rule > list_fm > fset_list_fm > $o ) > stream_rule > list_fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra1963046427795717555ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > produc6018962875968178549ist_fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Prover__Orule_001t__Prover__Orule,type,
    abstra538998172947062484e_rule: ( rule > rule > fset_rule > $o ) > stream_rule > rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Prover__Orule_001t__SeCaV__Ofm,type,
    abstra3486989597108156081ule_fm: ( rule > fm > fset_fm > $o ) > stream_rule > fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Prover__Orule_001t__SeCaV__Otm,type,
    abstra3486989597109074467ule_tm: ( rule > tm > fset_tm > $o ) > stream_rule > tm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__SeCaV__Ofm_001t__Prover__Orule,type,
    abstra6237254278925522007m_rule: ( fm > rule > fset_rule > $o ) > stream_fm > rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
    abstra2524456144953868014_fm_fm: ( fm > fm > fset_fm > $o ) > stream_fm > fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
    abstra2524456144954786400_fm_tm: ( fm > tm > fset_tm > $o ) > stream_fm > tm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__SeCaV__Otm_001t__Prover__Orule,type,
    abstra5677932923647598025m_rule: ( tm > rule > fset_rule > $o ) > stream_tm > rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
    abstra7084540462749491196_tm_fm: ( tm > fm > fset_fm > $o ) > stream_tm > fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
    abstra7084540462750409582_tm_tm: ( tm > tm > fset_tm > $o ) > stream_tm > tm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__List__Olist_It__SeCaV__Ofm_J_001t__Prover__Orule,type,
    abstra2231175174020358375m_rule: ( list_fm > rule > fset_rule > $o ) > stream_list_fm > set_rule > list_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra5255361903751151037ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > set_Pr5202636777678657877ist_fm > rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Prover__Orule_001t__Prover__Orule,type,
    abstra138634490569309770e_rule: ( rule > rule > fset_rule > $o ) > stream_rule > set_rule > rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Prover__Orule_001t__SeCaV__Ofm,type,
    abstract_per_rule_fm: ( rule > fm > fset_fm > $o ) > stream_rule > set_fm > rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__Prover__Orule_001t__SeCaV__Otm,type,
    abstract_per_rule_tm: ( rule > tm > fset_tm > $o ) > stream_rule > set_tm > rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__SeCaV__Ofm_001t__Prover__Orule,type,
    abstract_per_fm_rule: ( fm > rule > fset_rule > $o ) > stream_fm > set_rule > fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
    abstract_per_fm_fm: ( fm > fm > fset_fm > $o ) > stream_fm > set_fm > fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
    abstract_per_fm_tm: ( fm > tm > fset_tm > $o ) > stream_fm > set_tm > fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__SeCaV__Otm_001t__Prover__Orule,type,
    abstract_per_tm_rule: ( tm > rule > fset_rule > $o ) > stream_tm > set_rule > tm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
    abstract_per_tm_fm: ( tm > fm > fset_fm > $o ) > stream_tm > set_fm > tm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Oper_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
    abstract_per_tm_tm: ( tm > tm > fset_tm > $o ) > stream_tm > set_tm > tm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_001t__List__Olist_It__SeCaV__Ofm_J,type,
    abstract_pos_list_fm: stream_list_fm > list_fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    abstra4499547390127564210m_rule: stream2709947120125613254m_rule > produc340336539035504054m_rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_001t__Prover__Orule,type,
    abstract_pos_rule: stream_rule > rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_001t__SeCaV__Ofm,type,
    abstract_pos_fm: stream_fm > fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_001t__SeCaV__Otm,type,
    abstract_pos_tm: stream_tm > tm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_OSaturated_001t__Prover__Orule_001t__List__Olist_It__SeCaV__Ofm_J,type,
    abstra3959174058192036855ist_fm: ( rule > list_fm > fset_list_fm > $o ) > stream_rule > stream1460591043787736522m_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_OSaturated_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra6097777249025082867ist_fm: ( rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ) > stream_rule > stream2709947120125613254m_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_OSaturated_001t__Prover__Orule_001t__Prover__Orule,type,
    abstra8307935604394493588e_rule: ( rule > rule > fset_rule > $o ) > stream_rule > stream4385846686851721995e_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_OSaturated_001t__Prover__Orule_001t__SeCaV__Ofm,type,
    abstra2738778320242900721ule_fm: ( rule > fm > fset_fm > $o ) > stream_rule > stream6755863580823097540m_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem__Defs_OSaturated_001t__Prover__Orule_001t__SeCaV__Otm,type,
    abstra2738778320243819107ule_tm: ( rule > tm > fset_tm > $o ) > stream_rule > stream3258364598272819510m_rule > $o ).

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thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    fmembe5560845258961799315m_rule: abstra1818967566604834586m_rule > fset_A1018878633012886730m_rule > $o ).

thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__Prover__Orule_J_J,type,
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thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    fmembe1099044791951439339m_rule: abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o ).

thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    fmembe3420198702065081546m_rule: abstra8527173362768188419m_rule > fset_A9163128795023232105m_rule > $o ).

thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_J,type,
    fmembe6649473619981193638e_rule: abstra1868548086920761311e_rule > fset_A784838278677503813e_rule > $o ).

thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    fmembe3410169904382534669m_rule: abstra4034898751840576020m_rule > fset_A4040246727766634820m_rule > $o ).

thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__Prover__Orule_J_J,type,
    fmembe1361850161138308585m_rule: abstra781890097549642992m_rule > fset_A5145833822123714848m_rule > $o ).

thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__SeCaV__Otm_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    fmembe3189649771678221951m_rule: abstra3814378619136263302m_rule > fset_A5153381644976994998m_rule > $o ).

thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__SeCaV__Otm_Mt__Prover__Orule_J_J,type,
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thf(sy_c_FSet_Ofmember_001t__Abstract____Completeness__Otree_It__Prover__Orule_J,type,
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thf(sy_c_FSet_Ofmember_001t__List__Olist_It__SeCaV__Ofm_J,type,
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thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(sy_c_FSet_Ofmember_001t__Prover__Orule,type,
    fmember_rule: rule > fset_rule > $o ).

thf(sy_c_FSet_Ofmember_001t__SeCaV__Ofm,type,
    fmember_fm: fm > fset_fm > $o ).

thf(sy_c_FSet_Ofmember_001t__SeCaV__Otm,type,
    fmember_tm: tm > fset_tm > $o ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    comp_P5969275118686715022m_rule: ( produc340336539035504054m_rule > produc6018962875968178549ist_fm ) > ( abstra8520858721168826610m_rule > produc340336539035504054m_rule ) > abstra8520858721168826610m_rule > produc6018962875968178549ist_fm ).

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__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    minus_4573455687935125755m_rule: fset_A8849196716589201442m_rule > fset_A8849196716589201442m_rule > fset_A8849196716589201442m_rule ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    minus_7547787838945083330ist_fm: fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_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__List__Olist_It__SeCaV__Ofm_J_J,type,
    minus_639611354763871680ist_fm: set_list_fm > set_list_fm > set_list_fm ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    minus_7584274679010293948ist_fm: set_Pr5202636777678657877ist_fm > set_Pr5202636777678657877ist_fm > set_Pr5202636777678657877ist_fm ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Prover__Orule_J,type,
    minus_minus_set_rule: set_rule > set_rule > set_rule ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    minus_minus_set_fm: set_fm > set_fm > set_fm ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__SeCaV__Otm_J,type,
    minus_minus_set_tm: set_tm > set_tm > set_tm ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

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

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Hintikka_Oterms,type,
    terms: set_fm > set_tm ).

thf(sy_c_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    if_Abs1293773526126661548m_rule: $o > abstra8520858721168826610m_rule > abstra8520858721168826610m_rule > abstra8520858721168826610m_rule ).

thf(sy_c_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    if_fse9105678332773368284m_rule: $o > fset_A8849196716589201442m_rule > fset_A8849196716589201442m_rule > fset_A8849196716589201442m_rule ).

thf(sy_c_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    if_fse7999432387889793441ist_fm: $o > fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm ).

thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    if_Pro4760001780252510779ist_fm: $o > produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > produc6018962875968178549ist_fm ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Linear__Temporal__Logic__on__Streams_Oalw_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    linear4957294957385550906m_rule: ( stream2709947120125613254m_rule > $o ) > stream2709947120125613254m_rule > $o ).

thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    linear1816559245158234865m_rule: ( stream2709947120125613254m_rule > $o ) > stream2709947120125613254m_rule > $o ).

thf(sy_c_Linear__Temporal__Logic__on__Streams_Oholds_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    linear7925859179493631106m_rule: ( produc340336539035504054m_rule > $o ) > stream2709947120125613254m_rule > $o ).

thf(sy_c_List_Oappend_001t__SeCaV__Otm,type,
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thf(sy_c_List_Olist_Olist__all_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    list_a9038620553111681026m_rule: ( produc340336539035504054m_rule > $o ) > list_P2774625669004309958m_rule > $o ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Ofm_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Prover__Orule,type,
    set_rule2: list_rule > set_rule ).

thf(sy_c_List_Olist_Oset_001t__SeCaV__Ofm,type,
    set_fm2: list_fm > set_fm ).

thf(sy_c_List_Olist_Oset_001t__SeCaV__Otm,type,
    set_tm2: list_tm > set_tm ).

thf(sy_c_List_Oremdups_001t__SeCaV__Otm,type,
    remdups_tm: list_tm > list_tm ).

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
    bot_bot_list_fm_o: list_fm > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_M_Eo_J,type,
    bot_bo3088342962497934428t_fm_o: produc6018962875968178549ist_fm > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Prover__Orule_M_Eo_J,type,
    bot_bot_rule_o: rule > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__SeCaV__Ofm_M_Eo_J,type,
    bot_bot_fm_o: fm > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__SeCaV__Otm_M_Eo_J,type,
    bot_bot_tm_o: tm > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    bot_bo3860209539026414262m_rule: fset_A8849196716589201442m_rule ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    bot_bo6461889142629771335ist_fm: fset_P8989946509869081563ist_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__List__Olist_It__SeCaV__Ofm_J_J,type,
    bot_bot_set_list_fm: set_list_fm ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    bot_bo1798318783834493889ist_fm: set_Pr5202636777678657877ist_fm ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    bot_bo3426410157013319936m_rule: set_Pr1822751329126368876m_rule ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Prover__Orule_J,type,
    bot_bot_set_rule: set_rule ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    bot_bot_set_fm: set_fm ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__SeCaV__Otm_J,type,
    bot_bot_set_tm: set_tm ).

thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    ord_le6302656846121046646m_rule: fset_A8849196716589201442m_rule > fset_A8849196716589201442m_rule > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    ord_le7716396445699002503ist_fm: fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_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__List__Olist_It__SeCaV__Ofm_J_J,type,
    ord_less_set_list_fm: set_list_fm > set_list_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    ord_le9141428925376846849ist_fm: set_Pr5202636777678657877ist_fm > set_Pr5202636777678657877ist_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Prover__Orule_J,type,
    ord_less_set_rule: set_rule > set_rule > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    ord_less_set_fm: set_fm > set_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__SeCaV__Otm_J,type,
    ord_less_set_tm: set_tm > set_tm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
    ord_le59936386403465346m_rule: fset_A8849196716589201442m_rule > fset_A8849196716589201442m_rule > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    ord_le3986950534092794747ist_fm: fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_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__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    ord_le7838213414353715577ist_fm: set_list_fm > set_list_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    ord_le1771420097867575541ist_fm: set_Pr5202636777678657877ist_fm > set_Pr5202636777678657877ist_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__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    ord_le6390412330253371084m_rule: set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule > $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__Prover__Orule_J,type,
    ord_less_eq_set_rule: set_rule > set_rule > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    ord_less_eq_set_fm: set_fm > set_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Otm_J,type,
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thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    stl_Pr1506262294867171877ist_fm: stream4408948924543953275ist_fm > stream4408948924543953275ist_fm ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    stl_Pr12655793849929990m_rule: stream2709947120125613254m_rule > stream2709947120125613254m_rule ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    stl_Pr2679468901532635773m_rule: stream6210534828274662995m_rule > stream6210534828274662995m_rule ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Prover__Orule_J,type,
    stl_Pr497917230009538393e_rule: stream8099677779113257519e_rule > stream8099677779113257519e_rule ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    stl_Pr8529079933065201753m_rule: stream8953843411776101167m_rule > stream8953843411776101167m_rule ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
    stl_Pr2595822214614905141e_rule: stream4385846686851721995e_rule > stream4385846686851721995e_rule ).

thf(sy_c_Stream_Ostream_Ostl_001t__Prover__Orule,type,
    stl_rule: stream_rule > stream_rule ).

thf(sy_c_Stream_Ostream_Ostl_001t__SeCaV__Ofm,type,
    stl_fm: stream_fm > stream_fm ).

thf(sy_c_Stream_Ostream_Ostl_001t__SeCaV__Otm,type,
    stl_tm: stream_tm > stream_tm ).

thf(sy_c_Stream_Oszip_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
    szip_list_tm_list_fm: stream_list_tm > stream_list_fm > stream4408948924543953275ist_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__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
    szip_P2924820683901490861m_rule: stream4408948924543953275ist_fm > stream_rule > stream2709947120125613254m_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__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    szip_P811719526838699976m_rule: stream2709947120125613254m_rule > stream2709947120125613254m_rule > stream6210534828274662995m_rule ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Prover__Orule,type,
    szip_P2044787555563963556e_rule: stream2709947120125613254m_rule > stream_rule > stream8099677779113257519e_rule ).

thf(sy_c_Stream_Oszip_001t__Prover__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    szip_r6533552703345879972m_rule: stream_rule > stream2709947120125613254m_rule > stream8953843411776101167m_rule ).

thf(sy_c_Stream_Oszip_001t__Prover__Orule_001t__Prover__Orule,type,
    szip_rule_rule: stream_rule > stream_rule > stream4385846686851721995e_rule ).

thf(sy_c_member_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
    member5160799505749407497m_rule: abstra8520858721168826610m_rule > set_Ab3010718383674909480m_rule > $o ).

thf(sy_c_member_001t__List__Olist_It__SeCaV__Ofm_J,type,
    member_list_fm: list_fm > set_list_fm > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    member4699826688122452638ist_fm: produc6018962875968178549ist_fm > set_Pr5202636777678657877ist_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__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
    member7231649785386036813m_rule: produc340336539035504054m_rule > set_Pr1822751329126368876m_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__Prover__Orule,type,
    member_rule: rule > set_rule > $o ).

thf(sy_c_member_001t__SeCaV__Ofm,type,
    member_fm: fm > set_fm > $o ).

thf(sy_c_member_001t__SeCaV__Otm,type,
    member_tm: tm > set_tm > $o ).

thf(sy_v_k____,type,
    k: nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_n____,type,
    n: nat ).

thf(sy_v_p____,type,
    p: fm ).

thf(sy_v_pre____,type,
    pre: list_P2774625669004309958m_rule ).

thf(sy_v_steps,type,
    steps: stream2709947120125613254m_rule ).

thf(sy_v_suf____,type,
    suf: stream2709947120125613254m_rule ).

thf(sy_v_t____,type,
    t: tm ).

% Relevant facts (1265)
thf(fact_0_assms_I1_J,axiom,
    abstra6789711989322986974ist_fm @ eff @ rules @ steps ).

% assms(1)
thf(fact_1_assms_I2_J,axiom,
    abstra6097777249025082867ist_fm @ eff @ rules @ steps ).

% assms(2)
thf(fact_2_RuleSystem__Defs_Oepath_Ocong,axiom,
    abstra6789711989322986974ist_fm = abstra6789711989322986974ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_3_epath__sdrop,axiom,
    ! [Steps: stream2709947120125613254m_rule,N: nat] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( abstra6789711989322986974ist_fm @ eff @ rules @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ).

% epath_sdrop
thf(fact_4_wf__ipath__epath,axiom,
    ! [T: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra8167393772827826667ist_fm @ eff @ rules @ T )
     => ( ( abstra8836096083317999815m_rule @ T @ Steps )
       => ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps ) ) ) ).

% wf_ipath_epath
thf(fact_5_secavProver__def,axiom,
    ( secavProver
    = ( abstra7840189589880272446ist_fm @ eff @ rules ) ) ).

% secavProver_def
thf(fact_6_Saturated__sdrop,axiom,
    ! [Steps: stream2709947120125613254m_rule,N: nat] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
     => ( abstra6097777249025082867ist_fm @ eff @ rules @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ).

% Saturated_sdrop
thf(fact_7_PersistentRuleSystem__axioms,axiom,
    abstra3967921542344661089ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).

% PersistentRuleSystem_axioms
thf(fact_8__092_060open_062epath_A_Istl_A_Isdrop_Am_Asteps_J_J_092_060close_062,axiom,
    abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ ( sdrop_8169176516188972301m_rule @ m @ steps ) ) ).

% \<open>epath (stl (sdrop m steps))\<close>
thf(fact_9_RuleSystem__axioms,axiom,
    abstra5221733350967904376ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).

% RuleSystem_axioms
thf(fact_10_enabled__R,axiom,
    ! [S: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ? [X: rule] :
          ( ( member_rule @ X @ ( sset_rule @ rules ) )
          & ? [X_1: fset_P8989946509869081563ist_fm] : ( eff @ X @ S @ X_1 ) ) ) ).

% enabled_R
thf(fact_11_enabled__def,axiom,
    ! [R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
      = ( ? [X2: fset_P8989946509869081563ist_fm] : ( eff @ R @ S @ X2 ) ) ) ).

% enabled_def
thf(fact_12_fair__rules,axiom,
    abstra3799686578551160190r_rule @ rules @ rules ).

% fair_rules
thf(fact_13_PersistentRuleSystem_Oaxioms_I1_J,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm] :
      ( ( abstra3967921542344661089ist_fm @ Eff @ Rules @ S2 )
     => ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 ) ) ).

% PersistentRuleSystem.axioms(1)
thf(fact_14_i_Ofair__stl,axiom,
    ! [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule] :
      ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
     => ( abstra3665372904625986210m_rule @ Rules @ ( stl_Pr12655793849929990m_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_15_i_Ofair__stl,axiom,
    ! [Rules: stream_rule,Rs: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
     => ( abstra3799686578551160190r_rule @ Rules @ ( stl_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_16_i_Osdrop__fair,axiom,
    ! [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule,M: nat] :
      ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
     => ( abstra3665372904625986210m_rule @ Rules @ ( sdrop_8169176516188972301m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_17_i_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M: nat] :
      ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
     => ( abstra3799686578551160190r_rule @ Rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_18_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_rule] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_19_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S2: set_list_fm,S: list_fm] :
      ( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_list_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_20_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_tm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_21_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_22_RuleSystem_Oenabled__R,axiom,
    ! [Eff: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm] :
      ( ( abstra406286406349174356ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
            & ? [X_1: fset_P8989946509869081563ist_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_23_RuleSystem_Oenabled__R,axiom,
    ! [Eff: produc340336539035504054m_rule > rule > fset_rule > $o,Rules: stream2709947120125613254m_rule,S2: set_rule,S: rule] :
      ( ( abstra7792261008698710771e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
            & ? [X_1: fset_rule] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_24_RuleSystem_Oenabled__R,axiom,
    ! [Eff: produc340336539035504054m_rule > list_fm > fset_list_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_list_fm,S: list_fm] :
      ( ( abstra6353961794954871768ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
            & ? [X_1: fset_list_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_25_RuleSystem_Oenabled__R,axiom,
    ! [Eff: produc340336539035504054m_rule > tm > fset_tm > $o,Rules: stream2709947120125613254m_rule,S2: set_tm,S: tm] :
      ( ( abstra6746019703890151108ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
            & ? [X_1: fset_tm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_26_RuleSystem_Oenabled__R,axiom,
    ! [Eff: produc340336539035504054m_rule > fm > fset_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_fm,S: fm] :
      ( ( abstra6746019703889232722ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ? [X: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Rules ) )
            & ? [X_1: fset_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_27_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P8989946509869081563ist_fm] : ( Eff @ X @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_28_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra459280387713764508e_rule @ Eff @ Rules @ ( abstra2300331465871827593e_rule @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_29_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S2: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra2093743074600156143ist_fm @ Eff @ Rules @ ( abstra6869381243922927298ist_fm @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_30_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra1214582413480588891ule_tm @ Eff @ Rules @ ( abstra3300998584369155758ule_tm @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_31_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra1214582413479670505ule_fm @ Eff @ Rules @ ( abstra3300998584368237372ule_fm @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_32_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra8167393772827826667ist_fm @ Eff @ Rules @ ( abstra7840189589880272446ist_fm @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_33_RuleSystem__Defs_Owf_Ocong,axiom,
    abstra8167393772827826667ist_fm = abstra8167393772827826667ist_fm ).

% RuleSystem_Defs.wf.cong
thf(fact_34_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule] :
      ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
     => ( abstra3665372904625986210m_rule @ Rules @ ( stl_Pr12655793849929990m_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_35_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules: stream_rule,Rs: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
     => ( abstra3799686578551160190r_rule @ Rules @ ( stl_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_36_RuleSystem__Defs_Ofair_Ocong,axiom,
    abstra3799686578551160190r_rule = abstra3799686578551160190r_rule ).

% RuleSystem_Defs.fair.cong
thf(fact_37_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule,M: nat] :
      ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
     => ( abstra3665372904625986210m_rule @ Rules @ ( sdrop_8169176516188972301m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_38_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M: nat] :
      ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
     => ( abstra3799686578551160190r_rule @ Rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_39_RuleSystem__Defs_Oenabled__def,axiom,
    ( abstra1707737187183579335ist_fm
    = ( ^ [Eff2: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,R2: rule,S3: produc6018962875968178549ist_fm] :
        ? [X2: fset_P8989946509869081563ist_fm] : ( Eff2 @ R2 @ S3 @ X2 ) ) ) ).

% RuleSystem_Defs.enabled_def
thf(fact_40_RuleSystem__Defs_OmkTree_Ocong,axiom,
    abstra7840189589880272446ist_fm = abstra7840189589880272446ist_fm ).

% RuleSystem_Defs.mkTree.cong
thf(fact_41_RuleSystem__Defs_Oenabled_Ocong,axiom,
    abstra1707737187183579335ist_fm = abstra1707737187183579335ist_fm ).

% RuleSystem_Defs.enabled.cong
thf(fact_42_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream2709947120125613254m_rule,Steps: stream6742152135616053226m_rule,M: nat] :
      ( ( abstra406286406349174356ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra7926801140857401835m_rule @ ( abstra7211189027691227674ist_fm @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ S4 @ S2 )
                & ( abstra7926801140857401835m_rule @ ( abstra7211189027691227674ist_fm @ Eff @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_4865846802426565681m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_43_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc340336539035504054m_rule > rule > fset_rule > $o,Rules: stream2709947120125613254m_rule,S2: set_rule,S: rule,Rs: stream2709947120125613254m_rule,Steps: stream8953843411776101167m_rule,M: nat] :
      ( ( abstra7792261008698710771e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra7237224312902573144m_rule @ ( abstra5235605640745886637e_rule @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: rule] :
                ( ( member_rule @ S4 @ S2 )
                & ( abstra7237224312902573144m_rule @ ( abstra5235605640745886637e_rule @ Eff @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_746751008794826386m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_44_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc340336539035504054m_rule > list_fm > fset_list_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_list_fm,S: list_fm,Rs: stream2709947120125613254m_rule,Steps: stream8925945479932671214m_rule,M: nat] :
      ( ( abstra6353961794954871768ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra2822895227572703983m_rule @ ( abstra3177990756066841758ist_fm @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: list_fm] :
                ( ( member_list_fm @ S4 @ S2 )
                & ( abstra2822895227572703983m_rule @ ( abstra3177990756066841758ist_fm @ Eff @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_1184396275450712629m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_45_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc340336539035504054m_rule > tm > fset_tm > $o,Rules: stream2709947120125613254m_rule,S2: set_tm,S: tm,Rs: stream2709947120125613254m_rule,Steps: stream5028821900571319386m_rule,M: nat] :
      ( ( abstra6746019703890151108ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra7010922518557816411m_rule @ ( abstra57709018850439818ule_tm @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: tm] :
                ( ( member_tm @ S4 @ S2 )
                & ( abstra7010922518557816411m_rule @ ( abstra57709018850439818ule_tm @ Eff @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_6611066292851773601m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_46_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc340336539035504054m_rule > fm > fset_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_fm,S: fm,Rs: stream2709947120125613254m_rule,Steps: stream5249342033275632104m_rule,M: nat] :
      ( ( abstra6746019703889232722ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra2727389193241885417m_rule @ ( abstra57709018849521432ule_fm @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: fm] :
                ( ( member_fm @ S4 @ S2 )
                & ( abstra2727389193241885417m_rule @ ( abstra57709018849521432ule_fm @ Eff @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_2327532967535842607m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_47_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule,Steps: stream4385846686851721995e_rule,M: nat] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: rule] :
                ( ( member_rule @ S4 @ S2 )
                & ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_9113879250048157294e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_48_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S2: set_list_fm,S: list_fm,Rs: stream_rule,Steps: stream1460591043787736522m_rule,M: nat] :
      ( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra2882552375471187403m_rule @ ( abstra6869381243922927298ist_fm @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: list_fm] :
                ( ( member_list_fm @ S4 @ S2 )
                & ( abstra2882552375471187403m_rule @ ( abstra6869381243922927298ist_fm @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_779662454523830033m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_49_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule,Steps: stream3258364598272819510m_rule,M: nat] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra3841055140891909431m_rule @ ( abstra3300998584369155758ule_tm @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: tm] :
                ( ( member_tm @ S4 @ S2 )
                & ( abstra3841055140891909431m_rule @ ( abstra3300998584369155758ule_tm @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_3187986897935118717m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_50_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule,Steps: stream6755863580823097540m_rule,M: nat] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra4400376496169833413m_rule @ ( abstra3300998584368237372ule_fm @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: fm] :
                ( ( member_fm @ S4 @ S2 )
                & ( abstra4400376496169833413m_rule @ ( abstra3300998584368237372ule_fm @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_3747308253213042699m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_51_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule,M: nat] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ Eff @ Rs @ S ) @ Steps )
           => ? [N2: nat,S4: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ S4 @ S2 )
                & ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_8169176516188972301m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_52_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra6097777249025082867ist_fm = abstra6097777249025082867ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_53_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule,Steps: stream4385846686851721995e_rule] :
      ( ( abstra2971612018481630886e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff @ Rs @ S ) @ Steps )
           => ( abstra8307935604394493588e_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_54_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S2: set_list_fm,S: list_fm,Rs: stream_rule,Steps: stream1460591043787736522m_rule] :
      ( ( abstra3805608168546402149ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra2882552375471187403m_rule @ ( abstra6869381243922927298ist_fm @ Eff @ Rs @ S ) @ Steps )
           => ( abstra3959174058192036855ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_55_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule,Steps: stream3258364598272819510m_rule] :
      ( ( abstra3960359434151636177ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra3841055140891909431m_rule @ ( abstra3300998584369155758ule_tm @ Eff @ Rs @ S ) @ Steps )
           => ( abstra2738778320243819107ule_tm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_56_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule,Steps: stream6755863580823097540m_rule] :
      ( ( abstra3960359434150717791ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra4400376496169833413m_rule @ ( abstra3300998584368237372ule_fm @ Eff @ Rs @ S ) @ Steps )
           => ( abstra2738778320242900721ule_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_57_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra3967921542344661089ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ Eff @ Rs @ S ) @ Steps )
           => ( abstra6097777249025082867ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_58_wf__mkTree,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( abstra8167393772827826667ist_fm @ eff @ rules @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) ) ) ) ).

% wf_mkTree
thf(fact_59_ipath__mkTree__Saturated,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) @ Steps )
         => ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps ) ) ) ) ).

% ipath_mkTree_Saturated
thf(fact_60_RuleSystem__Defs_Owf__ipath__epath,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,T: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra8167393772827826667ist_fm @ Eff @ Rules @ T )
     => ( ( abstra8836096083317999815m_rule @ T @ Steps )
       => ( abstra6789711989322986974ist_fm @ Eff @ Rules @ Steps ) ) ) ).

% RuleSystem_Defs.wf_ipath_epath
thf(fact_61_k,axiom,
    ? [K: nat] :
      ( ( ord_less_nat @ m @ K )
      & ( suf
        = ( sdrop_8169176516188972301m_rule @ K @ steps ) ) ) ).

% k
thf(fact_62_ipath__mkTree__saturated,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule,R: rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) @ Steps )
         => ( ( member_rule @ R @ ( sset_rule @ rules ) )
           => ( abstra2533313685312581075ist_fm @ eff @ R @ Steps ) ) ) ) ) ).

% ipath_mkTree_saturated
thf(fact_63_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: tm > rule > fset_rule > $o,Rules: stream_tm,S2: set_rule,S: rule,Rs: stream_tm,Steps: stream8415551234601779672ule_tm,R: tm] :
      ( ( abstra6151302760690159735m_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra6979500642109889529air_tm @ Rules @ Rs )
         => ( ( abstra7530814456727015129ule_tm @ ( abstra5491941910907679316m_rule @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_tm @ R @ ( sset_tm @ Rules ) )
             => ( abstra2914376109873778153m_rule @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_64_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S2: set_rule,S: rule,Rs: stream_fm,Steps: stream8658065904927289446ule_fm,R: fm] :
      ( ( abstra6710624115968083717m_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra6979500642108971143air_fm @ Rules @ Rs )
         => ( ( abstra7526862431641379943ule_fm @ ( abstra6051263266185603298m_rule @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_fm @ R @ ( sset_fm @ Rules ) )
             => ( abstra3473697465151702135m_rule @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_65_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: tm > tm > fset_tm > $o,Rules: stream_tm,S2: set_tm,S: tm,Rs: stream_tm,Steps: stream5748647146893912941_tm_tm,R: tm] :
      ( ( abstra5645505808713051712_tm_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra6979500642109889529air_tm @ Rules @ Rs )
         => ( ( abstra2363653106310321558_tm_tm @ ( abstra4750730535116555939_tm_tm @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_tm @ R @ ( sset_tm @ Rules ) )
             => ( abstra589508014764448590_tm_tm @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_66_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > tm > fset_tm > $o,Rules: stream_fm,S2: set_tm,S: tm,Rs: stream_fm,Steps: stream5991161817219422715_tm_fm,R: fm] :
      ( ( abstra1085421490917428530_fm_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra6979500642108971143air_fm @ Rules @ Rs )
         => ( ( abstra2359701081224686372_tm_fm @ ( abstra190646217320932757_fm_tm @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_fm @ R @ ( sset_fm @ Rules ) )
             => ( abstra5252795733823601216_fm_tm @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_67_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: tm > fm > fset_fm > $o,Rules: stream_tm,S2: set_fm,S: fm,Rs: stream_tm,Steps: stream7757019309515125087_fm_tm,R: tm] :
      ( ( abstra5645505808712133326_tm_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra6979500642109889529air_tm @ Rules @ Rs )
         => ( ( abstra7026940825369474184_fm_tm @ ( abstra4750730535115637553_tm_fm @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_tm @ R @ ( sset_tm @ Rules ) )
             => ( abstra589508014763530204_tm_fm @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_68_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S2: set_fm,S: fm,Rs: stream_fm,Steps: stream7999533979840634861_fm_fm,R: fm] :
      ( ( abstra1085421490916510144_fm_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra6979500642108971143air_fm @ Rules @ Rs )
         => ( ( abstra7022988800283838998_fm_fm @ ( abstra190646217320014371_fm_fm @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_fm @ R @ ( sset_fm @ Rules ) )
             => ( abstra5252795733822682830_fm_fm @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_69_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule,Steps: stream4385846686851721995e_rule,R: rule] :
      ( ( abstra2971612018481630886e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
             => ( abstra1228885326027523764e_rule @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_70_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule,Steps: stream3258364598272819510m_rule,R: rule] :
      ( ( abstra3960359434151636177ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra3841055140891909431m_rule @ ( abstra3300998584369155758ule_tm @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
             => ( abstra723432783335254595ule_tm @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_71_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule,Steps: stream6755863580823097540m_rule,R: rule] :
      ( ( abstra3960359434150717791ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra4400376496169833413m_rule @ ( abstra3300998584368237372ule_fm @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
             => ( abstra723432783334336209ule_fm @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_72_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: list_fm > rule > fset_rule > $o,Rules: stream_list_fm,S2: set_rule,S: rule,Rs: stream_list_fm,Steps: stream5422195452554605548ist_fm,R: list_fm] :
      ( ( abstra1396063136455076235m_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra7078398738616236685ist_fm @ Rules @ Rs )
         => ( ( abstra9062661029539366125ist_fm @ ( abstra4459836211831601384m_rule @ Eff @ Rs @ S ) @ Steps )
           => ( ( member_list_fm @ R @ ( sset_list_fm @ Rules ) )
             => ( abstra252965116582954493m_rule @ Eff @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_73_fair__uu__def,axiom,
    abstra4598247580560492629u_rule = abstra3799686578551160190r_rule ).

% fair_uu_def
thf(fact_74_ipath__mkTree__sdrop,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule,M: nat] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) @ Steps )
         => ? [N2: nat,S4: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S4 @ top_to1730629564055774885ist_fm )
              & ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ ( sdrop_rule @ N2 @ Rs ) @ S4 ) @ ( sdrop_8169176516188972301m_rule @ M @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_sdrop
thf(fact_75_UNIV__I,axiom,
    ! [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ).

% UNIV_I
thf(fact_76_UNIV__I,axiom,
    ! [X3: list_fm] : ( member_list_fm @ X3 @ top_top_set_list_fm ) ).

% UNIV_I
thf(fact_77_UNIV__I,axiom,
    ! [X3: tm] : ( member_tm @ X3 @ top_top_set_tm ) ).

% UNIV_I
thf(fact_78_UNIV__I,axiom,
    ! [X3: fm] : ( member_fm @ X3 @ top_top_set_fm ) ).

% UNIV_I
thf(fact_79_UNIV__I,axiom,
    ! [X3: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X3 @ top_to1730629564055774885ist_fm ) ).

% UNIV_I
thf(fact_80_iso__tuple__UNIV__I,axiom,
    ! [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ).

% iso_tuple_UNIV_I
thf(fact_81_iso__tuple__UNIV__I,axiom,
    ! [X3: list_fm] : ( member_list_fm @ X3 @ top_top_set_list_fm ) ).

% iso_tuple_UNIV_I
thf(fact_82_iso__tuple__UNIV__I,axiom,
    ! [X3: tm] : ( member_tm @ X3 @ top_top_set_tm ) ).

% iso_tuple_UNIV_I
thf(fact_83_iso__tuple__UNIV__I,axiom,
    ! [X3: fm] : ( member_fm @ X3 @ top_top_set_fm ) ).

% iso_tuple_UNIV_I
thf(fact_84_iso__tuple__UNIV__I,axiom,
    ! [X3: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X3 @ top_to1730629564055774885ist_fm ) ).

% iso_tuple_UNIV_I
thf(fact_85_Saturated__def,axiom,
    ! [Steps: stream2709947120125613254m_rule] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
      = ( ! [X4: rule] :
            ( ( member_rule @ X4 @ ( sset_rule @ rules ) )
           => ( abstra2533313685312581075ist_fm @ eff @ X4 @ Steps ) ) ) ) ).

% Saturated_def
thf(fact_86_all__rules__persistent,axiom,
    ! [R3: rule] :
      ( ( member_rule @ R3 @ ( sset_rule @ rules ) )
     => ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R3 ) ) ).

% all_rules_persistent
thf(fact_87_per,axiom,
    ! [R: rule] :
      ( ( member_rule @ R @ ( sset_rule @ rules ) )
     => ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R ) ) ).

% per
thf(fact_88_sdrop__stl,axiom,
    ! [N: nat,S: stream2709947120125613254m_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N @ ( stl_Pr12655793849929990m_rule @ S ) )
      = ( stl_Pr12655793849929990m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_89_sdrop__stl,axiom,
    ! [N: nat,S: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_90_order__less__imp__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_91_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_92_mem__Collect__eq,axiom,
    ! [A: produc6018962875968178549ist_fm,P: produc6018962875968178549ist_fm > $o] :
      ( ( member4699826688122452638ist_fm @ A @ ( collec5845330005437105888ist_fm @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
    ! [A: rule,P: rule > $o] :
      ( ( member_rule @ A @ ( collect_rule @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
    ! [A: list_fm,P: list_fm > $o] :
      ( ( member_list_fm @ A @ ( collect_list_fm @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
    ! [A: tm,P: tm > $o] :
      ( ( member_tm @ A @ ( collect_tm @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
    ! [A: fm,P: fm > $o] :
      ( ( member_fm @ A @ ( collect_fm @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_97_Collect__mem__eq,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ( collec5845330005437105888ist_fm
        @ ^ [X4: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
    ! [A2: set_rule] :
      ( ( collect_rule
        @ ^ [X4: rule] : ( member_rule @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
    ! [A2: set_list_fm] :
      ( ( collect_list_fm
        @ ^ [X4: list_fm] : ( member_list_fm @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
    ! [A2: set_tm] :
      ( ( collect_tm
        @ ^ [X4: tm] : ( member_tm @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
    ! [A2: set_fm] :
      ( ( collect_fm
        @ ^ [X4: fm] : ( member_fm @ X4 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_102_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_103_linorder__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
      | ( X3 = Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_less_linear
thf(fact_104_order__less__imp__triv,axiom,
    ! [X3: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_105_order__less__not__sym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_not_sym
thf(fact_106_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_107_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_108_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_109_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_110_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_111_order__less__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_less_trans
thf(fact_112_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_113_linorder__neq__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_114_order__less__asym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_asym
thf(fact_115_linorder__neqE,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE
thf(fact_116_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_117_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_118_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_119_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ( ord_less_nat @ Y @ X3 )
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_120_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_121_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( ord_less_nat @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B2: nat] :
              ( ( P @ B2 @ A3 )
             => ( P @ A3 @ B2 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_122_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X5: nat] : ( P2 @ X5 ) )
    = ( ^ [P3: nat > $o] :
        ? [N3: nat] :
          ( ( P3 @ N3 )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ~ ( P3 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_123_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_124_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_125_linorder__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( X3 != Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_cases
thf(fact_126_antisym__conv3,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv3
thf(fact_127_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X: nat] :
          ( ! [Y3: nat] :
              ( ( ord_less_nat @ Y3 @ X )
             => ( P @ Y3 ) )
         => ( P @ X ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_128_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_129_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_130_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_131_less__imp__neq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% less_imp_neq
thf(fact_132_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_133_RuleSystem__Defs_Osaturated_Ocong,axiom,
    abstra2533313685312581075ist_fm = abstra2533313685312581075ist_fm ).

% RuleSystem_Defs.saturated.cong
thf(fact_134_top__set__def,axiom,
    ( top_to1730629564055774885ist_fm
    = ( collec5845330005437105888ist_fm @ top_to4151594781713979000t_fm_o ) ) ).

% top_set_def
thf(fact_135_RuleSystem_Oper_Ocong,axiom,
    abstra5255361903751151037ist_fm = abstra5255361903751151037ist_fm ).

% RuleSystem.per.cong
thf(fact_136_top_Onot__eq__extremum,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] :
      ( ( A != top_to1730629564055774885ist_fm )
      = ( ord_le9141428925376846849ist_fm @ A @ top_to1730629564055774885ist_fm ) ) ).

% top.not_eq_extremum
thf(fact_137_top_Oextremum__strict,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] :
      ~ ( ord_le9141428925376846849ist_fm @ top_to1730629564055774885ist_fm @ A ) ).

% top.extremum_strict
thf(fact_138_fair__stl,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( abstra3799686578551160190r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).

% fair_stl
thf(fact_139_sdrop__fair,axiom,
    ! [Rs: stream_rule,M: nat] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( abstra3799686578551160190r_rule @ rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% sdrop_fair
thf(fact_140_UNIV__witness,axiom,
    ? [X: rule] : ( member_rule @ X @ top_top_set_rule ) ).

% UNIV_witness
thf(fact_141_UNIV__witness,axiom,
    ? [X: list_fm] : ( member_list_fm @ X @ top_top_set_list_fm ) ).

% UNIV_witness
thf(fact_142_UNIV__witness,axiom,
    ? [X: tm] : ( member_tm @ X @ top_top_set_tm ) ).

% UNIV_witness
thf(fact_143_UNIV__witness,axiom,
    ? [X: fm] : ( member_fm @ X @ top_top_set_fm ) ).

% UNIV_witness
thf(fact_144_UNIV__witness,axiom,
    ? [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ top_to1730629564055774885ist_fm ) ).

% UNIV_witness
thf(fact_145_UNIV__eq__I,axiom,
    ! [A2: set_rule] :
      ( ! [X: rule] : ( member_rule @ X @ A2 )
     => ( top_top_set_rule = A2 ) ) ).

% UNIV_eq_I
thf(fact_146_UNIV__eq__I,axiom,
    ! [A2: set_list_fm] :
      ( ! [X: list_fm] : ( member_list_fm @ X @ A2 )
     => ( top_top_set_list_fm = A2 ) ) ).

% UNIV_eq_I
thf(fact_147_UNIV__eq__I,axiom,
    ! [A2: set_tm] :
      ( ! [X: tm] : ( member_tm @ X @ A2 )
     => ( top_top_set_tm = A2 ) ) ).

% UNIV_eq_I
thf(fact_148_UNIV__eq__I,axiom,
    ! [A2: set_fm] :
      ( ! [X: fm] : ( member_fm @ X @ A2 )
     => ( top_top_set_fm = A2 ) ) ).

% UNIV_eq_I
thf(fact_149_UNIV__eq__I,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ! [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ A2 )
     => ( top_to1730629564055774885ist_fm = A2 ) ) ).

% UNIV_eq_I
thf(fact_150_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra6097777249025082867ist_fm
    = ( ^ [Eff2: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,Steps2: stream2709947120125613254m_rule] :
        ! [X4: rule] :
          ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
         => ( abstra2533313685312581075ist_fm @ Eff2 @ X4 @ Steps2 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_151_PersistentRuleSystem_Oper,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,R: rule] :
      ( ( abstra3967921542344661089ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
       => ( abstra5255361903751151037ist_fm @ Eff @ Rules @ S2 @ R ) ) ) ).

% PersistentRuleSystem.per
thf(fact_152_stl__sset,axiom,
    ! [X3: produc6018962875968178549ist_fm,A: stream4408948924543953275ist_fm] :
      ( ( member4699826688122452638ist_fm @ X3 @ ( sset_P5379989128061332361ist_fm @ ( stl_Pr1506262294867171877ist_fm @ A ) ) )
     => ( member4699826688122452638ist_fm @ X3 @ ( sset_P5379989128061332361ist_fm @ A ) ) ) ).

% stl_sset
thf(fact_153_stl__sset,axiom,
    ! [X3: list_fm,A: stream_list_fm] :
      ( ( member_list_fm @ X3 @ ( sset_list_fm @ ( stl_list_fm @ A ) ) )
     => ( member_list_fm @ X3 @ ( sset_list_fm @ A ) ) ) ).

% stl_sset
thf(fact_154_stl__sset,axiom,
    ! [X3: tm,A: stream_tm] :
      ( ( member_tm @ X3 @ ( sset_tm @ ( stl_tm @ A ) ) )
     => ( member_tm @ X3 @ ( sset_tm @ A ) ) ) ).

% stl_sset
thf(fact_155_stl__sset,axiom,
    ! [X3: fm,A: stream_fm] :
      ( ( member_fm @ X3 @ ( sset_fm @ ( stl_fm @ A ) ) )
     => ( member_fm @ X3 @ ( sset_fm @ A ) ) ) ).

% stl_sset
thf(fact_156_stl__sset,axiom,
    ! [X3: rule,A: stream_rule] :
      ( ( member_rule @ X3 @ ( sset_rule @ ( stl_rule @ A ) ) )
     => ( member_rule @ X3 @ ( sset_rule @ A ) ) ) ).

% stl_sset
thf(fact_157_stl__sset,axiom,
    ! [X3: produc340336539035504054m_rule,A: stream2709947120125613254m_rule] :
      ( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ ( stl_Pr12655793849929990m_rule @ A ) ) )
     => ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ A ) ) ) ).

% stl_sset
thf(fact_158_minWait__ex,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ? [N2: nat] : ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ).

% minWait_ex
thf(fact_159_PersistentRuleSystem__def,axiom,
    ( abstra3967921542344661089ist_fm
    = ( ^ [Eff2: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm] :
          ( ( abstra5221733350967904376ist_fm @ Eff2 @ Rules2 @ S5 )
          & ( abstra9069655090277518462ist_fm @ Eff2 @ Rules2 @ S5 ) ) ) ) ).

% PersistentRuleSystem_def
thf(fact_160_PersistentRuleSystem_Ointro,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( abstra9069655090277518462ist_fm @ Eff @ Rules @ S2 )
       => ( abstra3967921542344661089ist_fm @ Eff @ Rules @ S2 ) ) ) ).

% PersistentRuleSystem.intro
thf(fact_161_PersistentRuleSystem__axioms__def,axiom,
    ( abstra9069655090277518462ist_fm
    = ( ^ [Eff2: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm] :
        ! [R2: rule] :
          ( ( member_rule @ R2 @ ( sset_rule @ Rules2 ) )
         => ( abstra5255361903751151037ist_fm @ Eff2 @ Rules2 @ S5 @ R2 ) ) ) ) ).

% PersistentRuleSystem_axioms_def
thf(fact_162_PersistentRuleSystem__axioms_Ointro,axiom,
    ! [Rules: stream_rule,Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,S2: set_Pr5202636777678657877ist_fm] :
      ( ! [R4: rule] :
          ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
         => ( abstra5255361903751151037ist_fm @ Eff @ Rules @ S2 @ R4 ) )
     => ( abstra9069655090277518462ist_fm @ Eff @ Rules @ S2 ) ) ).

% PersistentRuleSystem_axioms.intro
thf(fact_163_sdrop__szip,axiom,
    ! [N: nat,S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
      ( ( sdrop_7192298464603511222m_rule @ N @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) )
      = ( szip_P811719526838699976m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_164_sdrop__szip,axiom,
    ! [N: nat,S1: stream2709947120125613254m_rule,S22: stream_rule] :
      ( ( sdrop_1938960342593938834e_rule @ N @ ( szip_P2044787555563963556e_rule @ S1 @ S22 ) )
      = ( szip_P2044787555563963556e_rule @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_165_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S22: stream2709947120125613254m_rule] :
      ( ( sdrop_746751008794826386m_rule @ N @ ( szip_r6533552703345879972m_rule @ S1 @ S22 ) )
      = ( szip_r6533552703345879972m_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_166_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S22: stream_rule] :
      ( ( sdrop_9113879250048157294e_rule @ N @ ( szip_rule_rule @ S1 @ S22 ) )
      = ( szip_rule_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_167_sdrop__szip,axiom,
    ! [N: nat,S1: stream4408948924543953275ist_fm,S22: stream_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) )
      = ( szip_P2924820683901490861m_rule @ ( sdrop_9176333610110415838ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_168_trim__fair,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( abstra3799686578551160190r_rule @ rules @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) ) ).

% trim_fair
thf(fact_169_eff__S,axiom,
    ! [S: produc6018962875968178549ist_fm,R: rule,Sl: fset_P8989946509869081563ist_fm,S6: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( member_rule @ R @ ( sset_rule @ rules ) )
       => ( ( eff @ R @ S @ Sl )
         => ( ( fmembe3754813877001230652ist_fm @ S6 @ Sl )
           => ( member4699826688122452638ist_fm @ S6 @ top_to1730629564055774885ist_fm ) ) ) ) ) ).

% eff_S
thf(fact_170_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra406286406349174356ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra467781697624067491ist_fm @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_171_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc340336539035504054m_rule > rule > fset_rule > $o,Rules: stream2709947120125613254m_rule,S2: set_rule,S: rule,Rs: stream2709947120125613254m_rule] :
      ( ( abstra7792261008698710771e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra1957041411669721828e_rule @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_172_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc340336539035504054m_rule > list_fm > fset_list_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_list_fm,S: list_fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6353961794954871768ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra1132159929896080295ist_fm @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_173_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc340336539035504054m_rule > tm > fset_tm > $o,Rules: stream2709947120125613254m_rule,S2: set_tm,S: tm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6746019703890151108ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra1468671940392962579ule_tm @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_174_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc340336539035504054m_rule > fm > fset_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_fm,S: fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6746019703889232722ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra1468671940392044193ule_fm @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_175_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7234149737463204544e_rule @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_176_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S2: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra5712636991756272843ist_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_177_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra8423890812398148919ule_tm @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_178_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra8423890812397230533ule_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_179_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra1707737187183579335ist_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_180_fair__fenum,axiom,
    abstra3799686578551160190r_rule @ rules @ ( abstra745658567949189203m_rule @ rules ) ).

% fair_fenum
thf(fact_181_sset__fenum,axiom,
    ( ( sset_rule @ ( abstra745658567949189203m_rule @ rules ) )
    = ( sset_rule @ rules ) ) ).

% sset_fenum
thf(fact_182_RuleSystem__Defs_Otrim_Ocong,axiom,
    abstra7806236140005899928ist_fm = abstra7806236140005899928ist_fm ).

% RuleSystem_Defs.trim.cong
thf(fact_183_RuleSystem__Defs_Ofenum_Ocong,axiom,
    abstra745658567949189203m_rule = abstra745658567949189203m_rule ).

% RuleSystem_Defs.fenum.cong
thf(fact_184_stream_Oexpand,axiom,
    ! [Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
      ( ( ( ( shd_Pr4562317740776619530m_rule @ Stream )
          = ( shd_Pr4562317740776619530m_rule @ Stream2 ) )
        & ( ( stl_Pr12655793849929990m_rule @ Stream )
          = ( stl_Pr12655793849929990m_rule @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_185_stream_Oexpand,axiom,
    ! [Stream: stream_rule,Stream2: stream_rule] :
      ( ( ( ( shd_rule @ Stream )
          = ( shd_rule @ Stream2 ) )
        & ( ( stl_rule @ Stream )
          = ( stl_rule @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_186_stream_Ocoinduct,axiom,
    ! [R5: stream2709947120125613254m_rule > stream2709947120125613254m_rule > $o,Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
      ( ( R5 @ Stream @ Stream2 )
     => ( ! [Stream3: stream2709947120125613254m_rule,Stream4: stream2709947120125613254m_rule] :
            ( ( R5 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr4562317740776619530m_rule @ Stream3 )
                = ( shd_Pr4562317740776619530m_rule @ Stream4 ) )
              & ( R5 @ ( stl_Pr12655793849929990m_rule @ Stream3 ) @ ( stl_Pr12655793849929990m_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_187_stream_Ocoinduct,axiom,
    ! [R5: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R5 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R5 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( R5 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_188_stream_Ocoinduct__strong,axiom,
    ! [R5: stream2709947120125613254m_rule > stream2709947120125613254m_rule > $o,Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
      ( ( R5 @ Stream @ Stream2 )
     => ( ! [Stream3: stream2709947120125613254m_rule,Stream4: stream2709947120125613254m_rule] :
            ( ( R5 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr4562317740776619530m_rule @ Stream3 )
                = ( shd_Pr4562317740776619530m_rule @ Stream4 ) )
              & ( ( R5 @ ( stl_Pr12655793849929990m_rule @ Stream3 ) @ ( stl_Pr12655793849929990m_rule @ Stream4 ) )
                | ( ( stl_Pr12655793849929990m_rule @ Stream3 )
                  = ( stl_Pr12655793849929990m_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_189_stream_Ocoinduct__strong,axiom,
    ! [R5: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R5 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R5 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( ( R5 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) )
                | ( ( stl_rule @ Stream3 )
                  = ( stl_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_190_shd__sset,axiom,
    ! [A: stream4408948924543953275ist_fm] : ( member4699826688122452638ist_fm @ ( shd_Pr3211216682057661985ist_fm @ A ) @ ( sset_P5379989128061332361ist_fm @ A ) ) ).

% shd_sset
thf(fact_191_shd__sset,axiom,
    ! [A: stream_list_fm] : ( member_list_fm @ ( shd_list_fm @ A ) @ ( sset_list_fm @ A ) ) ).

% shd_sset
thf(fact_192_shd__sset,axiom,
    ! [A: stream_tm] : ( member_tm @ ( shd_tm @ A ) @ ( sset_tm @ A ) ) ).

% shd_sset
thf(fact_193_shd__sset,axiom,
    ! [A: stream_fm] : ( member_fm @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).

% shd_sset
thf(fact_194_shd__sset,axiom,
    ! [A: stream_rule] : ( member_rule @ ( shd_rule @ A ) @ ( sset_rule @ A ) ) ).

% shd_sset
thf(fact_195_shd__sset,axiom,
    ! [A: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) @ ( sset_P4484857331586881186m_rule @ A ) ) ).

% shd_sset
thf(fact_196_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: tm > rule > fset_rule > $o,Rules: stream_tm,S2: set_rule,S: rule,Rs: stream_tm] :
      ( ( abstra7099963851359259790m_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra6979500642109889529air_tm @ Rules @ Rs )
         => ( member_tm @ ( shd_tm @ ( abstra7005108030455562926m_rule @ Eff @ Rs @ S ) ) @ ( sset_tm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_197_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S2: set_rule,S: rule,Rs: stream_fm] :
      ( ( abstra7659285206637183772m_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra6979500642108971143air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra7564429385733486908m_rule @ Eff @ Rs @ S ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_198_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: tm > tm > fset_tm > $o,Rules: stream_tm,S2: set_tm,S: tm,Rs: stream_tm] :
      ( ( abstra5526243810835382505_tm_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra6979500642109889529air_tm @ Rules @ Rs )
         => ( member_tm @ ( shd_tm @ ( abstra3994774178628530889_tm_tm @ Eff @ Rs @ S ) ) @ ( sset_tm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_199_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > tm > fset_tm > $o,Rules: stream_fm,S2: set_tm,S: tm,Rs: stream_fm] :
      ( ( abstra966159493039759323_fm_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra6979500642108971143air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra8658061897687683515_fm_tm @ Eff @ Rs @ S ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_200_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: tm > fm > fset_fm > $o,Rules: stream_tm,S2: set_fm,S: fm,Rs: stream_tm] :
      ( ( abstra5526243810834464119_tm_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra6979500642109889529air_tm @ Rules @ Rs )
         => ( member_tm @ ( shd_tm @ ( abstra3994774178627612503_tm_fm @ Eff @ Rs @ S ) ) @ ( sset_tm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_201_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S2: set_fm,S: fm,Rs: stream_fm] :
      ( ( abstra966159493038840937_fm_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra6979500642108971143air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra8658061897686765129_fm_fm @ Eff @ Rs @ S ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_202_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra2680930950814381487e_rule @ Eff @ Rs @ S ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_203_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra4814164703917039368ule_tm @ Eff @ Rs @ S ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_204_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra4814164703916120982ule_fm @ Eff @ Rs @ S ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_205_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: list_fm > rule > fset_rule > $o,Rules: stream_list_fm,S2: set_rule,S: rule,Rs: stream_list_fm] :
      ( ( abstra6744069043672362530m_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra7078398738616236685ist_fm @ Rules @ Rs )
         => ( member_list_fm @ ( shd_list_fm @ ( abstra7407046389438567490m_rule @ Eff @ Rs @ S ) ) @ ( sset_list_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_206_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra406286406349174356ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( abstra467781697624067491ist_fm @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( abstra1355475359686397044ist_fm @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_207_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc340336539035504054m_rule > rule > fset_rule > $o,Rules: stream2709947120125613254m_rule,S2: set_rule,S: rule,Rs: stream2709947120125613254m_rule] :
      ( ( abstra7792261008698710771e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( abstra1957041411669721828e_rule @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( abstra6343601422196104403e_rule @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_208_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc340336539035504054m_rule > list_fm > fset_list_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_list_fm,S: list_fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6353961794954871768ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( abstra1132159929896080295ist_fm @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( abstra4870899028808501752ist_fm @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_209_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc340336539035504054m_rule > tm > fset_tm > $o,Rules: stream2709947120125613254m_rule,S2: set_tm,S: tm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6746019703890151108ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( abstra1468671940392962579ule_tm @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( abstra1639899932327861476ule_tm @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_210_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc340336539035504054m_rule > fm > fset_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_fm,S: fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6746019703889232722ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( abstra1468671940392044193ule_fm @ Eff @ ( shd_Pr4562317740776619530m_rule @ ( abstra1639899932326943090ule_fm @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_211_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra7234149737463204544e_rule @ Eff @ ( shd_rule @ ( abstra2680930950814381487e_rule @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_212_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S2: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra5712636991756272843ist_fm @ Eff @ ( shd_rule @ ( abstra593219384675117596ist_fm @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_213_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra8423890812398148919ule_tm @ Eff @ ( shd_rule @ ( abstra4814164703917039368ule_tm @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_214_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra8423890812397230533ule_fm @ Eff @ ( shd_rule @ ( abstra4814164703916120982ule_fm @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_215_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra1707737187183579335ist_fm @ Eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_216_sset__induct,axiom,
    ! [Y: produc6018962875968178549ist_fm,S: stream4408948924543953275ist_fm,P: produc6018962875968178549ist_fm > stream4408948924543953275ist_fm > $o] :
      ( ( member4699826688122452638ist_fm @ Y @ ( sset_P5379989128061332361ist_fm @ S ) )
     => ( ! [S7: stream4408948924543953275ist_fm] : ( P @ ( shd_Pr3211216682057661985ist_fm @ S7 ) @ S7 )
       => ( ! [S7: stream4408948924543953275ist_fm,Y2: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ Y2 @ ( sset_P5379989128061332361ist_fm @ ( stl_Pr1506262294867171877ist_fm @ S7 ) ) )
             => ( ( P @ Y2 @ ( stl_Pr1506262294867171877ist_fm @ S7 ) )
               => ( P @ Y2 @ S7 ) ) )
         => ( P @ Y @ S ) ) ) ) ).

% sset_induct
thf(fact_217_sset__induct,axiom,
    ! [Y: list_fm,S: stream_list_fm,P: list_fm > stream_list_fm > $o] :
      ( ( member_list_fm @ Y @ ( sset_list_fm @ S ) )
     => ( ! [S7: stream_list_fm] : ( P @ ( shd_list_fm @ S7 ) @ S7 )
       => ( ! [S7: stream_list_fm,Y2: list_fm] :
              ( ( member_list_fm @ Y2 @ ( sset_list_fm @ ( stl_list_fm @ S7 ) ) )
             => ( ( P @ Y2 @ ( stl_list_fm @ S7 ) )
               => ( P @ Y2 @ S7 ) ) )
         => ( P @ Y @ S ) ) ) ) ).

% sset_induct
thf(fact_218_sset__induct,axiom,
    ! [Y: tm,S: stream_tm,P: tm > stream_tm > $o] :
      ( ( member_tm @ Y @ ( sset_tm @ S ) )
     => ( ! [S7: stream_tm] : ( P @ ( shd_tm @ S7 ) @ S7 )
       => ( ! [S7: stream_tm,Y2: tm] :
              ( ( member_tm @ Y2 @ ( sset_tm @ ( stl_tm @ S7 ) ) )
             => ( ( P @ Y2 @ ( stl_tm @ S7 ) )
               => ( P @ Y2 @ S7 ) ) )
         => ( P @ Y @ S ) ) ) ) ).

% sset_induct
thf(fact_219_sset__induct,axiom,
    ! [Y: fm,S: stream_fm,P: fm > stream_fm > $o] :
      ( ( member_fm @ Y @ ( sset_fm @ S ) )
     => ( ! [S7: stream_fm] : ( P @ ( shd_fm @ S7 ) @ S7 )
       => ( ! [S7: stream_fm,Y2: fm] :
              ( ( member_fm @ Y2 @ ( sset_fm @ ( stl_fm @ S7 ) ) )
             => ( ( P @ Y2 @ ( stl_fm @ S7 ) )
               => ( P @ Y2 @ S7 ) ) )
         => ( P @ Y @ S ) ) ) ) ).

% sset_induct
thf(fact_220_sset__induct,axiom,
    ! [Y: rule,S: stream_rule,P: rule > stream_rule > $o] :
      ( ( member_rule @ Y @ ( sset_rule @ S ) )
     => ( ! [S7: stream_rule] : ( P @ ( shd_rule @ S7 ) @ S7 )
       => ( ! [S7: stream_rule,Y2: rule] :
              ( ( member_rule @ Y2 @ ( sset_rule @ ( stl_rule @ S7 ) ) )
             => ( ( P @ Y2 @ ( stl_rule @ S7 ) )
               => ( P @ Y2 @ S7 ) ) )
         => ( P @ Y @ S ) ) ) ) ).

% sset_induct
thf(fact_221_sset__induct,axiom,
    ! [Y: produc340336539035504054m_rule,S: stream2709947120125613254m_rule,P: produc340336539035504054m_rule > stream2709947120125613254m_rule > $o] :
      ( ( member7231649785386036813m_rule @ Y @ ( sset_P4484857331586881186m_rule @ S ) )
     => ( ! [S7: stream2709947120125613254m_rule] : ( P @ ( shd_Pr4562317740776619530m_rule @ S7 ) @ S7 )
       => ( ! [S7: stream2709947120125613254m_rule,Y2: produc340336539035504054m_rule] :
              ( ( member7231649785386036813m_rule @ Y2 @ ( sset_P4484857331586881186m_rule @ ( stl_Pr12655793849929990m_rule @ S7 ) ) )
             => ( ( P @ Y2 @ ( stl_Pr12655793849929990m_rule @ S7 ) )
               => ( P @ Y2 @ S7 ) ) )
         => ( P @ Y @ S ) ) ) ) ).

% sset_induct
thf(fact_222_RuleSystem_Oeff__S,axiom,
    ! [Eff: tm > rule > fset_rule > $o,Rules: stream_tm,S2: set_rule,S: rule,R: tm,Sl: fset_rule,S6: rule] :
      ( ( abstra7099963851359259790m_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( member_tm @ R @ ( sset_tm @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_rule @ S6 @ Sl )
             => ( member_rule @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_223_RuleSystem_Oeff__S,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S2: set_rule,S: rule,R: fm,Sl: fset_rule,S6: rule] :
      ( ( abstra7659285206637183772m_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( member_fm @ R @ ( sset_fm @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_rule @ S6 @ Sl )
             => ( member_rule @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_224_RuleSystem_Oeff__S,axiom,
    ! [Eff: tm > tm > fset_tm > $o,Rules: stream_tm,S2: set_tm,S: tm,R: tm,Sl: fset_tm,S6: tm] :
      ( ( abstra5526243810835382505_tm_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( member_tm @ R @ ( sset_tm @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_tm @ S6 @ Sl )
             => ( member_tm @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_225_RuleSystem_Oeff__S,axiom,
    ! [Eff: fm > tm > fset_tm > $o,Rules: stream_fm,S2: set_tm,S: tm,R: fm,Sl: fset_tm,S6: tm] :
      ( ( abstra966159493039759323_fm_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( member_fm @ R @ ( sset_fm @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_tm @ S6 @ Sl )
             => ( member_tm @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_226_RuleSystem_Oeff__S,axiom,
    ! [Eff: tm > fm > fset_fm > $o,Rules: stream_tm,S2: set_fm,S: fm,R: tm,Sl: fset_fm,S6: fm] :
      ( ( abstra5526243810834464119_tm_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( member_tm @ R @ ( sset_tm @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_fm @ S6 @ Sl )
             => ( member_fm @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_227_RuleSystem_Oeff__S,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S2: set_fm,S: fm,R: fm,Sl: fset_fm,S6: fm] :
      ( ( abstra966159493038840937_fm_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( member_fm @ R @ ( sset_fm @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_fm @ S6 @ Sl )
             => ( member_fm @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_228_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,R: rule,Sl: fset_rule,S6: rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_rule @ S6 @ Sl )
             => ( member_rule @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_229_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,R: rule,Sl: fset_tm,S6: tm] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_tm @ S6 @ Sl )
             => ( member_tm @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_230_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,R: rule,Sl: fset_fm,S6: fm] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_fm @ S6 @ Sl )
             => ( member_fm @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_231_RuleSystem_Oeff__S,axiom,
    ! [Eff: list_fm > rule > fset_rule > $o,Rules: stream_list_fm,S2: set_rule,S: rule,R: list_fm,Sl: fset_rule,S6: rule] :
      ( ( abstra6744069043672362530m_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( member_list_fm @ R @ ( sset_list_fm @ Rules ) )
         => ( ( Eff @ R @ S @ Sl )
           => ( ( fmember_rule @ S6 @ Sl )
             => ( member_rule @ S6 @ S2 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_232_RuleSystem_Ointro,axiom,
    ! [S2: set_rule,Rules: stream_tm,Eff: tm > rule > fset_rule > $o] :
      ( ! [S7: rule] :
          ( ( member_rule @ S7 @ S2 )
         => ! [R4: tm] :
              ( ( member_tm @ R4 @ ( sset_tm @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: rule] :
                      ( ( fmember_rule @ S4 @ Sl2 )
                     => ( member_rule @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: rule] :
            ( ( member_rule @ S7 @ S2 )
           => ? [X6: tm] :
                ( ( member_tm @ X6 @ ( sset_tm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra7099963851359259790m_rule @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_233_RuleSystem_Ointro,axiom,
    ! [S2: set_rule,Rules: stream_fm,Eff: fm > rule > fset_rule > $o] :
      ( ! [S7: rule] :
          ( ( member_rule @ S7 @ S2 )
         => ! [R4: fm] :
              ( ( member_fm @ R4 @ ( sset_fm @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: rule] :
                      ( ( fmember_rule @ S4 @ Sl2 )
                     => ( member_rule @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: rule] :
            ( ( member_rule @ S7 @ S2 )
           => ? [X6: fm] :
                ( ( member_fm @ X6 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra7659285206637183772m_rule @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_234_RuleSystem_Ointro,axiom,
    ! [S2: set_tm,Rules: stream_tm,Eff: tm > tm > fset_tm > $o] :
      ( ! [S7: tm] :
          ( ( member_tm @ S7 @ S2 )
         => ! [R4: tm] :
              ( ( member_tm @ R4 @ ( sset_tm @ Rules ) )
             => ! [Sl2: fset_tm] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: tm] :
                      ( ( fmember_tm @ S4 @ Sl2 )
                     => ( member_tm @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: tm] :
            ( ( member_tm @ S7 @ S2 )
           => ? [X6: tm] :
                ( ( member_tm @ X6 @ ( sset_tm @ Rules ) )
                & ? [X_12: fset_tm] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra5526243810835382505_tm_tm @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_235_RuleSystem_Ointro,axiom,
    ! [S2: set_tm,Rules: stream_fm,Eff: fm > tm > fset_tm > $o] :
      ( ! [S7: tm] :
          ( ( member_tm @ S7 @ S2 )
         => ! [R4: fm] :
              ( ( member_fm @ R4 @ ( sset_fm @ Rules ) )
             => ! [Sl2: fset_tm] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: tm] :
                      ( ( fmember_tm @ S4 @ Sl2 )
                     => ( member_tm @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: tm] :
            ( ( member_tm @ S7 @ S2 )
           => ? [X6: fm] :
                ( ( member_fm @ X6 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_tm] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra966159493039759323_fm_tm @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_236_RuleSystem_Ointro,axiom,
    ! [S2: set_fm,Rules: stream_tm,Eff: tm > fm > fset_fm > $o] :
      ( ! [S7: fm] :
          ( ( member_fm @ S7 @ S2 )
         => ! [R4: tm] :
              ( ( member_tm @ R4 @ ( sset_tm @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: fm] :
                      ( ( fmember_fm @ S4 @ Sl2 )
                     => ( member_fm @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: fm] :
            ( ( member_fm @ S7 @ S2 )
           => ? [X6: tm] :
                ( ( member_tm @ X6 @ ( sset_tm @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra5526243810834464119_tm_fm @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_237_RuleSystem_Ointro,axiom,
    ! [S2: set_fm,Rules: stream_fm,Eff: fm > fm > fset_fm > $o] :
      ( ! [S7: fm] :
          ( ( member_fm @ S7 @ S2 )
         => ! [R4: fm] :
              ( ( member_fm @ R4 @ ( sset_fm @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: fm] :
                      ( ( fmember_fm @ S4 @ Sl2 )
                     => ( member_fm @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: fm] :
            ( ( member_fm @ S7 @ S2 )
           => ? [X6: fm] :
                ( ( member_fm @ X6 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra966159493038840937_fm_fm @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_238_RuleSystem_Ointro,axiom,
    ! [S2: set_rule,Rules: stream_rule,Eff: rule > rule > fset_rule > $o] :
      ( ! [S7: rule] :
          ( ( member_rule @ S7 @ S2 )
         => ! [R4: rule] :
              ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: rule] :
                      ( ( fmember_rule @ S4 @ Sl2 )
                     => ( member_rule @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: rule] :
            ( ( member_rule @ S7 @ S2 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_239_RuleSystem_Ointro,axiom,
    ! [S2: set_tm,Rules: stream_rule,Eff: rule > tm > fset_tm > $o] :
      ( ! [S7: tm] :
          ( ( member_tm @ S7 @ S2 )
         => ! [R4: rule] :
              ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_tm] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: tm] :
                      ( ( fmember_tm @ S4 @ Sl2 )
                     => ( member_tm @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: tm] :
            ( ( member_tm @ S7 @ S2 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_tm] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_240_RuleSystem_Ointro,axiom,
    ! [S2: set_fm,Rules: stream_rule,Eff: rule > fm > fset_fm > $o] :
      ( ! [S7: fm] :
          ( ( member_fm @ S7 @ S2 )
         => ! [R4: rule] :
              ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: fm] :
                      ( ( fmember_fm @ S4 @ Sl2 )
                     => ( member_fm @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: fm] :
            ( ( member_fm @ S7 @ S2 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_241_RuleSystem_Ointro,axiom,
    ! [S2: set_rule,Rules: stream_list_fm,Eff: list_fm > rule > fset_rule > $o] :
      ( ! [S7: rule] :
          ( ( member_rule @ S7 @ S2 )
         => ! [R4: list_fm] :
              ( ( member_list_fm @ R4 @ ( sset_list_fm @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R4 @ S7 @ Sl2 )
                 => ! [S4: rule] :
                      ( ( fmember_rule @ S4 @ Sl2 )
                     => ( member_rule @ S4 @ S2 ) ) ) ) )
     => ( ! [S7: rule] :
            ( ( member_rule @ S7 @ S2 )
           => ? [X6: list_fm] :
                ( ( member_list_fm @ X6 @ ( sset_list_fm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S7 @ X_12 ) ) )
       => ( abstra6744069043672362530m_rule @ Eff @ Rules @ S2 ) ) ) ).

% RuleSystem.intro
thf(fact_242_RuleSystem__def,axiom,
    ( abstra7099963851359259790m_rule
    = ( ^ [Eff2: tm > rule > fset_rule > $o,Rules2: stream_tm,S5: set_rule] :
          ( ! [S3: rule] :
              ( ( member_rule @ S3 @ S5 )
             => ! [R2: tm] :
                  ( ( member_tm @ R2 @ ( sset_tm @ Rules2 ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl3 )
                         => ( member_rule @ S8 @ S5 ) ) ) ) )
          & ! [S3: rule] :
              ( ( member_rule @ S3 @ S5 )
             => ? [X4: tm] :
                  ( ( member_tm @ X4 @ ( sset_tm @ Rules2 ) )
                  & ? [X2: fset_rule] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_243_RuleSystem__def,axiom,
    ( abstra7659285206637183772m_rule
    = ( ^ [Eff2: fm > rule > fset_rule > $o,Rules2: stream_fm,S5: set_rule] :
          ( ! [S3: rule] :
              ( ( member_rule @ S3 @ S5 )
             => ! [R2: fm] :
                  ( ( member_fm @ R2 @ ( sset_fm @ Rules2 ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl3 )
                         => ( member_rule @ S8 @ S5 ) ) ) ) )
          & ! [S3: rule] :
              ( ( member_rule @ S3 @ S5 )
             => ? [X4: fm] :
                  ( ( member_fm @ X4 @ ( sset_fm @ Rules2 ) )
                  & ? [X2: fset_rule] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_244_RuleSystem__def,axiom,
    ( abstra5526243810835382505_tm_tm
    = ( ^ [Eff2: tm > tm > fset_tm > $o,Rules2: stream_tm,S5: set_tm] :
          ( ! [S3: tm] :
              ( ( member_tm @ S3 @ S5 )
             => ! [R2: tm] :
                  ( ( member_tm @ R2 @ ( sset_tm @ Rules2 ) )
                 => ! [Sl3: fset_tm] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: tm] :
                          ( ( fmember_tm @ S8 @ Sl3 )
                         => ( member_tm @ S8 @ S5 ) ) ) ) )
          & ! [S3: tm] :
              ( ( member_tm @ S3 @ S5 )
             => ? [X4: tm] :
                  ( ( member_tm @ X4 @ ( sset_tm @ Rules2 ) )
                  & ? [X2: fset_tm] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_245_RuleSystem__def,axiom,
    ( abstra966159493039759323_fm_tm
    = ( ^ [Eff2: fm > tm > fset_tm > $o,Rules2: stream_fm,S5: set_tm] :
          ( ! [S3: tm] :
              ( ( member_tm @ S3 @ S5 )
             => ! [R2: fm] :
                  ( ( member_fm @ R2 @ ( sset_fm @ Rules2 ) )
                 => ! [Sl3: fset_tm] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: tm] :
                          ( ( fmember_tm @ S8 @ Sl3 )
                         => ( member_tm @ S8 @ S5 ) ) ) ) )
          & ! [S3: tm] :
              ( ( member_tm @ S3 @ S5 )
             => ? [X4: fm] :
                  ( ( member_fm @ X4 @ ( sset_fm @ Rules2 ) )
                  & ? [X2: fset_tm] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_246_RuleSystem__def,axiom,
    ( abstra5526243810834464119_tm_fm
    = ( ^ [Eff2: tm > fm > fset_fm > $o,Rules2: stream_tm,S5: set_fm] :
          ( ! [S3: fm] :
              ( ( member_fm @ S3 @ S5 )
             => ! [R2: tm] :
                  ( ( member_tm @ R2 @ ( sset_tm @ Rules2 ) )
                 => ! [Sl3: fset_fm] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl3 )
                         => ( member_fm @ S8 @ S5 ) ) ) ) )
          & ! [S3: fm] :
              ( ( member_fm @ S3 @ S5 )
             => ? [X4: tm] :
                  ( ( member_tm @ X4 @ ( sset_tm @ Rules2 ) )
                  & ? [X2: fset_fm] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_247_RuleSystem__def,axiom,
    ( abstra966159493038840937_fm_fm
    = ( ^ [Eff2: fm > fm > fset_fm > $o,Rules2: stream_fm,S5: set_fm] :
          ( ! [S3: fm] :
              ( ( member_fm @ S3 @ S5 )
             => ! [R2: fm] :
                  ( ( member_fm @ R2 @ ( sset_fm @ Rules2 ) )
                 => ! [Sl3: fset_fm] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl3 )
                         => ( member_fm @ S8 @ S5 ) ) ) ) )
          & ! [S3: fm] :
              ( ( member_fm @ S3 @ S5 )
             => ? [X4: fm] :
                  ( ( member_fm @ X4 @ ( sset_fm @ Rules2 ) )
                  & ? [X2: fset_fm] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_248_RuleSystem__def,axiom,
    ( abstra6805424310982398927e_rule
    = ( ^ [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule] :
          ( ! [S3: rule] :
              ( ( member_rule @ S3 @ S5 )
             => ! [R2: rule] :
                  ( ( member_rule @ R2 @ ( sset_rule @ Rules2 ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl3 )
                         => ( member_rule @ S8 @ S5 ) ) ) ) )
          & ! [S3: rule] :
              ( ( member_rule @ S3 @ S5 )
             => ? [X4: rule] :
                  ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
                  & ? [X2: fset_rule] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_249_RuleSystem__def,axiom,
    ( abstra4909020524820736232ule_tm
    = ( ^ [Eff2: rule > tm > fset_tm > $o,Rules2: stream_rule,S5: set_tm] :
          ( ! [S3: tm] :
              ( ( member_tm @ S3 @ S5 )
             => ! [R2: rule] :
                  ( ( member_rule @ R2 @ ( sset_rule @ Rules2 ) )
                 => ! [Sl3: fset_tm] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: tm] :
                          ( ( fmember_tm @ S8 @ Sl3 )
                         => ( member_tm @ S8 @ S5 ) ) ) ) )
          & ! [S3: tm] :
              ( ( member_tm @ S3 @ S5 )
             => ? [X4: rule] :
                  ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
                  & ? [X2: fset_tm] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_250_RuleSystem__def,axiom,
    ( abstra4909020524819817846ule_fm
    = ( ^ [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm] :
          ( ! [S3: fm] :
              ( ( member_fm @ S3 @ S5 )
             => ! [R2: rule] :
                  ( ( member_rule @ R2 @ ( sset_rule @ Rules2 ) )
                 => ! [Sl3: fset_fm] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl3 )
                         => ( member_fm @ S8 @ S5 ) ) ) ) )
          & ! [S3: fm] :
              ( ( member_fm @ S3 @ S5 )
             => ? [X4: rule] :
                  ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
                  & ? [X2: fset_fm] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_251_RuleSystem__def,axiom,
    ( abstra6744069043672362530m_rule
    = ( ^ [Eff2: list_fm > rule > fset_rule > $o,Rules2: stream_list_fm,S5: set_rule] :
          ( ! [S3: rule] :
              ( ( member_rule @ S3 @ S5 )
             => ! [R2: list_fm] :
                  ( ( member_list_fm @ R2 @ ( sset_list_fm @ Rules2 ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff2 @ R2 @ S3 @ Sl3 )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl3 )
                         => ( member_rule @ S8 @ S5 ) ) ) ) )
          & ! [S3: rule] :
              ( ( member_rule @ S3 @ S5 )
             => ? [X4: list_fm] :
                  ( ( member_list_fm @ X4 @ ( sset_list_fm @ Rules2 ) )
                  & ? [X2: fset_rule] : ( Eff2 @ X4 @ S3 @ X2 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_252_i_Osset__fenum,axiom,
    ! [Rules: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ ( abstra4452268713091674743m_rule @ Rules ) )
      = ( sset_P4484857331586881186m_rule @ Rules ) ) ).

% i.sset_fenum
thf(fact_253_i_Osset__fenum,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ ( abstra745658567949189203m_rule @ Rules ) )
      = ( sset_rule @ Rules ) ) ).

% i.sset_fenum
thf(fact_254_RuleSystem__Defs_Osset__fenum,axiom,
    ! [Rules: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ ( abstra4452268713091674743m_rule @ Rules ) )
      = ( sset_P4484857331586881186m_rule @ Rules ) ) ).

% RuleSystem_Defs.sset_fenum
thf(fact_255_RuleSystem__Defs_Osset__fenum,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ ( abstra745658567949189203m_rule @ Rules ) )
      = ( sset_rule @ Rules ) ) ).

% RuleSystem_Defs.sset_fenum
thf(fact_256_i_Ofair__fenum,axiom,
    ! [Rules: stream_rule] : ( abstra3799686578551160190r_rule @ Rules @ ( abstra745658567949189203m_rule @ Rules ) ) ).

% i.fair_fenum
thf(fact_257_RuleSystem__Defs_Ofair__fenum,axiom,
    ! [Rules: stream_rule] : ( abstra3799686578551160190r_rule @ Rules @ ( abstra745658567949189203m_rule @ Rules ) ) ).

% RuleSystem_Defs.fair_fenum
thf(fact_258_trim__in__R,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( member_rule @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ ( sset_rule @ rules ) ) ) ) ).

% trim_in_R
thf(fact_259_trim__enabled,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S ) ) ) ).

% trim_enabled
thf(fact_260_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules @ ( abstra2680930950814381487e_rule @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_261_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S2: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules @ ( abstra593219384675117596ist_fm @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_262_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules @ ( abstra4814164703917039368ule_tm @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_263_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules @ ( abstra4814164703916120982ule_fm @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_264_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules @ ( abstra7806236140005899928ist_fm @ Eff @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_265_PersistentRuleSystem_Oaxioms_I2_J,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm] :
      ( ( abstra3967921542344661089ist_fm @ Eff @ Rules @ S2 )
     => ( abstra9069655090277518462ist_fm @ Eff @ Rules @ S2 ) ) ).

% PersistentRuleSystem.axioms(2)
thf(fact_266_szip_Osimps_I2_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
      ( ( stl_Pr2679468901532635773m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) )
      = ( szip_P811719526838699976m_rule @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_267_szip_Osimps_I2_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S22: stream_rule] :
      ( ( stl_Pr497917230009538393e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S22 ) )
      = ( szip_P2044787555563963556e_rule @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_268_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S22: stream2709947120125613254m_rule] :
      ( ( stl_Pr8529079933065201753m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S22 ) )
      = ( szip_r6533552703345879972m_rule @ ( stl_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_269_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( stl_Pr2595822214614905141e_rule @ ( szip_rule_rule @ S1 @ S22 ) )
      = ( szip_rule_rule @ ( stl_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_270_szip_Osimps_I2_J,axiom,
    ! [S1: stream4408948924543953275ist_fm,S22: stream_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) )
      = ( szip_P2924820683901490861m_rule @ ( stl_Pr1506262294867171877ist_fm @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_271_trim__alt,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( ( abstra7806236140005899928ist_fm @ eff @ Rs @ S )
          = ( sdrop_rule @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ Rs ) ) ) ) ).

% trim_alt
thf(fact_272_RuleSystem_Opos,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,Rs: stream_rule,R: rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
       => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
         => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
            = R ) ) ) ) ).

% RuleSystem.pos
thf(fact_273_fenum__uu__def,axiom,
    abstra1582897422107675196u_rule = abstra745658567949189203m_rule ).

% fenum_uu_def
thf(fact_274_pos,axiom,
    ! [Rs: stream_rule,R: rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( ( member_rule @ R @ ( sset_rule @ rules ) )
       => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
          = R ) ) ) ).

% pos
thf(fact_275_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc340336539035504054m_rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra406286406349174356ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra1355475359686397044ist_fm @ Eff @ Rs @ S )
            = ( sdrop_8169176516188972301m_rule @ ( abstra7932678668315420303ist_fm @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_276_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc340336539035504054m_rule > rule > fset_rule > $o,Rules: stream2709947120125613254m_rule,S2: set_rule,S: rule,Rs: stream2709947120125613254m_rule] :
      ( ( abstra7792261008698710771e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra6343601422196104403e_rule @ Eff @ Rs @ S )
            = ( sdrop_8169176516188972301m_rule @ ( abstra898225168387205880e_rule @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_277_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc340336539035504054m_rule > list_fm > fset_list_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_list_fm,S: list_fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6353961794954871768ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra4870899028808501752ist_fm @ Eff @ Rs @ S )
            = ( sdrop_8169176516188972301m_rule @ ( abstra3212481807891478163ist_fm @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_278_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc340336539035504054m_rule > tm > fset_tm > $o,Rules: stream2709947120125613254m_rule,S2: set_tm,S: tm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6746019703890151108ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra1639899932327861476ule_tm @ Eff @ Rs @ S )
            = ( sdrop_8169176516188972301m_rule @ ( abstra5959417719433974527ule_tm @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_279_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc340336539035504054m_rule > fm > fset_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_fm,S: fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6746019703889232722ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( abstra1639899932326943090ule_fm @ Eff @ Rs @ S )
            = ( sdrop_8169176516188972301m_rule @ ( abstra5959417719433056141ule_fm @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_280_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra2680930950814381487e_rule @ Eff @ Rs @ S )
            = ( sdrop_rule @ ( abstra538998172947062484e_rule @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_281_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S2: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra593219384675117596ist_fm @ Eff @ Rs @ S )
            = ( sdrop_rule @ ( abstra101201851149073335ist_fm @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_282_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra4814164703917039368ule_tm @ Eff @ Rs @ S )
            = ( sdrop_rule @ ( abstra3486989597109074467ule_tm @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_283_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra4814164703916120982ule_fm @ Eff @ Rs @ S )
            = ( sdrop_rule @ ( abstra3486989597108156081ule_fm @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_284_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( abstra7806236140005899928ist_fm @ Eff @ Rs @ S )
            = ( sdrop_rule @ ( abstra1963046427795717555ist_fm @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_285_top__empty__eq,axiom,
    ( top_top_rule_o
    = ( ^ [X4: rule] : ( member_rule @ X4 @ top_top_set_rule ) ) ) ).

% top_empty_eq
thf(fact_286_top__empty__eq,axiom,
    ( top_top_list_fm_o
    = ( ^ [X4: list_fm] : ( member_list_fm @ X4 @ top_top_set_list_fm ) ) ) ).

% top_empty_eq
thf(fact_287_top__empty__eq,axiom,
    ( top_top_tm_o
    = ( ^ [X4: tm] : ( member_tm @ X4 @ top_top_set_tm ) ) ) ).

% top_empty_eq
thf(fact_288_top__empty__eq,axiom,
    ( top_top_fm_o
    = ( ^ [X4: fm] : ( member_fm @ X4 @ top_top_set_fm ) ) ) ).

% top_empty_eq
thf(fact_289_top__empty__eq,axiom,
    ( top_to4151594781713979000t_fm_o
    = ( ^ [X4: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X4 @ top_to1730629564055774885ist_fm ) ) ) ).

% top_empty_eq
thf(fact_290_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,S: rule,Rs: stream_rule,T2: abstra1868548086920761311e_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( fmembe6649473619981193638e_rule @ T2 @ ( abstra342551825319314182e_rule @ ( abstra2300331465871827593e_rule @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_rule,S4: rule] :
                ( ( member_rule @ S4 @ S2 )
                & ( Eff @ ( shd_rule @ ( abstra2680930950814381487e_rule @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmember_rule @ S4 @ Sl4 )
                & ( T2
                  = ( abstra2300331465871827593e_rule @ Eff @ ( stl_rule @ ( abstra2680930950814381487e_rule @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_291_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,S: tm,Rs: stream_rule,T2: abstra6507763151854140770m_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( fmembe7087723215442806363m_rule @ T2 @ ( abstra5584344027378090725m_rule @ ( abstra3300998584369155758ule_tm @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_tm,S4: tm] :
                ( ( member_tm @ S4 @ S2 )
                & ( Eff @ ( shd_rule @ ( abstra4814164703917039368ule_tm @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmember_tm @ S4 @ Sl4 )
                & ( T2
                  = ( abstra3300998584369155758ule_tm @ Eff @ ( stl_rule @ ( abstra4814164703917039368ule_tm @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_292_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,S: fm,Rs: stream_rule,T2: abstra781890097549642992m_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( fmembe1361850161138308585m_rule @ T2 @ ( abstra6143665382656014707m_rule @ ( abstra3300998584368237372ule_fm @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_fm,S4: fm] :
                ( ( member_fm @ S4 @ S2 )
                & ( Eff @ ( shd_rule @ ( abstra4814164703916120982ule_fm @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmember_fm @ S4 @ Sl4 )
                & ( T2
                  = ( abstra3300998584368237372ule_fm @ Eff @ ( stl_rule @ ( abstra4814164703916120982ule_fm @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_293_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > list_fm > fset_list_fm > $o,Rules: stream_rule,S2: set_list_fm,S: list_fm,Rs: stream_rule,T2: abstra8131174881113258998m_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( fmembe6179286811934061679m_rule @ T2 @ ( abstra5150248878800018553m_rule @ ( abstra6869381243922927298ist_fm @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_list_fm,S4: list_fm] :
                ( ( member_list_fm @ S4 @ S2 )
                & ( Eff @ ( shd_rule @ ( abstra593219384675117596ist_fm @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmember_list_fm @ S4 @ Sl4 )
                & ( T2
                  = ( abstra6869381243922927298ist_fm @ Eff @ ( stl_rule @ ( abstra593219384675117596ist_fm @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_294_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule,T2: abstra8520858721168826610m_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( member4699826688122452638ist_fm @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( fmembe1099044791951439339m_rule @ T2 @ ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_P8989946509869081563ist_fm,S4: produc6018962875968178549ist_fm] :
                ( ( member4699826688122452638ist_fm @ S4 @ S2 )
                & ( Eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmembe3754813877001230652ist_fm @ S4 @ Sl4 )
                & ( T2
                  = ( abstra7840189589880272446ist_fm @ Eff @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_295_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc340336539035504054m_rule > rule > fset_rule > $o,Rules: stream2709947120125613254m_rule,S2: set_rule,S: rule,Rs: stream2709947120125613254m_rule,T2: abstra8527173362768188419m_rule] :
      ( ( abstra7792261008698710771e_rule @ Eff @ Rules @ S2 )
     => ( ( member_rule @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( fmembe3420198702065081546m_rule @ T2 @ ( abstra6199799721713675050m_rule @ ( abstra5235605640745886637e_rule @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_rule,S4: rule] :
                ( ( member_rule @ S4 @ S2 )
                & ( Eff @ ( shd_Pr4562317740776619530m_rule @ ( abstra6343601422196104403e_rule @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmember_rule @ S4 @ Sl4 )
                & ( T2
                  = ( abstra5235605640745886637e_rule @ Eff @ ( stl_Pr12655793849929990m_rule @ ( abstra6343601422196104403e_rule @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_296_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc340336539035504054m_rule > tm > fset_tm > $o,Rules: stream2709947120125613254m_rule,S2: set_tm,S: tm,Rs: stream2709947120125613254m_rule,T2: abstra3814378619136263302m_rule] :
      ( ( abstra6746019703890151108ule_tm @ Eff @ Rules @ S2 )
     => ( ( member_tm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( fmembe3189649771678221951m_rule @ T2 @ ( abstra4013857054756096521m_rule @ ( abstra57709018850439818ule_tm @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_tm,S4: tm] :
                ( ( member_tm @ S4 @ S2 )
                & ( Eff @ ( shd_Pr4562317740776619530m_rule @ ( abstra1639899932327861476ule_tm @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmember_tm @ S4 @ Sl4 )
                & ( T2
                  = ( abstra57709018850439818ule_tm @ Eff @ ( stl_Pr12655793849929990m_rule @ ( abstra1639899932327861476ule_tm @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_297_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc340336539035504054m_rule > fm > fset_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_fm,S: fm,Rs: stream2709947120125613254m_rule,T2: abstra4034898751840576020m_rule] :
      ( ( abstra6746019703889232722ule_fm @ Eff @ Rules @ S2 )
     => ( ( member_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( fmembe3410169904382534669m_rule @ T2 @ ( abstra8953695766294941335m_rule @ ( abstra57709018849521432ule_fm @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_fm,S4: fm] :
                ( ( member_fm @ S4 @ S2 )
                & ( Eff @ ( shd_Pr4562317740776619530m_rule @ ( abstra1639899932326943090ule_fm @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmember_fm @ S4 @ Sl4 )
                & ( T2
                  = ( abstra57709018849521432ule_fm @ Eff @ ( stl_Pr12655793849929990m_rule @ ( abstra1639899932326943090ule_fm @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_298_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc340336539035504054m_rule > list_fm > fset_list_fm > $o,Rules: stream2709947120125613254m_rule,S2: set_list_fm,S: list_fm,Rs: stream2709947120125613254m_rule,T2: abstra1818967566604834586m_rule] :
      ( ( abstra6353961794954871768ist_fm @ Eff @ Rules @ S2 )
     => ( ( member_list_fm @ S @ S2 )
       => ( ( abstra3665372904625986210m_rule @ Rules @ Rs )
         => ( ( fmembe5560845258961799315m_rule @ T2 @ ( abstra4388204207088007069m_rule @ ( abstra3177990756066841758ist_fm @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_list_fm,S4: list_fm] :
                ( ( member_list_fm @ S4 @ S2 )
                & ( Eff @ ( shd_Pr4562317740776619530m_rule @ ( abstra4870899028808501752ist_fm @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmember_list_fm @ S4 @ Sl4 )
                & ( T2
                  = ( abstra3177990756066841758ist_fm @ Eff @ ( stl_Pr12655793849929990m_rule @ ( abstra4870899028808501752ist_fm @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_299_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > abstra8520858721168826610m_rule > fset_A8849196716589201442m_rule > $o,Rules: stream_rule,S2: set_Ab3010718383674909480m_rule,S: abstra8520858721168826610m_rule,Rs: stream_rule,T2: abstra8987725909520199743e_rule] :
      ( ( abstra8554342375581516207m_rule @ Eff @ Rules @ S2 )
     => ( ( member5160799505749407497m_rule @ S @ S2 )
       => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
         => ( ( fmembe5599296573807765126e_rule @ T2 @ ( abstra2862371701884369510e_rule @ ( abstra7342323255560930153m_rule @ Eff @ Rs @ S ) ) )
           => ? [Sl4: fset_A8849196716589201442m_rule,S4: abstra8520858721168826610m_rule] :
                ( ( member5160799505749407497m_rule @ S4 @ S2 )
                & ( Eff @ ( shd_rule @ ( abstra2567749246238847887m_rule @ Eff @ Rs @ S ) ) @ S @ Sl4 )
                & ( fmembe1099044791951439339m_rule @ S4 @ Sl4 )
                & ( T2
                  = ( abstra7342323255560930153m_rule @ Eff @ ( stl_rule @ ( abstra2567749246238847887m_rule @ Eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_300_in__cont__mkTree,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule,T2: abstra8520858721168826610m_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( ( fmembe1099044791951439339m_rule @ T2 @ ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) ) )
         => ? [Sl4: fset_P8989946509869081563ist_fm,S4: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S4 @ top_to1730629564055774885ist_fm )
              & ( eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S @ Sl4 )
              & ( fmembe3754813877001230652ist_fm @ S4 @ Sl4 )
              & ( T2
                = ( abstra7840189589880272446ist_fm @ eff @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S4 ) ) ) ) ) ) ).

% in_cont_mkTree
thf(fact_301_RuleSystem_OminWait_Ocong,axiom,
    abstra1963046427795717555ist_fm = abstra1963046427795717555ist_fm ).

% RuleSystem.minWait.cong
thf(fact_302_minWait__le__pos,axiom,
    ! [Rs: stream_rule,R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( ( member_rule @ R @ ( sset_rule @ rules ) )
       => ( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
         => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) ) ) ) ) ).

% minWait_le_pos
thf(fact_303_RuleSystem_OminWait__le__pos,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,Rs: stream_rule,R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
       => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
         => ( ( abstra1707737187183579335ist_fm @ Eff @ R @ S )
           => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ Eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) ) ) ) ) ) ).

% RuleSystem.minWait_le_pos
thf(fact_304_minWait__least,axiom,
    ! [N: nat,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S )
     => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ N ) ) ).

% minWait_least
thf(fact_305_ipath_Ocoinduct,axiom,
    ! [X7: abstract_tree_rule > stream_rule > $o,X1: abstract_tree_rule,X22: stream_rule] :
      ( ( X7 @ X1 @ X22 )
     => ( ! [X12: abstract_tree_rule,X23: stream_rule] :
            ( ( X7 @ X12 @ X23 )
           => ? [T3: abstract_tree_rule,Steps3: stream_rule,T4: abstract_tree_rule] :
                ( ( X12 = T3 )
                & ( X23 = Steps3 )
                & ( ( abstract_root_rule @ T3 )
                  = ( shd_rule @ Steps3 ) )
                & ( fmembe4260254584891149511e_rule @ T4 @ ( abstract_cont_rule @ T3 ) )
                & ( ( X7 @ T4 @ ( stl_rule @ Steps3 ) )
                  | ( abstract_ipath_rule @ T4 @ ( stl_rule @ Steps3 ) ) ) ) )
       => ( abstract_ipath_rule @ X1 @ X22 ) ) ) ).

% ipath.coinduct
thf(fact_306_ipath_Ocoinduct,axiom,
    ! [X7: abstra8520858721168826610m_rule > stream2709947120125613254m_rule > $o,X1: abstra8520858721168826610m_rule,X22: stream2709947120125613254m_rule] :
      ( ( X7 @ X1 @ X22 )
     => ( ! [X12: abstra8520858721168826610m_rule,X23: stream2709947120125613254m_rule] :
            ( ( X7 @ X12 @ X23 )
           => ? [T3: abstra8520858721168826610m_rule,Steps3: stream2709947120125613254m_rule,T4: abstra8520858721168826610m_rule] :
                ( ( X12 = T3 )
                & ( X23 = Steps3 )
                & ( ( abstra5180277867158163941m_rule @ T3 )
                  = ( shd_Pr4562317740776619530m_rule @ Steps3 ) )
                & ( fmembe1099044791951439339m_rule @ T4 @ ( abstra2475333618431710837m_rule @ T3 ) )
                & ( ( X7 @ T4 @ ( stl_Pr12655793849929990m_rule @ Steps3 ) )
                  | ( abstra8836096083317999815m_rule @ T4 @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) ) )
       => ( abstra8836096083317999815m_rule @ X1 @ X22 ) ) ) ).

% ipath.coinduct
thf(fact_307_ipath_Ointros,axiom,
    ! [T: abstract_tree_rule,Steps: stream_rule,T2: abstract_tree_rule] :
      ( ( ( abstract_root_rule @ T )
        = ( shd_rule @ Steps ) )
     => ( ( fmembe4260254584891149511e_rule @ T2 @ ( abstract_cont_rule @ T ) )
       => ( ( abstract_ipath_rule @ T2 @ ( stl_rule @ Steps ) )
         => ( abstract_ipath_rule @ T @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_308_ipath_Ointros,axiom,
    ! [T: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule,T2: abstra8520858721168826610m_rule] :
      ( ( ( abstra5180277867158163941m_rule @ T )
        = ( shd_Pr4562317740776619530m_rule @ Steps ) )
     => ( ( fmembe1099044791951439339m_rule @ T2 @ ( abstra2475333618431710837m_rule @ T ) )
       => ( ( abstra8836096083317999815m_rule @ T2 @ ( stl_Pr12655793849929990m_rule @ Steps ) )
         => ( abstra8836096083317999815m_rule @ T @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_309_ipath_Osimps,axiom,
    ( abstract_ipath_rule
    = ( ^ [A1: abstract_tree_rule,A22: stream_rule] :
        ? [T5: abstract_tree_rule,Steps2: stream_rule,T6: abstract_tree_rule] :
          ( ( A1 = T5 )
          & ( A22 = Steps2 )
          & ( ( abstract_root_rule @ T5 )
            = ( shd_rule @ Steps2 ) )
          & ( fmembe4260254584891149511e_rule @ T6 @ ( abstract_cont_rule @ T5 ) )
          & ( abstract_ipath_rule @ T6 @ ( stl_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_310_ipath_Osimps,axiom,
    ( abstra8836096083317999815m_rule
    = ( ^ [A1: abstra8520858721168826610m_rule,A22: stream2709947120125613254m_rule] :
        ? [T5: abstra8520858721168826610m_rule,Steps2: stream2709947120125613254m_rule,T6: abstra8520858721168826610m_rule] :
          ( ( A1 = T5 )
          & ( A22 = Steps2 )
          & ( ( abstra5180277867158163941m_rule @ T5 )
            = ( shd_Pr4562317740776619530m_rule @ Steps2 ) )
          & ( fmembe1099044791951439339m_rule @ T6 @ ( abstra2475333618431710837m_rule @ T5 ) )
          & ( abstra8836096083317999815m_rule @ T6 @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_311_ipath_Ocases,axiom,
    ! [A12: abstract_tree_rule,A23: stream_rule] :
      ( ( abstract_ipath_rule @ A12 @ A23 )
     => ~ ( ( ( abstract_root_rule @ A12 )
            = ( shd_rule @ A23 ) )
         => ! [T7: abstract_tree_rule] :
              ( ( fmembe4260254584891149511e_rule @ T7 @ ( abstract_cont_rule @ A12 ) )
             => ~ ( abstract_ipath_rule @ T7 @ ( stl_rule @ A23 ) ) ) ) ) ).

% ipath.cases
thf(fact_312_ipath_Ocases,axiom,
    ! [A12: abstra8520858721168826610m_rule,A23: stream2709947120125613254m_rule] :
      ( ( abstra8836096083317999815m_rule @ A12 @ A23 )
     => ~ ( ( ( abstra5180277867158163941m_rule @ A12 )
            = ( shd_Pr4562317740776619530m_rule @ A23 ) )
         => ! [T7: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ T7 @ ( abstra2475333618431710837m_rule @ A12 ) )
             => ~ ( abstra8836096083317999815m_rule @ T7 @ ( stl_Pr12655793849929990m_rule @ A23 ) ) ) ) ) ).

% ipath.cases
thf(fact_313_stake__pos__minWait,axiom,
    ! [Rs: stream_rule,S: produc6018962875968178549ist_fm,R: rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( ( ord_less_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
       => ( ( member_rule @ R @ ( sset_rule @ rules ) )
         => ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
           => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ R )
              = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) ) ) ) ) ) ) ) ).

% stake_pos_minWait
thf(fact_314_RuleSystem_OminWait__least,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( abstra1707737187183579335ist_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S )
       => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ Eff @ Rs @ S ) @ N ) ) ) ).

% RuleSystem.minWait_least
thf(fact_315_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_316_dual__order_Orefl,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).

% dual_order.refl
thf(fact_317_dual__order_Orefl,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).

% dual_order.refl
thf(fact_318_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_319_order__refl,axiom,
    ! [X3: set_tm] : ( ord_less_eq_set_tm @ X3 @ X3 ) ).

% order_refl
thf(fact_320_order__refl,axiom,
    ! [X3: set_fm] : ( ord_less_eq_set_fm @ X3 @ X3 ) ).

% order_refl
thf(fact_321_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream2709947120125613254m_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) )
      = ( sdrop_8169176516188972301m_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_322_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_323_order__antisym__conv,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_324_order__antisym__conv,axiom,
    ! [Y: set_tm,X3: set_tm] :
      ( ( ord_less_eq_set_tm @ Y @ X3 )
     => ( ( ord_less_eq_set_tm @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_325_order__antisym__conv,axiom,
    ! [Y: set_fm,X3: set_fm] :
      ( ( ord_less_eq_set_fm @ Y @ X3 )
     => ( ( ord_less_eq_set_fm @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_326_linorder__le__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_327_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_328_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_329_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > set_fm,C: set_fm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_330_ord__le__eq__subst,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_331_ord__le__eq__subst,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_332_ord__le__eq__subst,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_fm,C: set_fm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_333_ord__le__eq__subst,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_334_ord__le__eq__subst,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_335_ord__le__eq__subst,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_336_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_337_ord__eq__le__subst,axiom,
    ! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_338_ord__eq__le__subst,axiom,
    ! [A: set_fm,F: nat > set_fm,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_339_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_340_ord__eq__le__subst,axiom,
    ! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_341_ord__eq__le__subst,axiom,
    ! [A: set_fm,F: set_tm > set_fm,B: set_tm,C: set_tm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_342_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_fm > nat,B: set_fm,C: set_fm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_343_ord__eq__le__subst,axiom,
    ! [A: set_tm,F: set_fm > set_tm,B: set_fm,C: set_fm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_344_ord__eq__le__subst,axiom,
    ! [A: set_fm,F: set_fm > set_fm,B: set_fm,C: set_fm] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_345_linorder__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_346_order__eq__refl,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_347_order__eq__refl,axiom,
    ! [X3: set_tm,Y: set_tm] :
      ( ( X3 = Y )
     => ( ord_less_eq_set_tm @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_348_order__eq__refl,axiom,
    ! [X3: set_fm,Y: set_fm] :
      ( ( X3 = Y )
     => ( ord_less_eq_set_fm @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_349_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_350_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_351_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_fm,C: set_fm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_352_order__subst2,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_353_order__subst2,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_354_order__subst2,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_fm,C: set_fm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F @ B ) @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_355_order__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_356_order__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_357_order__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F @ B ) @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_358_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_359_order__subst1,axiom,
    ! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_360_order__subst1,axiom,
    ! [A: nat,F: set_fm > nat,B: set_fm,C: set_fm] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_361_order__subst1,axiom,
    ! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_362_order__subst1,axiom,
    ! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_363_order__subst1,axiom,
    ! [A: set_tm,F: set_fm > set_tm,B: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_364_order__subst1,axiom,
    ! [A: set_fm,F: nat > set_fm,B: nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_365_order__subst1,axiom,
    ! [A: set_fm,F: set_tm > set_fm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_366_order__subst1,axiom,
    ! [A: set_fm,F: set_fm > set_fm,B: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_367_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_368_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_tm,Z2: set_tm] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_tm,B3: set_tm] :
          ( ( ord_less_eq_set_tm @ A4 @ B3 )
          & ( ord_less_eq_set_tm @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_369_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_fm,Z2: set_fm] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_fm,B3: set_fm] :
          ( ( ord_less_eq_set_fm @ A4 @ B3 )
          & ( ord_less_eq_set_fm @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_370_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_371_antisym,axiom,
    ! [A: set_tm,B: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_set_tm @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_372_antisym,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_373_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_374_dual__order_Otrans,axiom,
    ! [B: set_tm,A: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ B @ A )
     => ( ( ord_less_eq_set_tm @ C @ B )
       => ( ord_less_eq_set_tm @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_375_dual__order_Otrans,axiom,
    ! [B: set_fm,A: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ C @ B )
       => ( ord_less_eq_set_fm @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_376_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_377_dual__order_Oantisym,axiom,
    ! [B: set_tm,A: set_tm] :
      ( ( ord_less_eq_set_tm @ B @ A )
     => ( ( ord_less_eq_set_tm @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_378_dual__order_Oantisym,axiom,
    ! [B: set_fm,A: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_379_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_380_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_tm,Z2: set_tm] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_tm,B3: set_tm] :
          ( ( ord_less_eq_set_tm @ B3 @ A4 )
          & ( ord_less_eq_set_tm @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_381_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_fm,Z2: set_fm] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_fm,B3: set_fm] :
          ( ( ord_less_eq_set_fm @ B3 @ A4 )
          & ( ord_less_eq_set_fm @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_382_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: nat,B2: nat] :
            ( ( P @ B2 @ A3 )
           => ( P @ A3 @ B2 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_383_order__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_384_order__trans,axiom,
    ! [X3: set_tm,Y: set_tm,Z: set_tm] :
      ( ( ord_less_eq_set_tm @ X3 @ Y )
     => ( ( ord_less_eq_set_tm @ Y @ Z )
       => ( ord_less_eq_set_tm @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_385_order__trans,axiom,
    ! [X3: set_fm,Y: set_fm,Z: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y )
     => ( ( ord_less_eq_set_fm @ Y @ Z )
       => ( ord_less_eq_set_fm @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_386_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_387_order_Otrans,axiom,
    ! [A: set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ord_less_eq_set_tm @ A @ C ) ) ) ).

% order.trans
thf(fact_388_order_Otrans,axiom,
    ! [A: set_fm,B: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ord_less_eq_set_fm @ A @ C ) ) ) ).

% order.trans
thf(fact_389_order__antisym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_390_order__antisym,axiom,
    ! [X3: set_tm,Y: set_tm] :
      ( ( ord_less_eq_set_tm @ X3 @ Y )
     => ( ( ord_less_eq_set_tm @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_391_order__antisym,axiom,
    ! [X3: set_fm,Y: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y )
     => ( ( ord_less_eq_set_fm @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_392_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_393_ord__le__eq__trans,axiom,
    ! [A: set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_tm @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_394_ord__le__eq__trans,axiom,
    ! [A: set_fm,B: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_fm @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_395_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_396_ord__eq__le__trans,axiom,
    ! [A: set_tm,B: set_tm,C: set_tm] :
      ( ( A = B )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ord_less_eq_set_tm @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_397_ord__eq__le__trans,axiom,
    ! [A: set_fm,B: set_fm,C: set_fm] :
      ( ( A = B )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ord_less_eq_set_fm @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_398_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_399_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_tm,Z2: set_tm] : ( Y4 = Z2 ) )
    = ( ^ [X4: set_tm,Y5: set_tm] :
          ( ( ord_less_eq_set_tm @ X4 @ Y5 )
          & ( ord_less_eq_set_tm @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_400_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_401_le__cases3,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_402_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_403_leD,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y ) ) ).

% leD
thf(fact_404_leD,axiom,
    ! [Y: set_tm,X3: set_tm] :
      ( ( ord_less_eq_set_tm @ Y @ X3 )
     => ~ ( ord_less_set_tm @ X3 @ Y ) ) ).

% leD
thf(fact_405_leD,axiom,
    ! [Y: set_fm,X3: set_fm] :
      ( ( ord_less_eq_set_fm @ Y @ X3 )
     => ~ ( ord_less_set_fm @ X3 @ Y ) ) ).

% leD
thf(fact_406_leI,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% leI
thf(fact_407_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_408_nless__le,axiom,
    ! [A: set_tm,B: set_tm] :
      ( ( ~ ( ord_less_set_tm @ A @ B ) )
      = ( ~ ( ord_less_eq_set_tm @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_409_nless__le,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ~ ( ord_less_set_fm @ A @ B ) )
      = ( ~ ( ord_less_eq_set_fm @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_410_antisym__conv1,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_411_antisym__conv1,axiom,
    ! [X3: set_tm,Y: set_tm] :
      ( ~ ( ord_less_set_tm @ X3 @ Y )
     => ( ( ord_less_eq_set_tm @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_412_antisym__conv1,axiom,
    ! [X3: set_fm,Y: set_fm] :
      ( ~ ( ord_less_set_fm @ X3 @ Y )
     => ( ( ord_less_eq_set_fm @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_413_antisym__conv2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_414_antisym__conv2,axiom,
    ! [X3: set_tm,Y: set_tm] :
      ( ( ord_less_eq_set_tm @ X3 @ Y )
     => ( ( ~ ( ord_less_set_tm @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_415_antisym__conv2,axiom,
    ! [X3: set_fm,Y: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y )
     => ( ( ~ ( ord_less_set_fm @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_416_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_417_less__le__not__le,axiom,
    ( ord_less_set_tm
    = ( ^ [X4: set_tm,Y5: set_tm] :
          ( ( ord_less_eq_set_tm @ X4 @ Y5 )
          & ~ ( ord_less_eq_set_tm @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_418_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_419_not__le__imp__less,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X3 )
     => ( ord_less_nat @ X3 @ Y ) ) ).

% not_le_imp_less
thf(fact_420_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_421_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [A4: set_tm,B3: set_tm] :
          ( ( ord_less_set_tm @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_422_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A4: set_fm,B3: set_fm] :
          ( ( ord_less_set_fm @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_423_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_424_order_Ostrict__iff__order,axiom,
    ( ord_less_set_tm
    = ( ^ [A4: set_tm,B3: set_tm] :
          ( ( ord_less_eq_set_tm @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_425_order_Ostrict__iff__order,axiom,
    ( ord_less_set_fm
    = ( ^ [A4: set_fm,B3: set_fm] :
          ( ( ord_less_eq_set_fm @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_426_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_427_order_Ostrict__trans1,axiom,
    ! [A: set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_set_tm @ B @ C )
       => ( ord_less_set_tm @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_428_order_Ostrict__trans1,axiom,
    ! [A: set_fm,B: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_set_fm @ B @ C )
       => ( ord_less_set_fm @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_429_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_430_order_Ostrict__trans2,axiom,
    ! [A: set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_set_tm @ A @ B )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ord_less_set_tm @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_431_order_Ostrict__trans2,axiom,
    ! [A: set_fm,B: set_fm,C: set_fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ord_less_set_fm @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_432_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_433_order_Ostrict__iff__not,axiom,
    ( ord_less_set_tm
    = ( ^ [A4: set_tm,B3: set_tm] :
          ( ( ord_less_eq_set_tm @ A4 @ B3 )
          & ~ ( ord_less_eq_set_tm @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_434_order_Ostrict__iff__not,axiom,
    ( ord_less_set_fm
    = ( ^ [A4: set_fm,B3: set_fm] :
          ( ( ord_less_eq_set_fm @ A4 @ B3 )
          & ~ ( ord_less_eq_set_fm @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_435_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_nat @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_436_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [B3: set_tm,A4: set_tm] :
          ( ( ord_less_set_tm @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_437_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [B3: set_fm,A4: set_fm] :
          ( ( ord_less_set_fm @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_438_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_439_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_tm
    = ( ^ [B3: set_tm,A4: set_tm] :
          ( ( ord_less_eq_set_tm @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_440_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_fm
    = ( ^ [B3: set_fm,A4: set_fm] :
          ( ( ord_less_eq_set_fm @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_441_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_442_dual__order_Ostrict__trans1,axiom,
    ! [B: set_tm,A: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ B @ A )
     => ( ( ord_less_set_tm @ C @ B )
       => ( ord_less_set_tm @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_443_dual__order_Ostrict__trans1,axiom,
    ! [B: set_fm,A: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_set_fm @ C @ B )
       => ( ord_less_set_fm @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_444_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_445_dual__order_Ostrict__trans2,axiom,
    ! [B: set_tm,A: set_tm,C: set_tm] :
      ( ( ord_less_set_tm @ B @ A )
     => ( ( ord_less_eq_set_tm @ C @ B )
       => ( ord_less_set_tm @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_446_dual__order_Ostrict__trans2,axiom,
    ! [B: set_fm,A: set_fm,C: set_fm] :
      ( ( ord_less_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ C @ B )
       => ( ord_less_set_fm @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_447_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_448_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_tm
    = ( ^ [B3: set_tm,A4: set_tm] :
          ( ( ord_less_eq_set_tm @ B3 @ A4 )
          & ~ ( ord_less_eq_set_tm @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_449_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_fm
    = ( ^ [B3: set_fm,A4: set_fm] :
          ( ( ord_less_eq_set_fm @ B3 @ A4 )
          & ~ ( ord_less_eq_set_fm @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_450_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_451_order_Ostrict__implies__order,axiom,
    ! [A: set_tm,B: set_tm] :
      ( ( ord_less_set_tm @ A @ B )
     => ( ord_less_eq_set_tm @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_452_order_Ostrict__implies__order,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ( ord_less_eq_set_fm @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_453_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_454_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_tm,A: set_tm] :
      ( ( ord_less_set_tm @ B @ A )
     => ( ord_less_eq_set_tm @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_455_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_fm,A: set_fm] :
      ( ( ord_less_set_fm @ B @ A )
     => ( ord_less_eq_set_fm @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_456_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_457_order__le__less,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [X4: set_tm,Y5: set_tm] :
          ( ( ord_less_set_tm @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_458_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_459_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_460_order__less__le,axiom,
    ( ord_less_set_tm
    = ( ^ [X4: set_tm,Y5: set_tm] :
          ( ( ord_less_eq_set_tm @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_461_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_462_linorder__not__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
      = ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_not_le
thf(fact_463_linorder__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_not_less
thf(fact_464_order__less__imp__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_465_order__less__imp__le,axiom,
    ! [X3: set_tm,Y: set_tm] :
      ( ( ord_less_set_tm @ X3 @ Y )
     => ( ord_less_eq_set_tm @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_466_order__less__imp__le,axiom,
    ! [X3: set_fm,Y: set_fm] :
      ( ( ord_less_set_fm @ X3 @ Y )
     => ( ord_less_eq_set_fm @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_467_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_468_order__le__neq__trans,axiom,
    ! [A: set_tm,B: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_tm @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_469_order__le__neq__trans,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_fm @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_470_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_471_order__neq__le__trans,axiom,
    ! [A: set_tm,B: set_tm] :
      ( ( A != B )
     => ( ( ord_less_eq_set_tm @ A @ B )
       => ( ord_less_set_tm @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_472_order__neq__le__trans,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( A != B )
     => ( ( ord_less_eq_set_fm @ A @ B )
       => ( ord_less_set_fm @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_473_order__le__less__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_474_order__le__less__trans,axiom,
    ! [X3: set_tm,Y: set_tm,Z: set_tm] :
      ( ( ord_less_eq_set_tm @ X3 @ Y )
     => ( ( ord_less_set_tm @ Y @ Z )
       => ( ord_less_set_tm @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_475_order__le__less__trans,axiom,
    ! [X3: set_fm,Y: set_fm,Z: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y )
     => ( ( ord_less_set_fm @ Y @ Z )
       => ( ord_less_set_fm @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_476_order__less__le__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_477_order__less__le__trans,axiom,
    ! [X3: set_tm,Y: set_tm,Z: set_tm] :
      ( ( ord_less_set_tm @ X3 @ Y )
     => ( ( ord_less_eq_set_tm @ Y @ Z )
       => ( ord_less_set_tm @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_478_order__less__le__trans,axiom,
    ! [X3: set_fm,Y: set_fm,Z: set_fm] :
      ( ( ord_less_set_fm @ X3 @ Y )
     => ( ( ord_less_eq_set_fm @ Y @ Z )
       => ( ord_less_set_fm @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_479_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_480_order__le__less__subst1,axiom,
    ! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_481_order__le__less__subst1,axiom,
    ! [A: set_fm,F: nat > set_fm,B: nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_482_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_483_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_set_tm @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_484_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_fm,C: set_fm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_set_fm @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_485_order__le__less__subst2,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_486_order__le__less__subst2,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_set_tm @ ( F @ B ) @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_487_order__le__less__subst2,axiom,
    ! [A: set_tm,B: set_tm,F: set_tm > set_fm,C: set_fm] :
      ( ( ord_less_eq_set_tm @ A @ B )
     => ( ( ord_less_set_fm @ ( F @ B ) @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_488_order__le__less__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > nat,C: nat] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_489_order__le__less__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > set_tm,C: set_tm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_set_tm @ ( F @ B ) @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_490_order__le__less__subst2,axiom,
    ! [A: set_fm,B: set_fm,F: set_fm > set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_set_fm @ ( F @ B ) @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_491_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_492_order__less__le__subst1,axiom,
    ! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
      ( ( ord_less_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_493_order__less__le__subst1,axiom,
    ! [A: set_fm,F: nat > set_fm,B: nat,C: nat] :
      ( ( ord_less_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_494_order__less__le__subst1,axiom,
    ! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_495_order__less__le__subst1,axiom,
    ! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_496_order__less__le__subst1,axiom,
    ! [A: set_fm,F: set_tm > set_fm,B: set_tm,C: set_tm] :
      ( ( ord_less_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X: set_tm,Y2: set_tm] :
              ( ( ord_less_eq_set_tm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_497_order__less__le__subst1,axiom,
    ! [A: nat,F: set_fm > nat,B: set_fm,C: set_fm] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_498_order__less__le__subst1,axiom,
    ! [A: set_tm,F: set_fm > set_tm,B: set_fm,C: set_fm] :
      ( ( ord_less_set_tm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_499_order__less__le__subst1,axiom,
    ! [A: set_fm,F: set_fm > set_fm,B: set_fm,C: set_fm] :
      ( ( ord_less_set_fm @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ! [X: set_fm,Y2: set_fm] :
              ( ( ord_less_eq_set_fm @ X @ Y2 )
             => ( ord_less_eq_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_fm @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_500_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_501_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_set_tm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_502_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_fm,C: set_fm] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F @ B ) @ C )
       => ( ! [X: nat,Y2: nat] :
              ( ( ord_less_nat @ X @ Y2 )
             => ( ord_less_set_fm @ ( F @ X ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_fm @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_503_linorder__le__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_504_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_505_order__le__imp__less__or__eq,axiom,
    ! [X3: set_tm,Y: set_tm] :
      ( ( ord_less_eq_set_tm @ X3 @ Y )
     => ( ( ord_less_set_tm @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_506_order__le__imp__less__or__eq,axiom,
    ! [X3: set_fm,Y: set_fm] :
      ( ( ord_less_eq_set_fm @ X3 @ Y )
     => ( ( ord_less_set_fm @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_507_top__greatest,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] : ( ord_le1771420097867575541ist_fm @ A @ top_to1730629564055774885ist_fm ) ).

% top_greatest
thf(fact_508_top__greatest,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ A @ top_top_set_tm ) ).

% top_greatest
thf(fact_509_top__greatest,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ top_top_set_fm ) ).

% top_greatest
thf(fact_510_top_Oextremum__unique,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ top_to1730629564055774885ist_fm @ A )
      = ( A = top_to1730629564055774885ist_fm ) ) ).

% top.extremum_unique
thf(fact_511_top_Oextremum__unique,axiom,
    ! [A: set_tm] :
      ( ( ord_less_eq_set_tm @ top_top_set_tm @ A )
      = ( A = top_top_set_tm ) ) ).

% top.extremum_unique
thf(fact_512_top_Oextremum__unique,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A )
      = ( A = top_top_set_fm ) ) ).

% top.extremum_unique
thf(fact_513_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ top_to1730629564055774885ist_fm @ A )
     => ( A = top_to1730629564055774885ist_fm ) ) ).

% top.extremum_uniqueI
thf(fact_514_top_Oextremum__uniqueI,axiom,
    ! [A: set_tm] :
      ( ( ord_less_eq_set_tm @ top_top_set_tm @ A )
     => ( A = top_top_set_tm ) ) ).

% top.extremum_uniqueI
thf(fact_515_top_Oextremum__uniqueI,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A )
     => ( A = top_top_set_fm ) ) ).

% top.extremum_uniqueI
thf(fact_516_tree_Oexpand,axiom,
    ! [Tree: abstra8520858721168826610m_rule,Tree2: abstra8520858721168826610m_rule] :
      ( ( ( ( abstra5180277867158163941m_rule @ Tree )
          = ( abstra5180277867158163941m_rule @ Tree2 ) )
        & ( ( abstra2475333618431710837m_rule @ Tree )
          = ( abstra2475333618431710837m_rule @ Tree2 ) ) )
     => ( Tree = Tree2 ) ) ).

% tree.expand
thf(fact_517_sdrop_Osimps_I2_J,axiom,
    ! [N: nat,S: stream2709947120125613254m_rule] :
      ( ( sdrop_8169176516188972301m_rule @ ( suc @ N ) @ S )
      = ( sdrop_8169176516188972301m_rule @ N @ ( stl_Pr12655793849929990m_rule @ S ) ) ) ).

% sdrop.simps(2)
thf(fact_518_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_519_pos__least,axiom,
    ! [N: nat,Rs: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
      ( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_520_pos__least,axiom,
    ! [N: nat,Rs: stream_rule,R: rule] :
      ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_521_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: tm > rule > fset_rule > $o,Rules: stream_tm,S2: set_rule,Rs: stream_tm,S: rule,R: tm] :
      ( ( abstra7099963851359259790m_rule @ Eff @ Rules @ S2 )
     => ( ( abstra6979500642109889529air_tm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra5677932923647598025m_rule @ Eff @ Rs @ S ) @ ( abstract_pos_tm @ Rs @ R ) )
         => ( ( member_tm @ R @ ( sset_tm @ Rules ) )
           => ( ( member_rule @ S @ S2 )
             => ( ( abstract_pos_tm @ ( stl_tm @ ( abstra7005108030455562926m_rule @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_tm @ Rs @ R ) @ ( suc @ ( abstra5677932923647598025m_rule @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_522_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: tm > tm > fset_tm > $o,Rules: stream_tm,S2: set_tm,Rs: stream_tm,S: tm,R: tm] :
      ( ( abstra5526243810835382505_tm_tm @ Eff @ Rules @ S2 )
     => ( ( abstra6979500642109889529air_tm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra7084540462750409582_tm_tm @ Eff @ Rs @ S ) @ ( abstract_pos_tm @ Rs @ R ) )
         => ( ( member_tm @ R @ ( sset_tm @ Rules ) )
           => ( ( member_tm @ S @ S2 )
             => ( ( abstract_pos_tm @ ( stl_tm @ ( abstra3994774178628530889_tm_tm @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_tm @ Rs @ R ) @ ( suc @ ( abstra7084540462750409582_tm_tm @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_523_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: tm > fm > fset_fm > $o,Rules: stream_tm,S2: set_fm,Rs: stream_tm,S: fm,R: tm] :
      ( ( abstra5526243810834464119_tm_fm @ Eff @ Rules @ S2 )
     => ( ( abstra6979500642109889529air_tm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra7084540462749491196_tm_fm @ Eff @ Rs @ S ) @ ( abstract_pos_tm @ Rs @ R ) )
         => ( ( member_tm @ R @ ( sset_tm @ Rules ) )
           => ( ( member_fm @ S @ S2 )
             => ( ( abstract_pos_tm @ ( stl_tm @ ( abstra3994774178627612503_tm_fm @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_tm @ Rs @ R ) @ ( suc @ ( abstra7084540462749491196_tm_fm @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_524_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S2: set_rule,Rs: stream_fm,S: rule,R: fm] :
      ( ( abstra7659285206637183772m_rule @ Eff @ Rules @ S2 )
     => ( ( abstra6979500642108971143air_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra6237254278925522007m_rule @ Eff @ Rs @ S ) @ ( abstract_pos_fm @ Rs @ R ) )
         => ( ( member_fm @ R @ ( sset_fm @ Rules ) )
           => ( ( member_rule @ S @ S2 )
             => ( ( abstract_pos_fm @ ( stl_fm @ ( abstra7564429385733486908m_rule @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_fm @ Rs @ R ) @ ( suc @ ( abstra6237254278925522007m_rule @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_525_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: fm > tm > fset_tm > $o,Rules: stream_fm,S2: set_tm,Rs: stream_fm,S: tm,R: fm] :
      ( ( abstra966159493039759323_fm_tm @ Eff @ Rules @ S2 )
     => ( ( abstra6979500642108971143air_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra2524456144954786400_fm_tm @ Eff @ Rs @ S ) @ ( abstract_pos_fm @ Rs @ R ) )
         => ( ( member_fm @ R @ ( sset_fm @ Rules ) )
           => ( ( member_tm @ S @ S2 )
             => ( ( abstract_pos_fm @ ( stl_fm @ ( abstra8658061897687683515_fm_tm @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_fm @ Rs @ R ) @ ( suc @ ( abstra2524456144954786400_fm_tm @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_526_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S2: set_fm,Rs: stream_fm,S: fm,R: fm] :
      ( ( abstra966159493038840937_fm_fm @ Eff @ Rules @ S2 )
     => ( ( abstra6979500642108971143air_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra2524456144953868014_fm_fm @ Eff @ Rs @ S ) @ ( abstract_pos_fm @ Rs @ R ) )
         => ( ( member_fm @ R @ ( sset_fm @ Rules ) )
           => ( ( member_fm @ S @ S2 )
             => ( ( abstract_pos_fm @ ( stl_fm @ ( abstra8658061897686765129_fm_fm @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_fm @ Rs @ R ) @ ( suc @ ( abstra2524456144953868014_fm_fm @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_527_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,Rs: stream_rule,S: rule,R: rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra538998172947062484e_rule @ Eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
         => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
           => ( ( member_rule @ S @ S2 )
             => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra2680930950814381487e_rule @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra538998172947062484e_rule @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_528_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,Rs: stream_rule,S: tm,R: rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra3486989597109074467ule_tm @ Eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
         => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
           => ( ( member_tm @ S @ S2 )
             => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra4814164703917039368ule_tm @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra3486989597109074467ule_tm @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_529_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,Rs: stream_rule,S: fm,R: rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( abstra3799686578551160190r_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra3486989597108156081ule_fm @ Eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) )
         => ( ( member_rule @ R @ ( sset_rule @ Rules ) )
           => ( ( member_fm @ S @ S2 )
             => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra4814164703916120982ule_fm @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R ) @ ( suc @ ( abstra3486989597108156081ule_fm @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_530_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: list_fm > rule > fset_rule > $o,Rules: stream_list_fm,S2: set_rule,Rs: stream_list_fm,S: rule,R: list_fm] :
      ( ( abstra6744069043672362530m_rule @ Eff @ Rules @ S2 )
     => ( ( abstra7078398738616236685ist_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra6915028855912523229m_rule @ Eff @ Rs @ S ) @ ( abstract_pos_list_fm @ Rs @ R ) )
         => ( ( member_list_fm @ R @ ( sset_list_fm @ Rules ) )
           => ( ( member_rule @ S @ S2 )
             => ( ( abstract_pos_list_fm @ ( stl_list_fm @ ( abstra7407046389438567490m_rule @ Eff @ Rs @ S ) ) @ R )
                = ( minus_minus_nat @ ( abstract_pos_list_fm @ Rs @ R ) @ ( suc @ ( abstra6915028855912523229m_rule @ Eff @ Rs @ S ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_531_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_532_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S2: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream_rule,R: rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S2 )
     => ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_533_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_534_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_535_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).

% Suc_diff_diff
thf(fact_536_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_537_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_538_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_539_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_540_less__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_541_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_542_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_543_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_544_subset__UNIV,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] : ( ord_le1771420097867575541ist_fm @ A2 @ top_to1730629564055774885ist_fm ) ).

% subset_UNIV
thf(fact_545_subset__UNIV,axiom,
    ! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ top_top_set_tm ) ).

% subset_UNIV
thf(fact_546_subset__UNIV,axiom,
    ! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ top_top_set_fm ) ).

% subset_UNIV
thf(fact_547_sset__sdrop,axiom,
    ! [N: nat,S: stream_rule] : ( ord_less_eq_set_rule @ ( sset_rule @ ( sdrop_rule @ N @ S ) ) @ ( sset_rule @ S ) ) ).

% sset_sdrop
thf(fact_548_sset__sdrop,axiom,
    ! [N: nat,S: stream2709947120125613254m_rule] : ( ord_le6390412330253371084m_rule @ ( sset_P4484857331586881186m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).

% sset_sdrop
thf(fact_549_sset__sdrop,axiom,
    ! [N: nat,S: stream_tm] : ( ord_less_eq_set_tm @ ( sset_tm @ ( sdrop_tm @ N @ S ) ) @ ( sset_tm @ S ) ) ).

% sset_sdrop
thf(fact_550_sset__sdrop,axiom,
    ! [N: nat,S: stream_fm] : ( ord_less_eq_set_fm @ ( sset_fm @ ( sdrop_fm @ N @ S ) ) @ ( sset_fm @ S ) ) ).

% sset_sdrop
thf(fact_551_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_552_Suc__inject,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( suc @ X3 )
        = ( suc @ Y ) )
     => ( X3 = Y ) ) ).

% Suc_inject
thf(fact_553_linorder__neqE__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_554_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_555_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_556_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_557_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_558_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_559_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_560_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_561_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K2: nat,B: nat] :
      ( ( P @ K2 )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B ) )
       => ? [X: nat] :
            ( ( P @ X )
            & ! [Y3: nat] :
                ( ( P @ Y3 )
               => ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_562_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_563_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_564_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_565_le__trans,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K2 )
       => ( ord_less_eq_nat @ I @ K2 ) ) ) ).

% le_trans
thf(fact_566_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_567_diff__commute,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).

% diff_commute
thf(fact_568_Nat_OlessE,axiom,
    ! [I: nat,K2: nat] :
      ( ( ord_less_nat @ I @ K2 )
     => ( ( K2
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K2
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_569_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_570_Suc__lessE,axiom,
    ! [I: nat,K2: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K2 )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K2
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_571_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_572_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_573_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_574_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ N )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_575_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_576_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_577_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ N )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ I2 ) ) ) ) ).

% All_less_Suc
thf(fact_578_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M4: nat] :
            ( ( M
              = ( suc @ M4 ) )
            & ( ord_less_nat @ N @ M4 ) ) ) ) ).

% Suc_less_eq2
thf(fact_579_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_580_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_581_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K2 )
       => ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).

% less_trans_Suc
thf(fact_582_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K )
                   => ( P @ I3 @ K ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_583_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_584_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_585_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R5: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X: nat] : ( R5 @ X @ X )
       => ( ! [X: nat,Y2: nat,Z3: nat] :
              ( ( R5 @ X @ Y2 )
             => ( ( R5 @ Y2 @ Z3 )
               => ( R5 @ X @ Z3 ) ) )
         => ( ! [N2: nat] : ( R5 @ N2 @ ( suc @ N2 ) )
           => ( R5 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_586_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_587_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_588_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_589_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_590_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_591_Suc__le__D,axiom,
    ! [N: nat,M5: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
     => ? [M6: nat] :
          ( M5
          = ( suc @ M6 ) ) ) ).

% Suc_le_D
thf(fact_592_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_593_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_594_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_595_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_596_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_597_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_598_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N3: nat] :
          ( ( ord_less_nat @ M2 @ N3 )
          | ( M2 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_599_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_600_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M2 @ N3 )
          & ( M2 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_601_zero__induct__lemma,axiom,
    ! [P: nat > $o,K2: nat,I: nat] :
      ( ( P @ K2 )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_602_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_603_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,N: nat] :
      ( ( ord_less_nat @ J @ K2 )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).

% less_imp_diff_less
thf(fact_604_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_605_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_606_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_607_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_608_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_609_le__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_610_eq__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ( minus_minus_nat @ M @ K2 )
            = ( minus_minus_nat @ N @ K2 ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_611_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_612_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_tm,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_tm @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_tm @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_613_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_fm,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_fm @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_fm @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_614_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_615_lift__Suc__mono__le,axiom,
    ! [F: nat > set_tm,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_tm @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_tm @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_616_lift__Suc__mono__le,axiom,
    ! [F: nat > set_fm,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_fm @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_fm @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_617_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_618_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_619_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_620_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_621_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_622_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_623_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_624_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_625_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_626_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_627_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_628_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_629_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_630_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_631_fsubsetI,axiom,
    ! [A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
      ( ! [X: produc6018962875968178549ist_fm] :
          ( ( fmembe3754813877001230652ist_fm @ X @ A2 )
         => ( fmembe3754813877001230652ist_fm @ X @ B4 ) )
     => ( ord_le3986950534092794747ist_fm @ A2 @ B4 ) ) ).

% fsubsetI
thf(fact_632_fsubsetI,axiom,
    ! [A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
      ( ! [X: abstra8520858721168826610m_rule] :
          ( ( fmembe1099044791951439339m_rule @ X @ A2 )
         => ( fmembe1099044791951439339m_rule @ X @ B4 ) )
     => ( ord_le59936386403465346m_rule @ A2 @ B4 ) ) ).

% fsubsetI
thf(fact_633_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N @ K )
         => ( P @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_634_subset__antisym,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B4 )
     => ( ( ord_less_eq_set_tm @ B4 @ A2 )
       => ( A2 = B4 ) ) ) ).

% subset_antisym
thf(fact_635_subset__antisym,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B4 )
     => ( ( ord_less_eq_set_fm @ B4 @ A2 )
       => ( A2 = B4 ) ) ) ).

% subset_antisym
thf(fact_636_subsetI,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ! [X: produc6018962875968178549ist_fm] :
          ( ( member4699826688122452638ist_fm @ X @ A2 )
         => ( member4699826688122452638ist_fm @ X @ B4 ) )
     => ( ord_le1771420097867575541ist_fm @ A2 @ B4 ) ) ).

% subsetI
thf(fact_637_subsetI,axiom,
    ! [A2: set_rule,B4: set_rule] :
      ( ! [X: rule] :
          ( ( member_rule @ X @ A2 )
         => ( member_rule @ X @ B4 ) )
     => ( ord_less_eq_set_rule @ A2 @ B4 ) ) ).

% subsetI
thf(fact_638_subsetI,axiom,
    ! [A2: set_list_fm,B4: set_list_fm] :
      ( ! [X: list_fm] :
          ( ( member_list_fm @ X @ A2 )
         => ( member_list_fm @ X @ B4 ) )
     => ( ord_le7838213414353715577ist_fm @ A2 @ B4 ) ) ).

% subsetI
thf(fact_639_subsetI,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ! [X: tm] :
          ( ( member_tm @ X @ A2 )
         => ( member_tm @ X @ B4 ) )
     => ( ord_less_eq_set_tm @ A2 @ B4 ) ) ).

% subsetI
thf(fact_640_subsetI,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ! [X: fm] :
          ( ( member_fm @ X @ A2 )
         => ( member_fm @ X @ B4 ) )
     => ( ord_less_eq_set_fm @ A2 @ B4 ) ) ).

% subsetI
thf(fact_641_psubsetI,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B4 )
     => ( ( A2 != B4 )
       => ( ord_less_set_tm @ A2 @ B4 ) ) ) ).

% psubsetI
thf(fact_642_psubsetI,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B4 )
     => ( ( A2 != B4 )
       => ( ord_less_set_fm @ A2 @ B4 ) ) ) ).

% psubsetI
thf(fact_643_fminus__iff,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
      ( ( fmembe3754813877001230652ist_fm @ C @ ( minus_7547787838945083330ist_fm @ A2 @ B4 ) )
      = ( ( fmembe3754813877001230652ist_fm @ C @ A2 )
        & ~ ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ).

% fminus_iff
thf(fact_644_fminus__iff,axiom,
    ! [C: abstra8520858721168826610m_rule,A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
      ( ( fmembe1099044791951439339m_rule @ C @ ( minus_4573455687935125755m_rule @ A2 @ B4 ) )
      = ( ( fmembe1099044791951439339m_rule @ C @ A2 )
        & ~ ( fmembe1099044791951439339m_rule @ C @ B4 ) ) ) ).

% fminus_iff
thf(fact_645_fminusI,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
      ( ( fmembe3754813877001230652ist_fm @ C @ A2 )
     => ( ~ ( fmembe3754813877001230652ist_fm @ C @ B4 )
       => ( fmembe3754813877001230652ist_fm @ C @ ( minus_7547787838945083330ist_fm @ A2 @ B4 ) ) ) ) ).

% fminusI
thf(fact_646_fminusI,axiom,
    ! [C: abstra8520858721168826610m_rule,A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
      ( ( fmembe1099044791951439339m_rule @ C @ A2 )
     => ( ~ ( fmembe1099044791951439339m_rule @ C @ B4 )
       => ( fmembe1099044791951439339m_rule @ C @ ( minus_4573455687935125755m_rule @ A2 @ B4 ) ) ) ) ).

% fminusI
thf(fact_647_Collect__mono__iff,axiom,
    ! [P: tm > $o,Q: tm > $o] :
      ( ( ord_less_eq_set_tm @ ( collect_tm @ P ) @ ( collect_tm @ Q ) )
      = ( ! [X4: tm] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_648_Collect__mono__iff,axiom,
    ! [P: fm > $o,Q: fm > $o] :
      ( ( ord_less_eq_set_fm @ ( collect_fm @ P ) @ ( collect_fm @ Q ) )
      = ( ! [X4: fm] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_649_set__eq__subset,axiom,
    ( ( ^ [Y4: set_tm,Z2: set_tm] : ( Y4 = Z2 ) )
    = ( ^ [A5: set_tm,B5: set_tm] :
          ( ( ord_less_eq_set_tm @ A5 @ B5 )
          & ( ord_less_eq_set_tm @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_650_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_651_subset__trans,axiom,
    ! [A2: set_tm,B4: set_tm,C2: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B4 )
     => ( ( ord_less_eq_set_tm @ B4 @ C2 )
       => ( ord_less_eq_set_tm @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_652_subset__trans,axiom,
    ! [A2: set_fm,B4: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B4 )
     => ( ( ord_less_eq_set_fm @ B4 @ C2 )
       => ( ord_less_eq_set_fm @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_653_Collect__mono,axiom,
    ! [P: tm > $o,Q: tm > $o] :
      ( ! [X: tm] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_less_eq_set_tm @ ( collect_tm @ P ) @ ( collect_tm @ Q ) ) ) ).

% Collect_mono
thf(fact_654_Collect__mono,axiom,
    ! [P: fm > $o,Q: fm > $o] :
      ( ! [X: fm] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_less_eq_set_fm @ ( collect_fm @ P ) @ ( collect_fm @ Q ) ) ) ).

% Collect_mono
thf(fact_655_subset__refl,axiom,
    ! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ A2 ) ).

% subset_refl
thf(fact_656_subset__refl,axiom,
    ! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ A2 ) ).

% subset_refl
thf(fact_657_double__diff,axiom,
    ! [A2: set_rule,B4: set_rule,C2: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B4 )
     => ( ( ord_less_eq_set_rule @ B4 @ C2 )
       => ( ( minus_minus_set_rule @ B4 @ ( minus_minus_set_rule @ C2 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_658_double__diff,axiom,
    ! [A2: set_tm,B4: set_tm,C2: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B4 )
     => ( ( ord_less_eq_set_tm @ B4 @ C2 )
       => ( ( minus_minus_set_tm @ B4 @ ( minus_minus_set_tm @ C2 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_659_double__diff,axiom,
    ! [A2: set_fm,B4: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B4 )
     => ( ( ord_less_eq_set_fm @ B4 @ C2 )
       => ( ( minus_minus_set_fm @ B4 @ ( minus_minus_set_fm @ C2 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_660_Diff__subset,axiom,
    ! [A2: set_rule,B4: set_rule] : ( ord_less_eq_set_rule @ ( minus_minus_set_rule @ A2 @ B4 ) @ A2 ) ).

% Diff_subset
thf(fact_661_Diff__subset,axiom,
    ! [A2: set_tm,B4: set_tm] : ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A2 @ B4 ) @ A2 ) ).

% Diff_subset
thf(fact_662_Diff__subset,axiom,
    ! [A2: set_fm,B4: set_fm] : ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A2 @ B4 ) @ A2 ) ).

% Diff_subset
thf(fact_663_subset__iff,axiom,
    ( ord_le1771420097867575541ist_fm
    = ( ^ [A5: set_Pr5202636777678657877ist_fm,B5: set_Pr5202636777678657877ist_fm] :
        ! [T5: produc6018962875968178549ist_fm] :
          ( ( member4699826688122452638ist_fm @ T5 @ A5 )
         => ( member4699826688122452638ist_fm @ T5 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_664_subset__iff,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A5: set_rule,B5: set_rule] :
        ! [T5: rule] :
          ( ( member_rule @ T5 @ A5 )
         => ( member_rule @ T5 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_665_subset__iff,axiom,
    ( ord_le7838213414353715577ist_fm
    = ( ^ [A5: set_list_fm,B5: set_list_fm] :
        ! [T5: list_fm] :
          ( ( member_list_fm @ T5 @ A5 )
         => ( member_list_fm @ T5 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_666_subset__iff,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [A5: set_tm,B5: set_tm] :
        ! [T5: tm] :
          ( ( member_tm @ T5 @ A5 )
         => ( member_tm @ T5 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_667_subset__iff,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A5: set_fm,B5: set_fm] :
        ! [T5: fm] :
          ( ( member_fm @ T5 @ A5 )
         => ( member_fm @ T5 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_668_equalityD2,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ( A2 = B4 )
     => ( ord_less_eq_set_tm @ B4 @ A2 ) ) ).

% equalityD2
thf(fact_669_equalityD2,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ( A2 = B4 )
     => ( ord_less_eq_set_fm @ B4 @ A2 ) ) ).

% equalityD2
thf(fact_670_equalityD1,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ( A2 = B4 )
     => ( ord_less_eq_set_tm @ A2 @ B4 ) ) ).

% equalityD1
thf(fact_671_equalityD1,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ( A2 = B4 )
     => ( ord_less_eq_set_fm @ A2 @ B4 ) ) ).

% equalityD1
thf(fact_672_subset__eq,axiom,
    ( ord_le1771420097867575541ist_fm
    = ( ^ [A5: set_Pr5202636777678657877ist_fm,B5: set_Pr5202636777678657877ist_fm] :
        ! [X4: produc6018962875968178549ist_fm] :
          ( ( member4699826688122452638ist_fm @ X4 @ A5 )
         => ( member4699826688122452638ist_fm @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_673_subset__eq,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A5: set_rule,B5: set_rule] :
        ! [X4: rule] :
          ( ( member_rule @ X4 @ A5 )
         => ( member_rule @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_674_subset__eq,axiom,
    ( ord_le7838213414353715577ist_fm
    = ( ^ [A5: set_list_fm,B5: set_list_fm] :
        ! [X4: list_fm] :
          ( ( member_list_fm @ X4 @ A5 )
         => ( member_list_fm @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_675_subset__eq,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [A5: set_tm,B5: set_tm] :
        ! [X4: tm] :
          ( ( member_tm @ X4 @ A5 )
         => ( member_tm @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_676_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_677_equalityE,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ( A2 = B4 )
     => ~ ( ( ord_less_eq_set_tm @ A2 @ B4 )
         => ~ ( ord_less_eq_set_tm @ B4 @ A2 ) ) ) ).

% equalityE
thf(fact_678_equalityE,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ( A2 = B4 )
     => ~ ( ( ord_less_eq_set_fm @ A2 @ B4 )
         => ~ ( ord_less_eq_set_fm @ B4 @ A2 ) ) ) ).

% equalityE
thf(fact_679_Diff__mono,axiom,
    ! [A2: set_rule,C2: set_rule,D: set_rule,B4: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ C2 )
     => ( ( ord_less_eq_set_rule @ D @ B4 )
       => ( ord_less_eq_set_rule @ ( minus_minus_set_rule @ A2 @ B4 ) @ ( minus_minus_set_rule @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_680_Diff__mono,axiom,
    ! [A2: set_tm,C2: set_tm,D: set_tm,B4: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ C2 )
     => ( ( ord_less_eq_set_tm @ D @ B4 )
       => ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A2 @ B4 ) @ ( minus_minus_set_tm @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_681_Diff__mono,axiom,
    ! [A2: set_fm,C2: set_fm,D: set_fm,B4: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ C2 )
     => ( ( ord_less_eq_set_fm @ D @ B4 )
       => ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A2 @ B4 ) @ ( minus_minus_set_fm @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_682_subsetD,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm,C: produc6018962875968178549ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ A2 @ B4 )
     => ( ( member4699826688122452638ist_fm @ C @ A2 )
       => ( member4699826688122452638ist_fm @ C @ B4 ) ) ) ).

% subsetD
thf(fact_683_subsetD,axiom,
    ! [A2: set_rule,B4: set_rule,C: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B4 )
     => ( ( member_rule @ C @ A2 )
       => ( member_rule @ C @ B4 ) ) ) ).

% subsetD
thf(fact_684_subsetD,axiom,
    ! [A2: set_list_fm,B4: set_list_fm,C: list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ A2 @ B4 )
     => ( ( member_list_fm @ C @ A2 )
       => ( member_list_fm @ C @ B4 ) ) ) ).

% subsetD
thf(fact_685_subsetD,axiom,
    ! [A2: set_tm,B4: set_tm,C: tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B4 )
     => ( ( member_tm @ C @ A2 )
       => ( member_tm @ C @ B4 ) ) ) ).

% subsetD
thf(fact_686_subsetD,axiom,
    ! [A2: set_fm,B4: set_fm,C: fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B4 )
     => ( ( member_fm @ C @ A2 )
       => ( member_fm @ C @ B4 ) ) ) ).

% subsetD
thf(fact_687_in__mono,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm,X3: produc6018962875968178549ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ A2 @ B4 )
     => ( ( member4699826688122452638ist_fm @ X3 @ A2 )
       => ( member4699826688122452638ist_fm @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_688_in__mono,axiom,
    ! [A2: set_rule,B4: set_rule,X3: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B4 )
     => ( ( member_rule @ X3 @ A2 )
       => ( member_rule @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_689_in__mono,axiom,
    ! [A2: set_list_fm,B4: set_list_fm,X3: list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ A2 @ B4 )
     => ( ( member_list_fm @ X3 @ A2 )
       => ( member_list_fm @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_690_in__mono,axiom,
    ! [A2: set_tm,B4: set_tm,X3: tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B4 )
     => ( ( member_tm @ X3 @ A2 )
       => ( member_tm @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_691_in__mono,axiom,
    ! [A2: set_fm,B4: set_fm,X3: fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B4 )
     => ( ( member_fm @ X3 @ A2 )
       => ( member_fm @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_692_psubsetE,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ( ord_less_set_tm @ A2 @ B4 )
     => ~ ( ( ord_less_eq_set_tm @ A2 @ B4 )
         => ( ord_less_eq_set_tm @ B4 @ A2 ) ) ) ).

% psubsetE
thf(fact_693_psubsetE,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ( ord_less_set_fm @ A2 @ B4 )
     => ~ ( ( ord_less_eq_set_fm @ A2 @ B4 )
         => ( ord_less_eq_set_fm @ B4 @ A2 ) ) ) ).

% psubsetE
thf(fact_694_psubset__eq,axiom,
    ( ord_less_set_tm
    = ( ^ [A5: set_tm,B5: set_tm] :
          ( ( ord_less_eq_set_tm @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% psubset_eq
thf(fact_695_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_696_psubset__imp__subset,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ( ord_less_set_tm @ A2 @ B4 )
     => ( ord_less_eq_set_tm @ A2 @ B4 ) ) ).

% psubset_imp_subset
thf(fact_697_psubset__imp__subset,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ( ord_less_set_fm @ A2 @ B4 )
     => ( ord_less_eq_set_fm @ A2 @ B4 ) ) ).

% psubset_imp_subset
thf(fact_698_psubset__subset__trans,axiom,
    ! [A2: set_tm,B4: set_tm,C2: set_tm] :
      ( ( ord_less_set_tm @ A2 @ B4 )
     => ( ( ord_less_eq_set_tm @ B4 @ C2 )
       => ( ord_less_set_tm @ A2 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_699_psubset__subset__trans,axiom,
    ! [A2: set_fm,B4: set_fm,C2: set_fm] :
      ( ( ord_less_set_fm @ A2 @ B4 )
     => ( ( ord_less_eq_set_fm @ B4 @ C2 )
       => ( ord_less_set_fm @ A2 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_700_subset__not__subset__eq,axiom,
    ( ord_less_set_tm
    = ( ^ [A5: set_tm,B5: set_tm] :
          ( ( ord_less_eq_set_tm @ A5 @ B5 )
          & ~ ( ord_less_eq_set_tm @ B5 @ A5 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_701_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_702_subset__psubset__trans,axiom,
    ! [A2: set_tm,B4: set_tm,C2: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B4 )
     => ( ( ord_less_set_tm @ B4 @ C2 )
       => ( ord_less_set_tm @ A2 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_703_subset__psubset__trans,axiom,
    ! [A2: set_fm,B4: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B4 )
     => ( ( ord_less_set_fm @ B4 @ C2 )
       => ( ord_less_set_fm @ A2 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_704_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [A5: set_tm,B5: set_tm] :
          ( ( ord_less_set_tm @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_705_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_706_fminusD2,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
      ( ( fmembe3754813877001230652ist_fm @ C @ ( minus_7547787838945083330ist_fm @ A2 @ B4 ) )
     => ~ ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ).

% fminusD2
thf(fact_707_fminusD2,axiom,
    ! [C: abstra8520858721168826610m_rule,A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
      ( ( fmembe1099044791951439339m_rule @ C @ ( minus_4573455687935125755m_rule @ A2 @ B4 ) )
     => ~ ( fmembe1099044791951439339m_rule @ C @ B4 ) ) ).

% fminusD2
thf(fact_708_fminusD1,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
      ( ( fmembe3754813877001230652ist_fm @ C @ ( minus_7547787838945083330ist_fm @ A2 @ B4 ) )
     => ( fmembe3754813877001230652ist_fm @ C @ A2 ) ) ).

% fminusD1
thf(fact_709_fminusD1,axiom,
    ! [C: abstra8520858721168826610m_rule,A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
      ( ( fmembe1099044791951439339m_rule @ C @ ( minus_4573455687935125755m_rule @ A2 @ B4 ) )
     => ( fmembe1099044791951439339m_rule @ C @ A2 ) ) ).

% fminusD1
thf(fact_710_fminusE,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
      ( ( fmembe3754813877001230652ist_fm @ C @ ( minus_7547787838945083330ist_fm @ A2 @ B4 ) )
     => ~ ( ( fmembe3754813877001230652ist_fm @ C @ A2 )
         => ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ).

% fminusE
thf(fact_711_fminusE,axiom,
    ! [C: abstra8520858721168826610m_rule,A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
      ( ( fmembe1099044791951439339m_rule @ C @ ( minus_4573455687935125755m_rule @ A2 @ B4 ) )
     => ~ ( ( fmembe1099044791951439339m_rule @ C @ A2 )
         => ( fmembe1099044791951439339m_rule @ C @ B4 ) ) ) ).

% fminusE
thf(fact_712_pfsubset__imp__ex__fmem,axiom,
    ! [A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
      ( ( ord_le7716396445699002503ist_fm @ A2 @ B4 )
     => ? [B2: produc6018962875968178549ist_fm] : ( fmembe3754813877001230652ist_fm @ B2 @ ( minus_7547787838945083330ist_fm @ B4 @ A2 ) ) ) ).

% pfsubset_imp_ex_fmem
thf(fact_713_pfsubset__imp__ex__fmem,axiom,
    ! [A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
      ( ( ord_le6302656846121046646m_rule @ A2 @ B4 )
     => ? [B2: abstra8520858721168826610m_rule] : ( fmembe1099044791951439339m_rule @ B2 @ ( minus_4573455687935125755m_rule @ B4 @ A2 ) ) ) ).

% pfsubset_imp_ex_fmem
thf(fact_714_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_715_pfsubsetD,axiom,
    ! [A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
      ( ( ord_le7716396445699002503ist_fm @ A2 @ B4 )
     => ( ( fmembe3754813877001230652ist_fm @ C @ A2 )
       => ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ).

% pfsubsetD
thf(fact_716_pfsubsetD,axiom,
    ! [A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,C: abstra8520858721168826610m_rule] :
      ( ( ord_le6302656846121046646m_rule @ A2 @ B4 )
     => ( ( fmembe1099044791951439339m_rule @ C @ A2 )
       => ( fmembe1099044791951439339m_rule @ C @ B4 ) ) ) ).

% pfsubsetD
thf(fact_717_fset__eqI,axiom,
    ! [A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
      ( ! [X: produc6018962875968178549ist_fm] :
          ( ( fmembe3754813877001230652ist_fm @ X @ A2 )
          = ( fmembe3754813877001230652ist_fm @ X @ B4 ) )
     => ( A2 = B4 ) ) ).

% fset_eqI
thf(fact_718_fset__eqI,axiom,
    ! [A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule] :
      ( ! [X: abstra8520858721168826610m_rule] :
          ( ( fmembe1099044791951439339m_rule @ X @ A2 )
          = ( fmembe1099044791951439339m_rule @ X @ B4 ) )
     => ( A2 = B4 ) ) ).

% fset_eqI
thf(fact_719_fequalityCE,axiom,
    ! [A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
      ( ( A2 = B4 )
     => ( ( ( fmembe3754813877001230652ist_fm @ C @ A2 )
         => ~ ( fmembe3754813877001230652ist_fm @ C @ B4 ) )
       => ~ ( ~ ( fmembe3754813877001230652ist_fm @ C @ A2 )
           => ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_720_fequalityCE,axiom,
    ! [A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,C: abstra8520858721168826610m_rule] :
      ( ( A2 = B4 )
     => ( ( ( fmembe1099044791951439339m_rule @ C @ A2 )
         => ~ ( fmembe1099044791951439339m_rule @ C @ B4 ) )
       => ~ ( ~ ( fmembe1099044791951439339m_rule @ C @ A2 )
           => ( fmembe1099044791951439339m_rule @ C @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_721_eq__fmem__trans,axiom,
    ! [A: produc6018962875968178549ist_fm,B: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm] :
      ( ( A = B )
     => ( ( fmembe3754813877001230652ist_fm @ B @ A2 )
       => ( fmembe3754813877001230652ist_fm @ A @ A2 ) ) ) ).

% eq_fmem_trans
thf(fact_722_eq__fmem__trans,axiom,
    ! [A: abstra8520858721168826610m_rule,B: abstra8520858721168826610m_rule,A2: fset_A8849196716589201442m_rule] :
      ( ( A = B )
     => ( ( fmembe1099044791951439339m_rule @ B @ A2 )
       => ( fmembe1099044791951439339m_rule @ A @ A2 ) ) ) ).

% eq_fmem_trans
thf(fact_723_eqfset__imp__iff,axiom,
    ! [A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,X3: produc6018962875968178549ist_fm] :
      ( ( A2 = B4 )
     => ( ( fmembe3754813877001230652ist_fm @ X3 @ A2 )
        = ( fmembe3754813877001230652ist_fm @ X3 @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_724_eqfset__imp__iff,axiom,
    ! [A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,X3: abstra8520858721168826610m_rule] :
      ( ( A2 = B4 )
     => ( ( fmembe1099044791951439339m_rule @ X3 @ A2 )
        = ( fmembe1099044791951439339m_rule @ X3 @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_725_if__split__fmem1,axiom,
    ! [Q: $o,X3: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm,B: fset_P8989946509869081563ist_fm] :
      ( ( fmembe3754813877001230652ist_fm @ ( if_Pro4760001780252510779ist_fm @ Q @ X3 @ Y ) @ B )
      = ( ( Q
         => ( fmembe3754813877001230652ist_fm @ X3 @ B ) )
        & ( ~ Q
         => ( fmembe3754813877001230652ist_fm @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_726_if__split__fmem1,axiom,
    ! [Q: $o,X3: abstra8520858721168826610m_rule,Y: abstra8520858721168826610m_rule,B: fset_A8849196716589201442m_rule] :
      ( ( fmembe1099044791951439339m_rule @ ( if_Abs1293773526126661548m_rule @ Q @ X3 @ Y ) @ B )
      = ( ( Q
         => ( fmembe1099044791951439339m_rule @ X3 @ B ) )
        & ( ~ Q
         => ( fmembe1099044791951439339m_rule @ Y @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_727_if__split__fmem2,axiom,
    ! [A: produc6018962875968178549ist_fm,Q: $o,X3: fset_P8989946509869081563ist_fm,Y: fset_P8989946509869081563ist_fm] :
      ( ( fmembe3754813877001230652ist_fm @ A @ ( if_fse7999432387889793441ist_fm @ Q @ X3 @ Y ) )
      = ( ( Q
         => ( fmembe3754813877001230652ist_fm @ A @ X3 ) )
        & ( ~ Q
         => ( fmembe3754813877001230652ist_fm @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_728_if__split__fmem2,axiom,
    ! [A: abstra8520858721168826610m_rule,Q: $o,X3: fset_A8849196716589201442m_rule,Y: fset_A8849196716589201442m_rule] :
      ( ( fmembe1099044791951439339m_rule @ A @ ( if_fse9105678332773368284m_rule @ Q @ X3 @ Y ) )
      = ( ( Q
         => ( fmembe1099044791951439339m_rule @ A @ X3 ) )
        & ( ~ Q
         => ( fmembe1099044791951439339m_rule @ A @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_729_eqfelem__imp__iff,axiom,
    ! [X3: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm] :
      ( ( X3 = Y )
     => ( ( fmembe3754813877001230652ist_fm @ X3 @ A2 )
        = ( fmembe3754813877001230652ist_fm @ Y @ A2 ) ) ) ).

% eqfelem_imp_iff
thf(fact_730_eqfelem__imp__iff,axiom,
    ! [X3: abstra8520858721168826610m_rule,Y: abstra8520858721168826610m_rule,A2: fset_A8849196716589201442m_rule] :
      ( ( X3 = Y )
     => ( ( fmembe1099044791951439339m_rule @ X3 @ A2 )
        = ( fmembe1099044791951439339m_rule @ Y @ A2 ) ) ) ).

% eqfelem_imp_iff
thf(fact_731_fin__mono,axiom,
    ! [A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,X3: produc6018962875968178549ist_fm] :
      ( ( ord_le3986950534092794747ist_fm @ A2 @ B4 )
     => ( ( fmembe3754813877001230652ist_fm @ X3 @ A2 )
       => ( fmembe3754813877001230652ist_fm @ X3 @ B4 ) ) ) ).

% fin_mono
thf(fact_732_fin__mono,axiom,
    ! [A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,X3: abstra8520858721168826610m_rule] :
      ( ( ord_le59936386403465346m_rule @ A2 @ B4 )
     => ( ( fmembe1099044791951439339m_rule @ X3 @ A2 )
       => ( fmembe1099044791951439339m_rule @ X3 @ B4 ) ) ) ).

% fin_mono
thf(fact_733_fsubsetD,axiom,
    ! [A2: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
      ( ( ord_le3986950534092794747ist_fm @ A2 @ B4 )
     => ( ( fmembe3754813877001230652ist_fm @ C @ A2 )
       => ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ).

% fsubsetD
thf(fact_734_fsubsetD,axiom,
    ! [A2: fset_A8849196716589201442m_rule,B4: fset_A8849196716589201442m_rule,C: abstra8520858721168826610m_rule] :
      ( ( ord_le59936386403465346m_rule @ A2 @ B4 )
     => ( ( fmembe1099044791951439339m_rule @ C @ A2 )
       => ( fmembe1099044791951439339m_rule @ C @ B4 ) ) ) ).

% fsubsetD
thf(fact_735_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A @ C3 )
              & ( ord_less_eq_nat @ C3 @ B )
              & ! [X6: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X6 )
                    & ( ord_less_nat @ X6 @ C3 ) )
                 => ( P @ X6 ) )
              & ! [D2: nat] :
                  ( ! [X: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X )
                        & ( ord_less_nat @ X @ D2 ) )
                     => ( P @ X ) )
                 => ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_736_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_737_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_738_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ord_less_eq_nat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_739_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ord_less_eq_nat @ X6 @ T ) ) ).

% minf(6)
thf(fact_740_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).

% minf(8)
thf(fact_741_DiffI,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ C @ A2 )
     => ( ~ ( member4699826688122452638ist_fm @ C @ B4 )
       => ( member4699826688122452638ist_fm @ C @ ( minus_7584274679010293948ist_fm @ A2 @ B4 ) ) ) ) ).

% DiffI
thf(fact_742_DiffI,axiom,
    ! [C: list_fm,A2: set_list_fm,B4: set_list_fm] :
      ( ( member_list_fm @ C @ A2 )
     => ( ~ ( member_list_fm @ C @ B4 )
       => ( member_list_fm @ C @ ( minus_639611354763871680ist_fm @ A2 @ B4 ) ) ) ) ).

% DiffI
thf(fact_743_DiffI,axiom,
    ! [C: tm,A2: set_tm,B4: set_tm] :
      ( ( member_tm @ C @ A2 )
     => ( ~ ( member_tm @ C @ B4 )
       => ( member_tm @ C @ ( minus_minus_set_tm @ A2 @ B4 ) ) ) ) ).

% DiffI
thf(fact_744_DiffI,axiom,
    ! [C: fm,A2: set_fm,B4: set_fm] :
      ( ( member_fm @ C @ A2 )
     => ( ~ ( member_fm @ C @ B4 )
       => ( member_fm @ C @ ( minus_minus_set_fm @ A2 @ B4 ) ) ) ) ).

% DiffI
thf(fact_745_DiffI,axiom,
    ! [C: rule,A2: set_rule,B4: set_rule] :
      ( ( member_rule @ C @ A2 )
     => ( ~ ( member_rule @ C @ B4 )
       => ( member_rule @ C @ ( minus_minus_set_rule @ A2 @ B4 ) ) ) ) ).

% DiffI
thf(fact_746_Diff__iff,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ C @ ( minus_7584274679010293948ist_fm @ A2 @ B4 ) )
      = ( ( member4699826688122452638ist_fm @ C @ A2 )
        & ~ ( member4699826688122452638ist_fm @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_747_Diff__iff,axiom,
    ! [C: list_fm,A2: set_list_fm,B4: set_list_fm] :
      ( ( member_list_fm @ C @ ( minus_639611354763871680ist_fm @ A2 @ B4 ) )
      = ( ( member_list_fm @ C @ A2 )
        & ~ ( member_list_fm @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_748_Diff__iff,axiom,
    ! [C: tm,A2: set_tm,B4: set_tm] :
      ( ( member_tm @ C @ ( minus_minus_set_tm @ A2 @ B4 ) )
      = ( ( member_tm @ C @ A2 )
        & ~ ( member_tm @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_749_Diff__iff,axiom,
    ! [C: fm,A2: set_fm,B4: set_fm] :
      ( ( member_fm @ C @ ( minus_minus_set_fm @ A2 @ B4 ) )
      = ( ( member_fm @ C @ A2 )
        & ~ ( member_fm @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_750_Diff__iff,axiom,
    ! [C: rule,A2: set_rule,B4: set_rule] :
      ( ( member_rule @ C @ ( minus_minus_set_rule @ A2 @ B4 ) )
      = ( ( member_rule @ C @ A2 )
        & ~ ( member_rule @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_751_Diff__idemp,axiom,
    ! [A2: set_rule,B4: set_rule] :
      ( ( minus_minus_set_rule @ ( minus_minus_set_rule @ A2 @ B4 ) @ B4 )
      = ( minus_minus_set_rule @ A2 @ B4 ) ) ).

% Diff_idemp
thf(fact_752_psubset__imp__ex__mem,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le9141428925376846849ist_fm @ A2 @ B4 )
     => ? [B2: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ B2 @ ( minus_7584274679010293948ist_fm @ B4 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_753_psubset__imp__ex__mem,axiom,
    ! [A2: set_list_fm,B4: set_list_fm] :
      ( ( ord_less_set_list_fm @ A2 @ B4 )
     => ? [B2: list_fm] : ( member_list_fm @ B2 @ ( minus_639611354763871680ist_fm @ B4 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_754_psubset__imp__ex__mem,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ( ord_less_set_tm @ A2 @ B4 )
     => ? [B2: tm] : ( member_tm @ B2 @ ( minus_minus_set_tm @ B4 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_755_psubset__imp__ex__mem,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ( ord_less_set_fm @ A2 @ B4 )
     => ? [B2: fm] : ( member_fm @ B2 @ ( minus_minus_set_fm @ B4 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_756_psubset__imp__ex__mem,axiom,
    ! [A2: set_rule,B4: set_rule] :
      ( ( ord_less_set_rule @ A2 @ B4 )
     => ? [B2: rule] : ( member_rule @ B2 @ ( minus_minus_set_rule @ B4 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_757_DiffE,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ C @ ( minus_7584274679010293948ist_fm @ A2 @ B4 ) )
     => ~ ( ( member4699826688122452638ist_fm @ C @ A2 )
         => ( member4699826688122452638ist_fm @ C @ B4 ) ) ) ).

% DiffE
thf(fact_758_DiffE,axiom,
    ! [C: list_fm,A2: set_list_fm,B4: set_list_fm] :
      ( ( member_list_fm @ C @ ( minus_639611354763871680ist_fm @ A2 @ B4 ) )
     => ~ ( ( member_list_fm @ C @ A2 )
         => ( member_list_fm @ C @ B4 ) ) ) ).

% DiffE
thf(fact_759_DiffE,axiom,
    ! [C: tm,A2: set_tm,B4: set_tm] :
      ( ( member_tm @ C @ ( minus_minus_set_tm @ A2 @ B4 ) )
     => ~ ( ( member_tm @ C @ A2 )
         => ( member_tm @ C @ B4 ) ) ) ).

% DiffE
thf(fact_760_DiffE,axiom,
    ! [C: fm,A2: set_fm,B4: set_fm] :
      ( ( member_fm @ C @ ( minus_minus_set_fm @ A2 @ B4 ) )
     => ~ ( ( member_fm @ C @ A2 )
         => ( member_fm @ C @ B4 ) ) ) ).

% DiffE
thf(fact_761_DiffE,axiom,
    ! [C: rule,A2: set_rule,B4: set_rule] :
      ( ( member_rule @ C @ ( minus_minus_set_rule @ A2 @ B4 ) )
     => ~ ( ( member_rule @ C @ A2 )
         => ( member_rule @ C @ B4 ) ) ) ).

% DiffE
thf(fact_762_DiffD1,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ C @ ( minus_7584274679010293948ist_fm @ A2 @ B4 ) )
     => ( member4699826688122452638ist_fm @ C @ A2 ) ) ).

% DiffD1
thf(fact_763_DiffD1,axiom,
    ! [C: list_fm,A2: set_list_fm,B4: set_list_fm] :
      ( ( member_list_fm @ C @ ( minus_639611354763871680ist_fm @ A2 @ B4 ) )
     => ( member_list_fm @ C @ A2 ) ) ).

% DiffD1
thf(fact_764_DiffD1,axiom,
    ! [C: tm,A2: set_tm,B4: set_tm] :
      ( ( member_tm @ C @ ( minus_minus_set_tm @ A2 @ B4 ) )
     => ( member_tm @ C @ A2 ) ) ).

% DiffD1
thf(fact_765_DiffD1,axiom,
    ! [C: fm,A2: set_fm,B4: set_fm] :
      ( ( member_fm @ C @ ( minus_minus_set_fm @ A2 @ B4 ) )
     => ( member_fm @ C @ A2 ) ) ).

% DiffD1
thf(fact_766_DiffD1,axiom,
    ! [C: rule,A2: set_rule,B4: set_rule] :
      ( ( member_rule @ C @ ( minus_minus_set_rule @ A2 @ B4 ) )
     => ( member_rule @ C @ A2 ) ) ).

% DiffD1
thf(fact_767_DiffD2,axiom,
    ! [C: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ C @ ( minus_7584274679010293948ist_fm @ A2 @ B4 ) )
     => ~ ( member4699826688122452638ist_fm @ C @ B4 ) ) ).

% DiffD2
thf(fact_768_DiffD2,axiom,
    ! [C: list_fm,A2: set_list_fm,B4: set_list_fm] :
      ( ( member_list_fm @ C @ ( minus_639611354763871680ist_fm @ A2 @ B4 ) )
     => ~ ( member_list_fm @ C @ B4 ) ) ).

% DiffD2
thf(fact_769_DiffD2,axiom,
    ! [C: tm,A2: set_tm,B4: set_tm] :
      ( ( member_tm @ C @ ( minus_minus_set_tm @ A2 @ B4 ) )
     => ~ ( member_tm @ C @ B4 ) ) ).

% DiffD2
thf(fact_770_DiffD2,axiom,
    ! [C: fm,A2: set_fm,B4: set_fm] :
      ( ( member_fm @ C @ ( minus_minus_set_fm @ A2 @ B4 ) )
     => ~ ( member_fm @ C @ B4 ) ) ).

% DiffD2
thf(fact_771_DiffD2,axiom,
    ! [C: rule,A2: set_rule,B4: set_rule] :
      ( ( member_rule @ C @ ( minus_minus_set_rule @ A2 @ B4 ) )
     => ~ ( member_rule @ C @ B4 ) ) ).

% DiffD2
thf(fact_772_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_773_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_774_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_775_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_776_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_777_pinf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ Z4 @ X )
         => ( ( P @ X )
            = ( P4 @ X ) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ Z4 @ X )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_778_pinf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ Z4 @ X )
         => ( ( P @ X )
            = ( P4 @ X ) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ Z4 @ X )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_779_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_780_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_781_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ~ ( ord_less_nat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_782_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ord_less_nat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_783_minf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ X @ Z4 )
         => ( ( P @ X )
            = ( P4 @ X ) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ X @ Z4 )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_784_minf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ X @ Z4 )
         => ( ( P @ X )
            = ( P4 @ X ) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ X @ Z4 )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P4 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_785_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_786_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_787_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ord_less_nat @ X6 @ T ) ) ).

% minf(5)
thf(fact_788_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ~ ( ord_less_nat @ T @ X6 ) ) ).

% minf(7)
thf(fact_789_mkTree_Osimps_I1_J,axiom,
    ! [Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra5180277867158163941m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) )
      = ( produc1733806532565653680m_rule @ S @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) ) ).

% mkTree.simps(1)
thf(fact_790_konig_Osimps_I1_J,axiom,
    ! [T: abstract_tree_rule] :
      ( ( shd_rule @ ( abstract_konig_rule @ T ) )
      = ( abstract_root_rule @ T ) ) ).

% konig.simps(1)
thf(fact_791_konig_Osimps_I1_J,axiom,
    ! [T: abstra8520858721168826610m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( abstra9098142006323500257m_rule @ T ) )
      = ( abstra5180277867158163941m_rule @ T ) ) ).

% konig.simps(1)
thf(fact_792_top_Oordering__top__axioms,axiom,
    orderi4677956558335502598ist_fm @ ord_le1771420097867575541ist_fm @ ord_le9141428925376846849ist_fm @ top_to1730629564055774885ist_fm ).

% top.ordering_top_axioms
thf(fact_793_top_Oordering__top__axioms,axiom,
    ordering_top_set_tm @ ord_less_eq_set_tm @ ord_less_set_tm @ top_top_set_tm ).

% top.ordering_top_axioms
thf(fact_794_top_Oordering__top__axioms,axiom,
    ordering_top_set_fm @ ord_less_eq_set_fm @ ord_less_set_fm @ top_top_set_fm ).

% top.ordering_top_axioms
thf(fact_795_per__def,axiom,
    ! [R: rule] :
      ( ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R )
      = ( ! [S3: produc6018962875968178549ist_fm,R1: rule,Sl5: fset_P8989946509869081563ist_fm,S8: produc6018962875968178549ist_fm] :
            ( ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
              & ( abstra1707737187183579335ist_fm @ eff @ R @ S3 )
              & ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ rules ) @ ( insert_rule @ R @ bot_bot_set_rule ) ) )
              & ( eff @ R1 @ S3 @ Sl5 )
              & ( fmembe3754813877001230652ist_fm @ S8 @ Sl5 ) )
           => ( abstra1707737187183579335ist_fm @ eff @ R @ S8 ) ) ) ) ).

% per_def
thf(fact_796__092_060open_062_092_060forall_062k_092_060ge_062m_O_At_A_092_060in_062_Aset_A_Iptms_A_Ishd_A_Istl_A_Isdrop_Ak_Asteps_J_J_J_J_092_060close_062,axiom,
    ! [K3: nat] :
      ( ( ord_less_eq_nat @ m @ K3 )
     => ( member_tm @ t @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ ( sdrop_8169176516188972301m_rule @ K3 @ steps ) ) ) ) ) ) ) ).

% \<open>\<forall>k\<ge>m. t \<in> set (ptms (shd (stl (sdrop k steps))))\<close>
thf(fact_797_empty__Collect__eq,axiom,
    ! [P: rule > $o] :
      ( ( bot_bot_set_rule
        = ( collect_rule @ P ) )
      = ( ! [X4: rule] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_798_Collect__empty__eq,axiom,
    ! [P: rule > $o] :
      ( ( ( collect_rule @ P )
        = bot_bot_set_rule )
      = ( ! [X4: rule] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_799_all__not__in__conv,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ( ! [X4: produc6018962875968178549ist_fm] :
            ~ ( member4699826688122452638ist_fm @ X4 @ A2 ) )
      = ( A2 = bot_bo1798318783834493889ist_fm ) ) ).

% all_not_in_conv
thf(fact_800_all__not__in__conv,axiom,
    ! [A2: set_list_fm] :
      ( ( ! [X4: list_fm] :
            ~ ( member_list_fm @ X4 @ A2 ) )
      = ( A2 = bot_bot_set_list_fm ) ) ).

% all_not_in_conv
thf(fact_801_all__not__in__conv,axiom,
    ! [A2: set_tm] :
      ( ( ! [X4: tm] :
            ~ ( member_tm @ X4 @ A2 ) )
      = ( A2 = bot_bot_set_tm ) ) ).

% all_not_in_conv
thf(fact_802_all__not__in__conv,axiom,
    ! [A2: set_fm] :
      ( ( ! [X4: fm] :
            ~ ( member_fm @ X4 @ A2 ) )
      = ( A2 = bot_bot_set_fm ) ) ).

% all_not_in_conv
thf(fact_803_all__not__in__conv,axiom,
    ! [A2: set_rule] :
      ( ( ! [X4: rule] :
            ~ ( member_rule @ X4 @ A2 ) )
      = ( A2 = bot_bot_set_rule ) ) ).

% all_not_in_conv
thf(fact_804_empty__iff,axiom,
    ! [C: produc6018962875968178549ist_fm] :
      ~ ( member4699826688122452638ist_fm @ C @ bot_bo1798318783834493889ist_fm ) ).

% empty_iff
thf(fact_805_empty__iff,axiom,
    ! [C: list_fm] :
      ~ ( member_list_fm @ C @ bot_bot_set_list_fm ) ).

% empty_iff
thf(fact_806_empty__iff,axiom,
    ! [C: tm] :
      ~ ( member_tm @ C @ bot_bot_set_tm ) ).

% empty_iff
thf(fact_807_empty__iff,axiom,
    ! [C: fm] :
      ~ ( member_fm @ C @ bot_bot_set_fm ) ).

% empty_iff
thf(fact_808_empty__iff,axiom,
    ! [C: rule] :
      ~ ( member_rule @ C @ bot_bot_set_rule ) ).

% empty_iff
thf(fact_809_insert__absorb2,axiom,
    ! [X3: rule,A2: set_rule] :
      ( ( insert_rule @ X3 @ ( insert_rule @ X3 @ A2 ) )
      = ( insert_rule @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_810_insert__iff,axiom,
    ! [A: produc6018962875968178549ist_fm,B: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ A @ ( insert4380798862274524997ist_fm @ B @ A2 ) )
      = ( ( A = B )
        | ( member4699826688122452638ist_fm @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_811_insert__iff,axiom,
    ! [A: rule,B: rule,A2: set_rule] :
      ( ( member_rule @ A @ ( insert_rule @ B @ A2 ) )
      = ( ( A = B )
        | ( member_rule @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_812_insert__iff,axiom,
    ! [A: list_fm,B: list_fm,A2: set_list_fm] :
      ( ( member_list_fm @ A @ ( insert_list_fm @ B @ A2 ) )
      = ( ( A = B )
        | ( member_list_fm @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_813_insert__iff,axiom,
    ! [A: tm,B: tm,A2: set_tm] :
      ( ( member_tm @ A @ ( insert_tm @ B @ A2 ) )
      = ( ( A = B )
        | ( member_tm @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_814_insert__iff,axiom,
    ! [A: fm,B: fm,A2: set_fm] :
      ( ( member_fm @ A @ ( insert_fm @ B @ A2 ) )
      = ( ( A = B )
        | ( member_fm @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_815_insertCI,axiom,
    ! [A: produc6018962875968178549ist_fm,B4: set_Pr5202636777678657877ist_fm,B: produc6018962875968178549ist_fm] :
      ( ( ~ ( member4699826688122452638ist_fm @ A @ B4 )
       => ( A = B ) )
     => ( member4699826688122452638ist_fm @ A @ ( insert4380798862274524997ist_fm @ B @ B4 ) ) ) ).

% insertCI
thf(fact_816_insertCI,axiom,
    ! [A: rule,B4: set_rule,B: rule] :
      ( ( ~ ( member_rule @ A @ B4 )
       => ( A = B ) )
     => ( member_rule @ A @ ( insert_rule @ B @ B4 ) ) ) ).

% insertCI
thf(fact_817_insertCI,axiom,
    ! [A: list_fm,B4: set_list_fm,B: list_fm] :
      ( ( ~ ( member_list_fm @ A @ B4 )
       => ( A = B ) )
     => ( member_list_fm @ A @ ( insert_list_fm @ B @ B4 ) ) ) ).

% insertCI
thf(fact_818_insertCI,axiom,
    ! [A: tm,B4: set_tm,B: tm] :
      ( ( ~ ( member_tm @ A @ B4 )
       => ( A = B ) )
     => ( member_tm @ A @ ( insert_tm @ B @ B4 ) ) ) ).

% insertCI
thf(fact_819_insertCI,axiom,
    ! [A: fm,B4: set_fm,B: fm] :
      ( ( ~ ( member_fm @ A @ B4 )
       => ( A = B ) )
     => ( member_fm @ A @ ( insert_fm @ B @ B4 ) ) ) ).

% insertCI
thf(fact_820_empty__subsetI,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ bot_bot_set_rule @ A2 ) ).

% empty_subsetI
thf(fact_821_empty__subsetI,axiom,
    ! [A2: set_tm] : ( ord_less_eq_set_tm @ bot_bot_set_tm @ A2 ) ).

% empty_subsetI
thf(fact_822_empty__subsetI,axiom,
    ! [A2: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A2 ) ).

% empty_subsetI
thf(fact_823_subset__empty,axiom,
    ! [A2: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ bot_bot_set_rule )
      = ( A2 = bot_bot_set_rule ) ) ).

% subset_empty
thf(fact_824_subset__empty,axiom,
    ! [A2: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ bot_bot_set_tm )
      = ( A2 = bot_bot_set_tm ) ) ).

% subset_empty
thf(fact_825_subset__empty,axiom,
    ! [A2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
      = ( A2 = bot_bot_set_fm ) ) ).

% subset_empty
thf(fact_826_singletonI,axiom,
    ! [A: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ A @ ( insert4380798862274524997ist_fm @ A @ bot_bo1798318783834493889ist_fm ) ) ).

% singletonI
thf(fact_827_singletonI,axiom,
    ! [A: list_fm] : ( member_list_fm @ A @ ( insert_list_fm @ A @ bot_bot_set_list_fm ) ) ).

% singletonI
thf(fact_828_singletonI,axiom,
    ! [A: tm] : ( member_tm @ A @ ( insert_tm @ A @ bot_bot_set_tm ) ) ).

% singletonI
thf(fact_829_singletonI,axiom,
    ! [A: fm] : ( member_fm @ A @ ( insert_fm @ A @ bot_bot_set_fm ) ) ).

% singletonI
thf(fact_830_singletonI,axiom,
    ! [A: rule] : ( member_rule @ A @ ( insert_rule @ A @ bot_bot_set_rule ) ) ).

% singletonI
thf(fact_831_insert__subset,axiom,
    ! [X3: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ ( insert4380798862274524997ist_fm @ X3 @ A2 ) @ B4 )
      = ( ( member4699826688122452638ist_fm @ X3 @ B4 )
        & ( ord_le1771420097867575541ist_fm @ A2 @ B4 ) ) ) ).

% insert_subset
thf(fact_832_insert__subset,axiom,
    ! [X3: rule,A2: set_rule,B4: set_rule] :
      ( ( ord_less_eq_set_rule @ ( insert_rule @ X3 @ A2 ) @ B4 )
      = ( ( member_rule @ X3 @ B4 )
        & ( ord_less_eq_set_rule @ A2 @ B4 ) ) ) ).

% insert_subset
thf(fact_833_insert__subset,axiom,
    ! [X3: list_fm,A2: set_list_fm,B4: set_list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ ( insert_list_fm @ X3 @ A2 ) @ B4 )
      = ( ( member_list_fm @ X3 @ B4 )
        & ( ord_le7838213414353715577ist_fm @ A2 @ B4 ) ) ) ).

% insert_subset
thf(fact_834_insert__subset,axiom,
    ! [X3: tm,A2: set_tm,B4: set_tm] :
      ( ( ord_less_eq_set_tm @ ( insert_tm @ X3 @ A2 ) @ B4 )
      = ( ( member_tm @ X3 @ B4 )
        & ( ord_less_eq_set_tm @ A2 @ B4 ) ) ) ).

% insert_subset
thf(fact_835_insert__subset,axiom,
    ! [X3: fm,A2: set_fm,B4: set_fm] :
      ( ( ord_less_eq_set_fm @ ( insert_fm @ X3 @ A2 ) @ B4 )
      = ( ( member_fm @ X3 @ B4 )
        & ( ord_less_eq_set_fm @ A2 @ B4 ) ) ) ).

% insert_subset
thf(fact_836_Diff__cancel,axiom,
    ! [A2: set_rule] :
      ( ( minus_minus_set_rule @ A2 @ A2 )
      = bot_bot_set_rule ) ).

% Diff_cancel
thf(fact_837_empty__Diff,axiom,
    ! [A2: set_rule] :
      ( ( minus_minus_set_rule @ bot_bot_set_rule @ A2 )
      = bot_bot_set_rule ) ).

% empty_Diff
thf(fact_838_Diff__empty,axiom,
    ! [A2: set_rule] :
      ( ( minus_minus_set_rule @ A2 @ bot_bot_set_rule )
      = A2 ) ).

% Diff_empty
thf(fact_839_insert__Diff1,axiom,
    ! [X3: produc6018962875968178549ist_fm,B4: set_Pr5202636777678657877ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ X3 @ B4 )
     => ( ( minus_7584274679010293948ist_fm @ ( insert4380798862274524997ist_fm @ X3 @ A2 ) @ B4 )
        = ( minus_7584274679010293948ist_fm @ A2 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_840_insert__Diff1,axiom,
    ! [X3: list_fm,B4: set_list_fm,A2: set_list_fm] :
      ( ( member_list_fm @ X3 @ B4 )
     => ( ( minus_639611354763871680ist_fm @ ( insert_list_fm @ X3 @ A2 ) @ B4 )
        = ( minus_639611354763871680ist_fm @ A2 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_841_insert__Diff1,axiom,
    ! [X3: tm,B4: set_tm,A2: set_tm] :
      ( ( member_tm @ X3 @ B4 )
     => ( ( minus_minus_set_tm @ ( insert_tm @ X3 @ A2 ) @ B4 )
        = ( minus_minus_set_tm @ A2 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_842_insert__Diff1,axiom,
    ! [X3: fm,B4: set_fm,A2: set_fm] :
      ( ( member_fm @ X3 @ B4 )
     => ( ( minus_minus_set_fm @ ( insert_fm @ X3 @ A2 ) @ B4 )
        = ( minus_minus_set_fm @ A2 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_843_insert__Diff1,axiom,
    ! [X3: rule,B4: set_rule,A2: set_rule] :
      ( ( member_rule @ X3 @ B4 )
     => ( ( minus_minus_set_rule @ ( insert_rule @ X3 @ A2 ) @ B4 )
        = ( minus_minus_set_rule @ A2 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_844_Diff__insert0,axiom,
    ! [X3: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ~ ( member4699826688122452638ist_fm @ X3 @ A2 )
     => ( ( minus_7584274679010293948ist_fm @ A2 @ ( insert4380798862274524997ist_fm @ X3 @ B4 ) )
        = ( minus_7584274679010293948ist_fm @ A2 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_845_Diff__insert0,axiom,
    ! [X3: list_fm,A2: set_list_fm,B4: set_list_fm] :
      ( ~ ( member_list_fm @ X3 @ A2 )
     => ( ( minus_639611354763871680ist_fm @ A2 @ ( insert_list_fm @ X3 @ B4 ) )
        = ( minus_639611354763871680ist_fm @ A2 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_846_Diff__insert0,axiom,
    ! [X3: tm,A2: set_tm,B4: set_tm] :
      ( ~ ( member_tm @ X3 @ A2 )
     => ( ( minus_minus_set_tm @ A2 @ ( insert_tm @ X3 @ B4 ) )
        = ( minus_minus_set_tm @ A2 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_847_Diff__insert0,axiom,
    ! [X3: fm,A2: set_fm,B4: set_fm] :
      ( ~ ( member_fm @ X3 @ A2 )
     => ( ( minus_minus_set_fm @ A2 @ ( insert_fm @ X3 @ B4 ) )
        = ( minus_minus_set_fm @ A2 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_848_Diff__insert0,axiom,
    ! [X3: rule,A2: set_rule,B4: set_rule] :
      ( ~ ( member_rule @ X3 @ A2 )
     => ( ( minus_minus_set_rule @ A2 @ ( insert_rule @ X3 @ B4 ) )
        = ( minus_minus_set_rule @ A2 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_849_singleton__insert__inj__eq,axiom,
    ! [B: rule,A: rule,A2: set_rule] :
      ( ( ( insert_rule @ B @ bot_bot_set_rule )
        = ( insert_rule @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_rule @ A2 @ ( insert_rule @ B @ bot_bot_set_rule ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_850_singleton__insert__inj__eq,axiom,
    ! [B: tm,A: tm,A2: set_tm] :
      ( ( ( insert_tm @ B @ bot_bot_set_tm )
        = ( insert_tm @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_tm @ A2 @ ( insert_tm @ B @ bot_bot_set_tm ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_851_singleton__insert__inj__eq,axiom,
    ! [B: fm,A: fm,A2: set_fm] :
      ( ( ( insert_fm @ B @ bot_bot_set_fm )
        = ( insert_fm @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_fm @ A2 @ ( insert_fm @ B @ bot_bot_set_fm ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_852_singleton__insert__inj__eq_H,axiom,
    ! [A: rule,A2: set_rule,B: rule] :
      ( ( ( insert_rule @ A @ A2 )
        = ( insert_rule @ B @ bot_bot_set_rule ) )
      = ( ( A = B )
        & ( ord_less_eq_set_rule @ A2 @ ( insert_rule @ B @ bot_bot_set_rule ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_853_singleton__insert__inj__eq_H,axiom,
    ! [A: tm,A2: set_tm,B: tm] :
      ( ( ( insert_tm @ A @ A2 )
        = ( insert_tm @ B @ bot_bot_set_tm ) )
      = ( ( A = B )
        & ( ord_less_eq_set_tm @ A2 @ ( insert_tm @ B @ bot_bot_set_tm ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_854_singleton__insert__inj__eq_H,axiom,
    ! [A: fm,A2: set_fm,B: fm] :
      ( ( ( insert_fm @ A @ A2 )
        = ( insert_fm @ B @ bot_bot_set_fm ) )
      = ( ( A = B )
        & ( ord_less_eq_set_fm @ A2 @ ( insert_fm @ B @ bot_bot_set_fm ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_855_Diff__UNIV,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ( minus_7584274679010293948ist_fm @ A2 @ top_to1730629564055774885ist_fm )
      = bot_bo1798318783834493889ist_fm ) ).

% Diff_UNIV
thf(fact_856_Diff__UNIV,axiom,
    ! [A2: set_rule] :
      ( ( minus_minus_set_rule @ A2 @ top_top_set_rule )
      = bot_bot_set_rule ) ).

% Diff_UNIV
thf(fact_857_Diff__eq__empty__iff,axiom,
    ! [A2: set_rule,B4: set_rule] :
      ( ( ( minus_minus_set_rule @ A2 @ B4 )
        = bot_bot_set_rule )
      = ( ord_less_eq_set_rule @ A2 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_858_Diff__eq__empty__iff,axiom,
    ! [A2: set_tm,B4: set_tm] :
      ( ( ( minus_minus_set_tm @ A2 @ B4 )
        = bot_bot_set_tm )
      = ( ord_less_eq_set_tm @ A2 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_859_Diff__eq__empty__iff,axiom,
    ! [A2: set_fm,B4: set_fm] :
      ( ( ( minus_minus_set_fm @ A2 @ B4 )
        = bot_bot_set_fm )
      = ( ord_less_eq_set_fm @ A2 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_860_insert__Diff__single,axiom,
    ! [A: rule,A2: set_rule] :
      ( ( insert_rule @ A @ ( minus_minus_set_rule @ A2 @ ( insert_rule @ A @ bot_bot_set_rule ) ) )
      = ( insert_rule @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_861_above,axiom,
    ! [K3: nat] :
      ( ( ord_less_nat @ m @ K3 )
     => ( member_tm @ t @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ K3 @ steps ) ) ) ) ) ) ).

% above
thf(fact_862__092_060open_062t_A_092_060in_062_Aset_A_Iptms_A_Ishd_A_Istl_A_Isdrop_Am_Asteps_J_J_J_J_092_060close_062,axiom,
    member_tm @ t @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ ( sdrop_8169176516188972301m_rule @ m @ steps ) ) ) ) ) ).

% \<open>t \<in> set (ptms (shd (stl (sdrop m steps))))\<close>
thf(fact_863_t,axiom,
    member_tm @ t @ ( terms @ ( tree_fms @ steps ) ) ).

% t
thf(fact_864_Diff__single__insert,axiom,
    ! [A2: set_rule,X3: rule,B4: set_rule] :
      ( ( ord_less_eq_set_rule @ ( minus_minus_set_rule @ A2 @ ( insert_rule @ X3 @ bot_bot_set_rule ) ) @ B4 )
     => ( ord_less_eq_set_rule @ A2 @ ( insert_rule @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_865_Diff__single__insert,axiom,
    ! [A2: set_tm,X3: tm,B4: set_tm] :
      ( ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A2 @ ( insert_tm @ X3 @ bot_bot_set_tm ) ) @ B4 )
     => ( ord_less_eq_set_tm @ A2 @ ( insert_tm @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_866_Diff__single__insert,axiom,
    ! [A2: set_fm,X3: fm,B4: set_fm] :
      ( ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A2 @ ( insert_fm @ X3 @ bot_bot_set_fm ) ) @ B4 )
     => ( ord_less_eq_set_fm @ A2 @ ( insert_fm @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_867_subset__insert__iff,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,X3: produc6018962875968178549ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ A2 @ ( insert4380798862274524997ist_fm @ X3 @ B4 ) )
      = ( ( ( member4699826688122452638ist_fm @ X3 @ A2 )
         => ( ord_le1771420097867575541ist_fm @ ( minus_7584274679010293948ist_fm @ A2 @ ( insert4380798862274524997ist_fm @ X3 @ bot_bo1798318783834493889ist_fm ) ) @ B4 ) )
        & ( ~ ( member4699826688122452638ist_fm @ X3 @ A2 )
         => ( ord_le1771420097867575541ist_fm @ A2 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_868_subset__insert__iff,axiom,
    ! [A2: set_list_fm,X3: list_fm,B4: set_list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ A2 @ ( insert_list_fm @ X3 @ B4 ) )
      = ( ( ( member_list_fm @ X3 @ A2 )
         => ( ord_le7838213414353715577ist_fm @ ( minus_639611354763871680ist_fm @ A2 @ ( insert_list_fm @ X3 @ bot_bot_set_list_fm ) ) @ B4 ) )
        & ( ~ ( member_list_fm @ X3 @ A2 )
         => ( ord_le7838213414353715577ist_fm @ A2 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_869_subset__insert__iff,axiom,
    ! [A2: set_rule,X3: rule,B4: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ ( insert_rule @ X3 @ B4 ) )
      = ( ( ( member_rule @ X3 @ A2 )
         => ( ord_less_eq_set_rule @ ( minus_minus_set_rule @ A2 @ ( insert_rule @ X3 @ bot_bot_set_rule ) ) @ B4 ) )
        & ( ~ ( member_rule @ X3 @ A2 )
         => ( ord_less_eq_set_rule @ A2 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_870_subset__insert__iff,axiom,
    ! [A2: set_tm,X3: tm,B4: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ ( insert_tm @ X3 @ B4 ) )
      = ( ( ( member_tm @ X3 @ A2 )
         => ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A2 @ ( insert_tm @ X3 @ bot_bot_set_tm ) ) @ B4 ) )
        & ( ~ ( member_tm @ X3 @ A2 )
         => ( ord_less_eq_set_tm @ A2 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_871_subset__insert__iff,axiom,
    ! [A2: set_fm,X3: fm,B4: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ ( insert_fm @ X3 @ B4 ) )
      = ( ( ( member_fm @ X3 @ A2 )
         => ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A2 @ ( insert_fm @ X3 @ bot_bot_set_fm ) ) @ B4 ) )
        & ( ~ ( member_fm @ X3 @ A2 )
         => ( ord_less_eq_set_fm @ A2 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_872_Diff__insert,axiom,
    ! [A2: set_rule,A: rule,B4: set_rule] :
      ( ( minus_minus_set_rule @ A2 @ ( insert_rule @ A @ B4 ) )
      = ( minus_minus_set_rule @ ( minus_minus_set_rule @ A2 @ B4 ) @ ( insert_rule @ A @ bot_bot_set_rule ) ) ) ).

% Diff_insert
thf(fact_873_insert__Diff,axiom,
    ! [A: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ A @ A2 )
     => ( ( insert4380798862274524997ist_fm @ A @ ( minus_7584274679010293948ist_fm @ A2 @ ( insert4380798862274524997ist_fm @ A @ bot_bo1798318783834493889ist_fm ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_874_insert__Diff,axiom,
    ! [A: list_fm,A2: set_list_fm] :
      ( ( member_list_fm @ A @ A2 )
     => ( ( insert_list_fm @ A @ ( minus_639611354763871680ist_fm @ A2 @ ( insert_list_fm @ A @ bot_bot_set_list_fm ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_875_insert__Diff,axiom,
    ! [A: tm,A2: set_tm] :
      ( ( member_tm @ A @ A2 )
     => ( ( insert_tm @ A @ ( minus_minus_set_tm @ A2 @ ( insert_tm @ A @ bot_bot_set_tm ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_876_insert__Diff,axiom,
    ! [A: fm,A2: set_fm] :
      ( ( member_fm @ A @ A2 )
     => ( ( insert_fm @ A @ ( minus_minus_set_fm @ A2 @ ( insert_fm @ A @ bot_bot_set_fm ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_877_insert__Diff,axiom,
    ! [A: rule,A2: set_rule] :
      ( ( member_rule @ A @ A2 )
     => ( ( insert_rule @ A @ ( minus_minus_set_rule @ A2 @ ( insert_rule @ A @ bot_bot_set_rule ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_878_Diff__insert2,axiom,
    ! [A2: set_rule,A: rule,B4: set_rule] :
      ( ( minus_minus_set_rule @ A2 @ ( insert_rule @ A @ B4 ) )
      = ( minus_minus_set_rule @ ( minus_minus_set_rule @ A2 @ ( insert_rule @ A @ bot_bot_set_rule ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_879_Diff__insert__absorb,axiom,
    ! [X3: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ~ ( member4699826688122452638ist_fm @ X3 @ A2 )
     => ( ( minus_7584274679010293948ist_fm @ ( insert4380798862274524997ist_fm @ X3 @ A2 ) @ ( insert4380798862274524997ist_fm @ X3 @ bot_bo1798318783834493889ist_fm ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_880_Diff__insert__absorb,axiom,
    ! [X3: list_fm,A2: set_list_fm] :
      ( ~ ( member_list_fm @ X3 @ A2 )
     => ( ( minus_639611354763871680ist_fm @ ( insert_list_fm @ X3 @ A2 ) @ ( insert_list_fm @ X3 @ bot_bot_set_list_fm ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_881_Diff__insert__absorb,axiom,
    ! [X3: tm,A2: set_tm] :
      ( ~ ( member_tm @ X3 @ A2 )
     => ( ( minus_minus_set_tm @ ( insert_tm @ X3 @ A2 ) @ ( insert_tm @ X3 @ bot_bot_set_tm ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_882_Diff__insert__absorb,axiom,
    ! [X3: fm,A2: set_fm] :
      ( ~ ( member_fm @ X3 @ A2 )
     => ( ( minus_minus_set_fm @ ( insert_fm @ X3 @ A2 ) @ ( insert_fm @ X3 @ bot_bot_set_fm ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_883_Diff__insert__absorb,axiom,
    ! [X3: rule,A2: set_rule] :
      ( ~ ( member_rule @ X3 @ A2 )
     => ( ( minus_minus_set_rule @ ( insert_rule @ X3 @ A2 ) @ ( insert_rule @ X3 @ bot_bot_set_rule ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_884_bot_Oextremum,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ bot_bot_set_rule @ A ) ).

% bot.extremum
thf(fact_885_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_886_bot_Oextremum,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ bot_bot_set_tm @ A ) ).

% bot.extremum
thf(fact_887_bot_Oextremum,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A ) ).

% bot.extremum
thf(fact_888_bot_Oextremum__unique,axiom,
    ! [A: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ bot_bot_set_rule )
      = ( A = bot_bot_set_rule ) ) ).

% bot.extremum_unique
thf(fact_889_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_890_bot_Oextremum__unique,axiom,
    ! [A: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ bot_bot_set_tm )
      = ( A = bot_bot_set_tm ) ) ).

% bot.extremum_unique
thf(fact_891_bot_Oextremum__unique,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ bot_bot_set_fm )
      = ( A = bot_bot_set_fm ) ) ).

% bot.extremum_unique
thf(fact_892_bot_Oextremum__uniqueI,axiom,
    ! [A: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ bot_bot_set_rule )
     => ( A = bot_bot_set_rule ) ) ).

% bot.extremum_uniqueI
thf(fact_893_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_894_bot_Oextremum__uniqueI,axiom,
    ! [A: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ bot_bot_set_tm )
     => ( A = bot_bot_set_tm ) ) ).

% bot.extremum_uniqueI
thf(fact_895_bot_Oextremum__uniqueI,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ bot_bot_set_fm )
     => ( A = bot_bot_set_fm ) ) ).

% bot.extremum_uniqueI
thf(fact_896_bot_Onot__eq__extremum,axiom,
    ! [A: set_rule] :
      ( ( A != bot_bot_set_rule )
      = ( ord_less_set_rule @ bot_bot_set_rule @ A ) ) ).

% bot.not_eq_extremum
thf(fact_897_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_898_bot_Oextremum__strict,axiom,
    ! [A: set_rule] :
      ~ ( ord_less_set_rule @ A @ bot_bot_set_rule ) ).

% bot.extremum_strict
thf(fact_899_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_900_insert__UNIV,axiom,
    ! [X3: rule] :
      ( ( insert_rule @ X3 @ top_top_set_rule )
      = top_top_set_rule ) ).

% insert_UNIV
thf(fact_901_insert__UNIV,axiom,
    ! [X3: produc6018962875968178549ist_fm] :
      ( ( insert4380798862274524997ist_fm @ X3 @ top_to1730629564055774885ist_fm )
      = top_to1730629564055774885ist_fm ) ).

% insert_UNIV
thf(fact_902_insert__mono,axiom,
    ! [C2: set_rule,D: set_rule,A: rule] :
      ( ( ord_less_eq_set_rule @ C2 @ D )
     => ( ord_less_eq_set_rule @ ( insert_rule @ A @ C2 ) @ ( insert_rule @ A @ D ) ) ) ).

% insert_mono
thf(fact_903_insert__mono,axiom,
    ! [C2: set_tm,D: set_tm,A: tm] :
      ( ( ord_less_eq_set_tm @ C2 @ D )
     => ( ord_less_eq_set_tm @ ( insert_tm @ A @ C2 ) @ ( insert_tm @ A @ D ) ) ) ).

% insert_mono
thf(fact_904_insert__mono,axiom,
    ! [C2: set_fm,D: set_fm,A: fm] :
      ( ( ord_less_eq_set_fm @ C2 @ D )
     => ( ord_less_eq_set_fm @ ( insert_fm @ A @ C2 ) @ ( insert_fm @ A @ D ) ) ) ).

% insert_mono
thf(fact_905_subset__insert,axiom,
    ! [X3: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ~ ( member4699826688122452638ist_fm @ X3 @ A2 )
     => ( ( ord_le1771420097867575541ist_fm @ A2 @ ( insert4380798862274524997ist_fm @ X3 @ B4 ) )
        = ( ord_le1771420097867575541ist_fm @ A2 @ B4 ) ) ) ).

% subset_insert
thf(fact_906_subset__insert,axiom,
    ! [X3: rule,A2: set_rule,B4: set_rule] :
      ( ~ ( member_rule @ X3 @ A2 )
     => ( ( ord_less_eq_set_rule @ A2 @ ( insert_rule @ X3 @ B4 ) )
        = ( ord_less_eq_set_rule @ A2 @ B4 ) ) ) ).

% subset_insert
thf(fact_907_subset__insert,axiom,
    ! [X3: list_fm,A2: set_list_fm,B4: set_list_fm] :
      ( ~ ( member_list_fm @ X3 @ A2 )
     => ( ( ord_le7838213414353715577ist_fm @ A2 @ ( insert_list_fm @ X3 @ B4 ) )
        = ( ord_le7838213414353715577ist_fm @ A2 @ B4 ) ) ) ).

% subset_insert
thf(fact_908_subset__insert,axiom,
    ! [X3: tm,A2: set_tm,B4: set_tm] :
      ( ~ ( member_tm @ X3 @ A2 )
     => ( ( ord_less_eq_set_tm @ A2 @ ( insert_tm @ X3 @ B4 ) )
        = ( ord_less_eq_set_tm @ A2 @ B4 ) ) ) ).

% subset_insert
thf(fact_909_subset__insert,axiom,
    ! [X3: fm,A2: set_fm,B4: set_fm] :
      ( ~ ( member_fm @ X3 @ A2 )
     => ( ( ord_less_eq_set_fm @ A2 @ ( insert_fm @ X3 @ B4 ) )
        = ( ord_less_eq_set_fm @ A2 @ B4 ) ) ) ).

% subset_insert
thf(fact_910_subset__insertI,axiom,
    ! [B4: set_rule,A: rule] : ( ord_less_eq_set_rule @ B4 @ ( insert_rule @ A @ B4 ) ) ).

% subset_insertI
thf(fact_911_subset__insertI,axiom,
    ! [B4: set_tm,A: tm] : ( ord_less_eq_set_tm @ B4 @ ( insert_tm @ A @ B4 ) ) ).

% subset_insertI
thf(fact_912_subset__insertI,axiom,
    ! [B4: set_fm,A: fm] : ( ord_less_eq_set_fm @ B4 @ ( insert_fm @ A @ B4 ) ) ).

% subset_insertI
thf(fact_913_subset__insertI2,axiom,
    ! [A2: set_rule,B4: set_rule,B: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B4 )
     => ( ord_less_eq_set_rule @ A2 @ ( insert_rule @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_914_subset__insertI2,axiom,
    ! [A2: set_tm,B4: set_tm,B: tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B4 )
     => ( ord_less_eq_set_tm @ A2 @ ( insert_tm @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_915_subset__insertI2,axiom,
    ! [A2: set_fm,B4: set_fm,B: fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B4 )
     => ( ord_less_eq_set_fm @ A2 @ ( insert_fm @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_916_mk__disjoint__insert,axiom,
    ! [A: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ A @ A2 )
     => ? [B7: set_Pr5202636777678657877ist_fm] :
          ( ( A2
            = ( insert4380798862274524997ist_fm @ A @ B7 ) )
          & ~ ( member4699826688122452638ist_fm @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_917_mk__disjoint__insert,axiom,
    ! [A: rule,A2: set_rule] :
      ( ( member_rule @ A @ A2 )
     => ? [B7: set_rule] :
          ( ( A2
            = ( insert_rule @ A @ B7 ) )
          & ~ ( member_rule @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_918_mk__disjoint__insert,axiom,
    ! [A: list_fm,A2: set_list_fm] :
      ( ( member_list_fm @ A @ A2 )
     => ? [B7: set_list_fm] :
          ( ( A2
            = ( insert_list_fm @ A @ B7 ) )
          & ~ ( member_list_fm @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_919_mk__disjoint__insert,axiom,
    ! [A: tm,A2: set_tm] :
      ( ( member_tm @ A @ A2 )
     => ? [B7: set_tm] :
          ( ( A2
            = ( insert_tm @ A @ B7 ) )
          & ~ ( member_tm @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_920_mk__disjoint__insert,axiom,
    ! [A: fm,A2: set_fm] :
      ( ( member_fm @ A @ A2 )
     => ? [B7: set_fm] :
          ( ( A2
            = ( insert_fm @ A @ B7 ) )
          & ~ ( member_fm @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_921_singleton__inject,axiom,
    ! [A: rule,B: rule] :
      ( ( ( insert_rule @ A @ bot_bot_set_rule )
        = ( insert_rule @ B @ bot_bot_set_rule ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_922_insert__not__empty,axiom,
    ! [A: rule,A2: set_rule] :
      ( ( insert_rule @ A @ A2 )
     != bot_bot_set_rule ) ).

% insert_not_empty
thf(fact_923_doubleton__eq__iff,axiom,
    ! [A: rule,B: rule,C: rule,D3: rule] :
      ( ( ( insert_rule @ A @ ( insert_rule @ B @ bot_bot_set_rule ) )
        = ( insert_rule @ C @ ( insert_rule @ D3 @ bot_bot_set_rule ) ) )
      = ( ( ( A = C )
          & ( B = D3 ) )
        | ( ( A = D3 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_924_insert__commute,axiom,
    ! [X3: rule,Y: rule,A2: set_rule] :
      ( ( insert_rule @ X3 @ ( insert_rule @ Y @ A2 ) )
      = ( insert_rule @ Y @ ( insert_rule @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_925_singleton__iff,axiom,
    ! [B: produc6018962875968178549ist_fm,A: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ B @ ( insert4380798862274524997ist_fm @ A @ bot_bo1798318783834493889ist_fm ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_926_singleton__iff,axiom,
    ! [B: list_fm,A: list_fm] :
      ( ( member_list_fm @ B @ ( insert_list_fm @ A @ bot_bot_set_list_fm ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_927_singleton__iff,axiom,
    ! [B: tm,A: tm] :
      ( ( member_tm @ B @ ( insert_tm @ A @ bot_bot_set_tm ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_928_singleton__iff,axiom,
    ! [B: fm,A: fm] :
      ( ( member_fm @ B @ ( insert_fm @ A @ bot_bot_set_fm ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_929_singleton__iff,axiom,
    ! [B: rule,A: rule] :
      ( ( member_rule @ B @ ( insert_rule @ A @ bot_bot_set_rule ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_930_insert__eq__iff,axiom,
    ! [A: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B: produc6018962875968178549ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ~ ( member4699826688122452638ist_fm @ A @ A2 )
     => ( ~ ( member4699826688122452638ist_fm @ B @ B4 )
       => ( ( ( insert4380798862274524997ist_fm @ A @ A2 )
            = ( insert4380798862274524997ist_fm @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A2 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_Pr5202636777678657877ist_fm] :
                  ( ( A2
                    = ( insert4380798862274524997ist_fm @ B @ C4 ) )
                  & ~ ( member4699826688122452638ist_fm @ B @ C4 )
                  & ( B4
                    = ( insert4380798862274524997ist_fm @ A @ C4 ) )
                  & ~ ( member4699826688122452638ist_fm @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_931_insert__eq__iff,axiom,
    ! [A: rule,A2: set_rule,B: rule,B4: set_rule] :
      ( ~ ( member_rule @ A @ A2 )
     => ( ~ ( member_rule @ B @ B4 )
       => ( ( ( insert_rule @ A @ A2 )
            = ( insert_rule @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A2 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_rule] :
                  ( ( A2
                    = ( insert_rule @ B @ C4 ) )
                  & ~ ( member_rule @ B @ C4 )
                  & ( B4
                    = ( insert_rule @ A @ C4 ) )
                  & ~ ( member_rule @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_932_insert__eq__iff,axiom,
    ! [A: list_fm,A2: set_list_fm,B: list_fm,B4: set_list_fm] :
      ( ~ ( member_list_fm @ A @ A2 )
     => ( ~ ( member_list_fm @ B @ B4 )
       => ( ( ( insert_list_fm @ A @ A2 )
            = ( insert_list_fm @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A2 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_list_fm] :
                  ( ( A2
                    = ( insert_list_fm @ B @ C4 ) )
                  & ~ ( member_list_fm @ B @ C4 )
                  & ( B4
                    = ( insert_list_fm @ A @ C4 ) )
                  & ~ ( member_list_fm @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_933_insert__eq__iff,axiom,
    ! [A: tm,A2: set_tm,B: tm,B4: set_tm] :
      ( ~ ( member_tm @ A @ A2 )
     => ( ~ ( member_tm @ B @ B4 )
       => ( ( ( insert_tm @ A @ A2 )
            = ( insert_tm @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A2 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_tm] :
                  ( ( A2
                    = ( insert_tm @ B @ C4 ) )
                  & ~ ( member_tm @ B @ C4 )
                  & ( B4
                    = ( insert_tm @ A @ C4 ) )
                  & ~ ( member_tm @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_934_insert__eq__iff,axiom,
    ! [A: fm,A2: set_fm,B: fm,B4: set_fm] :
      ( ~ ( member_fm @ A @ A2 )
     => ( ~ ( member_fm @ B @ B4 )
       => ( ( ( insert_fm @ A @ A2 )
            = ( insert_fm @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A2 = B4 ) )
            & ( ( A != B )
             => ? [C4: set_fm] :
                  ( ( A2
                    = ( insert_fm @ B @ C4 ) )
                  & ~ ( member_fm @ B @ C4 )
                  & ( B4
                    = ( insert_fm @ A @ C4 ) )
                  & ~ ( member_fm @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_935_insert__absorb,axiom,
    ! [A: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ A @ A2 )
     => ( ( insert4380798862274524997ist_fm @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_936_insert__absorb,axiom,
    ! [A: rule,A2: set_rule] :
      ( ( member_rule @ A @ A2 )
     => ( ( insert_rule @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_937_insert__absorb,axiom,
    ! [A: list_fm,A2: set_list_fm] :
      ( ( member_list_fm @ A @ A2 )
     => ( ( insert_list_fm @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_938_insert__absorb,axiom,
    ! [A: tm,A2: set_tm] :
      ( ( member_tm @ A @ A2 )
     => ( ( insert_tm @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_939_insert__absorb,axiom,
    ! [A: fm,A2: set_fm] :
      ( ( member_fm @ A @ A2 )
     => ( ( insert_fm @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_940_insert__ident,axiom,
    ! [X3: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ~ ( member4699826688122452638ist_fm @ X3 @ A2 )
     => ( ~ ( member4699826688122452638ist_fm @ X3 @ B4 )
       => ( ( ( insert4380798862274524997ist_fm @ X3 @ A2 )
            = ( insert4380798862274524997ist_fm @ X3 @ B4 ) )
          = ( A2 = B4 ) ) ) ) ).

% insert_ident
thf(fact_941_insert__ident,axiom,
    ! [X3: rule,A2: set_rule,B4: set_rule] :
      ( ~ ( member_rule @ X3 @ A2 )
     => ( ~ ( member_rule @ X3 @ B4 )
       => ( ( ( insert_rule @ X3 @ A2 )
            = ( insert_rule @ X3 @ B4 ) )
          = ( A2 = B4 ) ) ) ) ).

% insert_ident
thf(fact_942_insert__ident,axiom,
    ! [X3: list_fm,A2: set_list_fm,B4: set_list_fm] :
      ( ~ ( member_list_fm @ X3 @ A2 )
     => ( ~ ( member_list_fm @ X3 @ B4 )
       => ( ( ( insert_list_fm @ X3 @ A2 )
            = ( insert_list_fm @ X3 @ B4 ) )
          = ( A2 = B4 ) ) ) ) ).

% insert_ident
thf(fact_943_insert__ident,axiom,
    ! [X3: tm,A2: set_tm,B4: set_tm] :
      ( ~ ( member_tm @ X3 @ A2 )
     => ( ~ ( member_tm @ X3 @ B4 )
       => ( ( ( insert_tm @ X3 @ A2 )
            = ( insert_tm @ X3 @ B4 ) )
          = ( A2 = B4 ) ) ) ) ).

% insert_ident
thf(fact_944_insert__ident,axiom,
    ! [X3: fm,A2: set_fm,B4: set_fm] :
      ( ~ ( member_fm @ X3 @ A2 )
     => ( ~ ( member_fm @ X3 @ B4 )
       => ( ( ( insert_fm @ X3 @ A2 )
            = ( insert_fm @ X3 @ B4 ) )
          = ( A2 = B4 ) ) ) ) ).

% insert_ident
thf(fact_945_singletonD,axiom,
    ! [B: produc6018962875968178549ist_fm,A: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ B @ ( insert4380798862274524997ist_fm @ A @ bot_bo1798318783834493889ist_fm ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_946_singletonD,axiom,
    ! [B: list_fm,A: list_fm] :
      ( ( member_list_fm @ B @ ( insert_list_fm @ A @ bot_bot_set_list_fm ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_947_singletonD,axiom,
    ! [B: tm,A: tm] :
      ( ( member_tm @ B @ ( insert_tm @ A @ bot_bot_set_tm ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_948_singletonD,axiom,
    ! [B: fm,A: fm] :
      ( ( member_fm @ B @ ( insert_fm @ A @ bot_bot_set_fm ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_949_singletonD,axiom,
    ! [B: rule,A: rule] :
      ( ( member_rule @ B @ ( insert_rule @ A @ bot_bot_set_rule ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_950_Set_Oset__insert,axiom,
    ! [X3: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ X3 @ A2 )
     => ~ ! [B7: set_Pr5202636777678657877ist_fm] :
            ( ( A2
              = ( insert4380798862274524997ist_fm @ X3 @ B7 ) )
           => ( member4699826688122452638ist_fm @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_951_Set_Oset__insert,axiom,
    ! [X3: rule,A2: set_rule] :
      ( ( member_rule @ X3 @ A2 )
     => ~ ! [B7: set_rule] :
            ( ( A2
              = ( insert_rule @ X3 @ B7 ) )
           => ( member_rule @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_952_Set_Oset__insert,axiom,
    ! [X3: list_fm,A2: set_list_fm] :
      ( ( member_list_fm @ X3 @ A2 )
     => ~ ! [B7: set_list_fm] :
            ( ( A2
              = ( insert_list_fm @ X3 @ B7 ) )
           => ( member_list_fm @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_953_Set_Oset__insert,axiom,
    ! [X3: tm,A2: set_tm] :
      ( ( member_tm @ X3 @ A2 )
     => ~ ! [B7: set_tm] :
            ( ( A2
              = ( insert_tm @ X3 @ B7 ) )
           => ( member_tm @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_954_Set_Oset__insert,axiom,
    ! [X3: fm,A2: set_fm] :
      ( ( member_fm @ X3 @ A2 )
     => ~ ! [B7: set_fm] :
            ( ( A2
              = ( insert_fm @ X3 @ B7 ) )
           => ( member_fm @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_955_ex__in__conv,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ( ? [X4: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X4 @ A2 ) )
      = ( A2 != bot_bo1798318783834493889ist_fm ) ) ).

% ex_in_conv
thf(fact_956_ex__in__conv,axiom,
    ! [A2: set_list_fm] :
      ( ( ? [X4: list_fm] : ( member_list_fm @ X4 @ A2 ) )
      = ( A2 != bot_bot_set_list_fm ) ) ).

% ex_in_conv
thf(fact_957_ex__in__conv,axiom,
    ! [A2: set_tm] :
      ( ( ? [X4: tm] : ( member_tm @ X4 @ A2 ) )
      = ( A2 != bot_bot_set_tm ) ) ).

% ex_in_conv
thf(fact_958_ex__in__conv,axiom,
    ! [A2: set_fm] :
      ( ( ? [X4: fm] : ( member_fm @ X4 @ A2 ) )
      = ( A2 != bot_bot_set_fm ) ) ).

% ex_in_conv
thf(fact_959_ex__in__conv,axiom,
    ! [A2: set_rule] :
      ( ( ? [X4: rule] : ( member_rule @ X4 @ A2 ) )
      = ( A2 != bot_bot_set_rule ) ) ).

% ex_in_conv
thf(fact_960_insertI2,axiom,
    ! [A: produc6018962875968178549ist_fm,B4: set_Pr5202636777678657877ist_fm,B: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ A @ B4 )
     => ( member4699826688122452638ist_fm @ A @ ( insert4380798862274524997ist_fm @ B @ B4 ) ) ) ).

% insertI2
thf(fact_961_insertI2,axiom,
    ! [A: rule,B4: set_rule,B: rule] :
      ( ( member_rule @ A @ B4 )
     => ( member_rule @ A @ ( insert_rule @ B @ B4 ) ) ) ).

% insertI2
thf(fact_962_insertI2,axiom,
    ! [A: list_fm,B4: set_list_fm,B: list_fm] :
      ( ( member_list_fm @ A @ B4 )
     => ( member_list_fm @ A @ ( insert_list_fm @ B @ B4 ) ) ) ).

% insertI2
thf(fact_963_insertI2,axiom,
    ! [A: tm,B4: set_tm,B: tm] :
      ( ( member_tm @ A @ B4 )
     => ( member_tm @ A @ ( insert_tm @ B @ B4 ) ) ) ).

% insertI2
thf(fact_964_insertI2,axiom,
    ! [A: fm,B4: set_fm,B: fm] :
      ( ( member_fm @ A @ B4 )
     => ( member_fm @ A @ ( insert_fm @ B @ B4 ) ) ) ).

% insertI2
thf(fact_965_insertI1,axiom,
    ! [A: produc6018962875968178549ist_fm,B4: set_Pr5202636777678657877ist_fm] : ( member4699826688122452638ist_fm @ A @ ( insert4380798862274524997ist_fm @ A @ B4 ) ) ).

% insertI1
thf(fact_966_insertI1,axiom,
    ! [A: rule,B4: set_rule] : ( member_rule @ A @ ( insert_rule @ A @ B4 ) ) ).

% insertI1
thf(fact_967_insertI1,axiom,
    ! [A: list_fm,B4: set_list_fm] : ( member_list_fm @ A @ ( insert_list_fm @ A @ B4 ) ) ).

% insertI1
thf(fact_968_insertI1,axiom,
    ! [A: tm,B4: set_tm] : ( member_tm @ A @ ( insert_tm @ A @ B4 ) ) ).

% insertI1
thf(fact_969_insertI1,axiom,
    ! [A: fm,B4: set_fm] : ( member_fm @ A @ ( insert_fm @ A @ B4 ) ) ).

% insertI1
thf(fact_970_equals0I,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ! [Y2: produc6018962875968178549ist_fm] :
          ~ ( member4699826688122452638ist_fm @ Y2 @ A2 )
     => ( A2 = bot_bo1798318783834493889ist_fm ) ) ).

% equals0I
thf(fact_971_equals0I,axiom,
    ! [A2: set_list_fm] :
      ( ! [Y2: list_fm] :
          ~ ( member_list_fm @ Y2 @ A2 )
     => ( A2 = bot_bot_set_list_fm ) ) ).

% equals0I
thf(fact_972_equals0I,axiom,
    ! [A2: set_tm] :
      ( ! [Y2: tm] :
          ~ ( member_tm @ Y2 @ A2 )
     => ( A2 = bot_bot_set_tm ) ) ).

% equals0I
thf(fact_973_equals0I,axiom,
    ! [A2: set_fm] :
      ( ! [Y2: fm] :
          ~ ( member_fm @ Y2 @ A2 )
     => ( A2 = bot_bot_set_fm ) ) ).

% equals0I
thf(fact_974_equals0I,axiom,
    ! [A2: set_rule] :
      ( ! [Y2: rule] :
          ~ ( member_rule @ Y2 @ A2 )
     => ( A2 = bot_bot_set_rule ) ) ).

% equals0I
thf(fact_975_equals0D,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,A: produc6018962875968178549ist_fm] :
      ( ( A2 = bot_bo1798318783834493889ist_fm )
     => ~ ( member4699826688122452638ist_fm @ A @ A2 ) ) ).

% equals0D
thf(fact_976_equals0D,axiom,
    ! [A2: set_list_fm,A: list_fm] :
      ( ( A2 = bot_bot_set_list_fm )
     => ~ ( member_list_fm @ A @ A2 ) ) ).

% equals0D
thf(fact_977_equals0D,axiom,
    ! [A2: set_tm,A: tm] :
      ( ( A2 = bot_bot_set_tm )
     => ~ ( member_tm @ A @ A2 ) ) ).

% equals0D
thf(fact_978_equals0D,axiom,
    ! [A2: set_fm,A: fm] :
      ( ( A2 = bot_bot_set_fm )
     => ~ ( member_fm @ A @ A2 ) ) ).

% equals0D
thf(fact_979_equals0D,axiom,
    ! [A2: set_rule,A: rule] :
      ( ( A2 = bot_bot_set_rule )
     => ~ ( member_rule @ A @ A2 ) ) ).

% equals0D
thf(fact_980_insertE,axiom,
    ! [A: produc6018962875968178549ist_fm,B: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ A @ ( insert4380798862274524997ist_fm @ B @ A2 ) )
     => ( ( A != B )
       => ( member4699826688122452638ist_fm @ A @ A2 ) ) ) ).

% insertE
thf(fact_981_insertE,axiom,
    ! [A: rule,B: rule,A2: set_rule] :
      ( ( member_rule @ A @ ( insert_rule @ B @ A2 ) )
     => ( ( A != B )
       => ( member_rule @ A @ A2 ) ) ) ).

% insertE
thf(fact_982_insertE,axiom,
    ! [A: list_fm,B: list_fm,A2: set_list_fm] :
      ( ( member_list_fm @ A @ ( insert_list_fm @ B @ A2 ) )
     => ( ( A != B )
       => ( member_list_fm @ A @ A2 ) ) ) ).

% insertE
thf(fact_983_insertE,axiom,
    ! [A: tm,B: tm,A2: set_tm] :
      ( ( member_tm @ A @ ( insert_tm @ B @ A2 ) )
     => ( ( A != B )
       => ( member_tm @ A @ A2 ) ) ) ).

% insertE
thf(fact_984_insertE,axiom,
    ! [A: fm,B: fm,A2: set_fm] :
      ( ( member_fm @ A @ ( insert_fm @ B @ A2 ) )
     => ( ( A != B )
       => ( member_fm @ A @ A2 ) ) ) ).

% insertE
thf(fact_985_emptyE,axiom,
    ! [A: produc6018962875968178549ist_fm] :
      ~ ( member4699826688122452638ist_fm @ A @ bot_bo1798318783834493889ist_fm ) ).

% emptyE
thf(fact_986_emptyE,axiom,
    ! [A: list_fm] :
      ~ ( member_list_fm @ A @ bot_bot_set_list_fm ) ).

% emptyE
thf(fact_987_emptyE,axiom,
    ! [A: tm] :
      ~ ( member_tm @ A @ bot_bot_set_tm ) ).

% emptyE
thf(fact_988_emptyE,axiom,
    ! [A: fm] :
      ~ ( member_fm @ A @ bot_bot_set_fm ) ).

% emptyE
thf(fact_989_emptyE,axiom,
    ! [A: rule] :
      ~ ( member_rule @ A @ bot_bot_set_rule ) ).

% emptyE
thf(fact_990_ordering__top_Oextremum__uniqueI,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
       => ( A = Top ) ) ) ).

% ordering_top.extremum_uniqueI
thf(fact_991_ordering__top_Onot__eq__extremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( A != Top )
        = ( Less @ A @ Top ) ) ) ).

% ordering_top.not_eq_extremum
thf(fact_992_ordering__top_Oextremum__unique,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
        = ( A = Top ) ) ) ).

% ordering_top.extremum_unique
thf(fact_993_ordering__top_Oextremum__strict,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ~ ( Less @ Top @ A ) ) ).

% ordering_top.extremum_strict
thf(fact_994_ordering__top_Oextremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( Less_eq @ A @ Top ) ) ).

% ordering_top.extremum
thf(fact_995_subset__singletonD,axiom,
    ! [A2: set_rule,X3: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ ( insert_rule @ X3 @ bot_bot_set_rule ) )
     => ( ( A2 = bot_bot_set_rule )
        | ( A2
          = ( insert_rule @ X3 @ bot_bot_set_rule ) ) ) ) ).

% subset_singletonD
thf(fact_996_subset__singletonD,axiom,
    ! [A2: set_tm,X3: tm] :
      ( ( ord_less_eq_set_tm @ A2 @ ( insert_tm @ X3 @ bot_bot_set_tm ) )
     => ( ( A2 = bot_bot_set_tm )
        | ( A2
          = ( insert_tm @ X3 @ bot_bot_set_tm ) ) ) ) ).

% subset_singletonD
thf(fact_997_subset__singletonD,axiom,
    ! [A2: set_fm,X3: fm] :
      ( ( ord_less_eq_set_fm @ A2 @ ( insert_fm @ X3 @ bot_bot_set_fm ) )
     => ( ( A2 = bot_bot_set_fm )
        | ( A2
          = ( insert_fm @ X3 @ bot_bot_set_fm ) ) ) ) ).

% subset_singletonD
thf(fact_998_subset__singleton__iff,axiom,
    ! [X7: set_rule,A: rule] :
      ( ( ord_less_eq_set_rule @ X7 @ ( insert_rule @ A @ bot_bot_set_rule ) )
      = ( ( X7 = bot_bot_set_rule )
        | ( X7
          = ( insert_rule @ A @ bot_bot_set_rule ) ) ) ) ).

% subset_singleton_iff
thf(fact_999_subset__singleton__iff,axiom,
    ! [X7: set_tm,A: tm] :
      ( ( ord_less_eq_set_tm @ X7 @ ( insert_tm @ A @ bot_bot_set_tm ) )
      = ( ( X7 = bot_bot_set_tm )
        | ( X7
          = ( insert_tm @ A @ bot_bot_set_tm ) ) ) ) ).

% subset_singleton_iff
thf(fact_1000_subset__singleton__iff,axiom,
    ! [X7: set_fm,A: fm] :
      ( ( ord_less_eq_set_fm @ X7 @ ( insert_fm @ A @ bot_bot_set_fm ) )
      = ( ( X7 = bot_bot_set_fm )
        | ( X7
          = ( insert_fm @ A @ bot_bot_set_fm ) ) ) ) ).

% subset_singleton_iff
thf(fact_1001_subrelI,axiom,
    ! [R: set_Pr1822751329126368876m_rule,S: set_Pr1822751329126368876m_rule] :
      ( ! [X: produc6018962875968178549ist_fm,Y2: rule] :
          ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y2 ) @ R )
         => ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X @ Y2 ) @ S ) )
     => ( ord_le6390412330253371084m_rule @ R @ S ) ) ).

% subrelI
thf(fact_1002_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y2: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% subrelI
thf(fact_1003_subrelI,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y2 ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y2 ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% subrelI
thf(fact_1004_subrelI,axiom,
    ! [R: set_Pr5202636777678657877ist_fm,S: set_Pr5202636777678657877ist_fm] :
      ( ! [X: list_tm,Y2: list_fm] :
          ( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X @ Y2 ) @ R )
         => ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X @ Y2 ) @ S ) )
     => ( ord_le1771420097867575541ist_fm @ R @ S ) ) ).

% subrelI
thf(fact_1005_insert__Diff__if,axiom,
    ! [X3: produc6018962875968178549ist_fm,B4: set_Pr5202636777678657877ist_fm,A2: set_Pr5202636777678657877ist_fm] :
      ( ( ( member4699826688122452638ist_fm @ X3 @ B4 )
       => ( ( minus_7584274679010293948ist_fm @ ( insert4380798862274524997ist_fm @ X3 @ A2 ) @ B4 )
          = ( minus_7584274679010293948ist_fm @ A2 @ B4 ) ) )
      & ( ~ ( member4699826688122452638ist_fm @ X3 @ B4 )
       => ( ( minus_7584274679010293948ist_fm @ ( insert4380798862274524997ist_fm @ X3 @ A2 ) @ B4 )
          = ( insert4380798862274524997ist_fm @ X3 @ ( minus_7584274679010293948ist_fm @ A2 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1006_insert__Diff__if,axiom,
    ! [X3: list_fm,B4: set_list_fm,A2: set_list_fm] :
      ( ( ( member_list_fm @ X3 @ B4 )
       => ( ( minus_639611354763871680ist_fm @ ( insert_list_fm @ X3 @ A2 ) @ B4 )
          = ( minus_639611354763871680ist_fm @ A2 @ B4 ) ) )
      & ( ~ ( member_list_fm @ X3 @ B4 )
       => ( ( minus_639611354763871680ist_fm @ ( insert_list_fm @ X3 @ A2 ) @ B4 )
          = ( insert_list_fm @ X3 @ ( minus_639611354763871680ist_fm @ A2 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1007_insert__Diff__if,axiom,
    ! [X3: tm,B4: set_tm,A2: set_tm] :
      ( ( ( member_tm @ X3 @ B4 )
       => ( ( minus_minus_set_tm @ ( insert_tm @ X3 @ A2 ) @ B4 )
          = ( minus_minus_set_tm @ A2 @ B4 ) ) )
      & ( ~ ( member_tm @ X3 @ B4 )
       => ( ( minus_minus_set_tm @ ( insert_tm @ X3 @ A2 ) @ B4 )
          = ( insert_tm @ X3 @ ( minus_minus_set_tm @ A2 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1008_insert__Diff__if,axiom,
    ! [X3: fm,B4: set_fm,A2: set_fm] :
      ( ( ( member_fm @ X3 @ B4 )
       => ( ( minus_minus_set_fm @ ( insert_fm @ X3 @ A2 ) @ B4 )
          = ( minus_minus_set_fm @ A2 @ B4 ) ) )
      & ( ~ ( member_fm @ X3 @ B4 )
       => ( ( minus_minus_set_fm @ ( insert_fm @ X3 @ A2 ) @ B4 )
          = ( insert_fm @ X3 @ ( minus_minus_set_fm @ A2 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1009_insert__Diff__if,axiom,
    ! [X3: rule,B4: set_rule,A2: set_rule] :
      ( ( ( member_rule @ X3 @ B4 )
       => ( ( minus_minus_set_rule @ ( insert_rule @ X3 @ A2 ) @ B4 )
          = ( minus_minus_set_rule @ A2 @ B4 ) ) )
      & ( ~ ( member_rule @ X3 @ B4 )
       => ( ( minus_minus_set_rule @ ( insert_rule @ X3 @ A2 ) @ B4 )
          = ( insert_rule @ X3 @ ( minus_minus_set_rule @ A2 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1010_empty__not__UNIV,axiom,
    bot_bot_set_rule != top_top_set_rule ).

% empty_not_UNIV
thf(fact_1011_empty__not__UNIV,axiom,
    bot_bo1798318783834493889ist_fm != top_to1730629564055774885ist_fm ).

% empty_not_UNIV
thf(fact_1012_not__psubset__empty,axiom,
    ! [A2: set_rule] :
      ~ ( ord_less_set_rule @ A2 @ bot_bot_set_rule ) ).

% not_psubset_empty
thf(fact_1013_i_ONE__R,axiom,
    ! [Rules: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ Rules )
     != bot_bo3426410157013319936m_rule ) ).

% i.NE_R
thf(fact_1014_i_ONE__R,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ Rules )
     != bot_bot_set_rule ) ).

% i.NE_R
thf(fact_1015_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ Rules )
     != bot_bo3426410157013319936m_rule ) ).

% RuleSystem_Defs.NE_R
thf(fact_1016_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ Rules )
     != bot_bot_set_rule ) ).

% RuleSystem_Defs.NE_R
thf(fact_1017_psubset__insert__iff,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,X3: produc6018962875968178549ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le9141428925376846849ist_fm @ A2 @ ( insert4380798862274524997ist_fm @ X3 @ B4 ) )
      = ( ( ( member4699826688122452638ist_fm @ X3 @ B4 )
         => ( ord_le9141428925376846849ist_fm @ A2 @ B4 ) )
        & ( ~ ( member4699826688122452638ist_fm @ X3 @ B4 )
         => ( ( ( member4699826688122452638ist_fm @ X3 @ A2 )
             => ( ord_le9141428925376846849ist_fm @ ( minus_7584274679010293948ist_fm @ A2 @ ( insert4380798862274524997ist_fm @ X3 @ bot_bo1798318783834493889ist_fm ) ) @ B4 ) )
            & ( ~ ( member4699826688122452638ist_fm @ X3 @ A2 )
             => ( ord_le1771420097867575541ist_fm @ A2 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1018_psubset__insert__iff,axiom,
    ! [A2: set_list_fm,X3: list_fm,B4: set_list_fm] :
      ( ( ord_less_set_list_fm @ A2 @ ( insert_list_fm @ X3 @ B4 ) )
      = ( ( ( member_list_fm @ X3 @ B4 )
         => ( ord_less_set_list_fm @ A2 @ B4 ) )
        & ( ~ ( member_list_fm @ X3 @ B4 )
         => ( ( ( member_list_fm @ X3 @ A2 )
             => ( ord_less_set_list_fm @ ( minus_639611354763871680ist_fm @ A2 @ ( insert_list_fm @ X3 @ bot_bot_set_list_fm ) ) @ B4 ) )
            & ( ~ ( member_list_fm @ X3 @ A2 )
             => ( ord_le7838213414353715577ist_fm @ A2 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1019_psubset__insert__iff,axiom,
    ! [A2: set_rule,X3: rule,B4: set_rule] :
      ( ( ord_less_set_rule @ A2 @ ( insert_rule @ X3 @ B4 ) )
      = ( ( ( member_rule @ X3 @ B4 )
         => ( ord_less_set_rule @ A2 @ B4 ) )
        & ( ~ ( member_rule @ X3 @ B4 )
         => ( ( ( member_rule @ X3 @ A2 )
             => ( ord_less_set_rule @ ( minus_minus_set_rule @ A2 @ ( insert_rule @ X3 @ bot_bot_set_rule ) ) @ B4 ) )
            & ( ~ ( member_rule @ X3 @ A2 )
             => ( ord_less_eq_set_rule @ A2 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1020_psubset__insert__iff,axiom,
    ! [A2: set_tm,X3: tm,B4: set_tm] :
      ( ( ord_less_set_tm @ A2 @ ( insert_tm @ X3 @ B4 ) )
      = ( ( ( member_tm @ X3 @ B4 )
         => ( ord_less_set_tm @ A2 @ B4 ) )
        & ( ~ ( member_tm @ X3 @ B4 )
         => ( ( ( member_tm @ X3 @ A2 )
             => ( ord_less_set_tm @ ( minus_minus_set_tm @ A2 @ ( insert_tm @ X3 @ bot_bot_set_tm ) ) @ B4 ) )
            & ( ~ ( member_tm @ X3 @ A2 )
             => ( ord_less_eq_set_tm @ A2 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1021_psubset__insert__iff,axiom,
    ! [A2: set_fm,X3: fm,B4: set_fm] :
      ( ( ord_less_set_fm @ A2 @ ( insert_fm @ X3 @ B4 ) )
      = ( ( ( member_fm @ X3 @ B4 )
         => ( ord_less_set_fm @ A2 @ B4 ) )
        & ( ~ ( member_fm @ X3 @ B4 )
         => ( ( ( member_fm @ X3 @ A2 )
             => ( ord_less_set_fm @ ( minus_minus_set_fm @ A2 @ ( insert_fm @ X3 @ bot_bot_set_fm ) ) @ B4 ) )
            & ( ~ ( member_fm @ X3 @ A2 )
             => ( ord_less_eq_set_fm @ A2 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1022_subset__Diff__insert,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm,X3: produc6018962875968178549ist_fm,C2: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ A2 @ ( minus_7584274679010293948ist_fm @ B4 @ ( insert4380798862274524997ist_fm @ X3 @ C2 ) ) )
      = ( ( ord_le1771420097867575541ist_fm @ A2 @ ( minus_7584274679010293948ist_fm @ B4 @ C2 ) )
        & ~ ( member4699826688122452638ist_fm @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_1023_subset__Diff__insert,axiom,
    ! [A2: set_list_fm,B4: set_list_fm,X3: list_fm,C2: set_list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ A2 @ ( minus_639611354763871680ist_fm @ B4 @ ( insert_list_fm @ X3 @ C2 ) ) )
      = ( ( ord_le7838213414353715577ist_fm @ A2 @ ( minus_639611354763871680ist_fm @ B4 @ C2 ) )
        & ~ ( member_list_fm @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_1024_subset__Diff__insert,axiom,
    ! [A2: set_rule,B4: set_rule,X3: rule,C2: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ ( minus_minus_set_rule @ B4 @ ( insert_rule @ X3 @ C2 ) ) )
      = ( ( ord_less_eq_set_rule @ A2 @ ( minus_minus_set_rule @ B4 @ C2 ) )
        & ~ ( member_rule @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_1025_subset__Diff__insert,axiom,
    ! [A2: set_tm,B4: set_tm,X3: tm,C2: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ ( minus_minus_set_tm @ B4 @ ( insert_tm @ X3 @ C2 ) ) )
      = ( ( ord_less_eq_set_tm @ A2 @ ( minus_minus_set_tm @ B4 @ C2 ) )
        & ~ ( member_tm @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_1026_subset__Diff__insert,axiom,
    ! [A2: set_fm,B4: set_fm,X3: fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ ( minus_minus_set_fm @ B4 @ ( insert_fm @ X3 @ C2 ) ) )
      = ( ( ord_less_eq_set_fm @ A2 @ ( minus_minus_set_fm @ B4 @ C2 ) )
        & ~ ( member_fm @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_1027_szip_Osimps_I1_J,axiom,
    ! [S1: stream_nat,S22: stream_nat] :
      ( ( shd_Pr4260400998323988397at_nat @ ( szip_nat_nat @ S1 @ S22 ) )
      = ( product_Pair_nat_nat @ ( shd_nat @ S1 ) @ ( shd_nat @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_1028_szip_Osimps_I1_J,axiom,
    ! [S1: stream6724221391990029191at_nat,S22: stream6724221391990029191at_nat] :
      ( ( shd_Pr8412153233960533267at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S22 ) )
      = ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S1 ) @ ( shd_Pr4260400998323988397at_nat @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_1029_szip_Osimps_I1_J,axiom,
    ! [S1: stream_list_tm,S22: stream_list_fm] :
      ( ( shd_Pr3211216682057661985ist_fm @ ( szip_list_tm_list_fm @ S1 @ S22 ) )
      = ( produc1414352766439514085ist_fm @ ( shd_list_tm @ S1 ) @ ( shd_list_fm @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_1030_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( shd_Pr2264621979884435249e_rule @ ( szip_rule_rule @ S1 @ S22 ) )
      = ( produc5849431337705160893e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_1031_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream2709947120125613254m_rule] :
      ( ( shd_Pr1208753253167450709m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S22 ) )
      = ( produc97041875958848737m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_1032_szip_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S22: stream_rule] :
      ( ( shd_Pr2400962586966563157e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S22 ) )
      = ( produc4831648765031708129e_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_1033_szip_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
      ( ( shd_Pr4461660664618831993m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) )
      = ( produc6261311607089640965m_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_1034_szip_Osimps_I1_J,axiom,
    ! [S1: stream4408948924543953275ist_fm,S22: stream_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) )
      = ( produc1733806532565653680m_rule @ ( shd_Pr3211216682057661985ist_fm @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_1035_NE__R,axiom,
    ( ( sset_rule @ rules )
   != bot_bot_set_rule ) ).

% NE_R
thf(fact_1036_epath__sdrop__ptms,axiom,
    ! [Steps: stream2709947120125613254m_rule,N: nat] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( ord_less_eq_set_tm @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) ) @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ) ) ) ).

% epath_sdrop_ptms
thf(fact_1037_RuleSystem_Oper__def,axiom,
    ! [Eff: tm > rule > fset_rule > $o,Rules: stream_tm,S2: set_rule,R: tm] :
      ( ( abstra7099963851359259790m_rule @ Eff @ Rules @ S2 )
     => ( ( abstract_per_tm_rule @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: rule,R1: tm,Sl5: fset_rule,S8: rule] :
              ( ( ( member_rule @ S3 @ S2 )
                & ( abstra1391462102081896669m_rule @ Eff @ R @ S3 )
                & ( member_tm @ R1 @ ( minus_minus_set_tm @ ( sset_tm @ Rules ) @ ( insert_tm @ R @ bot_bot_set_tm ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_rule @ S8 @ Sl5 ) )
             => ( abstra1391462102081896669m_rule @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1038_RuleSystem_Oper__def,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S2: set_rule,R: fm] :
      ( ( abstra7659285206637183772m_rule @ Eff @ Rules @ S2 )
     => ( ( abstract_per_fm_rule @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: rule,R1: fm,Sl5: fset_rule,S8: rule] :
              ( ( ( member_rule @ S3 @ S2 )
                & ( abstra1950783457359820651m_rule @ Eff @ R @ S3 )
                & ( member_fm @ R1 @ ( minus_minus_set_fm @ ( sset_fm @ Rules ) @ ( insert_fm @ R @ bot_bot_set_fm ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_rule @ S8 @ Sl5 ) )
             => ( abstra1950783457359820651m_rule @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1039_RuleSystem_Oper__def,axiom,
    ! [Eff: tm > tm > fset_tm > $o,Rules: stream_tm,S2: set_tm,R: tm] :
      ( ( abstra5526243810835382505_tm_tm @ Eff @ Rules @ S2 )
     => ( ( abstract_per_tm_tm @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: tm,R1: tm,Sl5: fset_tm,S8: tm] :
              ( ( ( member_tm @ S3 @ S2 )
                & ( abstra249629297200918874_tm_tm @ Eff @ R @ S3 )
                & ( member_tm @ R1 @ ( minus_minus_set_tm @ ( sset_tm @ Rules ) @ ( insert_tm @ R @ bot_bot_set_tm ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_tm @ S8 @ Sl5 ) )
             => ( abstra249629297200918874_tm_tm @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1040_RuleSystem_Oper__def,axiom,
    ! [Eff: fm > tm > fset_tm > $o,Rules: stream_fm,S2: set_tm,R: fm] :
      ( ( abstra966159493039759323_fm_tm @ Eff @ Rules @ S2 )
     => ( ( abstract_per_fm_tm @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: tm,R1: fm,Sl5: fset_tm,S8: tm] :
              ( ( ( member_tm @ S3 @ S2 )
                & ( abstra4912917016260071500_fm_tm @ Eff @ R @ S3 )
                & ( member_fm @ R1 @ ( minus_minus_set_fm @ ( sset_fm @ Rules ) @ ( insert_fm @ R @ bot_bot_set_fm ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_tm @ S8 @ Sl5 ) )
             => ( abstra4912917016260071500_fm_tm @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1041_RuleSystem_Oper__def,axiom,
    ! [Eff: tm > fm > fset_fm > $o,Rules: stream_tm,S2: set_fm,R: tm] :
      ( ( abstra5526243810834464119_tm_fm @ Eff @ Rules @ S2 )
     => ( ( abstract_per_tm_fm @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: fm,R1: tm,Sl5: fset_fm,S8: fm] :
              ( ( ( member_fm @ S3 @ S2 )
                & ( abstra249629297200000488_tm_fm @ Eff @ R @ S3 )
                & ( member_tm @ R1 @ ( minus_minus_set_tm @ ( sset_tm @ Rules ) @ ( insert_tm @ R @ bot_bot_set_tm ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_fm @ S8 @ Sl5 ) )
             => ( abstra249629297200000488_tm_fm @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1042_RuleSystem_Oper__def,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S2: set_fm,R: fm] :
      ( ( abstra966159493038840937_fm_fm @ Eff @ Rules @ S2 )
     => ( ( abstract_per_fm_fm @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: fm,R1: fm,Sl5: fset_fm,S8: fm] :
              ( ( ( member_fm @ S3 @ S2 )
                & ( abstra4912917016259153114_fm_fm @ Eff @ R @ S3 )
                & ( member_fm @ R1 @ ( minus_minus_set_fm @ ( sset_fm @ Rules ) @ ( insert_fm @ R @ bot_bot_set_fm ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_fm @ S8 @ Sl5 ) )
             => ( abstra4912917016259153114_fm_fm @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1043_RuleSystem_Oper__def,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S2: set_rule,R: rule] :
      ( ( abstra6805424310982398927e_rule @ Eff @ Rules @ S2 )
     => ( ( abstra138634490569309770e_rule @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: rule,R1: rule,Sl5: fset_rule,S8: rule] :
              ( ( ( member_rule @ S3 @ S2 )
                & ( abstra7234149737463204544e_rule @ Eff @ R @ S3 )
                & ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ Rules ) @ ( insert_rule @ R @ bot_bot_set_rule ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_rule @ S8 @ Sl5 ) )
             => ( abstra7234149737463204544e_rule @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1044_RuleSystem_Oper__def,axiom,
    ! [Eff: rule > tm > fset_tm > $o,Rules: stream_rule,S2: set_tm,R: rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff @ Rules @ S2 )
     => ( ( abstract_per_rule_tm @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: tm,R1: rule,Sl5: fset_tm,S8: tm] :
              ( ( ( member_tm @ S3 @ S2 )
                & ( abstra8423890812398148919ule_tm @ Eff @ R @ S3 )
                & ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ Rules ) @ ( insert_rule @ R @ bot_bot_set_rule ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_tm @ S8 @ Sl5 ) )
             => ( abstra8423890812398148919ule_tm @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1045_RuleSystem_Oper__def,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S2: set_fm,R: rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff @ Rules @ S2 )
     => ( ( abstract_per_rule_fm @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: fm,R1: rule,Sl5: fset_fm,S8: fm] :
              ( ( ( member_fm @ S3 @ S2 )
                & ( abstra8423890812397230533ule_fm @ Eff @ R @ S3 )
                & ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ Rules ) @ ( insert_rule @ R @ bot_bot_set_rule ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_fm @ S8 @ Sl5 ) )
             => ( abstra8423890812397230533ule_fm @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1046_RuleSystem_Oper__def,axiom,
    ! [Eff: list_fm > rule > fset_rule > $o,Rules: stream_list_fm,S2: set_rule,R: list_fm] :
      ( ( abstra6744069043672362530m_rule @ Eff @ Rules @ S2 )
     => ( ( abstra2231175174020358375m_rule @ Eff @ Rules @ S2 @ R )
        = ( ! [S3: rule,R1: list_fm,Sl5: fset_rule,S8: rule] :
              ( ( ( member_rule @ S3 @ S2 )
                & ( abstra3303091959664946929m_rule @ Eff @ R @ S3 )
                & ( member_list_fm @ R1 @ ( minus_639611354763871680ist_fm @ ( sset_list_fm @ Rules ) @ ( insert_list_fm @ R @ bot_bot_set_list_fm ) ) )
                & ( Eff @ R1 @ S3 @ Sl5 )
                & ( fmember_rule @ S8 @ Sl5 ) )
             => ( abstra3303091959664946929m_rule @ Eff @ R @ S8 ) ) ) ) ) ).

% RuleSystem.per_def
thf(fact_1047_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: nat > nat > fset_nat > $o,Rs: stream_nat,S: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ Eff @ Rs @ S ) )
      = ( product_Pair_nat_nat @ S @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff @ Rs @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_1048_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs: stream6724221391990029191at_nat,S: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ Eff @ Rs @ S ) )
      = ( produc6161850002892822231at_nat @ S @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff @ Rs @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_1049_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: list_fm > list_tm > fset_list_tm > $o,Rs: stream_list_fm,S: list_tm] :
      ( ( abstra2656519403522841862ist_fm @ ( abstra4388168957404043029ist_tm @ Eff @ Rs @ S ) )
      = ( produc1414352766439514085ist_fm @ S @ ( shd_list_fm @ ( abstra1372434334096485691ist_tm @ Eff @ Rs @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_1050_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra5180277867158163941m_rule @ ( abstra7840189589880272446ist_fm @ Eff @ Rs @ S ) )
      = ( produc1733806532565653680m_rule @ S @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff @ Rs @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_1051__092_060open_062t_A_092_060in_062_Aset_A_Isubterms_A_Ipseq_A_Ishd_A_Isdrop_Am_Asteps_J_J_J_J_092_060close_062,axiom,
    member_tm @ t @ ( set_tm2 @ ( subterms @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ m @ steps ) ) ) ) ) ).

% \<open>t \<in> set (subterms (pseq (shd (sdrop m steps))))\<close>
thf(fact_1052__092_060open_062_092_060exists_062m_O_At_A_092_060in_062_Aset_A_Isubterms_A_Ipseq_A_Ishd_A_Isdrop_Am_Asteps_J_J_J_J_092_060close_062,axiom,
    ? [M6: nat] : ( member_tm @ t @ ( set_tm2 @ ( subterms @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ M6 @ steps ) ) ) ) ) ) ).

% \<open>\<exists>m. t \<in> set (subterms (pseq (shd (sdrop m steps))))\<close>
thf(fact_1053__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_At_A_092_060in_062_Aset_A_Isubterms_A_Ipseq_A_Ishd_A_Isdrop_Am_Asteps_J_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [M6: nat] :
        ~ ( member_tm @ t @ ( set_tm2 @ ( subterms @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ M6 @ steps ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>m. t \<in> set (subterms (pseq (shd (sdrop m steps)))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1054_diff__shunt__var,axiom,
    ! [X3: set_rule,Y: set_rule] :
      ( ( ( minus_minus_set_rule @ X3 @ Y )
        = bot_bot_set_rule )
      = ( ord_less_eq_set_rule @ X3 @ Y ) ) ).

% diff_shunt_var
thf(fact_1055_diff__shunt__var,axiom,
    ! [X3: set_tm,Y: set_tm] :
      ( ( ( minus_minus_set_tm @ X3 @ Y )
        = bot_bot_set_tm )
      = ( ord_less_eq_set_tm @ X3 @ Y ) ) ).

% diff_shunt_var
thf(fact_1056_diff__shunt__var,axiom,
    ! [X3: set_fm,Y: set_fm] :
      ( ( ( minus_minus_set_fm @ X3 @ Y )
        = bot_bot_set_fm )
      = ( ord_less_eq_set_fm @ X3 @ Y ) ) ).

% diff_shunt_var
thf(fact_1057_fempty__iff,axiom,
    ! [C: produc6018962875968178549ist_fm] :
      ~ ( fmembe3754813877001230652ist_fm @ C @ bot_bo6461889142629771335ist_fm ) ).

% fempty_iff
thf(fact_1058_fempty__iff,axiom,
    ! [C: abstra8520858721168826610m_rule] :
      ~ ( fmembe1099044791951439339m_rule @ C @ bot_bo3860209539026414262m_rule ) ).

% fempty_iff
thf(fact_1059_all__not__fin__conv,axiom,
    ! [A2: fset_P8989946509869081563ist_fm] :
      ( ( ! [X4: produc6018962875968178549ist_fm] :
            ~ ( fmembe3754813877001230652ist_fm @ X4 @ A2 ) )
      = ( A2 = bot_bo6461889142629771335ist_fm ) ) ).

% all_not_fin_conv
thf(fact_1060_all__not__fin__conv,axiom,
    ! [A2: fset_A8849196716589201442m_rule] :
      ( ( ! [X4: abstra8520858721168826610m_rule] :
            ~ ( fmembe1099044791951439339m_rule @ X4 @ A2 ) )
      = ( A2 = bot_bo3860209539026414262m_rule ) ) ).

% all_not_fin_conv
thf(fact_1061_bot__empty__eq,axiom,
    ( bot_bo3088342962497934428t_fm_o
    = ( ^ [X4: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X4 @ bot_bo1798318783834493889ist_fm ) ) ) ).

% bot_empty_eq
thf(fact_1062_bot__empty__eq,axiom,
    ( bot_bot_list_fm_o
    = ( ^ [X4: list_fm] : ( member_list_fm @ X4 @ bot_bot_set_list_fm ) ) ) ).

% bot_empty_eq
thf(fact_1063_bot__empty__eq,axiom,
    ( bot_bot_tm_o
    = ( ^ [X4: tm] : ( member_tm @ X4 @ bot_bot_set_tm ) ) ) ).

% bot_empty_eq
thf(fact_1064_bot__empty__eq,axiom,
    ( bot_bot_fm_o
    = ( ^ [X4: fm] : ( member_fm @ X4 @ bot_bot_set_fm ) ) ) ).

% bot_empty_eq
thf(fact_1065_bot__empty__eq,axiom,
    ( bot_bot_rule_o
    = ( ^ [X4: rule] : ( member_rule @ X4 @ bot_bot_set_rule ) ) ) ).

% bot_empty_eq
thf(fact_1066_bot__set__def,axiom,
    ( bot_bot_set_rule
    = ( collect_rule @ bot_bot_rule_o ) ) ).

% bot_set_def
thf(fact_1067_equalsffemptyI,axiom,
    ! [A2: fset_P8989946509869081563ist_fm] :
      ( ! [Y2: produc6018962875968178549ist_fm] :
          ~ ( fmembe3754813877001230652ist_fm @ Y2 @ A2 )
     => ( A2 = bot_bo6461889142629771335ist_fm ) ) ).

% equalsffemptyI
thf(fact_1068_equalsffemptyI,axiom,
    ! [A2: fset_A8849196716589201442m_rule] :
      ( ! [Y2: abstra8520858721168826610m_rule] :
          ~ ( fmembe1099044791951439339m_rule @ Y2 @ A2 )
     => ( A2 = bot_bo3860209539026414262m_rule ) ) ).

% equalsffemptyI
thf(fact_1069_equalsffemptyD,axiom,
    ! [A2: fset_P8989946509869081563ist_fm,A: produc6018962875968178549ist_fm] :
      ( ( A2 = bot_bo6461889142629771335ist_fm )
     => ~ ( fmembe3754813877001230652ist_fm @ A @ A2 ) ) ).

% equalsffemptyD
thf(fact_1070_equalsffemptyD,axiom,
    ! [A2: fset_A8849196716589201442m_rule,A: abstra8520858721168826610m_rule] :
      ( ( A2 = bot_bo3860209539026414262m_rule )
     => ~ ( fmembe1099044791951439339m_rule @ A @ A2 ) ) ).

% equalsffemptyD
thf(fact_1071_ex__fin__conv,axiom,
    ! [A2: fset_P8989946509869081563ist_fm] :
      ( ( ? [X4: produc6018962875968178549ist_fm] : ( fmembe3754813877001230652ist_fm @ X4 @ A2 ) )
      = ( A2 != bot_bo6461889142629771335ist_fm ) ) ).

% ex_fin_conv
thf(fact_1072_ex__fin__conv,axiom,
    ! [A2: fset_A8849196716589201442m_rule] :
      ( ( ? [X4: abstra8520858721168826610m_rule] : ( fmembe1099044791951439339m_rule @ X4 @ A2 ) )
      = ( A2 != bot_bo3860209539026414262m_rule ) ) ).

% ex_fin_conv
thf(fact_1073_femptyE,axiom,
    ! [A: produc6018962875968178549ist_fm] :
      ~ ( fmembe3754813877001230652ist_fm @ A @ bot_bo6461889142629771335ist_fm ) ).

% femptyE
thf(fact_1074_femptyE,axiom,
    ! [A: abstra8520858721168826610m_rule] :
      ~ ( fmembe1099044791951439339m_rule @ A @ bot_bo3860209539026414262m_rule ) ).

% femptyE
thf(fact_1075_is__singletonI,axiom,
    ! [X3: rule] : ( is_singleton_rule @ ( insert_rule @ X3 @ bot_bot_set_rule ) ) ).

% is_singletonI
thf(fact_1076_subset__code_I1_J,axiom,
    ! [Xs: list_P2887561121880082555ist_fm,B4: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ ( set_Pr5051287786238754058ist_fm @ Xs ) @ B4 )
      = ( ! [X4: produc6018962875968178549ist_fm] :
            ( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
           => ( member4699826688122452638ist_fm @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1077_subset__code_I1_J,axiom,
    ! [Xs: list_rule,B4: set_rule] :
      ( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ B4 )
      = ( ! [X4: rule] :
            ( ( member_rule @ X4 @ ( set_rule2 @ Xs ) )
           => ( member_rule @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1078_subset__code_I1_J,axiom,
    ! [Xs: list_list_fm,B4: set_list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ B4 )
      = ( ! [X4: list_fm] :
            ( ( member_list_fm @ X4 @ ( set_list_fm2 @ Xs ) )
           => ( member_list_fm @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1079_subset__code_I1_J,axiom,
    ! [Xs: list_tm,B4: set_tm] :
      ( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B4 )
      = ( ! [X4: tm] :
            ( ( member_tm @ X4 @ ( set_tm2 @ Xs ) )
           => ( member_tm @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1080_subset__code_I1_J,axiom,
    ! [Xs: list_fm,B4: set_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B4 )
      = ( ! [X4: fm] :
            ( ( member_fm @ X4 @ ( set_fm2 @ Xs ) )
           => ( member_fm @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1081_insert__subsetI,axiom,
    ! [X3: produc6018962875968178549ist_fm,A2: set_Pr5202636777678657877ist_fm,X7: set_Pr5202636777678657877ist_fm] :
      ( ( member4699826688122452638ist_fm @ X3 @ A2 )
     => ( ( ord_le1771420097867575541ist_fm @ X7 @ A2 )
       => ( ord_le1771420097867575541ist_fm @ ( insert4380798862274524997ist_fm @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_1082_insert__subsetI,axiom,
    ! [X3: rule,A2: set_rule,X7: set_rule] :
      ( ( member_rule @ X3 @ A2 )
     => ( ( ord_less_eq_set_rule @ X7 @ A2 )
       => ( ord_less_eq_set_rule @ ( insert_rule @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_1083_insert__subsetI,axiom,
    ! [X3: list_fm,A2: set_list_fm,X7: set_list_fm] :
      ( ( member_list_fm @ X3 @ A2 )
     => ( ( ord_le7838213414353715577ist_fm @ X7 @ A2 )
       => ( ord_le7838213414353715577ist_fm @ ( insert_list_fm @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_1084_insert__subsetI,axiom,
    ! [X3: tm,A2: set_tm,X7: set_tm] :
      ( ( member_tm @ X3 @ A2 )
     => ( ( ord_less_eq_set_tm @ X7 @ A2 )
       => ( ord_less_eq_set_tm @ ( insert_tm @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_1085_insert__subsetI,axiom,
    ! [X3: fm,A2: set_fm,X7: set_fm] :
      ( ( member_fm @ X3 @ A2 )
     => ( ( ord_less_eq_set_fm @ X7 @ A2 )
       => ( ord_less_eq_set_fm @ ( insert_fm @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_1086_subset__emptyI,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ! [X: produc6018962875968178549ist_fm] :
          ~ ( member4699826688122452638ist_fm @ X @ A2 )
     => ( ord_le1771420097867575541ist_fm @ A2 @ bot_bo1798318783834493889ist_fm ) ) ).

% subset_emptyI
thf(fact_1087_subset__emptyI,axiom,
    ! [A2: set_list_fm] :
      ( ! [X: list_fm] :
          ~ ( member_list_fm @ X @ A2 )
     => ( ord_le7838213414353715577ist_fm @ A2 @ bot_bot_set_list_fm ) ) ).

% subset_emptyI
thf(fact_1088_subset__emptyI,axiom,
    ! [A2: set_rule] :
      ( ! [X: rule] :
          ~ ( member_rule @ X @ A2 )
     => ( ord_less_eq_set_rule @ A2 @ bot_bot_set_rule ) ) ).

% subset_emptyI
thf(fact_1089_subset__emptyI,axiom,
    ! [A2: set_tm] :
      ( ! [X: tm] :
          ~ ( member_tm @ X @ A2 )
     => ( ord_less_eq_set_tm @ A2 @ bot_bot_set_tm ) ) ).

% subset_emptyI
thf(fact_1090_subset__emptyI,axiom,
    ! [A2: set_fm] :
      ( ! [X: fm] :
          ~ ( member_fm @ X @ A2 )
     => ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm ) ) ).

% subset_emptyI
thf(fact_1091_is__singletonI_H,axiom,
    ! [A2: set_tm] :
      ( ( A2 != bot_bot_set_tm )
     => ( ! [X: tm,Y2: tm] :
            ( ( member_tm @ X @ A2 )
           => ( ( member_tm @ Y2 @ A2 )
             => ( X = Y2 ) ) )
       => ( is_singleton_tm @ A2 ) ) ) ).

% is_singletonI'
thf(fact_1092_is__singletonI_H,axiom,
    ! [A2: set_fm] :
      ( ( A2 != bot_bot_set_fm )
     => ( ! [X: fm,Y2: fm] :
            ( ( member_fm @ X @ A2 )
           => ( ( member_fm @ Y2 @ A2 )
             => ( X = Y2 ) ) )
       => ( is_singleton_fm @ A2 ) ) ) ).

% is_singletonI'
thf(fact_1093_is__singletonI_H,axiom,
    ! [A2: set_rule] :
      ( ( A2 != bot_bot_set_rule )
     => ( ! [X: rule,Y2: rule] :
            ( ( member_rule @ X @ A2 )
           => ( ( member_rule @ Y2 @ A2 )
             => ( X = Y2 ) ) )
       => ( is_singleton_rule @ A2 ) ) ) ).

% is_singletonI'
thf(fact_1094_pseq__in__tree__fms,axiom,
    ! [X3: produc340336539035504054m_rule,Steps: stream2709947120125613254m_rule,P5: fm] :
      ( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Steps ) )
     => ( ( member_fm @ P5 @ ( set_fm2 @ ( pseq @ X3 ) ) )
       => ( member_fm @ P5 @ ( tree_fms @ Steps ) ) ) ) ).

% pseq_in_tree_fms
thf(fact_1095_prod__decode__aux_Oelims,axiom,
    ! [X3: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa @ X3 )
         => ( Y
            = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X3 @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa @ X3 )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa @ ( suc @ X3 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_1096_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K4: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K4 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K4 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus_nat @ M2 @ ( suc @ K4 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_1097_epath__completeness__Saturated,axiom,
    ! [S: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ? [T8: abstra8520858721168826610m_rule] :
            ( ( ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T8 ) )
              = S )
            & ( abstra8167393772827826667ist_fm @ eff @ rules @ T8 )
            & ( abstra3822255059661934556m_rule @ T8 ) )
        | ? [Steps4: stream2709947120125613254m_rule] :
            ( ( ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps4 ) )
              = S )
            & ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps4 )
            & ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps4 ) ) ) ) ).

% epath_completeness_Saturated
thf(fact_1098_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1099_pair__less__iff1,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( product_Pair_nat_nat @ X3 @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z ) ) ).

% pair_less_iff1
thf(fact_1100_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_1101_epath_Ocases,axiom,
    ! [A: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ A )
     => ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ A ) ) ) @ Sl2 )
             => ( ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ A ) ) ) ) ) ) ).

% epath.cases
thf(fact_1102_epath_Osimps,axiom,
    ! [A: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ A )
      = ( ? [Steps2: stream2709947120125613254m_rule,Sl3: fset_P8989946509869081563ist_fm] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( sset_rule @ rules ) )
            & ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) @ Sl3 )
            & ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps2 ) ) @ Sl3 )
            & ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).

% epath.simps
thf(fact_1103_epath_Ointros,axiom,
    ! [Steps: stream2709947120125613254m_rule,Sl: fset_P8989946509869081563ist_fm] :
      ( ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( sset_rule @ rules ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) @ Sl )
       => ( ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ Sl )
         => ( ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps ) )
           => ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps ) ) ) ) ) ).

% epath.intros
thf(fact_1104_epath_Ocoinduct,axiom,
    ! [X7: stream2709947120125613254m_rule > $o,X3: stream2709947120125613254m_rule] :
      ( ( X7 @ X3 )
     => ( ! [X: stream2709947120125613254m_rule] :
            ( ( X7 @ X )
           => ? [Steps3: stream2709947120125613254m_rule,Sl6: fset_P8989946509869081563ist_fm] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ ( sset_rule @ rules ) )
                & ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) @ Sl6 )
                & ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps3 ) ) @ Sl6 )
                & ( ( X7 @ ( stl_Pr12655793849929990m_rule @ Steps3 ) )
                  | ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) ) )
       => ( abstra6789711989322986974ist_fm @ eff @ rules @ X3 ) ) ) ).

% epath.coinduct
thf(fact_1105_ptms__def,axiom,
    ( ptms
    = ( ^ [Z5: produc340336539035504054m_rule] : ( produc4874987732838623697ist_fm @ ( produc993771546023551940m_rule @ Z5 ) ) ) ) ).

% ptms_def
thf(fact_1106_epath__eff,axiom,
    ! [Steps: stream2709947120125613254m_rule,Ss: fset_P8989946509869081563ist_fm] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ Ss )
       => ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) @ Ss ) ) ) ).

% epath_eff
thf(fact_1107_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1108_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_1109_pseq__def,axiom,
    ( pseq
    = ( ^ [Z5: produc340336539035504054m_rule] : ( produc3242453064496107283ist_fm @ ( produc993771546023551940m_rule @ Z5 ) ) ) ) ).

% pseq_def
thf(fact_1110_wf_Ocases,axiom,
    ! [A: abstra8520858721168826610m_rule] :
      ( ( abstra8167393772827826667ist_fm @ eff @ rules @ A )
     => ~ ( ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ( ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ A ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ A ) ) )
           => ~ ! [T4: abstra8520858721168826610m_rule] :
                  ( ( fmembe1099044791951439339m_rule @ T4 @ ( abstra2475333618431710837m_rule @ A ) )
                 => ( abstra8167393772827826667ist_fm @ eff @ rules @ T4 ) ) ) ) ) ).

% wf.cases
thf(fact_1111_wf_Osimps,axiom,
    ! [A: abstra8520858721168826610m_rule] :
      ( ( abstra8167393772827826667ist_fm @ eff @ rules @ A )
      = ( ? [T5: abstra8520858721168826610m_rule] :
            ( ( A = T5 )
            & ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T5 ) ) @ ( sset_rule @ rules ) )
            & ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T5 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T5 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T5 ) ) )
            & ! [X4: abstra8520858721168826610m_rule] :
                ( ( fmembe1099044791951439339m_rule @ X4 @ ( abstra2475333618431710837m_rule @ T5 ) )
               => ( abstra8167393772827826667ist_fm @ eff @ rules @ X4 ) ) ) ) ) ).

% wf.simps
thf(fact_1112_wf_Ocoinduct,axiom,
    ! [X7: abstra8520858721168826610m_rule > $o,X3: abstra8520858721168826610m_rule] :
      ( ( X7 @ X3 )
     => ( ! [X: abstra8520858721168826610m_rule] :
            ( ( X7 @ X )
           => ? [T3: abstra8520858721168826610m_rule] :
                ( ( X = T3 )
                & ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( sset_rule @ rules ) )
                & ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T3 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T3 ) ) )
                & ! [Xa2: abstra8520858721168826610m_rule] :
                    ( ( fmembe1099044791951439339m_rule @ Xa2 @ ( abstra2475333618431710837m_rule @ T3 ) )
                   => ( ( X7 @ Xa2 )
                      | ( abstra8167393772827826667ist_fm @ eff @ rules @ Xa2 ) ) ) ) )
       => ( abstra8167393772827826667ist_fm @ eff @ rules @ X3 ) ) ) ).

% wf.coinduct
thf(fact_1113_wf_Ointros,axiom,
    ! [T: abstra8520858721168826610m_rule] :
      ( ( member_rule @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T ) ) @ ( sset_rule @ rules ) )
     => ( ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T ) ) )
       => ( ! [T7: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ T7 @ ( abstra2475333618431710837m_rule @ T ) )
             => ( abstra8167393772827826667ist_fm @ eff @ rules @ T7 ) )
         => ( abstra8167393772827826667ist_fm @ eff @ rules @ T ) ) ) ) ).

% wf.intros
thf(fact_1114_mkTree_Osimps_I2_J,axiom,
    ! [Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) )
      = ( fimage2815761261218872234m_rule @ ( abstra7840189589880272446ist_fm @ eff @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) @ ( abstra3309574903381802724ist_fm @ eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S ) ) ) ).

% mkTree.simps(2)
thf(fact_1115_pickEff,axiom,
    ! [R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
     => ( eff @ R @ S @ ( abstra3309574903381802724ist_fm @ eff @ R @ S ) ) ) ).

% pickEff
thf(fact_1116_mkTree_Ocode,axiom,
    ! [Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra7840189589880272446ist_fm @ eff @ Rs @ S )
      = ( abstra5937941257295885381m_rule @ ( produc1733806532565653680m_rule @ S @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) @ ( fimage2815761261218872234m_rule @ ( abstra7840189589880272446ist_fm @ eff @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) @ ( abstra3309574903381802724ist_fm @ eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S ) ) ) ) ).

% mkTree.code
thf(fact_1117_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1118_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_1119_ori_I2_J,axiom,
    ( suf
    = ( sdrop_8169176516188972301m_rule @ k @ ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ ( plus_plus_nat @ n @ m ) @ one_one_nat ) @ steps ) ) ) ).

% ori(2)
thf(fact_1120_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_1121_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_1122_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_1123_diff__diff__left,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% diff_diff_left
thf(fact_1124_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_1125_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1126_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1127_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1128_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1129_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_1130_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1131_nat__arith_Osuc1,axiom,
    ! [A2: nat,K2: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_1132_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_1133_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_1134_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_1135_add__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_less_mono1
thf(fact_1136_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_1137_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_1138_add__less__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K2 @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_1139_add__lessD1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
     => ( ord_less_nat @ I @ K2 ) ) ).

% add_lessD1
thf(fact_1140_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_1141_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_1142_diff__cancel2,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_1143_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_1144_add__leE,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).

% add_leE
thf(fact_1145_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_1146_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_1147_add__leD1,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_1148_add__leD2,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
     => ( ord_less_eq_nat @ K2 @ N ) ) ).

% add_leD2
thf(fact_1149_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ K2 @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K2 @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_1150_add__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_1151_add__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_le_mono1
thf(fact_1152_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_1153_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_1154_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N3: nat] :
        ? [K4: nat] :
          ( N3
          = ( plus_plus_nat @ M2 @ K4 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1155_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1156_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N3: nat] :
        ? [K4: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M2 @ K4 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1157_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_1158_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_1159_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_1160_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K2: nat] :
      ( ! [M6: nat,N2: nat] :
          ( ( ord_less_nat @ M6 @ N2 )
         => ( ord_less_nat @ ( F @ M6 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1161_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_1162_less__diff__conv,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).

% less_diff_conv
thf(fact_1163_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K2 )
        = ( J
          = ( plus_plus_nat @ K2 @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1164_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1165_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1166_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1167_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).

% le_diff_conv
thf(fact_1168_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1169_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1170_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1171_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).

% less_diff_conv2
thf(fact_1172_ori_I1_J,axiom,
    ( pre
    = ( stake_5421812949518764133m_rule @ k @ ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ ( plus_plus_nat @ n @ m ) @ one_one_nat ) @ steps ) ) ) ).

% ori(1)
thf(fact_1173_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_1174_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M7: nat] :
      ( ( P @ X3 )
     => ( ! [X: nat] :
            ( ( P @ X )
           => ( ord_less_eq_nat @ X @ M7 ) )
       => ~ ! [M6: nat] :
              ( ( P @ M6 )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M6 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1175__092_060open_062Neg_A_IUni_Ap_J_A_092_060in_062_Aset_A_Ipseq_A_Ishd_A_Isdrop_A_In_A_L_Am_A_L_A1_J_Asteps_J_J_J_092_060close_062,axiom,
    member_fm @ ( neg @ ( uni @ p ) ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ ( plus_plus_nat @ n @ m ) @ one_one_nat ) @ steps ) ) ) ) ).

% \<open>Neg (Uni p) \<in> set (pseq (shd (sdrop (n + m + 1) steps)))\<close>
thf(fact_1176__092_060open_062Neg_A_IUni_Ap_J_A_092_060in_062_Atree__fms_Asteps_092_060close_062,axiom,
    member_fm @ ( neg @ ( uni @ p ) ) @ ( tree_fms @ steps ) ).

% \<open>Neg (Uni p) \<in> tree_fms steps\<close>
thf(fact_1177_n,axiom,
    member_fm @ ( neg @ ( uni @ p ) ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ) ) ).

% n
thf(fact_1178_calculation,axiom,
    member_fm @ ( neg @ ( uni @ p ) ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ suf ) ) ) ).

% calculation
thf(fact_1179__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_ANeg_A_IUni_Ap_J_A_092_060in_062_Aset_A_Ipseq_A_Ishd_A_Isdrop_An_Asteps_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [N2: nat] :
        ~ ( member_fm @ ( neg @ ( uni @ p ) ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ steps ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>n. Neg (Uni p) \<in> set (pseq (shd (sdrop n steps))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1180__C_K_C,axiom,
    ! [R: rule] :
      ~ ( affects @ R @ ( neg @ ( uni @ p ) ) ) ).

% "*"
thf(fact_1181_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_1182_epath__stl__ptms,axiom,
    ! [Steps: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( ( ptms @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) )
        = ( remdups_tm @ ( append_tm @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) @ ( append_tm @ ( subterms @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) ) @ ( subterms @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) ) ) ) ) ) ) ) ) ).

% epath_stl_ptms
thf(fact_1183_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1184_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1185_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1186_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1187_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1188_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1189_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1190_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1191_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1192_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_1193_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_1194_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_1195_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_1196_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1197_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_1198_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1199_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M2: nat] :
            ( N
            = ( suc @ M2 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1200_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1201_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M6: nat] :
          ( N
          = ( suc @ M6 ) ) ) ).

% gr0_implies_Suc
thf(fact_1202_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_1203_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ~ ( P @ I4 ) )
            & ( P @ K ) ) ) ) ).

% ex_least_nat_le
thf(fact_1204_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_1205_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_1206_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K )
          & ( ( plus_plus_nat @ I @ K )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1207_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_1208_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1209_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_1210_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1211_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_1212_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1213_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1214_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1215_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1216_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1217_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_1218_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1219_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1220_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1221_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1222_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N2 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_1223_list__decode_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X3
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_1224_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1225_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1226_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1227_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1228_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1229_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1230_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X: nat] : ( P @ X @ zero_zero_nat )
     => ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
       => ( ! [X: nat,Y2: nat] :
              ( ( P @ X @ Y2 )
             => ( P @ ( suc @ X ) @ ( suc @ Y2 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1231_zero__induct,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( P @ K2 )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1232_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1233_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1234_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1235_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M6: nat] :
          ( N
          = ( suc @ M6 ) ) ) ).

% not0_implies_Suc
thf(fact_1236_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_nat @ K @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1237_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_1238_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1239_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D4: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D4 ) )
                & ~ ( P @ D4 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1240_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D4: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D4 ) )
           => ( P @ D4 ) ) ) ) ).

% nat_diff_split
thf(fact_1241_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_1242_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_1243_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_1244_epath__effect,axiom,
    ! [Steps: stream2709947120125613254m_rule,A2: list_tm,Z: list_fm,R: rule] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( ( ( shd_Pr4562317740776619530m_rule @ Steps )
          = ( produc1733806532565653680m_rule @ ( produc1414352766439514085ist_fm @ A2 @ Z ) @ R ) )
       => ? [B7: list_tm,Z6: list_fm,R6: rule] :
            ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B7 @ Z6 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) ) )
            & ( ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) )
              = ( produc1733806532565653680m_rule @ ( produc1414352766439514085ist_fm @ B7 @ Z6 ) @ R6 ) )
            & ( B7
              = ( remdups_tm @ ( append_tm @ A2 @ ( append_tm @ ( subterms @ Z ) @ ( subterms @ Z6 ) ) ) ) ) ) ) ) ).

% epath_effect
thf(fact_1245_effect__preserves__unaffected,axiom,
    ! [P5: fm,Z: list_fm,R: rule,B4: list_tm,Z7: list_fm,A2: list_tm] :
      ( ( member_fm @ P5 @ ( set_fm2 @ Z ) )
     => ( ~ ( affects @ R @ P5 )
       => ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z7 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) ) )
         => ( member_fm @ P5 @ ( set_fm2 @ Z7 ) ) ) ) ) ).

% effect_preserves_unaffected
thf(fact_1246_effect__tms,axiom,
    ! [B4: list_tm,Z7: list_fm,R: rule,A2: list_tm,Z: list_fm] :
      ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z7 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) ) )
     => ( B4
        = ( remdups_tm @ ( append_tm @ A2 @ ( append_tm @ ( subterms @ Z ) @ ( subterms @ Z7 ) ) ) ) ) ) ).

% effect_tms
thf(fact_1247_parts__in__effect,axiom,
    ! [P5: fm,Z: list_fm,B4: list_tm,Z7: list_fm,R: rule,A2: list_tm] :
      ( ( member_fm @ P5 @ ( set_fm2 @ Z ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z7 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) ) )
       => ? [C5: list_tm,Xs2: list_fm] :
            ( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ C5 ) )
            & ( member_list_fm @ Xs2 @ ( set_list_fm2 @ ( parts @ C5 @ R @ P5 ) ) )
            & ( ord_less_eq_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Z7 ) ) ) ) ) ).

% parts_in_effect
thf(fact_1248_parts__preserves__unaffected,axiom,
    ! [R: rule,P5: fm,Z7: list_fm,A2: list_tm] :
      ( ~ ( affects @ R @ P5 )
     => ( ( member_list_fm @ Z7 @ ( set_list_fm2 @ ( parts @ A2 @ R @ P5 ) ) )
       => ( member_fm @ P5 @ ( set_fm2 @ Z7 ) ) ) ) ).

% parts_preserves_unaffected
thf(fact_1249_parts__in__children,axiom,
    ! [P5: fm,Z: list_fm,Z7: list_fm,A2: list_tm,R: rule] :
      ( ( member_fm @ P5 @ ( set_fm2 @ Z ) )
     => ( ( member_list_fm @ Z7 @ ( set_list_fm2 @ ( children @ A2 @ R @ Z ) ) )
       => ? [B7: list_tm,Xs2: list_fm] :
            ( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ B7 ) )
            & ( member_list_fm @ Xs2 @ ( set_list_fm2 @ ( parts @ B7 @ R @ P5 ) ) )
            & ( ord_less_eq_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Z7 ) ) ) ) ) ).

% parts_in_children
thf(fact_1250_children__preserves__unaffected,axiom,
    ! [P5: fm,Z: list_fm,R: rule,Z7: list_fm,A2: list_tm] :
      ( ( member_fm @ P5 @ ( set_fm2 @ Z ) )
     => ( ~ ( affects @ R @ P5 )
       => ( ( member_list_fm @ Z7 @ ( set_list_fm2 @ ( children @ A2 @ R @ Z ) ) )
         => ( member_fm @ P5 @ ( set_fm2 @ Z7 ) ) ) ) ) ).

% children_preserves_unaffected
thf(fact_1251_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1252__092_060open_062list__all_A_I_092_060lambda_062xs_O_A_092_060not_062_Aaffects_A_Isnd_Axs_J_A_INeg_A_IUni_Ap_J_J_J_A_Istake_A_In_A_L_Am_A_L_A1_J_Asteps_J_092_060close_062,axiom,
    ( list_a9038620553111681026m_rule
    @ ^ [Xs3: produc340336539035504054m_rule] :
        ~ ( affects @ ( produc6514711208036230658m_rule @ Xs3 ) @ ( neg @ ( uni @ p ) ) )
    @ ( stake_5421812949518764133m_rule @ ( plus_plus_nat @ ( plus_plus_nat @ n @ m ) @ one_one_nat ) @ steps ) ) ).

% \<open>list_all (\<lambda>xs. \<not> affects (snd xs) (Neg (Uni p))) (stake (n + m + 1) steps)\<close>
thf(fact_1253_fm_Osize__gen_I6_J,axiom,
    ! [X62: fm] :
      ( ( size_fm @ ( uni @ X62 ) )
      = ( plus_plus_nat @ ( size_fm @ X62 ) @ ( suc @ zero_zero_nat ) ) ) ).

% fm.size_gen(6)
thf(fact_1254_pre,axiom,
    ( list_a9038620553111681026m_rule
    @ ^ [Xs3: produc340336539035504054m_rule] :
        ( ( produc6514711208036230658m_rule @ Xs3 )
       != gammaUni )
    @ pre ) ).

% pre
thf(fact_1255_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X4: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X4 )
    @ ^ [X4: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X4 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_1256_eff__def,axiom,
    ( eff
    = ( ^ [R2: rule,S3: produc6018962875968178549ist_fm] :
          ( ^ [Y4: fset_P8989946509869081563ist_fm,Z2: fset_P8989946509869081563ist_fm] : ( Y4 = Z2 )
          @ ( effect @ R2 @ S3 ) ) ) ) ).

% eff_def
thf(fact_1257_fm_Osize__gen_I7_J,axiom,
    ! [X72: fm] :
      ( ( size_fm @ ( neg @ X72 ) )
      = ( plus_plus_nat @ ( size_fm @ X72 ) @ ( suc @ zero_zero_nat ) ) ) ).

% fm.size_gen(7)
thf(fact_1258_mkTree__unfold,axiom,
    ! [Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra7840189589880272446ist_fm @ eff @ Rs @ S )
      = ( case_s7672397449011031346m_rule
        @ ^ [R2: rule,S8: stream_rule] : ( abstra5937941257295885381m_rule @ ( produc1733806532565653680m_rule @ S @ R2 ) @ ( fimage2815761261218872234m_rule @ ( abstra7840189589880272446ist_fm @ eff @ S8 ) @ ( abstra3309574903381802724ist_fm @ eff @ R2 @ S ) ) )
        @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) ).

% mkTree_unfold
thf(fact_1259_suf,axiom,
    ( linear7925859179493631106m_rule
    @ ^ [Step: produc340336539035504054m_rule] :
        ( ( produc6514711208036230658m_rule @ Step )
        = gammaUni )
    @ suf ) ).

% suf
thf(fact_1260__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pre_Asuf_Ak_O_A_092_060lbrakk_062list__all_A_I_092_060lambda_062xs_O_Asnd_Axs_A_092_060noteq_062_AGammaUni_J_Apre_059_Aholds_A_Iis__rule_AGammaUni_J_Asuf_059_Apre_A_061_Astake_Ak_A_Isdrop_A_In_A_L_Am_A_L_A1_J_Asteps_J_059_Asuf_A_061_Asdrop_Ak_A_Isdrop_A_In_A_L_Am_A_L_A1_J_Asteps_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Pre: list_P2774625669004309958m_rule] :
        ( ( list_a9038620553111681026m_rule
          @ ^ [Xs3: produc340336539035504054m_rule] :
              ( ( produc6514711208036230658m_rule @ Xs3 )
             != gammaUni )
          @ Pre )
       => ! [Suf: stream2709947120125613254m_rule] :
            ( ( linear7925859179493631106m_rule
              @ ^ [Step: produc340336539035504054m_rule] :
                  ( ( produc6514711208036230658m_rule @ Step )
                  = gammaUni )
              @ Suf )
           => ! [K: nat] :
                ( ( Pre
                  = ( stake_5421812949518764133m_rule @ K @ ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ ( plus_plus_nat @ n @ m ) @ one_one_nat ) @ steps ) ) )
               => ( Suf
                 != ( sdrop_8169176516188972301m_rule @ K @ ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ ( plus_plus_nat @ n @ m ) @ one_one_nat ) @ steps ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>pre suf k. \<lbrakk>list_all (\<lambda>xs. snd xs \<noteq> GammaUni) pre; holds (is_rule GammaUni) suf; pre = stake k (sdrop (n + m + 1) steps); suf = sdrop k (sdrop (n + m + 1) steps)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1261_epath__preserves__unaffected,axiom,
    ! [P5: fm,Steps: stream2709947120125613254m_rule,Pre2: list_P2774625669004309958m_rule,Suf2: stream2709947120125613254m_rule] :
      ( ( member_fm @ P5 @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) ) )
     => ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
       => ( ( Steps
            = ( shift_2334844276748245581m_rule @ Pre2 @ Suf2 ) )
         => ( ( list_a9038620553111681026m_rule
              @ ^ [Xs3: produc340336539035504054m_rule] :
                  ~ ( affects @ ( produc6514711208036230658m_rule @ Xs3 ) @ P5 )
              @ Pre2 )
           => ( member_fm @ P5 @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ Suf2 ) ) ) ) ) ) ) ) ).

% epath_preserves_unaffected
thf(fact_1262__092_060open_062ev_A_Iholds_A_Iis__rule_AGammaUni_J_J_A_Isdrop_A_In_A_L_Am_A_L_A1_J_Asteps_J_092_060close_062,axiom,
    ( linear1816559245158234865m_rule
    @ ( linear7925859179493631106m_rule
      @ ^ [Step: produc340336539035504054m_rule] :
          ( ( produc6514711208036230658m_rule @ Step )
          = gammaUni ) )
    @ ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ ( plus_plus_nat @ n @ m ) @ one_one_nat ) @ steps ) ) ).

% \<open>ev (holds (is_rule GammaUni)) (sdrop (n + m + 1) steps)\<close>
thf(fact_1263_Saturated__ev__rule,axiom,
    ! [Steps: stream2709947120125613254m_rule,R: rule,N: nat] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
     => ( linear1816559245158234865m_rule
        @ ( linear7925859179493631106m_rule
          @ ^ [Step: produc340336539035504054m_rule] :
              ( ( produc6514711208036230658m_rule @ Step )
              = R ) )
        @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ).

% Saturated_ev_rule
thf(fact_1264_ipath__mkTree__ev,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule,R: rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) @ Steps )
         => ( ( member_rule @ R @ ( sset_rule @ rules ) )
           => ( ( linear4957294957385550906m_rule
                @ ( linear7925859179493631106m_rule
                  @ ^ [Step: produc340336539035504054m_rule] : ( abstra1707737187183579335ist_fm @ eff @ R @ ( produc993771546023551940m_rule @ Step ) ) )
                @ Steps )
             => ( linear1816559245158234865m_rule
                @ ( linear7925859179493631106m_rule
                  @ ^ [Step: produc340336539035504054m_rule] :
                      ( ( produc6514711208036230658m_rule @ Step )
                      = R ) )
                @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_ev

% Helper facts (13)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( if_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( if_nat @ $true @ X3 @ Y )
      = 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,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
    ! [X3: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm] :
      ( ( if_Pro4760001780252510779ist_fm @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
    ! [X3: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm] :
      ( ( if_Pro4760001780252510779ist_fm @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
    ! [X3: fset_P8989946509869081563ist_fm,Y: fset_P8989946509869081563ist_fm] :
      ( ( if_fse7999432387889793441ist_fm @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
    ! [X3: fset_P8989946509869081563ist_fm,Y: fset_P8989946509869081563ist_fm] :
      ( ( if_fse7999432387889793441ist_fm @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
    ! [X3: abstra8520858721168826610m_rule,Y: abstra8520858721168826610m_rule] :
      ( ( if_Abs1293773526126661548m_rule @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
    ! [X3: abstra8520858721168826610m_rule,Y: abstra8520858721168826610m_rule] :
      ( ( if_Abs1293773526126661548m_rule @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_3_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J_T,axiom,
    ! [X3: fset_A8849196716589201442m_rule,Y: fset_A8849196716589201442m_rule] :
      ( ( if_fse9105678332773368284m_rule @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J_T,axiom,
    ! [X3: fset_A8849196716589201442m_rule,Y: fset_A8849196716589201442m_rule] :
      ( ( if_fse9105678332773368284m_rule @ $true @ X3 @ Y )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    abstra6789711989322986974ist_fm @ eff @ rules @ suf ).

%------------------------------------------------------------------------------