TPTP Problem File: SLH0868^1.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_00598_024782__13278336_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1963 ( 668 unt; 672 typ;   0 def)
%            Number of atoms       : 3605 (1121 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 11654 ( 268   ~;  37   |; 249   &;9565   @)
%                                         (   0 <=>;1535  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   7 avg)
%            Number of types       :  114 ( 113 usr)
%            Number of type conns  : 2306 (2306   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  562 ( 559 usr;  32 con; 0-4 aty)
%            Number of variables   : 3913 ( 268   ^;3462   !; 183   ?;3913   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 08:45:48.848
%------------------------------------------------------------------------------
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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__List__Olist_It__Nat__Onat_J,type,
    list_nat: $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__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Extended____Nat__Oenat,type,
    extended_enat: $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__Num__Onum,type,
    num: $tType ).

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

% Explicit typings (559)
thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Nat__Onat_001t__Prover__Orule,type,
    abstra8880880309027573988t_rule: ( nat > rule > fset_rule > $o ) > stream_nat > set_rule > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Nat__Onat_001t__SeCaV__Ofm,type,
    abstra3975341336035342497nat_fm: ( nat > fm > fset_fm > $o ) > stream_nat > set_fm > $o ).

thf(sy_c_Abstract__Completeness_OPersistentRuleSystem_001t__Nat__Onat_001t__SeCaV__Otm,type,
    abstra3975341336036260883nat_tm: ( nat > tm > fset_tm > $o ) > stream_nat > set_tm > $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__Nat__Onat,type,
    abstra4367803203942586852le_nat: ( rule > nat > fset_nat > $o ) > stream_rule > set_nat > $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__Nat__Onat,type,
    abstra2169370910433742005tm_nat: ( tm > nat > fset_nat > $o ) > stream_tm > set_nat > $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__List__Olist_It__SeCaV__Otm_J,type,
    abstra4205323827771120475ist_tm: ( list_fm > list_tm > fset_list_tm > $o ) > stream_list_fm > set_list_tm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Nat__Onat_001t__Prover__Orule,type,
    abstra7777025902706499341t_rule: ( nat > rule > fset_rule > $o ) > stream_nat > set_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Nat__Onat_001t__SeCaV__Ofm,type,
    abstra733049861594131384nat_fm: ( nat > fm > fset_fm > $o ) > stream_nat > set_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Nat__Onat_001t__SeCaV__Otm,type,
    abstra733049861595049770nat_tm: ( nat > tm > fset_tm > $o ) > stream_nat > set_tm > $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__Nat__Onat,type,
    abstra2021474039004664625le_nat: ( produc340336539035504054m_rule > nat > fset_nat > $o ) > stream2709947120125613254m_rule > set_nat > $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__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
    abstra3101592633059391453ist_fm: ( produc7694839378271647877e_rule > list_fm > fset_list_fm > $o ) > stream4385846686851721995e_rule > set_list_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__Nat__Onat,type,
    abstra5123640173530818412le_nat: ( produc7694839378271647877e_rule > nat > fset_nat > $o ) > stream4385846686851721995e_rule > set_nat > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__Prover__Orule,type,
    abstra2905114889036010542e_rule: ( produc7694839378271647877e_rule > rule > fset_rule > $o ) > stream4385846686851721995e_rule > set_rule > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__SeCaV__Ofm,type,
    abstra1261657321278078423ule_fm: ( produc7694839378271647877e_rule > fm > fset_fm > $o ) > stream4385846686851721995e_rule > set_fm > $o ).

thf(sy_c_Abstract__Completeness_ORuleSystem_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__SeCaV__Otm,type,
    abstra1261657321278996809ule_tm: ( produc7694839378271647877e_rule > tm > fset_tm > $o ) > stream4385846686851721995e_rule > set_tm > $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__Nat__Onat,type,
    abstra3263948797621512205le_nat: ( rule > nat > fset_nat > $o ) > stream_rule > set_nat > $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__Nat__Onat,type,
    abstra8150451472847306700tm_nat: ( tm > nat > fset_nat > $o ) > stream_tm > set_nat > $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__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
    abstra6387871064559822178ist_fm: ( produc7694839378271647877e_rule > list_fm > fset_list_fm > $o ) > stream4385846686851721995e_rule > list_fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__Nat__Onat,type,
    abstra8209949696089825831le_nat: ( produc7694839378271647877e_rule > nat > fset_nat > $o ) > stream4385846686851721995e_rule > nat > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__Prover__Orule,type,
    abstra6973468861704762345e_rule: ( produc7694839378271647877e_rule > rule > fset_rule > $o ) > stream4385846686851721995e_rule > rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__SeCaV__Ofm,type,
    abstra7063818707276771804ule_fm: ( produc7694839378271647877e_rule > fm > fset_fm > $o ) > stream4385846686851721995e_rule > fm > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_OminWait_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__SeCaV__Otm,type,
    abstra7063818707277690190ule_tm: ( produc7694839378271647877e_rule > tm > fset_tm > $o ) > stream4385846686851721995e_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__Nat__Onat,type,
    abstra6265646195202037778le_nat: ( rule > nat > fset_nat > $o ) > stream_rule > nat > 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_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_Opos_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    abstra2497064141961437305ist_fm: stream4408948924543953275ist_fm > produc6018962875968178549ist_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__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,
    abstra1175832030460196561m_rule: stream6210534828274662995m_rule > produc8828831911945107917m_rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_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,
    abstra3497755757335914413e_rule: stream8099677779113257519e_rule > produc9112364199808626345e_rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_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,
    abstra2305546423536801965m_rule: stream8953843411776101167m_rule > produc7920154866009513897m_rule > nat ).

thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
    abstra7798328242894269833e_rule: stream4385846686851721995e_rule > produc7694839378271647877e_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__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__Nat__Onat,type,
    abstra3870052345063948242le_nat: ( rule > nat > fset_nat > $o ) > stream_rule > stream8912087984624402761t_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 ).

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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    bNF_Ca4389690986151823996et_nat: set_Pr1261947904930325089at_nat > ( nat > set_nat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    bNF_Ca7035579180170703325set_fm: set_Pr1261947904930325089at_nat > ( nat > set_fm ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Set__Oset_It__SeCaV__Otm_J,type,
    bNF_Ca7039531205256338511set_tm: set_Pr1261947904930325089at_nat > ( nat > set_tm ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Extended____Nat__Oenat,type,
    bNF_Ca3721535472040450505d_enat: set_Pr8693737435421807431at_nat > ( product_prod_nat_nat > extended_enat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    bNF_Ca8308629720386654381at_nat: set_Pr8693737435421807431at_nat > ( product_prod_nat_nat > nat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    bNF_Ca100218009143674979et_nat: set_Pr8693737435421807431at_nat > ( product_prod_nat_nat > set_nat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    bNF_Ca8531860788103687606set_fm: set_Pr8693737435421807431at_nat > ( product_prod_nat_nat > set_fm ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__SeCaV__Otm_J,type,
    bNF_Ca8535812813189322792set_tm: set_Pr8693737435421807431at_nat > ( product_prod_nat_nat > set_tm ) > $o ).

thf(sy_c_EPathHintikka_Opseq,type,
    pseq: produc340336539035504054m_rule > list_fm ).

thf(sy_c_EPathHintikka_Optms,type,
    ptms: produc340336539035504054m_rule > list_tm ).

thf(sy_c_EPathHintikka_Otree__fms,type,
    tree_fms: stream2709947120125613254m_rule > set_fm ).

thf(sy_c_Extended__Nat_OeSuc,type,
    extended_eSuc: extended_enat > extended_enat ).

thf(sy_c_Extended__Nat_Oenat,type,
    extended_enat2: nat > extended_enat ).

thf(sy_c_FSet_Ofimage_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_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    fimage3741683731962187332ist_fm: ( abstra8520858721168826610m_rule > produc6018962875968178549ist_fm ) > fset_A8849196716589201442m_rule > fset_P8989946509869081563ist_fm ).

thf(sy_c_FSet_Ofimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    fimage4743371125182381497ist_fm: ( list_fm > produc6018962875968178549ist_fm ) > fset_list_fm > fset_P8989946509869081563ist_fm ).

thf(sy_c_FSet_Ofimage_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,
    fimage2815761261218872234m_rule: ( produc6018962875968178549ist_fm > abstra8520858721168826610m_rule ) > fset_P8989946509869081563ist_fm > fset_A8849196716589201442m_rule ).

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__Prover__Orule_J,type,
    fmembe4260254584891149511e_rule: abstract_tree_rule > fset_A3695954389205439742e_rule > $o ).

thf(sy_c_FSet_Ofmember_001t__Nat__Onat,type,
    fmember_nat: nat > fset_nat > $o ).

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,
    fmembe3754813877001230652ist_fm: produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ).

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_FSet_Ofset__of__list_001t__List__Olist_It__SeCaV__Ofm_J,type,
    fset_of_list_list_fm: list_list_fm > fset_list_fm ).

thf(sy_c_FSet_Ofset__of__list_001t__Nat__Onat,type,
    fset_of_list_nat: list_nat > fset_nat ).

thf(sy_c_FSet_Ofset__of__list_001t__SeCaV__Ofm,type,
    fset_of_list_fm: list_fm > fset_fm ).

thf(sy_c_FSet_Ofset__of__list_001t__SeCaV__Otm,type,
    fset_of_list_tm: list_tm > fset_tm ).

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__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__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__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_Hintikka_Oterms,type,
    terms: set_fm > set_tm ).

thf(sy_c_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    if_list_list_fm: $o > list_list_fm > list_list_fm > list_list_fm ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__List__Olist_It__Prover__Orule_J,type,
    if_list_rule: $o > list_rule > list_rule > list_rule ).

thf(sy_c_If_001t__List__Olist_It__SeCaV__Ofm_J,type,
    if_list_fm: $o > list_fm > list_fm > list_fm ).

thf(sy_c_If_001t__List__Olist_It__SeCaV__Otm_J,type,
    if_list_tm: $o > list_tm > list_tm > list_tm ).

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_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Ofm_J,type,
    append_list_fm: list_list_fm > list_list_fm > list_list_fm ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Oappend_001t__SeCaV__Ofm,type,
    append_fm: list_fm > list_fm > list_fm ).

thf(sy_c_List_Oappend_001t__SeCaV__Otm,type,
    append_tm: list_tm > list_tm > list_tm ).

thf(sy_c_List_Ocan__select_001t__List__Olist_It__SeCaV__Ofm_J,type,
    can_select_list_fm: ( list_fm > $o ) > set_list_fm > $o ).

thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
    can_select_nat: ( nat > $o ) > set_nat > $o ).

thf(sy_c_List_Ocan__select_001t__Prover__Orule,type,
    can_select_rule: ( rule > $o ) > set_rule > $o ).

thf(sy_c_List_Ocan__select_001t__SeCaV__Ofm,type,
    can_select_fm: ( fm > $o ) > set_fm > $o ).

thf(sy_c_List_Ocan__select_001t__SeCaV__Otm,type,
    can_select_tm: ( tm > $o ) > set_tm > $o ).

thf(sy_c_List_Oconcat_001t__SeCaV__Otm,type,
    concat_tm: list_list_tm > list_tm ).

thf(sy_c_List_Oinsert_001t__List__Olist_It__SeCaV__Ofm_J,type,
    insert_list_fm: list_fm > list_list_fm > list_list_fm ).

thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oinsert_001t__Prover__Orule,type,
    insert_rule: rule > list_rule > list_rule ).

thf(sy_c_List_Oinsert_001t__SeCaV__Ofm,type,
    insert_fm: fm > list_fm > list_fm ).

thf(sy_c_List_Oinsert_001t__SeCaV__Otm,type,
    insert_tm: tm > list_tm > list_tm ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__SeCaV__Ofm_J,type,
    cons_list_fm: list_fm > list_list_fm > list_list_fm ).

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

thf(sy_c_List_Olist_OCons_001t__Prover__Orule,type,
    cons_rule: rule > list_rule > list_rule ).

thf(sy_c_List_Olist_OCons_001t__SeCaV__Ofm,type,
    cons_fm: fm > list_fm > list_fm ).

thf(sy_c_List_Olist_OCons_001t__SeCaV__Otm,type,
    cons_tm: tm > list_tm > list_tm ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Ofm_J,type,
    nil_list_fm: list_list_fm ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_ONil_001t__SeCaV__Ofm,type,
    nil_fm: list_fm ).

thf(sy_c_List_Olist_ONil_001t__SeCaV__Otm,type,
    nil_tm: list_tm ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Otm_J,type,
    map_fm_list_tm: ( fm > list_tm ) > list_fm > list_list_tm ).

thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
    map_tm_list_tm: ( tm > list_tm ) > list_tm > list_list_tm ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Ofm_J,type,
    set_list_fm2: list_list_fm > set_list_fm ).

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

thf(sy_c_List_Olist_Oset_001t__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_Olist__ex1_001t__List__Olist_It__SeCaV__Ofm_J,type,
    list_ex1_list_fm: ( list_fm > $o ) > list_list_fm > $o ).

thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
    list_ex1_nat: ( nat > $o ) > list_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__Prover__Orule,type,
    list_ex1_rule: ( rule > $o ) > list_rule > $o ).

thf(sy_c_List_Olist__ex1_001t__SeCaV__Ofm,type,
    list_ex1_fm: ( fm > $o ) > list_fm > $o ).

thf(sy_c_List_Olist__ex1_001t__SeCaV__Otm,type,
    list_ex1_tm: ( tm > $o ) > list_tm > $o ).

thf(sy_c_List_Omember_001t__List__Olist_It__SeCaV__Ofm_J,type,
    member_list_fm: list_list_fm > list_fm > $o ).

thf(sy_c_List_Omember_001t__Nat__Onat,type,
    member_nat: list_nat > nat > $o ).

thf(sy_c_List_Omember_001t__Prover__Orule,type,
    member_rule: list_rule > rule > $o ).

thf(sy_c_List_Omember_001t__SeCaV__Ofm,type,
    member_fm: list_fm > fm > $o ).

thf(sy_c_List_Omember_001t__SeCaV__Otm,type,
    member_tm: list_tm > tm > $o ).

thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

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

thf(sy_c_List_Oupt,type,
    upt: nat > nat > list_nat ).

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_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

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

thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

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__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

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__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $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__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__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__List__Olist_It__SeCaV__Ofm_J_J,type,
    ord_le3182162295460067583ist_fm: fset_list_fm > fset_list_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Nat__Onat_J,type,
    ord_less_eq_fset_nat: fset_nat > fset_nat > $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__FSet__Ofset_It__SeCaV__Ofm_J,type,
    ord_less_eq_fset_fm: fset_fm > fset_fm > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__SeCaV__Otm_J,type,
    ord_less_eq_fset_tm: fset_tm > fset_tm > $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__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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__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_J,type,
    ord_le8955734628938902733m_rule: set_Pr4971326047967503661m_rule > set_Pr4971326047967503661m_rule > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_J,type,
    ord_le184502606824660649e_rule: set_Pr2701264590931556105e_rule > set_Pr2701264590931556105e_rule > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_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,
    ord_le1038668239487504297m_rule: set_Pr3555430223594399753m_rule > set_Pr3555430223594399753m_rule > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_J,type,
    ord_le5312242768264209797e_rule: set_Pr7340090144965549541e_rule > set_Pr7340090144965549541e_rule > $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,
    ord_less_eq_set_tm: set_tm > set_tm > $o ).

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

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

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
    order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    order_4244789451125944108set_fm: ( set_fm > $o ) > set_fm ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__SeCaV__Otm_J,type,
    order_4248741476211579294set_tm: ( set_tm > $o ) > set_tm ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
    top_top_list_fm_o: list_fm > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
    top_top_nat_o: nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_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,
    top_to4151594781713979000t_fm_o: produc6018962875968178549ist_fm > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Prover__Orule_M_Eo_J,type,
    top_top_rule_o: rule > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__SeCaV__Ofm_M_Eo_J,type,
    top_top_fm_o: fm > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__SeCaV__Otm_M_Eo_J,type,
    top_top_tm_o: tm > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nat__Oenat,type,
    top_to3028658606643905974d_enat: extended_enat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
    top_top_set_list_fm: set_list_fm ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    top_to1730629564055774885ist_fm: set_Pr5202636777678657877ist_fm ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Prover__Orule_J,type,
    top_top_set_rule: set_rule ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__SeCaV__Ofm_J,type,
    top_top_set_fm: set_fm ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__SeCaV__Otm_J,type,
    top_top_set_tm: set_tm ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
    produc1414352766439514085ist_fm: list_tm > list_fm > produc6018962875968178549ist_fm ).

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

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
    produc1733806532565653680m_rule: produc6018962875968178549ist_fm > rule > produc340336539035504054m_rule ).

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

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__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,
    produc6261311607089640965m_rule: produc340336539035504054m_rule > produc340336539035504054m_rule > produc8828831911945107917m_rule ).

thf(sy_c_Product__Type_OPair_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,
    produc4831648765031708129e_rule: produc340336539035504054m_rule > rule > produc9112364199808626345e_rule ).

thf(sy_c_Product__Type_OPair_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,
    produc97041875958848737m_rule: rule > produc340336539035504054m_rule > produc7920154866009513897m_rule ).

thf(sy_c_Product__Type_OPair_001t__Prover__Orule_001t__Prover__Orule,type,
    produc5849431337705160893e_rule: rule > rule > produc7694839378271647877e_rule ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
    produc4874987732838623697ist_fm: produc6018962875968178549ist_fm > list_tm ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
    produc993771546023551940m_rule: produc340336539035504054m_rule > produc6018962875968178549ist_fm ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
    produc3242453064496107283ist_fm: produc6018962875968178549ist_fm > list_fm ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
    produc6514711208036230658m_rule: produc340336539035504054m_rule > rule ).

thf(sy_c_ProverLemmas_Oaffects,type,
    affects: rule > fm > $o ).

thf(sy_c_ProverLemmas_Opreds,type,
    preds: fm > set_fm ).

thf(sy_c_Prover_ObranchDone,type,
    branchDone: list_fm > $o ).

thf(sy_c_Prover_Ochildren,type,
    children: list_tm > rule > list_fm > list_list_fm ).

thf(sy_c_Prover_Oeff,type,
    eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ).

thf(sy_c_Prover_Oeffect,type,
    effect: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm ).

thf(sy_c_Prover_OlistFunTm,type,
    listFunTm: tm > list_nat ).

thf(sy_c_Prover_OlistFunTms,type,
    listFunTms: list_tm > list_nat ).

thf(sy_c_Prover_Oparts,type,
    parts: list_tm > rule > fm > list_list_fm ).

thf(sy_c_Prover_Orules,type,
    rules: stream_rule ).

thf(sy_c_Prover_OsecavProver,type,
    secavProver: produc6018962875968178549ist_fm > abstra8520858721168826610m_rule ).

thf(sy_c_Prover_OsubtermFm,type,
    subtermFm: fm > list_tm ).

thf(sy_c_Prover_OsubtermTm,type,
    subtermTm: tm > list_tm ).

thf(sy_c_Prover_Osubterms,type,
    subterms: list_fm > list_tm ).

thf(sy_c_SeCaV_Oext_001t__List__Olist_It__SeCaV__Ofm_J,type,
    ext_list_fm: list_list_fm > list_list_fm > $o ).

thf(sy_c_SeCaV_Oext_001t__Nat__Onat,type,
    ext_nat: list_nat > list_nat > $o ).

thf(sy_c_SeCaV_Oext_001t__SeCaV__Ofm,type,
    ext_fm: list_fm > list_fm > $o ).

thf(sy_c_SeCaV_Oext_001t__SeCaV__Otm,type,
    ext_tm: list_tm > list_tm > $o ).

thf(sy_c_SeCaV_Ofm_ONeg,type,
    neg: fm > fm ).

thf(sy_c_SeCaV_Ofm_OPre,type,
    pre: nat > list_tm > fm ).

thf(sy_c_SeCaV_Ofm_OUni,type,
    uni: fm > fm ).

thf(sy_c_SeCaV_Oinc__list,type,
    inc_list: list_tm > list_tm ).

thf(sy_c_SeCaV_Oinc__term,type,
    inc_term: tm > tm ).

thf(sy_c_SeCaV_Oliftt,type,
    liftt: tm > tm ).

thf(sy_c_SeCaV_Oliftts,type,
    liftts: list_tm > list_tm ).

thf(sy_c_SeCaV_Omember_001t__List__Olist_It__SeCaV__Ofm_J,type,
    member_list_fm2: list_fm > list_list_fm > $o ).

thf(sy_c_SeCaV_Omember_001t__Nat__Onat,type,
    member_nat2: nat > list_nat > $o ).

thf(sy_c_SeCaV_Omember_001t__Prover__Orule,type,
    member_rule2: rule > list_rule > $o ).

thf(sy_c_SeCaV_Omember_001t__SeCaV__Ofm,type,
    member_fm2: fm > list_fm > $o ).

thf(sy_c_SeCaV_Omember_001t__SeCaV__Otm,type,
    member_tm2: tm > list_tm > $o ).

thf(sy_c_SeCaV_Onew,type,
    new: nat > fm > $o ).

thf(sy_c_SeCaV_Onew__list,type,
    new_list: nat > list_tm > $o ).

thf(sy_c_SeCaV_Onew__term,type,
    new_term: nat > tm > $o ).

thf(sy_c_SeCaV_Oparams,type,
    params: fm > set_nat ).

thf(sy_c_SeCaV_Oparamst,type,
    paramst: tm > set_nat ).

thf(sy_c_SeCaV_Oparamst_H,type,
    paramst2: tm > set_nat ).

thf(sy_c_SeCaV_Oparamsts,type,
    paramsts: list_tm > set_nat ).

thf(sy_c_SeCaV_Osub,type,
    sub: nat > tm > fm > fm ).

thf(sy_c_SeCaV_Osub__list,type,
    sub_list: nat > tm > list_tm > list_tm ).

thf(sy_c_SeCaV_Osub__term,type,
    sub_term: nat > tm > tm > tm ).

thf(sy_c_SeCaV_Osubst,type,
    subst: fm > tm > nat > fm ).

thf(sy_c_SeCaV_Osubstt,type,
    substt: tm > tm > nat > tm ).

thf(sy_c_SeCaV_Osubstts,type,
    substts: list_tm > tm > nat > list_tm ).

thf(sy_c_SeCaV_Otm_OFun,type,
    fun: nat > list_tm > tm ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__SeCaV__Ofm_J,type,
    collect_list_fm: ( list_fm > $o ) > set_list_fm ).

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

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    collec5845330005437105888ist_fm: ( produc6018962875968178549ist_fm > $o ) > set_Pr5202636777678657877ist_fm ).

thf(sy_c_Set_OCollect_001t__Prover__Orule,type,
    collect_rule: ( rule > $o ) > set_rule ).

thf(sy_c_Set_OCollect_001t__SeCaV__Ofm,type,
    collect_fm: ( fm > $o ) > set_fm ).

thf(sy_c_Set_OCollect_001t__SeCaV__Otm,type,
    collect_tm: ( tm > $o ) > set_tm ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__SeCaV__Ofm_J,type,
    insert_list_fm2: list_fm > set_list_fm > set_list_fm ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat2: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    insert4380798862274524997ist_fm: produc6018962875968178549ist_fm > set_Pr5202636777678657877ist_fm > set_Pr5202636777678657877ist_fm ).

thf(sy_c_Set_Oinsert_001t__Prover__Orule,type,
    insert_rule2: rule > set_rule > set_rule ).

thf(sy_c_Set_Oinsert_001t__SeCaV__Ofm,type,
    insert_fm2: fm > set_fm > set_fm ).

thf(sy_c_Set_Oinsert_001t__SeCaV__Otm,type,
    insert_tm2: tm > set_tm > set_tm ).

thf(sy_c_Stream_Osdrop_001t__List__Olist_It__SeCaV__Ofm_J,type,
    sdrop_list_fm: nat > stream_list_fm > stream_list_fm ).

thf(sy_c_Stream_Osdrop_001t__List__Olist_It__SeCaV__Otm_J,type,
    sdrop_list_tm: nat > stream_list_tm > stream_list_tm ).

thf(sy_c_Stream_Osdrop_001t__Nat__Onat,type,
    sdrop_nat: nat > stream_nat > stream_nat ).

thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__Prover__Orule_J,type,
    sdrop_779662454523830033m_rule: nat > stream1460591043787736522m_rule > stream1460591043787736522m_rule ).

thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    sdrop_9176333610110415838ist_fm: nat > stream4408948924543953275ist_fm > stream4408948924543953275ist_fm ).

thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_J,type,
    sdrop_6661281195147388801e_rule: nat > stream791956382124094522e_rule > stream791956382124094522e_rule ).

thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Prover__Orule_J,type,
    sdrop_3122356783091591596t_rule: nat > stream8912087984624402761t_rule > stream8912087984624402761t_rule ).

thf(sy_c_Stream_Osdrop_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,
    sdrop_8169176516188972301m_rule: nat > stream2709947120125613254m_rule > stream2709947120125613254m_rule ).

thf(sy_c_Stream_Osdrop_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,
    sdrop_7192298464603511222m_rule: nat > stream6210534828274662995m_rule > stream6210534828274662995m_rule ).

thf(sy_c_Stream_Osdrop_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,
    sdrop_1938960342593938834e_rule: nat > stream8099677779113257519e_rule > stream8099677779113257519e_rule ).

thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_J,type,
    sdrop_7704051541046111184e_rule: nat > stream4681790265157167213e_rule > stream4681790265157167213e_rule ).

thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_Mt__Prover__Orule_J,type,
    sdrop_6431130323856442653e_rule: nat > stream128700168137323990e_rule > stream128700168137323990e_rule ).

thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
    sdrop_959413419342957231ist_fm: nat > stream4490188412977367400ist_fm > stream4490188412977367400ist_fm ).

thf(sy_c_Stream_Osdrop_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,
    sdrop_746751008794826386m_rule: nat > stream8953843411776101167m_rule > stream8953843411776101167m_rule ).

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    sset_P4484857331586881186m_rule: stream2709947120125613254m_rule > set_Pr1822751329126368876m_rule ).

thf(sy_c_Stream_Ostream_Osset_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,
    sset_P3768201806174902753m_rule: stream6210534828274662995m_rule > set_Pr4971326047967503661m_rule ).

thf(sy_c_Stream_Ostream_Osset_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,
    sset_P6986346827493259965e_rule: stream8099677779113257519e_rule > set_Pr2701264590931556105e_rule ).

thf(sy_c_Stream_Ostream_Osset_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,
    sset_P5794137493694147517m_rule: stream8953843411776101167m_rule > set_Pr3555430223594399753m_rule ).

thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
    sset_P1894962091411173529e_rule: stream4385846686851721995e_rule > set_Pr7340090144965549541e_rule ).

thf(sy_c_Stream_Ostream_Osset_001t__Prover__Orule,type,
    sset_rule: stream_rule > set_rule ).

thf(sy_c_Stream_Ostream_Osset_001t__SeCaV__Ofm,type,
    sset_fm: stream_fm > set_fm ).

thf(sy_c_Stream_Ostream_Osset_001t__SeCaV__Otm,type,
    sset_tm: stream_tm > set_tm ).

thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__SeCaV__Ofm_J,type,
    stl_list_fm: stream_list_fm > stream_list_fm ).

thf(sy_c_Stream_Ostream_Ostl_001t__Nat__Onat,type,
    stl_nat: stream_nat > stream_nat ).

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_Ostream__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,
    stream2134157564947672044m_rule: ( produc340336539035504054m_rule > $o ) > stream2709947120125613254m_rule > $o ).

thf(sy_c_Stream_Ostream__all_001t__Prover__Orule,type,
    stream_all_rule: ( rule > $o ) > stream_rule > $o ).

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__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
    szip_P1965874896742159130e_rule: stream4385846686851721995e_rule > stream4385846686851721995e_rule > stream4681790265157167213e_rule ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__Prover__Orule,type,
    szip_P4230510992641354557e_rule: stream4385846686851721995e_rule > stream_rule > stream128700168137323990e_rule ).

thf(sy_c_Stream_Oszip_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
    szip_r5451144443050508039ist_fm: stream_rule > stream4408948924543953275ist_fm > stream4490188412977367400ist_fm ).

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__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
    szip_r8329995387344033047e_rule: stream_rule > stream4385846686851721995e_rule > stream6316532330152022520e_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__List__Olist_It__SeCaV__Ofm_J,type,
    member_list_fm3: list_fm > set_list_fm > $o ).

thf(sy_c_member_001t__List__Olist_It__SeCaV__Otm_J,type,
    member_list_tm: list_tm > set_list_tm > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat3: nat > set_nat > $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_rule3: rule > set_rule > $o ).

thf(sy_c_member_001t__SeCaV__Ofm,type,
    member_fm3: fm > set_fm > $o ).

thf(sy_c_member_001t__SeCaV__Otm,type,
    member_tm3: tm > set_tm > $o ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_n____,type,
    n: nat ).

thf(sy_v_p____,type,
    p: fm ).

thf(sy_v_pt____,type,
    pt: fm ).

thf(sy_v_steps,type,
    steps: stream2709947120125613254m_rule ).

thf(sy_v_t____,type,
    t: tm ).

% Relevant facts (1277)
thf(fact_0_pt_I2_J,axiom,
    member_fm3 @ pt @ ( tree_fms @ steps ) ).

% pt(2)
thf(fact_1_m,axiom,
    member_fm3 @ pt @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ m @ steps ) ) ) ) ).

% m
thf(fact_2_pt_I1_J,axiom,
    member_tm3 @ t @ ( set_tm2 @ ( subtermFm @ pt ) ) ).

% pt(1)
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_Apt_A_092_060in_062_Aset_A_Ipseq_A_Ishd_A_Isdrop_Am_Asteps_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [M: nat] :
        ~ ( member_fm3 @ pt @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ M @ steps ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>m. pt \<in> set (pseq (shd (sdrop m steps))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_t,axiom,
    member_tm3 @ t @ ( terms @ ( tree_fms @ steps ) ) ).

% t
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pt_O_A_092_060lbrakk_062t_A_092_060in_062_Aset_A_IsubtermFm_Apt_J_059_Apt_A_092_060in_062_Atree__fms_Asteps_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Pt: fm] :
        ( ( member_tm3 @ t @ ( set_tm2 @ ( subtermFm @ Pt ) ) )
       => ~ ( member_fm3 @ Pt @ ( tree_fms @ steps ) ) ) ).

% \<open>\<And>thesis. (\<And>pt. \<lbrakk>t \<in> set (subtermFm pt); pt \<in> tree_fms steps\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_n,axiom,
    member_fm3 @ ( neg @ ( uni @ p ) ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ) ) ).

% n
thf(fact_7_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_9113879250048157294e_rule @ N @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_8_sdrop__szip,axiom,
    ! [N: nat,S1: stream4408948924543953275ist_fm,S2: stream_rule] :
      ( ( sdrop_8169176516188972301m_rule @ N @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
      = ( szip_P2924820683901490861m_rule @ ( sdrop_9176333610110415838ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_9_sdrop__szip,axiom,
    ! [N: nat,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( sdrop_1938960342593938834e_rule @ N @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
      = ( szip_P2044787555563963556e_rule @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_10_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_746751008794826386m_rule @ N @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
      = ( szip_r6533552703345879972m_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_11_sdrop__szip,axiom,
    ! [N: nat,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( sdrop_7192298464603511222m_rule @ N @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
      = ( szip_P811719526838699976m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_12_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream4385846686851721995e_rule] :
      ( ( sdrop_4667533458460693055e_rule @ N @ ( szip_r8329995387344033047e_rule @ S1 @ S2 ) )
      = ( szip_r8329995387344033047e_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_13_sdrop__szip,axiom,
    ! [N: nat,S1: stream4385846686851721995e_rule,S2: stream_rule] :
      ( ( sdrop_6431130323856442653e_rule @ N @ ( szip_P4230510992641354557e_rule @ S1 @ S2 ) )
      = ( szip_P4230510992641354557e_rule @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_14_sdrop__szip,axiom,
    ! [N: nat,S1: stream_list_tm,S2: stream_list_fm] :
      ( ( sdrop_9176333610110415838ist_fm @ N @ ( szip_list_tm_list_fm @ S1 @ S2 ) )
      = ( szip_list_tm_list_fm @ ( sdrop_list_tm @ N @ S1 ) @ ( sdrop_list_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_15_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream4408948924543953275ist_fm] :
      ( ( sdrop_959413419342957231ist_fm @ N @ ( szip_r5451144443050508039ist_fm @ S1 @ S2 ) )
      = ( szip_r5451144443050508039ist_fm @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9176333610110415838ist_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_16_sdrop__szip,axiom,
    ! [N: nat,S1: stream4385846686851721995e_rule,S2: stream4385846686851721995e_rule] :
      ( ( sdrop_7704051541046111184e_rule @ N @ ( szip_P1965874896742159130e_rule @ S1 @ S2 ) )
      = ( szip_P1965874896742159130e_rule @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_17__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_fm3 @ ( 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_18_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream4408948924543953275ist_fm] :
      ( ( shd_Pr3211216682057661985ist_fm @ ( sdrop_9176333610110415838ist_fm @ N @ S ) )
      = ( snth_P7093566783922538521ist_fm @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_19_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream4385846686851721995e_rule] :
      ( ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N @ S ) )
      = ( snth_P6178434775611311401e_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_20_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream8953843411776101167m_rule] :
      ( ( shd_Pr1208753253167450709m_rule @ ( sdrop_746751008794826386m_rule @ N @ S ) )
      = ( snth_P5203088247548055629m_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_21_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream8099677779113257519e_rule] :
      ( ( shd_Pr2400962586966563157e_rule @ ( sdrop_1938960342593938834e_rule @ N @ S ) )
      = ( snth_P6395297581347168077e_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_22_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream6210534828274662995m_rule] :
      ( ( shd_Pr4461660664618831993m_rule @ ( sdrop_7192298464603511222m_rule @ N @ S ) )
      = ( snth_P8853764340393315953m_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_23_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) )
      = ( snth_P6679518042731451922m_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_24_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream_rule] :
      ( ( shd_rule @ ( sdrop_rule @ N @ S ) )
      = ( snth_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_25_member,axiom,
    ( member_rule2
    = ( ^ [P: rule,Z: list_rule] : ( member_rule3 @ P @ ( set_rule2 @ Z ) ) ) ) ).

% member
thf(fact_26_member,axiom,
    ( member_tm2
    = ( ^ [P: tm,Z: list_tm] : ( member_tm3 @ P @ ( set_tm2 @ Z ) ) ) ) ).

% member
thf(fact_27_member,axiom,
    ( member_fm2
    = ( ^ [P: fm,Z: list_fm] : ( member_fm3 @ P @ ( set_fm2 @ Z ) ) ) ) ).

% member
thf(fact_28_member,axiom,
    ( member_nat2
    = ( ^ [P: nat,Z: list_nat] : ( member_nat3 @ P @ ( set_nat2 @ Z ) ) ) ) ).

% member
thf(fact_29_member,axiom,
    ( member_list_fm2
    = ( ^ [P: list_fm,Z: list_list_fm] : ( member_list_fm3 @ P @ ( set_list_fm2 @ Z ) ) ) ) ).

% member
thf(fact_30_in__set__member,axiom,
    ! [X: rule,Xs: list_rule] :
      ( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
      = ( member_rule @ Xs @ X ) ) ).

% in_set_member
thf(fact_31_in__set__member,axiom,
    ! [X: tm,Xs: list_tm] :
      ( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
      = ( member_tm @ Xs @ X ) ) ).

% in_set_member
thf(fact_32_in__set__member,axiom,
    ! [X: fm,Xs: list_fm] :
      ( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
      = ( member_fm @ Xs @ X ) ) ).

% in_set_member
thf(fact_33_in__set__member,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
      = ( member_nat @ Xs @ X ) ) ).

% in_set_member
thf(fact_34_in__set__member,axiom,
    ! [X: list_fm,Xs: list_list_fm] :
      ( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
      = ( member_list_fm @ Xs @ X ) ) ).

% in_set_member
thf(fact_35_assms_I2_J,axiom,
    abstra6097777249025082867ist_fm @ eff @ rules @ steps ).

% assms(2)
thf(fact_36_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
      = ( smap2_rule_rule_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_37_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc7694839378271647877e_rule > rule,S1: stream_rule,S2: stream4385846686851721995e_rule] :
      ( ( sdrop_rule @ N @ ( smap2_9025683351474882503e_rule @ F @ S1 @ S2 ) )
      = ( smap2_9025683351474882503e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_38_sdrop__smap2,axiom,
    ! [N: nat,F: produc7694839378271647877e_rule > rule > rule,S1: stream4385846686851721995e_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_3450944986098114541e_rule @ F @ S1 @ S2 ) )
      = ( smap2_3450944986098114541e_rule @ F @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_39_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc7694839378271647877e_rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_9113879250048157294e_rule @ N @ ( smap2_3901795709322785185e_rule @ F @ S1 @ S2 ) )
      = ( smap2_3901795709322785185e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_40_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc6018962875968178549ist_fm > rule,S1: stream_rule,S2: stream4408948924543953275ist_fm] :
      ( ( sdrop_rule @ N @ ( smap2_6567943727960754231m_rule @ F @ S1 @ S2 ) )
      = ( smap2_6567943727960754231m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9176333610110415838ist_fm @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_41_sdrop__smap2,axiom,
    ! [N: nat,F: produc6018962875968178549ist_fm > rule > rule,S1: stream4408948924543953275ist_fm,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_4866629018036412893e_rule @ F @ S1 @ S2 ) )
      = ( smap2_4866629018036412893e_rule @ F @ ( sdrop_9176333610110415838ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_42_sdrop__smap2,axiom,
    ! [N: nat,F: produc7694839378271647877e_rule > produc7694839378271647877e_rule > rule,S1: stream4385846686851721995e_rule,S2: stream4385846686851721995e_rule] :
      ( ( sdrop_rule @ N @ ( smap2_5087128713194948752e_rule @ F @ S1 @ S2 ) )
      = ( smap2_5087128713194948752e_rule @ F @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_43_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc6018962875968178549ist_fm,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_9176333610110415838ist_fm @ N @ ( smap2_9094267487109771409ist_fm @ F @ S1 @ S2 ) )
      = ( smap2_9094267487109771409ist_fm @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_44_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc7694839378271647877e_rule > produc7694839378271647877e_rule,S1: stream_rule,S2: stream4385846686851721995e_rule] :
      ( ( sdrop_9113879250048157294e_rule @ N @ ( smap2_8640882101725621264e_rule @ F @ S1 @ S2 ) )
      = ( smap2_8640882101725621264e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_45_sdrop__smap2,axiom,
    ! [N: nat,F: produc7694839378271647877e_rule > rule > produc7694839378271647877e_rule,S1: stream4385846686851721995e_rule,S2: stream_rule] :
      ( ( sdrop_9113879250048157294e_rule @ N @ ( smap2_9186613107897627242e_rule @ F @ S1 @ S2 ) )
      = ( smap2_9186613107897627242e_rule @ F @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_46_in__set__insert,axiom,
    ! [X: rule,Xs: list_rule] :
      ( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
     => ( ( insert_rule @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_47_in__set__insert,axiom,
    ! [X: tm,Xs: list_tm] :
      ( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
     => ( ( insert_tm @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_48_in__set__insert,axiom,
    ! [X: fm,Xs: list_fm] :
      ( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
     => ( ( insert_fm @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_49_in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_50_in__set__insert,axiom,
    ! [X: list_fm,Xs: list_list_fm] :
      ( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
     => ( ( insert_list_fm @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_51_list__ex1__iff,axiom,
    ( list_ex1_rule
    = ( ^ [P2: rule > $o,Xs2: list_rule] :
        ? [X2: rule] :
          ( ( member_rule3 @ X2 @ ( set_rule2 @ Xs2 ) )
          & ( P2 @ X2 )
          & ! [Y: rule] :
              ( ( ( member_rule3 @ Y @ ( set_rule2 @ Xs2 ) )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_52_list__ex1__iff,axiom,
    ( list_ex1_tm
    = ( ^ [P2: tm > $o,Xs2: list_tm] :
        ? [X2: tm] :
          ( ( member_tm3 @ X2 @ ( set_tm2 @ Xs2 ) )
          & ( P2 @ X2 )
          & ! [Y: tm] :
              ( ( ( member_tm3 @ Y @ ( set_tm2 @ Xs2 ) )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_53_list__ex1__iff,axiom,
    ( list_ex1_fm
    = ( ^ [P2: fm > $o,Xs2: list_fm] :
        ? [X2: fm] :
          ( ( member_fm3 @ X2 @ ( set_fm2 @ Xs2 ) )
          & ( P2 @ X2 )
          & ! [Y: fm] :
              ( ( ( member_fm3 @ Y @ ( set_fm2 @ Xs2 ) )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_54_list__ex1__iff,axiom,
    ( list_ex1_nat
    = ( ^ [P2: nat > $o,Xs2: list_nat] :
        ? [X2: nat] :
          ( ( member_nat3 @ X2 @ ( set_nat2 @ Xs2 ) )
          & ( P2 @ X2 )
          & ! [Y: nat] :
              ( ( ( member_nat3 @ Y @ ( set_nat2 @ Xs2 ) )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_55_list__ex1__iff,axiom,
    ( list_ex1_list_fm
    = ( ^ [P2: list_fm > $o,Xs2: list_list_fm] :
        ? [X2: list_fm] :
          ( ( member_list_fm3 @ X2 @ ( set_list_fm2 @ Xs2 ) )
          & ( P2 @ X2 )
          & ! [Y: list_fm] :
              ( ( ( member_list_fm3 @ Y @ ( set_list_fm2 @ Xs2 ) )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_56__092_060open_062Neg_A_IUni_Ap_J_A_092_060in_062_Atree__fms_Asteps_092_060close_062,axiom,
    member_fm3 @ ( neg @ ( uni @ p ) ) @ ( tree_fms @ steps ) ).

% \<open>Neg (Uni p) \<in> tree_fms steps\<close>
thf(fact_57_fm_Oinject_I7_J,axiom,
    ! [X7: fm,Y7: fm] :
      ( ( ( neg @ X7 )
        = ( neg @ Y7 ) )
      = ( X7 = Y7 ) ) ).

% fm.inject(7)
thf(fact_58_fm_Oinject_I6_J,axiom,
    ! [X6: fm,Y6: fm] :
      ( ( ( uni @ X6 )
        = ( uni @ Y6 ) )
      = ( X6 = Y6 ) ) ).

% fm.inject(6)
thf(fact_59_snth__smap2,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_60_snth__smap2,axiom,
    ! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule,N: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_61_snth__smap2,axiom,
    ! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule,N: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_62_snth__smap2,axiom,
    ! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule,N: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_63_snth__smap2,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N: nat] :
      ( ( snth_rule @ ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_64_snth__smap2,axiom,
    ! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule,N: nat] :
      ( ( snth_rule @ ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_65_snth__smap2,axiom,
    ! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule,N: nat] :
      ( ( snth_rule @ ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_66_snth__smap2,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule,N: nat] :
      ( ( snth_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) @ N )
      = ( F @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_smap2
thf(fact_67_assms_I1_J,axiom,
    abstra6789711989322986974ist_fm @ eff @ rules @ steps ).

% assms(1)
thf(fact_68_fm_Odistinct_I41_J,axiom,
    ! [X6: fm,X7: fm] :
      ( ( uni @ X6 )
     != ( neg @ X7 ) ) ).

% fm.distinct(41)
thf(fact_69_smap2__alt,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,S: stream2709947120125613254m_rule] :
      ( ( ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 )
        = S )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
            = ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_70_smap2__alt,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,S: stream_rule] :
      ( ( ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 )
        = S )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
            = ( snth_rule @ S @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_71_smap2__alt,axiom,
    ! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule,S: stream2709947120125613254m_rule] :
      ( ( ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 )
        = S )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_72_smap2__alt,axiom,
    ! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule,S: stream_rule] :
      ( ( ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 )
        = S )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_rule @ S @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_73_smap2__alt,axiom,
    ! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule,S: stream2709947120125613254m_rule] :
      ( ( ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 )
        = S )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
            = ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_74_smap2__alt,axiom,
    ! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule,S: stream_rule] :
      ( ( ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 )
        = S )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
            = ( snth_rule @ S @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_75_smap2__alt,axiom,
    ! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule,S: stream2709947120125613254m_rule] :
      ( ( ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 )
        = S )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_76_smap2__alt,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule,S: stream_rule] :
      ( ( ( smap2_rule_rule_rule @ F @ S1 @ S2 )
        = S )
      = ( ! [N3: nat] :
            ( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
            = ( snth_rule @ S @ N3 ) ) ) ) ).

% smap2_alt
thf(fact_77_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_78_tree__fms__in__pseq,axiom,
    ! [P3: fm,Steps: stream2709947120125613254m_rule] :
      ( ( member_fm3 @ P3 @ ( tree_fms @ Steps ) )
     => ? [N2: nat] : ( member_fm3 @ P3 @ ( set_fm2 @ ( pseq @ ( snth_P6679518042731451922m_rule @ Steps @ N2 ) ) ) ) ) ).

% tree_fms_in_pseq
thf(fact_79_smap2_Osimps_I1_J,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_80_smap2_Osimps_I1_J,axiom,
    ! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_81_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_82_smap2_Osimps_I1_J,axiom,
    ! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_83_smap2_Osimps_I1_J,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_rule @ ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_84_smap2_Osimps_I1_J,axiom,
    ! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_85_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_rule @ ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_86_smap2_Osimps_I1_J,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_87_can__select__set__list__ex1,axiom,
    ! [P4: tm > $o,A: list_tm] :
      ( ( can_select_tm @ P4 @ ( set_tm2 @ A ) )
      = ( list_ex1_tm @ P4 @ A ) ) ).

% can_select_set_list_ex1
thf(fact_88_can__select__set__list__ex1,axiom,
    ! [P4: fm > $o,A: list_fm] :
      ( ( can_select_fm @ P4 @ ( set_fm2 @ A ) )
      = ( list_ex1_fm @ P4 @ A ) ) ).

% can_select_set_list_ex1
thf(fact_89_can__select__set__list__ex1,axiom,
    ! [P4: nat > $o,A: list_nat] :
      ( ( can_select_nat @ P4 @ ( set_nat2 @ A ) )
      = ( list_ex1_nat @ P4 @ A ) ) ).

% can_select_set_list_ex1
thf(fact_90_can__select__set__list__ex1,axiom,
    ! [P4: list_fm > $o,A: list_list_fm] :
      ( ( can_select_list_fm @ P4 @ ( set_list_fm2 @ A ) )
      = ( list_ex1_list_fm @ P4 @ A ) ) ).

% can_select_set_list_ex1
thf(fact_91_subtermFm_Osimps_I6_J,axiom,
    ! [P3: fm] :
      ( ( subtermFm @ ( uni @ P3 ) )
      = ( subtermFm @ P3 ) ) ).

% subtermFm.simps(6)
thf(fact_92_subtermFm_Osimps_I7_J,axiom,
    ! [P3: fm] :
      ( ( subtermFm @ ( neg @ P3 ) )
      = ( subtermFm @ P3 ) ) ).

% subtermFm.simps(7)
thf(fact_93_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra6097777249025082867ist_fm = abstra6097777249025082867ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_94_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_95_rules__repeat__sdrop,axiom,
    ! [K: nat,R: rule] :
    ? [N2: nat] :
      ( ( snth_rule @ ( sdrop_rule @ K @ rules ) @ N2 )
      = R ) ).

% rules_repeat_sdrop
thf(fact_96_pseq__in__tree__fms,axiom,
    ! [X: produc340336539035504054m_rule,Steps: stream2709947120125613254m_rule,P3: fm] :
      ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Steps ) )
     => ( ( member_fm3 @ P3 @ ( set_fm2 @ ( pseq @ X ) ) )
       => ( member_fm3 @ P3 @ ( tree_fms @ Steps ) ) ) ) ).

% pseq_in_tree_fms
thf(fact_97_szip_Osimps_I1_J,axiom,
    ! [S1: stream_list_tm,S2: stream_list_fm] :
      ( ( shd_Pr3211216682057661985ist_fm @ ( szip_list_tm_list_fm @ S1 @ S2 ) )
      = ( produc1414352766439514085ist_fm @ ( shd_list_tm @ S1 ) @ ( shd_list_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_98_szip_Osimps_I1_J,axiom,
    ! [S1: stream_nat,S2: stream_nat] :
      ( ( shd_Pr4260400998323988397at_nat @ ( szip_nat_nat @ S1 @ S2 ) )
      = ( product_Pair_nat_nat @ ( shd_nat @ S1 ) @ ( shd_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_99_szip_Osimps_I1_J,axiom,
    ! [S1: stream6724221391990029191at_nat,S2: stream6724221391990029191at_nat] :
      ( ( shd_Pr8412153233960533267at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) )
      = ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S1 ) @ ( shd_Pr4260400998323988397at_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_100_szip_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr4461660664618831993m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
      = ( produc6261311607089640965m_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_101_szip_Osimps_I1_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( shd_Pr2400962586966563157e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
      = ( produc4831648765031708129e_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_102_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( shd_Pr1208753253167450709m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
      = ( produc97041875958848737m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_103_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr2264621979884435249e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( produc5849431337705160893e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_104_szip_Osimps_I1_J,axiom,
    ! [S1: stream4408948924543953275ist_fm,S2: stream_rule] :
      ( ( shd_Pr4562317740776619530m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
      = ( produc1733806532565653680m_rule @ ( shd_Pr3211216682057661985ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_105_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream_rule,N: nat] :
      ( ( snth_P6178434775611311401e_rule @ ( szip_rule_rule @ S1 @ S2 ) @ N )
      = ( produc5849431337705160893e_rule @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_106_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream2709947120125613254m_rule,N: nat] :
      ( ( snth_P5203088247548055629m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) @ N )
      = ( produc97041875958848737m_rule @ ( snth_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_107_snth__szip,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream_rule,N: nat] :
      ( ( snth_P6395297581347168077e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) @ N )
      = ( produc4831648765031708129e_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_108_snth__szip,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N: nat] :
      ( ( snth_P8853764340393315953m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) @ N )
      = ( produc6261311607089640965m_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_109_snth__szip,axiom,
    ! [S1: stream_list_tm,S2: stream_list_fm,N: nat] :
      ( ( snth_P7093566783922538521ist_fm @ ( szip_list_tm_list_fm @ S1 @ S2 ) @ N )
      = ( produc1414352766439514085ist_fm @ ( snth_list_tm @ S1 @ N ) @ ( snth_list_fm @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_110_snth__szip,axiom,
    ! [S1: stream4408948924543953275ist_fm,S2: stream_rule,N: nat] :
      ( ( snth_P6679518042731451922m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) @ N )
      = ( produc1733806532565653680m_rule @ ( snth_P7093566783922538521ist_fm @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_111_snth__szip,axiom,
    ! [S1: stream_nat,S2: stream_nat,N: nat] :
      ( ( snth_P9042041890044241317at_nat @ ( szip_nat_nat @ S1 @ S2 ) @ N )
      = ( product_Pair_nat_nat @ ( snth_nat @ S1 @ N ) @ ( snth_nat @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_112_snth__szip,axiom,
    ! [S1: stream6724221391990029191at_nat,S2: stream6724221391990029191at_nat,N: nat] :
      ( ( snth_P9098630209796499211at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) @ N )
      = ( produc6161850002892822231at_nat @ ( snth_P9042041890044241317at_nat @ S1 @ N ) @ ( snth_P9042041890044241317at_nat @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_113_not__in__set__insert,axiom,
    ! [X: rule,Xs: list_rule] :
      ( ~ ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
     => ( ( insert_rule @ X @ Xs )
        = ( cons_rule @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_114_not__in__set__insert,axiom,
    ! [X: tm,Xs: list_tm] :
      ( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
     => ( ( insert_tm @ X @ Xs )
        = ( cons_tm @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_115_not__in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = ( cons_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_116_not__in__set__insert,axiom,
    ! [X: list_fm,Xs: list_list_fm] :
      ( ~ ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
     => ( ( insert_list_fm @ X @ Xs )
        = ( cons_list_fm @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_117_not__in__set__insert,axiom,
    ! [X: fm,Xs: list_fm] :
      ( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
     => ( ( insert_fm @ X @ Xs )
        = ( cons_fm @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_118_List_Oset__insert,axiom,
    ! [X: rule,Xs: list_rule] :
      ( ( set_rule2 @ ( insert_rule @ X @ Xs ) )
      = ( insert_rule2 @ X @ ( set_rule2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_119_List_Oset__insert,axiom,
    ! [X: tm,Xs: list_tm] :
      ( ( set_tm2 @ ( insert_tm @ X @ Xs ) )
      = ( insert_tm2 @ X @ ( set_tm2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_120_List_Oset__insert,axiom,
    ! [X: fm,Xs: list_fm] :
      ( ( set_fm2 @ ( insert_fm @ X @ Xs ) )
      = ( insert_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_121_List_Oset__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
      = ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_122_List_Oset__insert,axiom,
    ! [X: list_fm,Xs: list_list_fm] :
      ( ( set_list_fm2 @ ( insert_list_fm @ X @ Xs ) )
      = ( insert_list_fm2 @ X @ ( set_list_fm2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_123_stream__all__def,axiom,
    ( stream2134157564947672044m_rule
    = ( ^ [P2: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
        ! [P: nat] : ( P2 @ ( snth_P6679518042731451922m_rule @ S3 @ P ) ) ) ) ).

% stream_all_def
thf(fact_124_stream__all__def,axiom,
    ( stream_all_rule
    = ( ^ [P2: rule > $o,S3: stream_rule] :
        ! [P: nat] : ( P2 @ ( snth_rule @ S3 @ P ) ) ) ) ).

% stream_all_def
thf(fact_125_list_Oinject,axiom,
    ! [X21: tm,X22: list_tm,Y21: tm,Y22: list_tm] :
      ( ( ( cons_tm @ X21 @ X22 )
        = ( cons_tm @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_126_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X22 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_127_list_Oinject,axiom,
    ! [X21: list_fm,X22: list_list_fm,Y21: list_fm,Y22: list_list_fm] :
      ( ( ( cons_list_fm @ X21 @ X22 )
        = ( cons_list_fm @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_128_list_Oinject,axiom,
    ! [X21: fm,X22: list_fm,Y21: fm,Y22: list_fm] :
      ( ( ( cons_fm @ X21 @ X22 )
        = ( cons_fm @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_129_list_Osimps_I15_J,axiom,
    ! [X21: rule,X22: list_rule] :
      ( ( set_rule2 @ ( cons_rule @ X21 @ X22 ) )
      = ( insert_rule2 @ X21 @ ( set_rule2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_130_list_Osimps_I15_J,axiom,
    ! [X21: tm,X22: list_tm] :
      ( ( set_tm2 @ ( cons_tm @ X21 @ X22 ) )
      = ( insert_tm2 @ X21 @ ( set_tm2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_131_list_Osimps_I15_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
      = ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_132_list_Osimps_I15_J,axiom,
    ! [X21: list_fm,X22: list_list_fm] :
      ( ( set_list_fm2 @ ( cons_list_fm @ X21 @ X22 ) )
      = ( insert_list_fm2 @ X21 @ ( set_list_fm2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_133_list_Osimps_I15_J,axiom,
    ! [X21: fm,X22: list_fm] :
      ( ( set_fm2 @ ( cons_fm @ X21 @ X22 ) )
      = ( insert_fm2 @ X21 @ ( set_fm2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_134_not__Cons__self2,axiom,
    ! [X: tm,Xs: list_tm] :
      ( ( cons_tm @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_135_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( cons_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_136_not__Cons__self2,axiom,
    ! [X: list_fm,Xs: list_list_fm] :
      ( ( cons_list_fm @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_137_not__Cons__self2,axiom,
    ! [X: fm,Xs: list_fm] :
      ( ( cons_fm @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_138_RuleSystem__Defs_Oepath_Ocong,axiom,
    abstra6789711989322986974ist_fm = abstra6789711989322986974ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_139_shd__sset,axiom,
    ! [A2: stream_list_fm] : ( member_list_fm3 @ ( shd_list_fm @ A2 ) @ ( sset_list_fm @ A2 ) ) ).

% shd_sset
thf(fact_140_shd__sset,axiom,
    ! [A2: stream_tm] : ( member_tm3 @ ( shd_tm @ A2 ) @ ( sset_tm @ A2 ) ) ).

% shd_sset
thf(fact_141_shd__sset,axiom,
    ! [A2: stream_fm] : ( member_fm3 @ ( shd_fm @ A2 ) @ ( sset_fm @ A2 ) ) ).

% shd_sset
thf(fact_142_shd__sset,axiom,
    ! [A2: stream_nat] : ( member_nat3 @ ( shd_nat @ A2 ) @ ( sset_nat @ A2 ) ) ).

% shd_sset
thf(fact_143_shd__sset,axiom,
    ! [A2: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) @ ( sset_P4484857331586881186m_rule @ A2 ) ) ).

% shd_sset
thf(fact_144_shd__sset,axiom,
    ! [A2: stream_rule] : ( member_rule3 @ ( shd_rule @ A2 ) @ ( sset_rule @ A2 ) ) ).

% shd_sset
thf(fact_145_list_Oset__intros_I2_J,axiom,
    ! [Y2: rule,X22: list_rule,X21: rule] :
      ( ( member_rule3 @ Y2 @ ( set_rule2 @ X22 ) )
     => ( member_rule3 @ Y2 @ ( set_rule2 @ ( cons_rule @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_146_list_Oset__intros_I2_J,axiom,
    ! [Y2: tm,X22: list_tm,X21: tm] :
      ( ( member_tm3 @ Y2 @ ( set_tm2 @ X22 ) )
     => ( member_tm3 @ Y2 @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_147_list_Oset__intros_I2_J,axiom,
    ! [Y2: nat,X22: list_nat,X21: nat] :
      ( ( member_nat3 @ Y2 @ ( set_nat2 @ X22 ) )
     => ( member_nat3 @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_148_list_Oset__intros_I2_J,axiom,
    ! [Y2: list_fm,X22: list_list_fm,X21: list_fm] :
      ( ( member_list_fm3 @ Y2 @ ( set_list_fm2 @ X22 ) )
     => ( member_list_fm3 @ Y2 @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_149_list_Oset__intros_I2_J,axiom,
    ! [Y2: fm,X22: list_fm,X21: fm] :
      ( ( member_fm3 @ Y2 @ ( set_fm2 @ X22 ) )
     => ( member_fm3 @ Y2 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_150_list_Oset__intros_I1_J,axiom,
    ! [X21: rule,X22: list_rule] : ( member_rule3 @ X21 @ ( set_rule2 @ ( cons_rule @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_151_list_Oset__intros_I1_J,axiom,
    ! [X21: tm,X22: list_tm] : ( member_tm3 @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_152_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : ( member_nat3 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_153_list_Oset__intros_I1_J,axiom,
    ! [X21: list_fm,X22: list_list_fm] : ( member_list_fm3 @ X21 @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_154_list_Oset__intros_I1_J,axiom,
    ! [X21: fm,X22: list_fm] : ( member_fm3 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_155_mem__Collect__eq,axiom,
    ! [A2: rule,P4: rule > $o] :
      ( ( member_rule3 @ A2 @ ( collect_rule @ P4 ) )
      = ( P4 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_156_mem__Collect__eq,axiom,
    ! [A2: list_fm,P4: list_fm > $o] :
      ( ( member_list_fm3 @ A2 @ ( collect_list_fm @ P4 ) )
      = ( P4 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_157_mem__Collect__eq,axiom,
    ! [A2: tm,P4: tm > $o] :
      ( ( member_tm3 @ A2 @ ( collect_tm @ P4 ) )
      = ( P4 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
    ! [A2: fm,P4: fm > $o] :
      ( ( member_fm3 @ A2 @ ( collect_fm @ P4 ) )
      = ( P4 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
    ! [A2: nat,P4: nat > $o] :
      ( ( member_nat3 @ A2 @ ( collect_nat @ P4 ) )
      = ( P4 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_160_Collect__mem__eq,axiom,
    ! [A: set_rule] :
      ( ( collect_rule
        @ ^ [X2: rule] : ( member_rule3 @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_161_Collect__mem__eq,axiom,
    ! [A: set_list_fm] :
      ( ( collect_list_fm
        @ ^ [X2: list_fm] : ( member_list_fm3 @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_162_Collect__mem__eq,axiom,
    ! [A: set_tm] :
      ( ( collect_tm
        @ ^ [X2: tm] : ( member_tm3 @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_163_Collect__mem__eq,axiom,
    ! [A: set_fm] :
      ( ( collect_fm
        @ ^ [X2: fm] : ( member_fm3 @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat3 @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_165_list_Oset__cases,axiom,
    ! [E: rule,A2: list_rule] :
      ( ( member_rule3 @ E @ ( set_rule2 @ A2 ) )
     => ( ! [Z2: list_rule] :
            ( A2
           != ( cons_rule @ E @ Z2 ) )
       => ~ ! [Z1: rule,Z2: list_rule] :
              ( ( A2
                = ( cons_rule @ Z1 @ Z2 ) )
             => ~ ( member_rule3 @ E @ ( set_rule2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_166_list_Oset__cases,axiom,
    ! [E: tm,A2: list_tm] :
      ( ( member_tm3 @ E @ ( set_tm2 @ A2 ) )
     => ( ! [Z2: list_tm] :
            ( A2
           != ( cons_tm @ E @ Z2 ) )
       => ~ ! [Z1: tm,Z2: list_tm] :
              ( ( A2
                = ( cons_tm @ Z1 @ Z2 ) )
             => ~ ( member_tm3 @ E @ ( set_tm2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_167_list_Oset__cases,axiom,
    ! [E: nat,A2: list_nat] :
      ( ( member_nat3 @ E @ ( set_nat2 @ A2 ) )
     => ( ! [Z2: list_nat] :
            ( A2
           != ( cons_nat @ E @ Z2 ) )
       => ~ ! [Z1: nat,Z2: list_nat] :
              ( ( A2
                = ( cons_nat @ Z1 @ Z2 ) )
             => ~ ( member_nat3 @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_168_list_Oset__cases,axiom,
    ! [E: list_fm,A2: list_list_fm] :
      ( ( member_list_fm3 @ E @ ( set_list_fm2 @ A2 ) )
     => ( ! [Z2: list_list_fm] :
            ( A2
           != ( cons_list_fm @ E @ Z2 ) )
       => ~ ! [Z1: list_fm,Z2: list_list_fm] :
              ( ( A2
                = ( cons_list_fm @ Z1 @ Z2 ) )
             => ~ ( member_list_fm3 @ E @ ( set_list_fm2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_169_list_Oset__cases,axiom,
    ! [E: fm,A2: list_fm] :
      ( ( member_fm3 @ E @ ( set_fm2 @ A2 ) )
     => ( ! [Z2: list_fm] :
            ( A2
           != ( cons_fm @ E @ Z2 ) )
       => ~ ! [Z1: fm,Z2: list_fm] :
              ( ( A2
                = ( cons_fm @ Z1 @ Z2 ) )
             => ~ ( member_fm3 @ E @ ( set_fm2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_170_set__ConsD,axiom,
    ! [Y2: rule,X: rule,Xs: list_rule] :
      ( ( member_rule3 @ Y2 @ ( set_rule2 @ ( cons_rule @ X @ Xs ) ) )
     => ( ( Y2 = X )
        | ( member_rule3 @ Y2 @ ( set_rule2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_171_set__ConsD,axiom,
    ! [Y2: tm,X: tm,Xs: list_tm] :
      ( ( member_tm3 @ Y2 @ ( set_tm2 @ ( cons_tm @ X @ Xs ) ) )
     => ( ( Y2 = X )
        | ( member_tm3 @ Y2 @ ( set_tm2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_172_set__ConsD,axiom,
    ! [Y2: nat,X: nat,Xs: list_nat] :
      ( ( member_nat3 @ Y2 @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
     => ( ( Y2 = X )
        | ( member_nat3 @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_173_set__ConsD,axiom,
    ! [Y2: list_fm,X: list_fm,Xs: list_list_fm] :
      ( ( member_list_fm3 @ Y2 @ ( set_list_fm2 @ ( cons_list_fm @ X @ Xs ) ) )
     => ( ( Y2 = X )
        | ( member_list_fm3 @ Y2 @ ( set_list_fm2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_174_set__ConsD,axiom,
    ! [Y2: fm,X: fm,Xs: list_fm] :
      ( ( member_fm3 @ Y2 @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) )
     => ( ( Y2 = X )
        | ( member_fm3 @ Y2 @ ( set_fm2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_175_can__select__def,axiom,
    ( can_select_rule
    = ( ^ [P2: rule > $o,A3: set_rule] :
        ? [X2: rule] :
          ( ( member_rule3 @ X2 @ A3 )
          & ( P2 @ X2 )
          & ! [Y: rule] :
              ( ( ( member_rule3 @ Y @ A3 )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_176_can__select__def,axiom,
    ( can_select_list_fm
    = ( ^ [P2: list_fm > $o,A3: set_list_fm] :
        ? [X2: list_fm] :
          ( ( member_list_fm3 @ X2 @ A3 )
          & ( P2 @ X2 )
          & ! [Y: list_fm] :
              ( ( ( member_list_fm3 @ Y @ A3 )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_177_can__select__def,axiom,
    ( can_select_tm
    = ( ^ [P2: tm > $o,A3: set_tm] :
        ? [X2: tm] :
          ( ( member_tm3 @ X2 @ A3 )
          & ( P2 @ X2 )
          & ! [Y: tm] :
              ( ( ( member_tm3 @ Y @ A3 )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_178_can__select__def,axiom,
    ( can_select_fm
    = ( ^ [P2: fm > $o,A3: set_fm] :
        ? [X2: fm] :
          ( ( member_fm3 @ X2 @ A3 )
          & ( P2 @ X2 )
          & ! [Y: fm] :
              ( ( ( member_fm3 @ Y @ A3 )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_179_can__select__def,axiom,
    ( can_select_nat
    = ( ^ [P2: nat > $o,A3: set_nat] :
        ? [X2: nat] :
          ( ( member_nat3 @ X2 @ A3 )
          & ( P2 @ X2 )
          & ! [Y: nat] :
              ( ( ( member_nat3 @ Y @ A3 )
                & ( P2 @ Y ) )
             => ( Y = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_180_snth__sset,axiom,
    ! [S: stream_list_fm,N: nat] : ( member_list_fm3 @ ( snth_list_fm @ S @ N ) @ ( sset_list_fm @ S ) ) ).

% snth_sset
thf(fact_181_snth__sset,axiom,
    ! [S: stream_tm,N: nat] : ( member_tm3 @ ( snth_tm @ S @ N ) @ ( sset_tm @ S ) ) ).

% snth_sset
thf(fact_182_snth__sset,axiom,
    ! [S: stream_fm,N: nat] : ( member_fm3 @ ( snth_fm @ S @ N ) @ ( sset_fm @ S ) ) ).

% snth_sset
thf(fact_183_snth__sset,axiom,
    ! [S: stream_nat,N: nat] : ( member_nat3 @ ( snth_nat @ S @ N ) @ ( sset_nat @ S ) ) ).

% snth_sset
thf(fact_184_snth__sset,axiom,
    ! [S: stream2709947120125613254m_rule,N: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ S @ N ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).

% snth_sset
thf(fact_185_snth__sset,axiom,
    ! [S: stream_rule,N: nat] : ( member_rule3 @ ( snth_rule @ S @ N ) @ ( sset_rule @ S ) ) ).

% snth_sset
thf(fact_186_SeCaV_Omember_Osimps_I2_J,axiom,
    ! [P3: tm,Q: tm,Z3: list_tm] :
      ( ( member_tm2 @ P3 @ ( cons_tm @ Q @ Z3 ) )
      = ( ( P3 != Q )
       => ( member_tm2 @ P3 @ Z3 ) ) ) ).

% SeCaV.member.simps(2)
thf(fact_187_SeCaV_Omember_Osimps_I2_J,axiom,
    ! [P3: nat,Q: nat,Z3: list_nat] :
      ( ( member_nat2 @ P3 @ ( cons_nat @ Q @ Z3 ) )
      = ( ( P3 != Q )
       => ( member_nat2 @ P3 @ Z3 ) ) ) ).

% SeCaV.member.simps(2)
thf(fact_188_SeCaV_Omember_Osimps_I2_J,axiom,
    ! [P3: list_fm,Q: list_fm,Z3: list_list_fm] :
      ( ( member_list_fm2 @ P3 @ ( cons_list_fm @ Q @ Z3 ) )
      = ( ( P3 != Q )
       => ( member_list_fm2 @ P3 @ Z3 ) ) ) ).

% SeCaV.member.simps(2)
thf(fact_189_SeCaV_Omember_Osimps_I2_J,axiom,
    ! [P3: fm,Q: fm,Z3: list_fm] :
      ( ( member_fm2 @ P3 @ ( cons_fm @ Q @ Z3 ) )
      = ( ( P3 != Q )
       => ( member_fm2 @ P3 @ Z3 ) ) ) ).

% SeCaV.member.simps(2)
thf(fact_190_member__rec_I1_J,axiom,
    ! [X: tm,Xs: list_tm,Y2: tm] :
      ( ( member_tm @ ( cons_tm @ X @ Xs ) @ Y2 )
      = ( ( X = Y2 )
        | ( member_tm @ Xs @ Y2 ) ) ) ).

% member_rec(1)
thf(fact_191_member__rec_I1_J,axiom,
    ! [X: nat,Xs: list_nat,Y2: nat] :
      ( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
      = ( ( X = Y2 )
        | ( member_nat @ Xs @ Y2 ) ) ) ).

% member_rec(1)
thf(fact_192_member__rec_I1_J,axiom,
    ! [X: list_fm,Xs: list_list_fm,Y2: list_fm] :
      ( ( member_list_fm @ ( cons_list_fm @ X @ Xs ) @ Y2 )
      = ( ( X = Y2 )
        | ( member_list_fm @ Xs @ Y2 ) ) ) ).

% member_rec(1)
thf(fact_193_member__rec_I1_J,axiom,
    ! [X: fm,Xs: list_fm,Y2: fm] :
      ( ( member_fm @ ( cons_fm @ X @ Xs ) @ Y2 )
      = ( ( X = Y2 )
        | ( member_fm @ Xs @ Y2 ) ) ) ).

% member_rec(1)
thf(fact_194_List_Oinsert__def,axiom,
    ( insert_rule
    = ( ^ [X2: rule,Xs2: list_rule] : ( if_list_rule @ ( member_rule3 @ X2 @ ( set_rule2 @ Xs2 ) ) @ Xs2 @ ( cons_rule @ X2 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_195_List_Oinsert__def,axiom,
    ( insert_tm
    = ( ^ [X2: tm,Xs2: list_tm] : ( if_list_tm @ ( member_tm3 @ X2 @ ( set_tm2 @ Xs2 ) ) @ Xs2 @ ( cons_tm @ X2 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_196_List_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [X2: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat3 @ X2 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_197_List_Oinsert__def,axiom,
    ( insert_list_fm
    = ( ^ [X2: list_fm,Xs2: list_list_fm] : ( if_list_list_fm @ ( member_list_fm3 @ X2 @ ( set_list_fm2 @ Xs2 ) ) @ Xs2 @ ( cons_list_fm @ X2 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_198_List_Oinsert__def,axiom,
    ( insert_fm
    = ( ^ [X2: fm,Xs2: list_fm] : ( if_list_fm @ ( member_fm3 @ X2 @ ( set_fm2 @ Xs2 ) ) @ Xs2 @ ( cons_fm @ X2 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_199_insert__absorb2,axiom,
    ! [X: rule,A: set_rule] :
      ( ( insert_rule2 @ X @ ( insert_rule2 @ X @ A ) )
      = ( insert_rule2 @ X @ A ) ) ).

% insert_absorb2
thf(fact_200_insert__absorb2,axiom,
    ! [X: fm,A: set_fm] :
      ( ( insert_fm2 @ X @ ( insert_fm2 @ X @ A ) )
      = ( insert_fm2 @ X @ A ) ) ).

% insert_absorb2
thf(fact_201_insert__absorb2,axiom,
    ! [X: nat,A: set_nat] :
      ( ( insert_nat2 @ X @ ( insert_nat2 @ X @ A ) )
      = ( insert_nat2 @ X @ A ) ) ).

% insert_absorb2
thf(fact_202_insert__iff,axiom,
    ! [A2: rule,B: rule,A: set_rule] :
      ( ( member_rule3 @ A2 @ ( insert_rule2 @ B @ A ) )
      = ( ( A2 = B )
        | ( member_rule3 @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_203_insert__iff,axiom,
    ! [A2: list_fm,B: list_fm,A: set_list_fm] :
      ( ( member_list_fm3 @ A2 @ ( insert_list_fm2 @ B @ A ) )
      = ( ( A2 = B )
        | ( member_list_fm3 @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_204_insert__iff,axiom,
    ! [A2: tm,B: tm,A: set_tm] :
      ( ( member_tm3 @ A2 @ ( insert_tm2 @ B @ A ) )
      = ( ( A2 = B )
        | ( member_tm3 @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_205_insert__iff,axiom,
    ! [A2: fm,B: fm,A: set_fm] :
      ( ( member_fm3 @ A2 @ ( insert_fm2 @ B @ A ) )
      = ( ( A2 = B )
        | ( member_fm3 @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_206_insert__iff,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( member_nat3 @ A2 @ ( insert_nat2 @ B @ A ) )
      = ( ( A2 = B )
        | ( member_nat3 @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_207_insertCI,axiom,
    ! [A2: rule,B2: set_rule,B: rule] :
      ( ( ~ ( member_rule3 @ A2 @ B2 )
       => ( A2 = B ) )
     => ( member_rule3 @ A2 @ ( insert_rule2 @ B @ B2 ) ) ) ).

% insertCI
thf(fact_208_insertCI,axiom,
    ! [A2: list_fm,B2: set_list_fm,B: list_fm] :
      ( ( ~ ( member_list_fm3 @ A2 @ B2 )
       => ( A2 = B ) )
     => ( member_list_fm3 @ A2 @ ( insert_list_fm2 @ B @ B2 ) ) ) ).

% insertCI
thf(fact_209_insertCI,axiom,
    ! [A2: tm,B2: set_tm,B: tm] :
      ( ( ~ ( member_tm3 @ A2 @ B2 )
       => ( A2 = B ) )
     => ( member_tm3 @ A2 @ ( insert_tm2 @ B @ B2 ) ) ) ).

% insertCI
thf(fact_210_insertCI,axiom,
    ! [A2: fm,B2: set_fm,B: fm] :
      ( ( ~ ( member_fm3 @ A2 @ B2 )
       => ( A2 = B ) )
     => ( member_fm3 @ A2 @ ( insert_fm2 @ B @ B2 ) ) ) ).

% insertCI
thf(fact_211_insertCI,axiom,
    ! [A2: nat,B2: set_nat,B: nat] :
      ( ( ~ ( member_nat3 @ A2 @ B2 )
       => ( A2 = B ) )
     => ( member_nat3 @ A2 @ ( insert_nat2 @ B @ B2 ) ) ) ).

% insertCI
thf(fact_212_old_Oprod_Oinject,axiom,
    ! [A2: list_tm,B: list_fm,A4: list_tm,B3: list_fm] :
      ( ( ( produc1414352766439514085ist_fm @ A2 @ B )
        = ( produc1414352766439514085ist_fm @ A4 @ B3 ) )
      = ( ( A2 = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_213_old_Oprod_Oinject,axiom,
    ! [A2: produc6018962875968178549ist_fm,B: rule,A4: produc6018962875968178549ist_fm,B3: rule] :
      ( ( ( produc1733806532565653680m_rule @ A2 @ B )
        = ( produc1733806532565653680m_rule @ A4 @ B3 ) )
      = ( ( A2 = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_214_old_Oprod_Oinject,axiom,
    ! [A2: nat,B: nat,A4: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A2 @ B )
        = ( product_Pair_nat_nat @ A4 @ B3 ) )
      = ( ( A2 = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_215_old_Oprod_Oinject,axiom,
    ! [A2: product_prod_nat_nat,B: product_prod_nat_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A2 @ B )
        = ( produc6161850002892822231at_nat @ A4 @ B3 ) )
      = ( ( A2 = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_216_prod_Oinject,axiom,
    ! [X1: list_tm,X23: list_fm,Y1: list_tm,Y23: list_fm] :
      ( ( ( produc1414352766439514085ist_fm @ X1 @ X23 )
        = ( produc1414352766439514085ist_fm @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_217_prod_Oinject,axiom,
    ! [X1: produc6018962875968178549ist_fm,X23: rule,Y1: produc6018962875968178549ist_fm,Y23: rule] :
      ( ( ( produc1733806532565653680m_rule @ X1 @ X23 )
        = ( produc1733806532565653680m_rule @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_218_prod_Oinject,axiom,
    ! [X1: nat,X23: nat,Y1: nat,Y23: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X23 )
        = ( product_Pair_nat_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_219_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X23: product_prod_nat_nat,Y1: product_prod_nat_nat,Y23: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X23 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_220_snth__sset__smerge,axiom,
    ! [Ss: stream6017534608192929797ist_fm,N: nat,M2: nat] : ( member_list_fm3 @ ( snth_list_fm @ ( snth_stream_list_fm @ Ss @ N ) @ M2 ) @ ( sset_list_fm @ ( smerge_list_fm @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_221_snth__sset__smerge,axiom,
    ! [Ss: stream_stream_tm,N: nat,M2: nat] : ( member_tm3 @ ( snth_tm @ ( snth_stream_tm @ Ss @ N ) @ M2 ) @ ( sset_tm @ ( smerge_tm @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_222_snth__sset__smerge,axiom,
    ! [Ss: stream_stream_fm,N: nat,M2: nat] : ( member_fm3 @ ( snth_fm @ ( snth_stream_fm @ Ss @ N ) @ M2 ) @ ( sset_fm @ ( smerge_fm @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_223_snth__sset__smerge,axiom,
    ! [Ss: stream_stream_nat,N: nat,M2: nat] : ( member_nat3 @ ( snth_nat @ ( snth_stream_nat @ Ss @ N ) @ M2 ) @ ( sset_nat @ ( smerge_nat @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_224_snth__sset__smerge,axiom,
    ! [Ss: stream3752074346242807894m_rule,N: nat,M2: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ ( snth_s6182113952396108578m_rule @ Ss @ N ) @ M2 ) @ ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_225_snth__sset__smerge,axiom,
    ! [Ss: stream_stream_rule,N: nat,M2: nat] : ( member_rule3 @ ( snth_rule @ ( snth_stream_rule @ Ss @ N ) @ M2 ) @ ( sset_rule @ ( smerge_rule @ Ss ) ) ) ).

% snth_sset_smerge
thf(fact_226_ext_Osimps_I2_J,axiom,
    ! [Y2: list_tm,P3: tm,Z3: list_tm] :
      ( ( ext_tm @ Y2 @ ( cons_tm @ P3 @ Z3 ) )
      = ( ( ( member_tm2 @ P3 @ Y2 )
         => ( ext_tm @ Y2 @ Z3 ) )
        & ( member_tm2 @ P3 @ Y2 ) ) ) ).

% ext.simps(2)
thf(fact_227_ext_Osimps_I2_J,axiom,
    ! [Y2: list_nat,P3: nat,Z3: list_nat] :
      ( ( ext_nat @ Y2 @ ( cons_nat @ P3 @ Z3 ) )
      = ( ( ( member_nat2 @ P3 @ Y2 )
         => ( ext_nat @ Y2 @ Z3 ) )
        & ( member_nat2 @ P3 @ Y2 ) ) ) ).

% ext.simps(2)
thf(fact_228_ext_Osimps_I2_J,axiom,
    ! [Y2: list_list_fm,P3: list_fm,Z3: list_list_fm] :
      ( ( ext_list_fm @ Y2 @ ( cons_list_fm @ P3 @ Z3 ) )
      = ( ( ( member_list_fm2 @ P3 @ Y2 )
         => ( ext_list_fm @ Y2 @ Z3 ) )
        & ( member_list_fm2 @ P3 @ Y2 ) ) ) ).

% ext.simps(2)
thf(fact_229_ext_Osimps_I2_J,axiom,
    ! [Y2: list_fm,P3: fm,Z3: list_fm] :
      ( ( ext_fm @ Y2 @ ( cons_fm @ P3 @ Z3 ) )
      = ( ( ( member_fm2 @ P3 @ Y2 )
         => ( ext_fm @ Y2 @ Z3 ) )
        & ( member_fm2 @ P3 @ Y2 ) ) ) ).

% ext.simps(2)
thf(fact_230_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_231_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra6097777249025082867ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,Steps2: stream2709947120125613254m_rule] :
        ! [X2: rule] :
          ( ( member_rule3 @ X2 @ ( sset_rule @ Rules ) )
         => ( abstra2533313685312581075ist_fm @ Eff @ X2 @ Steps2 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_232_Stream_Osmember__def,axiom,
    ( smember_list_fm
    = ( ^ [X2: list_fm,S3: stream_list_fm] : ( member_list_fm3 @ X2 @ ( sset_list_fm @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_233_Stream_Osmember__def,axiom,
    ( smember_tm
    = ( ^ [X2: tm,S3: stream_tm] : ( member_tm3 @ X2 @ ( sset_tm @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_234_Stream_Osmember__def,axiom,
    ( smember_fm
    = ( ^ [X2: fm,S3: stream_fm] : ( member_fm3 @ X2 @ ( sset_fm @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_235_Stream_Osmember__def,axiom,
    ( smember_nat
    = ( ^ [X2: nat,S3: stream_nat] : ( member_nat3 @ X2 @ ( sset_nat @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_236_Stream_Osmember__def,axiom,
    ( smembe4439892024482649336m_rule
    = ( ^ [X2: produc340336539035504054m_rule,S3: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ X2 @ ( sset_P4484857331586881186m_rule @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_237_Stream_Osmember__def,axiom,
    ( smember_rule
    = ( ^ [X2: rule,S3: stream_rule] : ( member_rule3 @ X2 @ ( sset_rule @ S3 ) ) ) ) ).

% Stream.smember_def
thf(fact_238_subtermFm__preds,axiom,
    ! [T: tm,P3: fm] :
      ( ( member_tm3 @ T @ ( set_tm2 @ ( subtermFm @ P3 ) ) )
      = ( ? [X2: fm] :
            ( ( member_fm3 @ X2 @ ( preds @ P3 ) )
            & ( member_tm3 @ T @ ( set_tm2 @ ( subtermFm @ X2 ) ) ) ) ) ) ).

% subtermFm_preds
thf(fact_239_subsetI,axiom,
    ! [A: set_rule,B2: set_rule] :
      ( ! [X3: rule] :
          ( ( member_rule3 @ X3 @ A )
         => ( member_rule3 @ X3 @ B2 ) )
     => ( ord_less_eq_set_rule @ A @ B2 ) ) ).

% subsetI
thf(fact_240_subsetI,axiom,
    ! [A: set_list_fm,B2: set_list_fm] :
      ( ! [X3: list_fm] :
          ( ( member_list_fm3 @ X3 @ A )
         => ( member_list_fm3 @ X3 @ B2 ) )
     => ( ord_le7838213414353715577ist_fm @ A @ B2 ) ) ).

% subsetI
thf(fact_241_subsetI,axiom,
    ! [A: set_tm,B2: set_tm] :
      ( ! [X3: tm] :
          ( ( member_tm3 @ X3 @ A )
         => ( member_tm3 @ X3 @ B2 ) )
     => ( ord_less_eq_set_tm @ A @ B2 ) ) ).

% subsetI
thf(fact_242_subsetI,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat3 @ X3 @ A )
         => ( member_nat3 @ X3 @ B2 ) )
     => ( ord_less_eq_set_nat @ A @ B2 ) ) ).

% subsetI
thf(fact_243_subsetI,axiom,
    ! [A: set_fm,B2: set_fm] :
      ( ! [X3: fm] :
          ( ( member_fm3 @ X3 @ A )
         => ( member_fm3 @ X3 @ B2 ) )
     => ( ord_less_eq_set_fm @ A @ B2 ) ) ).

% subsetI
thf(fact_244_subset__antisym,axiom,
    ! [A: set_tm,B2: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B2 )
     => ( ( ord_less_eq_set_tm @ B2 @ A )
       => ( A = B2 ) ) ) ).

% subset_antisym
thf(fact_245_subset__antisym,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% subset_antisym
thf(fact_246_subset__antisym,axiom,
    ! [A: set_fm,B2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B2 )
     => ( ( ord_less_eq_set_fm @ B2 @ A )
       => ( A = B2 ) ) ) ).

% subset_antisym
thf(fact_247_insert__subset,axiom,
    ! [X: rule,A: set_rule,B2: set_rule] :
      ( ( ord_less_eq_set_rule @ ( insert_rule2 @ X @ A ) @ B2 )
      = ( ( member_rule3 @ X @ B2 )
        & ( ord_less_eq_set_rule @ A @ B2 ) ) ) ).

% insert_subset
thf(fact_248_insert__subset,axiom,
    ! [X: list_fm,A: set_list_fm,B2: set_list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ ( insert_list_fm2 @ X @ A ) @ B2 )
      = ( ( member_list_fm3 @ X @ B2 )
        & ( ord_le7838213414353715577ist_fm @ A @ B2 ) ) ) ).

% insert_subset
thf(fact_249_insert__subset,axiom,
    ! [X: tm,A: set_tm,B2: set_tm] :
      ( ( ord_less_eq_set_tm @ ( insert_tm2 @ X @ A ) @ B2 )
      = ( ( member_tm3 @ X @ B2 )
        & ( ord_less_eq_set_tm @ A @ B2 ) ) ) ).

% insert_subset
thf(fact_250_insert__subset,axiom,
    ! [X: nat,A: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A ) @ B2 )
      = ( ( member_nat3 @ X @ B2 )
        & ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).

% insert_subset
thf(fact_251_insert__subset,axiom,
    ! [X: fm,A: set_fm,B2: set_fm] :
      ( ( ord_less_eq_set_fm @ ( insert_fm2 @ X @ A ) @ B2 )
      = ( ( member_fm3 @ X @ B2 )
        & ( ord_less_eq_set_fm @ A @ B2 ) ) ) ).

% insert_subset
thf(fact_252_SeCaV_Oext,axiom,
    ( ext_list_fm
    = ( ^ [Y: list_list_fm,Z: list_list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Z ) @ ( set_list_fm2 @ Y ) ) ) ) ).

% SeCaV.ext
thf(fact_253_SeCaV_Oext,axiom,
    ( ext_tm
    = ( ^ [Y: list_tm,Z: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Z ) @ ( set_tm2 @ Y ) ) ) ) ).

% SeCaV.ext
thf(fact_254_SeCaV_Oext,axiom,
    ( ext_nat
    = ( ^ [Y: list_nat,Z: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Z ) @ ( set_nat2 @ Y ) ) ) ) ).

% SeCaV.ext
thf(fact_255_SeCaV_Oext,axiom,
    ( ext_fm
    = ( ^ [Y: list_fm,Z: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Z ) @ ( set_fm2 @ Y ) ) ) ) ).

% SeCaV.ext
thf(fact_256_in__mono,axiom,
    ! [A: set_rule,B2: set_rule,X: rule] :
      ( ( ord_less_eq_set_rule @ A @ B2 )
     => ( ( member_rule3 @ X @ A )
       => ( member_rule3 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_257_in__mono,axiom,
    ! [A: set_list_fm,B2: set_list_fm,X: list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ A @ B2 )
     => ( ( member_list_fm3 @ X @ A )
       => ( member_list_fm3 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_258_in__mono,axiom,
    ! [A: set_tm,B2: set_tm,X: tm] :
      ( ( ord_less_eq_set_tm @ A @ B2 )
     => ( ( member_tm3 @ X @ A )
       => ( member_tm3 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_259_in__mono,axiom,
    ! [A: set_nat,B2: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( member_nat3 @ X @ A )
       => ( member_nat3 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_260_in__mono,axiom,
    ! [A: set_fm,B2: set_fm,X: fm] :
      ( ( ord_less_eq_set_fm @ A @ B2 )
     => ( ( member_fm3 @ X @ A )
       => ( member_fm3 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_261_subsetD,axiom,
    ! [A: set_rule,B2: set_rule,C: rule] :
      ( ( ord_less_eq_set_rule @ A @ B2 )
     => ( ( member_rule3 @ C @ A )
       => ( member_rule3 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_262_subsetD,axiom,
    ! [A: set_list_fm,B2: set_list_fm,C: list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ A @ B2 )
     => ( ( member_list_fm3 @ C @ A )
       => ( member_list_fm3 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_263_subsetD,axiom,
    ! [A: set_tm,B2: set_tm,C: tm] :
      ( ( ord_less_eq_set_tm @ A @ B2 )
     => ( ( member_tm3 @ C @ A )
       => ( member_tm3 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_264_subsetD,axiom,
    ! [A: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( member_nat3 @ C @ A )
       => ( member_nat3 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_265_subsetD,axiom,
    ! [A: set_fm,B2: set_fm,C: fm] :
      ( ( ord_less_eq_set_fm @ A @ B2 )
     => ( ( member_fm3 @ C @ A )
       => ( member_fm3 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_266_equalityE,axiom,
    ! [A: set_tm,B2: set_tm] :
      ( ( A = B2 )
     => ~ ( ( ord_less_eq_set_tm @ A @ B2 )
         => ~ ( ord_less_eq_set_tm @ B2 @ A ) ) ) ).

% equalityE
thf(fact_267_equalityE,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
     => ~ ( ( ord_less_eq_set_nat @ A @ B2 )
         => ~ ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).

% equalityE
thf(fact_268_equalityE,axiom,
    ! [A: set_fm,B2: set_fm] :
      ( ( A = B2 )
     => ~ ( ( ord_less_eq_set_fm @ A @ B2 )
         => ~ ( ord_less_eq_set_fm @ B2 @ A ) ) ) ).

% equalityE
thf(fact_269_subset__eq,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A3: set_rule,B4: set_rule] :
        ! [X2: rule] :
          ( ( member_rule3 @ X2 @ A3 )
         => ( member_rule3 @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_270_subset__eq,axiom,
    ( ord_le7838213414353715577ist_fm
    = ( ^ [A3: set_list_fm,B4: set_list_fm] :
        ! [X2: list_fm] :
          ( ( member_list_fm3 @ X2 @ A3 )
         => ( member_list_fm3 @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_271_subset__eq,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [A3: set_tm,B4: set_tm] :
        ! [X2: tm] :
          ( ( member_tm3 @ X2 @ A3 )
         => ( member_tm3 @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_272_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B4: set_nat] :
        ! [X2: nat] :
          ( ( member_nat3 @ X2 @ A3 )
         => ( member_nat3 @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_273_subset__eq,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A3: set_fm,B4: set_fm] :
        ! [X2: fm] :
          ( ( member_fm3 @ X2 @ A3 )
         => ( member_fm3 @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_274_equalityD1,axiom,
    ! [A: set_tm,B2: set_tm] :
      ( ( A = B2 )
     => ( ord_less_eq_set_tm @ A @ B2 ) ) ).

% equalityD1
thf(fact_275_equalityD1,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
     => ( ord_less_eq_set_nat @ A @ B2 ) ) ).

% equalityD1
thf(fact_276_equalityD1,axiom,
    ! [A: set_fm,B2: set_fm] :
      ( ( A = B2 )
     => ( ord_less_eq_set_fm @ A @ B2 ) ) ).

% equalityD1
thf(fact_277_equalityD2,axiom,
    ! [A: set_tm,B2: set_tm] :
      ( ( A = B2 )
     => ( ord_less_eq_set_tm @ B2 @ A ) ) ).

% equalityD2
thf(fact_278_equalityD2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
     => ( ord_less_eq_set_nat @ B2 @ A ) ) ).

% equalityD2
thf(fact_279_equalityD2,axiom,
    ! [A: set_fm,B2: set_fm] :
      ( ( A = B2 )
     => ( ord_less_eq_set_fm @ B2 @ A ) ) ).

% equalityD2
thf(fact_280_subset__iff,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A3: set_rule,B4: set_rule] :
        ! [T2: rule] :
          ( ( member_rule3 @ T2 @ A3 )
         => ( member_rule3 @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_281_subset__iff,axiom,
    ( ord_le7838213414353715577ist_fm
    = ( ^ [A3: set_list_fm,B4: set_list_fm] :
        ! [T2: list_fm] :
          ( ( member_list_fm3 @ T2 @ A3 )
         => ( member_list_fm3 @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_282_subset__iff,axiom,
    ( ord_less_eq_set_tm
    = ( ^ [A3: set_tm,B4: set_tm] :
        ! [T2: tm] :
          ( ( member_tm3 @ T2 @ A3 )
         => ( member_tm3 @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_283_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B4: set_nat] :
        ! [T2: nat] :
          ( ( member_nat3 @ T2 @ A3 )
         => ( member_nat3 @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_284_subset__iff,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A3: set_fm,B4: set_fm] :
        ! [T2: fm] :
          ( ( member_fm3 @ T2 @ A3 )
         => ( member_fm3 @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_285_subset__refl,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).

% subset_refl
thf(fact_286_subset__refl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% subset_refl
thf(fact_287_subset__refl,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).

% subset_refl
thf(fact_288_Collect__mono,axiom,
    ! [P4: tm > $o,Q2: tm > $o] :
      ( ! [X3: tm] :
          ( ( P4 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_less_eq_set_tm @ ( collect_tm @ P4 ) @ ( collect_tm @ Q2 ) ) ) ).

% Collect_mono
thf(fact_289_Collect__mono,axiom,
    ! [P4: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P4 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q2 ) ) ) ).

% Collect_mono
thf(fact_290_Collect__mono,axiom,
    ! [P4: fm > $o,Q2: fm > $o] :
      ( ! [X3: fm] :
          ( ( P4 @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_less_eq_set_fm @ ( collect_fm @ P4 ) @ ( collect_fm @ Q2 ) ) ) ).

% Collect_mono
thf(fact_291_subset__trans,axiom,
    ! [A: set_tm,B2: set_tm,C2: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B2 )
     => ( ( ord_less_eq_set_tm @ B2 @ C2 )
       => ( ord_less_eq_set_tm @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_292_subset__trans,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_293_subset__trans,axiom,
    ! [A: set_fm,B2: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B2 )
     => ( ( ord_less_eq_set_fm @ B2 @ C2 )
       => ( ord_less_eq_set_fm @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_294_set__eq__subset,axiom,
    ( ( ^ [Y3: set_tm,Z4: set_tm] : ( Y3 = Z4 ) )
    = ( ^ [A3: set_tm,B4: set_tm] :
          ( ( ord_less_eq_set_tm @ A3 @ B4 )
          & ( ord_less_eq_set_tm @ B4 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_295_set__eq__subset,axiom,
    ( ( ^ [Y3: set_nat,Z4: set_nat] : ( Y3 = Z4 ) )
    = ( ^ [A3: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B4 )
          & ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_296_set__eq__subset,axiom,
    ( ( ^ [Y3: set_fm,Z4: set_fm] : ( Y3 = Z4 ) )
    = ( ^ [A3: set_fm,B4: set_fm] :
          ( ( ord_less_eq_set_fm @ A3 @ B4 )
          & ( ord_less_eq_set_fm @ B4 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_297_Collect__mono__iff,axiom,
    ! [P4: tm > $o,Q2: tm > $o] :
      ( ( ord_less_eq_set_tm @ ( collect_tm @ P4 ) @ ( collect_tm @ Q2 ) )
      = ( ! [X2: tm] :
            ( ( P4 @ X2 )
           => ( Q2 @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_298_Collect__mono__iff,axiom,
    ! [P4: nat > $o,Q2: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q2 ) )
      = ( ! [X2: nat] :
            ( ( P4 @ X2 )
           => ( Q2 @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_299_Collect__mono__iff,axiom,
    ! [P4: fm > $o,Q2: fm > $o] :
      ( ( ord_less_eq_set_fm @ ( collect_fm @ P4 ) @ ( collect_fm @ Q2 ) )
      = ( ! [X2: fm] :
            ( ( P4 @ X2 )
           => ( Q2 @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_300_RuleSystem__Defs_Osaturated_Ocong,axiom,
    abstra2533313685312581075ist_fm = abstra2533313685312581075ist_fm ).

% RuleSystem_Defs.saturated.cong
thf(fact_301_insert__mono,axiom,
    ! [C2: set_rule,D: set_rule,A2: rule] :
      ( ( ord_less_eq_set_rule @ C2 @ D )
     => ( ord_less_eq_set_rule @ ( insert_rule2 @ A2 @ C2 ) @ ( insert_rule2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_302_insert__mono,axiom,
    ! [C2: set_tm,D: set_tm,A2: tm] :
      ( ( ord_less_eq_set_tm @ C2 @ D )
     => ( ord_less_eq_set_tm @ ( insert_tm2 @ A2 @ C2 ) @ ( insert_tm2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_303_insert__mono,axiom,
    ! [C2: set_nat,D: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ C2 @ D )
     => ( ord_less_eq_set_nat @ ( insert_nat2 @ A2 @ C2 ) @ ( insert_nat2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_304_insert__mono,axiom,
    ! [C2: set_fm,D: set_fm,A2: fm] :
      ( ( ord_less_eq_set_fm @ C2 @ D )
     => ( ord_less_eq_set_fm @ ( insert_fm2 @ A2 @ C2 ) @ ( insert_fm2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_305_subset__insert,axiom,
    ! [X: rule,A: set_rule,B2: set_rule] :
      ( ~ ( member_rule3 @ X @ A )
     => ( ( ord_less_eq_set_rule @ A @ ( insert_rule2 @ X @ B2 ) )
        = ( ord_less_eq_set_rule @ A @ B2 ) ) ) ).

% subset_insert
thf(fact_306_subset__insert,axiom,
    ! [X: list_fm,A: set_list_fm,B2: set_list_fm] :
      ( ~ ( member_list_fm3 @ X @ A )
     => ( ( ord_le7838213414353715577ist_fm @ A @ ( insert_list_fm2 @ X @ B2 ) )
        = ( ord_le7838213414353715577ist_fm @ A @ B2 ) ) ) ).

% subset_insert
thf(fact_307_subset__insert,axiom,
    ! [X: tm,A: set_tm,B2: set_tm] :
      ( ~ ( member_tm3 @ X @ A )
     => ( ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ X @ B2 ) )
        = ( ord_less_eq_set_tm @ A @ B2 ) ) ) ).

% subset_insert
thf(fact_308_subset__insert,axiom,
    ! [X: nat,A: set_nat,B2: set_nat] :
      ( ~ ( member_nat3 @ X @ A )
     => ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B2 ) )
        = ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).

% subset_insert
thf(fact_309_subset__insert,axiom,
    ! [X: fm,A: set_fm,B2: set_fm] :
      ( ~ ( member_fm3 @ X @ A )
     => ( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B2 ) )
        = ( ord_less_eq_set_fm @ A @ B2 ) ) ) ).

% subset_insert
thf(fact_310_subset__insertI,axiom,
    ! [B2: set_rule,A2: rule] : ( ord_less_eq_set_rule @ B2 @ ( insert_rule2 @ A2 @ B2 ) ) ).

% subset_insertI
thf(fact_311_subset__insertI,axiom,
    ! [B2: set_tm,A2: tm] : ( ord_less_eq_set_tm @ B2 @ ( insert_tm2 @ A2 @ B2 ) ) ).

% subset_insertI
thf(fact_312_subset__insertI,axiom,
    ! [B2: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat2 @ A2 @ B2 ) ) ).

% subset_insertI
thf(fact_313_subset__insertI,axiom,
    ! [B2: set_fm,A2: fm] : ( ord_less_eq_set_fm @ B2 @ ( insert_fm2 @ A2 @ B2 ) ) ).

% subset_insertI
thf(fact_314_subset__insertI2,axiom,
    ! [A: set_rule,B2: set_rule,B: rule] :
      ( ( ord_less_eq_set_rule @ A @ B2 )
     => ( ord_less_eq_set_rule @ A @ ( insert_rule2 @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_315_subset__insertI2,axiom,
    ! [A: set_tm,B2: set_tm,B: tm] :
      ( ( ord_less_eq_set_tm @ A @ B2 )
     => ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_316_subset__insertI2,axiom,
    ! [A: set_nat,B2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_317_subset__insertI2,axiom,
    ! [A: set_fm,B2: set_fm,B: fm] :
      ( ( ord_less_eq_set_fm @ A @ B2 )
     => ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_318_subset__code_I1_J,axiom,
    ! [Xs: list_rule,B2: set_rule] :
      ( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ B2 )
      = ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ ( set_rule2 @ Xs ) )
           => ( member_rule3 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_319_subset__code_I1_J,axiom,
    ! [Xs: list_list_fm,B2: set_list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ B2 )
      = ( ! [X2: list_fm] :
            ( ( member_list_fm3 @ X2 @ ( set_list_fm2 @ Xs ) )
           => ( member_list_fm3 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_320_subset__code_I1_J,axiom,
    ! [Xs: list_tm,B2: set_tm] :
      ( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B2 )
      = ( ! [X2: tm] :
            ( ( member_tm3 @ X2 @ ( set_tm2 @ Xs ) )
           => ( member_tm3 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_321_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
      = ( ! [X2: nat] :
            ( ( member_nat3 @ X2 @ ( set_nat2 @ Xs ) )
           => ( member_nat3 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_322_subset__code_I1_J,axiom,
    ! [Xs: list_fm,B2: set_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B2 )
      = ( ! [X2: fm] :
            ( ( member_fm3 @ X2 @ ( set_fm2 @ Xs ) )
           => ( member_fm3 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_323_preds_Osimps_I7_J,axiom,
    ! [P3: fm] :
      ( ( preds @ ( neg @ P3 ) )
      = ( preds @ P3 ) ) ).

% preds.simps(7)
thf(fact_324_preds_Osimps_I6_J,axiom,
    ! [P3: fm] :
      ( ( preds @ ( uni @ P3 ) )
      = ( preds @ P3 ) ) ).

% preds.simps(6)
thf(fact_325_Saturated__def,axiom,
    ! [Steps: stream2709947120125613254m_rule] :
      ( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
      = ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ ( sset_rule @ rules ) )
           => ( abstra2533313685312581075ist_fm @ eff @ X2 @ Steps ) ) ) ) ).

% Saturated_def
thf(fact_326_set__subset__Cons,axiom,
    ! [Xs: list_list_fm,X: list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ ( cons_list_fm @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_327_set__subset__Cons,axiom,
    ! [Xs: list_tm,X: tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ ( cons_tm @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_328_set__subset__Cons,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_329_set__subset__Cons,axiom,
    ! [Xs: list_fm,X: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_330_sset__sdrop,axiom,
    ! [N: nat,S: stream4408948924543953275ist_fm] : ( ord_le1771420097867575541ist_fm @ ( sset_P5379989128061332361ist_fm @ ( sdrop_9176333610110415838ist_fm @ N @ S ) ) @ ( sset_P5379989128061332361ist_fm @ S ) ) ).

% sset_sdrop
thf(fact_331_sset__sdrop,axiom,
    ! [N: nat,S: stream4385846686851721995e_rule] : ( ord_le5312242768264209797e_rule @ ( sset_P1894962091411173529e_rule @ ( sdrop_9113879250048157294e_rule @ N @ S ) ) @ ( sset_P1894962091411173529e_rule @ S ) ) ).

% sset_sdrop
thf(fact_332_sset__sdrop,axiom,
    ! [N: nat,S: stream8953843411776101167m_rule] : ( ord_le1038668239487504297m_rule @ ( sset_P5794137493694147517m_rule @ ( sdrop_746751008794826386m_rule @ N @ S ) ) @ ( sset_P5794137493694147517m_rule @ S ) ) ).

% sset_sdrop
thf(fact_333_sset__sdrop,axiom,
    ! [N: nat,S: stream8099677779113257519e_rule] : ( ord_le184502606824660649e_rule @ ( sset_P6986346827493259965e_rule @ ( sdrop_1938960342593938834e_rule @ N @ S ) ) @ ( sset_P6986346827493259965e_rule @ S ) ) ).

% sset_sdrop
thf(fact_334_sset__sdrop,axiom,
    ! [N: nat,S: stream6210534828274662995m_rule] : ( ord_le8955734628938902733m_rule @ ( sset_P3768201806174902753m_rule @ ( sdrop_7192298464603511222m_rule @ N @ S ) ) @ ( sset_P3768201806174902753m_rule @ S ) ) ).

% sset_sdrop
thf(fact_335_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_336_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_337_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_338_sset__sdrop,axiom,
    ! [N: nat,S: stream_nat] : ( ord_less_eq_set_nat @ ( sset_nat @ ( sdrop_nat @ N @ S ) ) @ ( sset_nat @ S ) ) ).

% sset_sdrop
thf(fact_339_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_340_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc6018962875968178549ist_fm] :
      ~ ! [A5: list_tm,B5: list_fm] :
          ( Y2
         != ( produc1414352766439514085ist_fm @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_341_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc340336539035504054m_rule] :
      ~ ! [A5: produc6018962875968178549ist_fm,B5: rule] :
          ( Y2
         != ( produc1733806532565653680m_rule @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_342_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_nat_nat] :
      ~ ! [A5: nat,B5: nat] :
          ( Y2
         != ( product_Pair_nat_nat @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_343_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
          ( Y2
         != ( produc6161850002892822231at_nat @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_344_surj__pair,axiom,
    ! [P3: produc6018962875968178549ist_fm] :
    ? [X3: list_tm,Y4: list_fm] :
      ( P3
      = ( produc1414352766439514085ist_fm @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_345_surj__pair,axiom,
    ! [P3: produc340336539035504054m_rule] :
    ? [X3: produc6018962875968178549ist_fm,Y4: rule] :
      ( P3
      = ( produc1733806532565653680m_rule @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_346_surj__pair,axiom,
    ! [P3: product_prod_nat_nat] :
    ? [X3: nat,Y4: nat] :
      ( P3
      = ( product_Pair_nat_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_347_surj__pair,axiom,
    ! [P3: produc859450856879609959at_nat] :
    ? [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] :
      ( P3
      = ( produc6161850002892822231at_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_348_prod__cases,axiom,
    ! [P4: produc6018962875968178549ist_fm > $o,P3: produc6018962875968178549ist_fm] :
      ( ! [A5: list_tm,B5: list_fm] : ( P4 @ ( produc1414352766439514085ist_fm @ A5 @ B5 ) )
     => ( P4 @ P3 ) ) ).

% prod_cases
thf(fact_349_prod__cases,axiom,
    ! [P4: produc340336539035504054m_rule > $o,P3: produc340336539035504054m_rule] :
      ( ! [A5: produc6018962875968178549ist_fm,B5: rule] : ( P4 @ ( produc1733806532565653680m_rule @ A5 @ B5 ) )
     => ( P4 @ P3 ) ) ).

% prod_cases
thf(fact_350_prod__cases,axiom,
    ! [P4: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
      ( ! [A5: nat,B5: nat] : ( P4 @ ( product_Pair_nat_nat @ A5 @ B5 ) )
     => ( P4 @ P3 ) ) ).

% prod_cases
thf(fact_351_prod__cases,axiom,
    ! [P4: produc859450856879609959at_nat > $o,P3: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B5: product_prod_nat_nat] : ( P4 @ ( produc6161850002892822231at_nat @ A5 @ B5 ) )
     => ( P4 @ P3 ) ) ).

% prod_cases
thf(fact_352_Pair__inject,axiom,
    ! [A2: list_tm,B: list_fm,A4: list_tm,B3: list_fm] :
      ( ( ( produc1414352766439514085ist_fm @ A2 @ B )
        = ( produc1414352766439514085ist_fm @ A4 @ B3 ) )
     => ~ ( ( A2 = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_353_Pair__inject,axiom,
    ! [A2: produc6018962875968178549ist_fm,B: rule,A4: produc6018962875968178549ist_fm,B3: rule] :
      ( ( ( produc1733806532565653680m_rule @ A2 @ B )
        = ( produc1733806532565653680m_rule @ A4 @ B3 ) )
     => ~ ( ( A2 = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_354_Pair__inject,axiom,
    ! [A2: nat,B: nat,A4: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A2 @ B )
        = ( product_Pair_nat_nat @ A4 @ B3 ) )
     => ~ ( ( A2 = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_355_Pair__inject,axiom,
    ! [A2: product_prod_nat_nat,B: product_prod_nat_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A2 @ B )
        = ( produc6161850002892822231at_nat @ A4 @ B3 ) )
     => ~ ( ( A2 = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_356_prod__cases3,axiom,
    ! [Y2: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B5: nat,C3: nat] :
          ( Y2
         != ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C3 ) ) ) ).

% prod_cases3
thf(fact_357_prod__induct3,axiom,
    ! [P4: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B5: nat,C3: nat] : ( P4 @ ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C3 ) ) )
     => ( P4 @ X ) ) ).

% prod_induct3
thf(fact_358_insertE,axiom,
    ! [A2: rule,B: rule,A: set_rule] :
      ( ( member_rule3 @ A2 @ ( insert_rule2 @ B @ A ) )
     => ( ( A2 != B )
       => ( member_rule3 @ A2 @ A ) ) ) ).

% insertE
thf(fact_359_insertE,axiom,
    ! [A2: list_fm,B: list_fm,A: set_list_fm] :
      ( ( member_list_fm3 @ A2 @ ( insert_list_fm2 @ B @ A ) )
     => ( ( A2 != B )
       => ( member_list_fm3 @ A2 @ A ) ) ) ).

% insertE
thf(fact_360_insertE,axiom,
    ! [A2: tm,B: tm,A: set_tm] :
      ( ( member_tm3 @ A2 @ ( insert_tm2 @ B @ A ) )
     => ( ( A2 != B )
       => ( member_tm3 @ A2 @ A ) ) ) ).

% insertE
thf(fact_361_insertE,axiom,
    ! [A2: fm,B: fm,A: set_fm] :
      ( ( member_fm3 @ A2 @ ( insert_fm2 @ B @ A ) )
     => ( ( A2 != B )
       => ( member_fm3 @ A2 @ A ) ) ) ).

% insertE
thf(fact_362_insertE,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( member_nat3 @ A2 @ ( insert_nat2 @ B @ A ) )
     => ( ( A2 != B )
       => ( member_nat3 @ A2 @ A ) ) ) ).

% insertE
thf(fact_363_insertI1,axiom,
    ! [A2: rule,B2: set_rule] : ( member_rule3 @ A2 @ ( insert_rule2 @ A2 @ B2 ) ) ).

% insertI1
thf(fact_364_insertI1,axiom,
    ! [A2: list_fm,B2: set_list_fm] : ( member_list_fm3 @ A2 @ ( insert_list_fm2 @ A2 @ B2 ) ) ).

% insertI1
thf(fact_365_insertI1,axiom,
    ! [A2: tm,B2: set_tm] : ( member_tm3 @ A2 @ ( insert_tm2 @ A2 @ B2 ) ) ).

% insertI1
thf(fact_366_insertI1,axiom,
    ! [A2: fm,B2: set_fm] : ( member_fm3 @ A2 @ ( insert_fm2 @ A2 @ B2 ) ) ).

% insertI1
thf(fact_367_insertI1,axiom,
    ! [A2: nat,B2: set_nat] : ( member_nat3 @ A2 @ ( insert_nat2 @ A2 @ B2 ) ) ).

% insertI1
thf(fact_368_insertI2,axiom,
    ! [A2: rule,B2: set_rule,B: rule] :
      ( ( member_rule3 @ A2 @ B2 )
     => ( member_rule3 @ A2 @ ( insert_rule2 @ B @ B2 ) ) ) ).

% insertI2
thf(fact_369_insertI2,axiom,
    ! [A2: list_fm,B2: set_list_fm,B: list_fm] :
      ( ( member_list_fm3 @ A2 @ B2 )
     => ( member_list_fm3 @ A2 @ ( insert_list_fm2 @ B @ B2 ) ) ) ).

% insertI2
thf(fact_370_insertI2,axiom,
    ! [A2: tm,B2: set_tm,B: tm] :
      ( ( member_tm3 @ A2 @ B2 )
     => ( member_tm3 @ A2 @ ( insert_tm2 @ B @ B2 ) ) ) ).

% insertI2
thf(fact_371_insertI2,axiom,
    ! [A2: fm,B2: set_fm,B: fm] :
      ( ( member_fm3 @ A2 @ B2 )
     => ( member_fm3 @ A2 @ ( insert_fm2 @ B @ B2 ) ) ) ).

% insertI2
thf(fact_372_insertI2,axiom,
    ! [A2: nat,B2: set_nat,B: nat] :
      ( ( member_nat3 @ A2 @ B2 )
     => ( member_nat3 @ A2 @ ( insert_nat2 @ B @ B2 ) ) ) ).

% insertI2
thf(fact_373_Set_Oset__insert,axiom,
    ! [X: rule,A: set_rule] :
      ( ( member_rule3 @ X @ A )
     => ~ ! [B6: set_rule] :
            ( ( A
              = ( insert_rule2 @ X @ B6 ) )
           => ( member_rule3 @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_374_Set_Oset__insert,axiom,
    ! [X: list_fm,A: set_list_fm] :
      ( ( member_list_fm3 @ X @ A )
     => ~ ! [B6: set_list_fm] :
            ( ( A
              = ( insert_list_fm2 @ X @ B6 ) )
           => ( member_list_fm3 @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_375_Set_Oset__insert,axiom,
    ! [X: tm,A: set_tm] :
      ( ( member_tm3 @ X @ A )
     => ~ ! [B6: set_tm] :
            ( ( A
              = ( insert_tm2 @ X @ B6 ) )
           => ( member_tm3 @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_376_Set_Oset__insert,axiom,
    ! [X: fm,A: set_fm] :
      ( ( member_fm3 @ X @ A )
     => ~ ! [B6: set_fm] :
            ( ( A
              = ( insert_fm2 @ X @ B6 ) )
           => ( member_fm3 @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_377_Set_Oset__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( ( member_nat3 @ X @ A )
     => ~ ! [B6: set_nat] :
            ( ( A
              = ( insert_nat2 @ X @ B6 ) )
           => ( member_nat3 @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_378_insert__ident,axiom,
    ! [X: rule,A: set_rule,B2: set_rule] :
      ( ~ ( member_rule3 @ X @ A )
     => ( ~ ( member_rule3 @ X @ B2 )
       => ( ( ( insert_rule2 @ X @ A )
            = ( insert_rule2 @ X @ B2 ) )
          = ( A = B2 ) ) ) ) ).

% insert_ident
thf(fact_379_insert__ident,axiom,
    ! [X: list_fm,A: set_list_fm,B2: set_list_fm] :
      ( ~ ( member_list_fm3 @ X @ A )
     => ( ~ ( member_list_fm3 @ X @ B2 )
       => ( ( ( insert_list_fm2 @ X @ A )
            = ( insert_list_fm2 @ X @ B2 ) )
          = ( A = B2 ) ) ) ) ).

% insert_ident
thf(fact_380_insert__ident,axiom,
    ! [X: tm,A: set_tm,B2: set_tm] :
      ( ~ ( member_tm3 @ X @ A )
     => ( ~ ( member_tm3 @ X @ B2 )
       => ( ( ( insert_tm2 @ X @ A )
            = ( insert_tm2 @ X @ B2 ) )
          = ( A = B2 ) ) ) ) ).

% insert_ident
thf(fact_381_insert__ident,axiom,
    ! [X: fm,A: set_fm,B2: set_fm] :
      ( ~ ( member_fm3 @ X @ A )
     => ( ~ ( member_fm3 @ X @ B2 )
       => ( ( ( insert_fm2 @ X @ A )
            = ( insert_fm2 @ X @ B2 ) )
          = ( A = B2 ) ) ) ) ).

% insert_ident
thf(fact_382_insert__ident,axiom,
    ! [X: nat,A: set_nat,B2: set_nat] :
      ( ~ ( member_nat3 @ X @ A )
     => ( ~ ( member_nat3 @ X @ B2 )
       => ( ( ( insert_nat2 @ X @ A )
            = ( insert_nat2 @ X @ B2 ) )
          = ( A = B2 ) ) ) ) ).

% insert_ident
thf(fact_383_insert__absorb,axiom,
    ! [A2: rule,A: set_rule] :
      ( ( member_rule3 @ A2 @ A )
     => ( ( insert_rule2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_384_insert__absorb,axiom,
    ! [A2: list_fm,A: set_list_fm] :
      ( ( member_list_fm3 @ A2 @ A )
     => ( ( insert_list_fm2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_385_insert__absorb,axiom,
    ! [A2: tm,A: set_tm] :
      ( ( member_tm3 @ A2 @ A )
     => ( ( insert_tm2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_386_insert__absorb,axiom,
    ! [A2: fm,A: set_fm] :
      ( ( member_fm3 @ A2 @ A )
     => ( ( insert_fm2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_387_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat3 @ A2 @ A )
     => ( ( insert_nat2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_388_insert__eq__iff,axiom,
    ! [A2: rule,A: set_rule,B: rule,B2: set_rule] :
      ( ~ ( member_rule3 @ A2 @ A )
     => ( ~ ( member_rule3 @ B @ B2 )
       => ( ( ( insert_rule2 @ A2 @ A )
            = ( insert_rule2 @ B @ B2 ) )
          = ( ( ( A2 = B )
             => ( A = B2 ) )
            & ( ( A2 != B )
             => ? [C4: set_rule] :
                  ( ( A
                    = ( insert_rule2 @ B @ C4 ) )
                  & ~ ( member_rule3 @ B @ C4 )
                  & ( B2
                    = ( insert_rule2 @ A2 @ C4 ) )
                  & ~ ( member_rule3 @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_389_insert__eq__iff,axiom,
    ! [A2: list_fm,A: set_list_fm,B: list_fm,B2: set_list_fm] :
      ( ~ ( member_list_fm3 @ A2 @ A )
     => ( ~ ( member_list_fm3 @ B @ B2 )
       => ( ( ( insert_list_fm2 @ A2 @ A )
            = ( insert_list_fm2 @ B @ B2 ) )
          = ( ( ( A2 = B )
             => ( A = B2 ) )
            & ( ( A2 != B )
             => ? [C4: set_list_fm] :
                  ( ( A
                    = ( insert_list_fm2 @ B @ C4 ) )
                  & ~ ( member_list_fm3 @ B @ C4 )
                  & ( B2
                    = ( insert_list_fm2 @ A2 @ C4 ) )
                  & ~ ( member_list_fm3 @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_390_insert__eq__iff,axiom,
    ! [A2: tm,A: set_tm,B: tm,B2: set_tm] :
      ( ~ ( member_tm3 @ A2 @ A )
     => ( ~ ( member_tm3 @ B @ B2 )
       => ( ( ( insert_tm2 @ A2 @ A )
            = ( insert_tm2 @ B @ B2 ) )
          = ( ( ( A2 = B )
             => ( A = B2 ) )
            & ( ( A2 != B )
             => ? [C4: set_tm] :
                  ( ( A
                    = ( insert_tm2 @ B @ C4 ) )
                  & ~ ( member_tm3 @ B @ C4 )
                  & ( B2
                    = ( insert_tm2 @ A2 @ C4 ) )
                  & ~ ( member_tm3 @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_391_insert__eq__iff,axiom,
    ! [A2: fm,A: set_fm,B: fm,B2: set_fm] :
      ( ~ ( member_fm3 @ A2 @ A )
     => ( ~ ( member_fm3 @ B @ B2 )
       => ( ( ( insert_fm2 @ A2 @ A )
            = ( insert_fm2 @ B @ B2 ) )
          = ( ( ( A2 = B )
             => ( A = B2 ) )
            & ( ( A2 != B )
             => ? [C4: set_fm] :
                  ( ( A
                    = ( insert_fm2 @ B @ C4 ) )
                  & ~ ( member_fm3 @ B @ C4 )
                  & ( B2
                    = ( insert_fm2 @ A2 @ C4 ) )
                  & ~ ( member_fm3 @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_392_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
      ( ~ ( member_nat3 @ A2 @ A )
     => ( ~ ( member_nat3 @ B @ B2 )
       => ( ( ( insert_nat2 @ A2 @ A )
            = ( insert_nat2 @ B @ B2 ) )
          = ( ( ( A2 = B )
             => ( A = B2 ) )
            & ( ( A2 != B )
             => ? [C4: set_nat] :
                  ( ( A
                    = ( insert_nat2 @ B @ C4 ) )
                  & ~ ( member_nat3 @ B @ C4 )
                  & ( B2
                    = ( insert_nat2 @ A2 @ C4 ) )
                  & ~ ( member_nat3 @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_393_insert__commute,axiom,
    ! [X: rule,Y2: rule,A: set_rule] :
      ( ( insert_rule2 @ X @ ( insert_rule2 @ Y2 @ A ) )
      = ( insert_rule2 @ Y2 @ ( insert_rule2 @ X @ A ) ) ) ).

% insert_commute
thf(fact_394_insert__commute,axiom,
    ! [X: fm,Y2: fm,A: set_fm] :
      ( ( insert_fm2 @ X @ ( insert_fm2 @ Y2 @ A ) )
      = ( insert_fm2 @ Y2 @ ( insert_fm2 @ X @ A ) ) ) ).

% insert_commute
thf(fact_395_insert__commute,axiom,
    ! [X: nat,Y2: nat,A: set_nat] :
      ( ( insert_nat2 @ X @ ( insert_nat2 @ Y2 @ A ) )
      = ( insert_nat2 @ Y2 @ ( insert_nat2 @ X @ A ) ) ) ).

% insert_commute
thf(fact_396_mk__disjoint__insert,axiom,
    ! [A2: rule,A: set_rule] :
      ( ( member_rule3 @ A2 @ A )
     => ? [B6: set_rule] :
          ( ( A
            = ( insert_rule2 @ A2 @ B6 ) )
          & ~ ( member_rule3 @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_397_mk__disjoint__insert,axiom,
    ! [A2: list_fm,A: set_list_fm] :
      ( ( member_list_fm3 @ A2 @ A )
     => ? [B6: set_list_fm] :
          ( ( A
            = ( insert_list_fm2 @ A2 @ B6 ) )
          & ~ ( member_list_fm3 @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_398_mk__disjoint__insert,axiom,
    ! [A2: tm,A: set_tm] :
      ( ( member_tm3 @ A2 @ A )
     => ? [B6: set_tm] :
          ( ( A
            = ( insert_tm2 @ A2 @ B6 ) )
          & ~ ( member_tm3 @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_399_mk__disjoint__insert,axiom,
    ! [A2: fm,A: set_fm] :
      ( ( member_fm3 @ A2 @ A )
     => ? [B6: set_fm] :
          ( ( A
            = ( insert_fm2 @ A2 @ B6 ) )
          & ~ ( member_fm3 @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_400_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat3 @ A2 @ A )
     => ? [B6: set_nat] :
          ( ( A
            = ( insert_nat2 @ A2 @ B6 ) )
          & ~ ( member_nat3 @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_401_order__refl,axiom,
    ! [X: set_tm] : ( ord_less_eq_set_tm @ X @ X ) ).

% order_refl
thf(fact_402_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_403_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_404_order__refl,axiom,
    ! [X: set_fm] : ( ord_less_eq_set_fm @ X @ X ) ).

% order_refl
thf(fact_405_order__refl,axiom,
    ! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ X ) ).

% order_refl
thf(fact_406_dual__order_Orefl,axiom,
    ! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_407_dual__order_Orefl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_408_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_409_dual__order_Orefl,axiom,
    ! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_410_dual__order_Orefl,axiom,
    ! [A2: extended_enat] : ( ord_le2932123472753598470d_enat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_411_insert__subsetI,axiom,
    ! [X: rule,A: set_rule,X4: set_rule] :
      ( ( member_rule3 @ X @ A )
     => ( ( ord_less_eq_set_rule @ X4 @ A )
       => ( ord_less_eq_set_rule @ ( insert_rule2 @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_412_insert__subsetI,axiom,
    ! [X: list_fm,A: set_list_fm,X4: set_list_fm] :
      ( ( member_list_fm3 @ X @ A )
     => ( ( ord_le7838213414353715577ist_fm @ X4 @ A )
       => ( ord_le7838213414353715577ist_fm @ ( insert_list_fm2 @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_413_insert__subsetI,axiom,
    ! [X: tm,A: set_tm,X4: set_tm] :
      ( ( member_tm3 @ X @ A )
     => ( ( ord_less_eq_set_tm @ X4 @ A )
       => ( ord_less_eq_set_tm @ ( insert_tm2 @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_414_insert__subsetI,axiom,
    ! [X: nat,A: set_nat,X4: set_nat] :
      ( ( member_nat3 @ X @ A )
     => ( ( ord_less_eq_set_nat @ X4 @ A )
       => ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_415_insert__subsetI,axiom,
    ! [X: fm,A: set_fm,X4: set_fm] :
      ( ( member_fm3 @ X @ A )
     => ( ( ord_less_eq_set_fm @ X4 @ A )
       => ( ord_less_eq_set_fm @ ( insert_fm2 @ X @ X4 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_416_subtermTm__le,axiom,
    ! [T: tm,S: tm] :
      ( ( member_tm3 @ T @ ( set_tm2 @ ( subtermTm @ S ) ) )
     => ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T ) ) @ ( set_tm2 @ ( subtermTm @ S ) ) ) ) ).

% subtermTm_le
thf(fact_417_relChain__def,axiom,
    ( bNF_Ca7039531205256338511set_tm
    = ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > set_tm] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_set_tm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_418_relChain__def,axiom,
    ( bNF_Ca8535812813189322792set_tm
    = ( ^ [R2: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > set_tm] :
        ! [I: product_prod_nat_nat,J: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_set_tm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_419_relChain__def,axiom,
    ( bNF_Ca4389690986151823996et_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > set_nat] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_set_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_420_relChain__def,axiom,
    ( bNF_Ca100218009143674979et_nat
    = ( ^ [R2: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > set_nat] :
        ! [I: product_prod_nat_nat,J: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_set_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_421_relChain__def,axiom,
    ( bNF_Ca968750328013420230at_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > nat] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_422_relChain__def,axiom,
    ( bNF_Ca8308629720386654381at_nat
    = ( ^ [R2: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > nat] :
        ! [I: product_prod_nat_nat,J: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_423_relChain__def,axiom,
    ( bNF_Ca7035579180170703325set_fm
    = ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > set_fm] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_set_fm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_424_relChain__def,axiom,
    ( bNF_Ca8531860788103687606set_fm
    = ( ^ [R2: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > set_fm] :
        ! [I: product_prod_nat_nat,J: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_set_fm @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_425_relChain__def,axiom,
    ( bNF_Ca1090507538293155056d_enat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > extended_enat] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
         => ( ord_le2932123472753598470d_enat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_426_relChain__def,axiom,
    ( bNF_Ca3721535472040450505d_enat
    = ( ^ [R2: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > extended_enat] :
        ! [I: product_prod_nat_nat,J: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R2 )
         => ( ord_le2932123472753598470d_enat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_427_subterm__Pre__refl,axiom,
    ! [Ts: list_tm,N: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ ( pre @ N @ Ts ) ) ) ) ).

% subterm_Pre_refl
thf(fact_428_fun__arguments__subterm,axiom,
    ! [N: nat,Ts: list_tm,P3: fm] :
      ( ( member_tm3 @ ( fun @ N @ Ts ) @ ( set_tm2 @ ( subtermFm @ P3 ) ) )
     => ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P3 ) ) ) ) ).

% fun_arguments_subterm
thf(fact_429_i_OSaturated__def,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra6097777249025082867ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ Steps )
      = ( ! [X2: rule] :
            ( ( member_rule3 @ X2 @ ( sset_rule @ Rules2 ) )
           => ( abstra2533313685312581075ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ X2 @ Steps ) ) ) ) ).

% i.Saturated_def
thf(fact_430_tm_Oinject_I1_J,axiom,
    ! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
      ( ( ( fun @ X11 @ X12 )
        = ( fun @ Y11 @ Y12 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 ) ) ) ).

% tm.inject(1)
thf(fact_431_fm_Oinject_I1_J,axiom,
    ! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
      ( ( ( pre @ X11 @ X12 )
        = ( pre @ Y11 @ Y12 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 ) ) ) ).

% fm.inject(1)
thf(fact_432_fm_Odistinct_I11_J,axiom,
    ! [X11: nat,X12: list_tm,X7: fm] :
      ( ( pre @ X11 @ X12 )
     != ( neg @ X7 ) ) ).

% fm.distinct(11)
thf(fact_433_fm_Odistinct_I9_J,axiom,
    ! [X11: nat,X12: list_tm,X6: fm] :
      ( ( pre @ X11 @ X12 )
     != ( uni @ X6 ) ) ).

% fm.distinct(9)
thf(fact_434_preds__shape,axiom,
    ! [Pre: fm,P3: fm] :
      ( ( member_fm3 @ Pre @ ( preds @ P3 ) )
     => ? [N2: nat,Ts2: list_tm] :
          ( Pre
          = ( pre @ N2 @ Ts2 ) ) ) ).

% preds_shape
thf(fact_435_subterm__Fun__refl,axiom,
    ! [Ts: list_tm,N: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermTm @ ( fun @ N @ Ts ) ) ) ) ).

% subterm_Fun_refl
thf(fact_436_subtermTm__refl,axiom,
    ! [T: tm] : ( member_tm3 @ T @ ( set_tm2 @ ( subtermTm @ T ) ) ) ).

% subtermTm_refl
thf(fact_437_order__antisym__conv,axiom,
    ! [Y2: set_tm,X: set_tm] :
      ( ( ord_less_eq_set_tm @ Y2 @ X )
     => ( ( ord_less_eq_set_tm @ X @ Y2 )
        = ( X = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_438_order__antisym__conv,axiom,
    ! [Y2: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y2 @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y2 )
        = ( X = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_439_order__antisym__conv,axiom,
    ! [Y2: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X )
     => ( ( ord_less_eq_nat @ X @ Y2 )
        = ( X = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_440_order__antisym__conv,axiom,
    ! [Y2: set_fm,X: set_fm] :
      ( ( ord_less_eq_set_fm @ Y2 @ X )
     => ( ( ord_less_eq_set_fm @ X @ Y2 )
        = ( X = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_441_order__antisym__conv,axiom,
    ! [Y2: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y2 @ X )
     => ( ( ord_le2932123472753598470d_enat @ X @ Y2 )
        = ( X = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_442_linorder__le__cases,axiom,
    ! [X: nat,Y2: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y2 )
     => ( ord_less_eq_nat @ Y2 @ X ) ) ).

% linorder_le_cases
thf(fact_443_linorder__le__cases,axiom,
    ! [X: extended_enat,Y2: extended_enat] :
      ( ~ ( ord_le2932123472753598470d_enat @ X @ Y2 )
     => ( ord_le2932123472753598470d_enat @ Y2 @ X ) ) ).

% linorder_le_cases
thf(fact_444_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_445_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_446_ord__le__eq__subst,axiom,
    ! [A2: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_447_ord__le__eq__subst,axiom,
    ! [A2: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_448_ord__le__eq__subst,axiom,
    ! [A2: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_449_ord__le__eq__subst,axiom,
    ! [A2: set_tm,B: set_tm,F: set_tm > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_tm @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_450_ord__le__eq__subst,axiom,
    ! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_451_ord__le__eq__subst,axiom,
    ! [A2: set_nat,B: set_nat,F: set_nat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_452_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: nat > set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_453_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_454_ord__eq__le__subst,axiom,
    ! [A2: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_455_ord__eq__le__subst,axiom,
    ! [A2: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_456_ord__eq__le__subst,axiom,
    ! [A2: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_457_ord__eq__le__subst,axiom,
    ! [A2: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_458_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X3: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_459_ord__eq__le__subst,axiom,
    ! [A2: extended_enat,F: set_tm > extended_enat,B: set_tm,C: set_tm] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X3: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_460_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X3: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_461_ord__eq__le__subst,axiom,
    ! [A2: extended_enat,F: set_nat > extended_enat,B: set_nat,C: set_nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X3: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_462_ord__eq__le__subst,axiom,
    ! [A2: set_tm,F: nat > set_tm,B: nat,C: nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_463_ord__eq__le__subst,axiom,
    ! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_464_linorder__linear,axiom,
    ! [X: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X @ Y2 )
      | ( ord_less_eq_nat @ Y2 @ X ) ) ).

% linorder_linear
thf(fact_465_linorder__linear,axiom,
    ! [X: extended_enat,Y2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y2 )
      | ( ord_le2932123472753598470d_enat @ Y2 @ X ) ) ).

% linorder_linear
thf(fact_466_order__eq__refl,axiom,
    ! [X: set_tm,Y2: set_tm] :
      ( ( X = Y2 )
     => ( ord_less_eq_set_tm @ X @ Y2 ) ) ).

% order_eq_refl
thf(fact_467_order__eq__refl,axiom,
    ! [X: set_nat,Y2: set_nat] :
      ( ( X = Y2 )
     => ( ord_less_eq_set_nat @ X @ Y2 ) ) ).

% order_eq_refl
thf(fact_468_order__eq__refl,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X = Y2 )
     => ( ord_less_eq_nat @ X @ Y2 ) ) ).

% order_eq_refl
thf(fact_469_order__eq__refl,axiom,
    ! [X: set_fm,Y2: set_fm] :
      ( ( X = Y2 )
     => ( ord_less_eq_set_fm @ X @ Y2 ) ) ).

% order_eq_refl
thf(fact_470_order__eq__refl,axiom,
    ! [X: extended_enat,Y2: extended_enat] :
      ( ( X = Y2 )
     => ( ord_le2932123472753598470d_enat @ X @ Y2 ) ) ).

% order_eq_refl
thf(fact_471_order__subst2,axiom,
    ! [A2: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_472_order__subst2,axiom,
    ! [A2: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_473_order__subst2,axiom,
    ! [A2: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_474_order__subst2,axiom,
    ! [A2: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_475_order__subst2,axiom,
    ! [A2: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_476_order__subst2,axiom,
    ! [A2: set_tm,B: set_tm,F: set_tm > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_tm @ A2 @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_477_order__subst2,axiom,
    ! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_478_order__subst2,axiom,
    ! [A2: set_nat,B: set_nat,F: set_nat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X3: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_479_order__subst2,axiom,
    ! [A2: nat,B: nat,F: nat > set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_480_order__subst2,axiom,
    ! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_481_order__subst1,axiom,
    ! [A2: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_482_order__subst1,axiom,
    ! [A2: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_483_order__subst1,axiom,
    ! [A2: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_484_order__subst1,axiom,
    ! [A2: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_le2932123472753598470d_enat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_485_order__subst1,axiom,
    ! [A2: set_tm,F: nat > set_tm,B: nat,C: nat] :
      ( ( ord_less_eq_set_tm @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_486_order__subst1,axiom,
    ! [A2: set_tm,F: extended_enat > set_tm,B: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_tm @ A2 @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_tm @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_487_order__subst1,axiom,
    ! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_488_order__subst1,axiom,
    ! [A2: set_nat,F: extended_enat > set_nat,B: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ! [X3: extended_enat,Y4: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_489_order__subst1,axiom,
    ! [A2: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ! [X3: set_tm,Y4: set_tm] :
              ( ( ord_less_eq_set_tm @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_490_order__subst1,axiom,
    ! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ! [X3: set_nat,Y4: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_491_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_tm,Z4: set_tm] : ( Y3 = Z4 ) )
    = ( ^ [A6: set_tm,B7: set_tm] :
          ( ( ord_less_eq_set_tm @ A6 @ B7 )
          & ( ord_less_eq_set_tm @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_492_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_nat,Z4: set_nat] : ( Y3 = Z4 ) )
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_493_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z4: nat] : ( Y3 = Z4 ) )
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A6 @ B7 )
          & ( ord_less_eq_nat @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_494_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_fm,Z4: set_fm] : ( Y3 = Z4 ) )
    = ( ^ [A6: set_fm,B7: set_fm] :
          ( ( ord_less_eq_set_fm @ A6 @ B7 )
          & ( ord_less_eq_set_fm @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_495_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: extended_enat,Z4: extended_enat] : ( Y3 = Z4 ) )
    = ( ^ [A6: extended_enat,B7: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A6 @ B7 )
          & ( ord_le2932123472753598470d_enat @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_496_antisym,axiom,
    ! [A2: set_tm,B: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B )
     => ( ( ord_less_eq_set_tm @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_497_antisym,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ord_less_eq_set_nat @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_498_antisym,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_499_antisym,axiom,
    ! [A2: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B )
     => ( ( ord_less_eq_set_fm @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_500_antisym,axiom,
    ! [A2: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B )
     => ( ( ord_le2932123472753598470d_enat @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_501_dual__order_Otrans,axiom,
    ! [B: set_tm,A2: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ B @ A2 )
     => ( ( ord_less_eq_set_tm @ C @ B )
       => ( ord_less_eq_set_tm @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_502_dual__order_Otrans,axiom,
    ! [B: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A2 )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_503_dual__order_Otrans,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_504_dual__order_Otrans,axiom,
    ! [B: set_fm,A2: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A2 )
     => ( ( ord_less_eq_set_fm @ C @ B )
       => ( ord_less_eq_set_fm @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_505_dual__order_Otrans,axiom,
    ! [B: extended_enat,A2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ C @ B )
       => ( ord_le2932123472753598470d_enat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_506_dual__order_Oantisym,axiom,
    ! [B: set_tm,A2: set_tm] :
      ( ( ord_less_eq_set_tm @ B @ A2 )
     => ( ( ord_less_eq_set_tm @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_507_dual__order_Oantisym,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_508_dual__order_Oantisym,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_509_dual__order_Oantisym,axiom,
    ! [B: set_fm,A2: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A2 )
     => ( ( ord_less_eq_set_fm @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_510_dual__order_Oantisym,axiom,
    ! [B: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_511_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_tm,Z4: set_tm] : ( Y3 = Z4 ) )
    = ( ^ [A6: set_tm,B7: set_tm] :
          ( ( ord_less_eq_set_tm @ B7 @ A6 )
          & ( ord_less_eq_set_tm @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_512_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_nat,Z4: set_nat] : ( Y3 = Z4 ) )
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A6 )
          & ( ord_less_eq_set_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_513_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: nat,Z4: nat] : ( Y3 = Z4 ) )
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_514_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_fm,Z4: set_fm] : ( Y3 = Z4 ) )
    = ( ^ [A6: set_fm,B7: set_fm] :
          ( ( ord_less_eq_set_fm @ B7 @ A6 )
          & ( ord_less_eq_set_fm @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_515_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: extended_enat,Z4: extended_enat] : ( Y3 = Z4 ) )
    = ( ^ [A6: extended_enat,B7: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B7 @ A6 )
          & ( ord_le2932123472753598470d_enat @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_516_linorder__wlog,axiom,
    ! [P4: nat > nat > $o,A2: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
         => ( P4 @ A5 @ B5 ) )
     => ( ! [A5: nat,B5: nat] :
            ( ( P4 @ B5 @ A5 )
           => ( P4 @ A5 @ B5 ) )
       => ( P4 @ A2 @ B ) ) ) ).

% linorder_wlog
thf(fact_517_linorder__wlog,axiom,
    ! [P4: extended_enat > extended_enat > $o,A2: extended_enat,B: extended_enat] :
      ( ! [A5: extended_enat,B5: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A5 @ B5 )
         => ( P4 @ A5 @ B5 ) )
     => ( ! [A5: extended_enat,B5: extended_enat] :
            ( ( P4 @ B5 @ A5 )
           => ( P4 @ A5 @ B5 ) )
       => ( P4 @ A2 @ B ) ) ) ).

% linorder_wlog
thf(fact_518_order__trans,axiom,
    ! [X: set_tm,Y2: set_tm,Z3: set_tm] :
      ( ( ord_less_eq_set_tm @ X @ Y2 )
     => ( ( ord_less_eq_set_tm @ Y2 @ Z3 )
       => ( ord_less_eq_set_tm @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_519_order__trans,axiom,
    ! [X: set_nat,Y2: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y2 )
     => ( ( ord_less_eq_set_nat @ Y2 @ Z3 )
       => ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_520_order__trans,axiom,
    ! [X: nat,Y2: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ Z3 )
       => ( ord_less_eq_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_521_order__trans,axiom,
    ! [X: set_fm,Y2: set_fm,Z3: set_fm] :
      ( ( ord_less_eq_set_fm @ X @ Y2 )
     => ( ( ord_less_eq_set_fm @ Y2 @ Z3 )
       => ( ord_less_eq_set_fm @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_522_order__trans,axiom,
    ! [X: extended_enat,Y2: extended_enat,Z3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y2 )
     => ( ( ord_le2932123472753598470d_enat @ Y2 @ Z3 )
       => ( ord_le2932123472753598470d_enat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_523_order_Otrans,axiom,
    ! [A2: set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ord_less_eq_set_tm @ A2 @ C ) ) ) ).

% order.trans
thf(fact_524_order_Otrans,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_525_order_Otrans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_526_order_Otrans,axiom,
    ! [A2: set_fm,B: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).

% order.trans
thf(fact_527_order_Otrans,axiom,
    ! [A2: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_528_order__antisym,axiom,
    ! [X: set_tm,Y2: set_tm] :
      ( ( ord_less_eq_set_tm @ X @ Y2 )
     => ( ( ord_less_eq_set_tm @ Y2 @ X )
       => ( X = Y2 ) ) ) ).

% order_antisym
thf(fact_529_order__antisym,axiom,
    ! [X: set_nat,Y2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y2 )
     => ( ( ord_less_eq_set_nat @ Y2 @ X )
       => ( X = Y2 ) ) ) ).

% order_antisym
thf(fact_530_order__antisym,axiom,
    ! [X: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ X )
       => ( X = Y2 ) ) ) ).

% order_antisym
thf(fact_531_order__antisym,axiom,
    ! [X: set_fm,Y2: set_fm] :
      ( ( ord_less_eq_set_fm @ X @ Y2 )
     => ( ( ord_less_eq_set_fm @ Y2 @ X )
       => ( X = Y2 ) ) ) ).

% order_antisym
thf(fact_532_order__antisym,axiom,
    ! [X: extended_enat,Y2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y2 )
     => ( ( ord_le2932123472753598470d_enat @ Y2 @ X )
       => ( X = Y2 ) ) ) ).

% order_antisym
thf(fact_533_ord__le__eq__trans,axiom,
    ! [A2: set_tm,B: set_tm,C: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_tm @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_534_ord__le__eq__trans,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_535_ord__le__eq__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_536_ord__le__eq__trans,axiom,
    ! [A2: set_fm,B: set_fm,C: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_537_ord__le__eq__trans,axiom,
    ! [A2: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B )
     => ( ( B = C )
       => ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_538_ord__eq__le__trans,axiom,
    ! [A2: set_tm,B: set_tm,C: set_tm] :
      ( ( A2 = B )
     => ( ( ord_less_eq_set_tm @ B @ C )
       => ( ord_less_eq_set_tm @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_539_ord__eq__le__trans,axiom,
    ! [A2: set_nat,B: set_nat,C: set_nat] :
      ( ( A2 = B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_540_ord__eq__le__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( A2 = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_541_ord__eq__le__trans,axiom,
    ! [A2: set_fm,B: set_fm,C: set_fm] :
      ( ( A2 = B )
     => ( ( ord_less_eq_set_fm @ B @ C )
       => ( ord_less_eq_set_fm @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_542_ord__eq__le__trans,axiom,
    ! [A2: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A2 = B )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ord_le2932123472753598470d_enat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_543_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_tm,Z4: set_tm] : ( Y3 = Z4 ) )
    = ( ^ [X2: set_tm,Y: set_tm] :
          ( ( ord_less_eq_set_tm @ X2 @ Y )
          & ( ord_less_eq_set_tm @ Y @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_544_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_nat,Z4: set_nat] : ( Y3 = Z4 ) )
    = ( ^ [X2: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y )
          & ( ord_less_eq_set_nat @ Y @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_545_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z4: nat] : ( Y3 = Z4 ) )
    = ( ^ [X2: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y )
          & ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_546_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_fm,Z4: set_fm] : ( Y3 = Z4 ) )
    = ( ^ [X2: set_fm,Y: set_fm] :
          ( ( ord_less_eq_set_fm @ X2 @ Y )
          & ( ord_less_eq_set_fm @ Y @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_547_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: extended_enat,Z4: extended_enat] : ( Y3 = Z4 ) )
    = ( ^ [X2: extended_enat,Y: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X2 @ Y )
          & ( ord_le2932123472753598470d_enat @ Y @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_548_le__cases3,axiom,
    ! [X: nat,Y2: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y2 )
       => ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y2 @ X )
         => ~ ( ord_less_eq_nat @ X @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
             => ~ ( ord_less_eq_nat @ Y2 @ X ) )
           => ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_549_le__cases3,axiom,
    ! [X: extended_enat,Y2: extended_enat,Z3: extended_enat] :
      ( ( ( ord_le2932123472753598470d_enat @ X @ Y2 )
       => ~ ( ord_le2932123472753598470d_enat @ Y2 @ Z3 ) )
     => ( ( ( ord_le2932123472753598470d_enat @ Y2 @ X )
         => ~ ( ord_le2932123472753598470d_enat @ X @ Z3 ) )
       => ( ( ( ord_le2932123472753598470d_enat @ X @ Z3 )
           => ~ ( ord_le2932123472753598470d_enat @ Z3 @ Y2 ) )
         => ( ( ( ord_le2932123472753598470d_enat @ Z3 @ Y2 )
             => ~ ( ord_le2932123472753598470d_enat @ Y2 @ X ) )
           => ( ( ( ord_le2932123472753598470d_enat @ Y2 @ Z3 )
               => ~ ( ord_le2932123472753598470d_enat @ Z3 @ X ) )
             => ~ ( ( ord_le2932123472753598470d_enat @ Z3 @ X )
                 => ~ ( ord_le2932123472753598470d_enat @ X @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_550_nle__le,axiom,
    ! [A2: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
      = ( ( ord_less_eq_nat @ B @ A2 )
        & ( B != A2 ) ) ) ).

% nle_le
thf(fact_551_nle__le,axiom,
    ! [A2: extended_enat,B: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ A2 @ B ) )
      = ( ( ord_le2932123472753598470d_enat @ B @ A2 )
        & ( B != A2 ) ) ) ).

% nle_le
thf(fact_552_ssubst__Pair__rhs,axiom,
    ! [R: list_tm,S: list_fm,R3: set_Pr5202636777678657877ist_fm,S4: list_fm] :
      ( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ R @ S ) @ R3 )
     => ( ( S4 = S )
       => ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ R @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_553_ssubst__Pair__rhs,axiom,
    ! [R: produc6018962875968178549ist_fm,S: rule,R3: set_Pr1822751329126368876m_rule,S4: rule] :
      ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ R @ S ) @ R3 )
     => ( ( S4 = S )
       => ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ R @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_554_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: nat,R3: set_Pr1261947904930325089at_nat,S4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R3 )
     => ( ( S4 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_555_ssubst__Pair__rhs,axiom,
    ! [R: product_prod_nat_nat,S: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S4: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S ) @ R3 )
     => ( ( S4 = S )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_556_saturated__effG__uu__def,axiom,
    ( abstra8695313511658867272ist_fm
    = ( ^ [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra2533313685312581075ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) ) ) ) ).

% saturated_effG_uu_def
thf(fact_557_Saturated__effG__uu__uu__def,axiom,
    ( abstra1395361944017196648ist_fm
    = ( ^ [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra6097777249025082867ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) ) ) ) ).

% Saturated_effG_uu_uu_def
thf(fact_558_paramst__subtermTm_I1_J,axiom,
    ! [T: tm,X5: nat] :
      ( ( member_nat3 @ X5 @ ( paramst @ T ) )
     => ? [L: list_tm] : ( member_tm3 @ ( fun @ X5 @ L ) @ ( set_tm2 @ ( subtermTm @ T ) ) ) ) ).

% paramst_subtermTm(1)
thf(fact_559_params__subtermFm,axiom,
    ! [P3: fm,X5: nat] :
      ( ( member_nat3 @ X5 @ ( params @ P3 ) )
     => ? [L: list_tm] : ( member_tm3 @ ( fun @ X5 @ L ) @ ( set_tm2 @ ( subtermFm @ P3 ) ) ) ) ).

% params_subtermFm
thf(fact_560_pos__least,axiom,
    ! [N: nat,Rs: stream4408948924543953275ist_fm,R: produc6018962875968178549ist_fm] :
      ( ( ( shd_Pr3211216682057661985ist_fm @ ( sdrop_9176333610110415838ist_fm @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra2497064141961437305ist_fm @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_561_pos__least,axiom,
    ! [N: nat,Rs: stream4385846686851721995e_rule,R: produc7694839378271647877e_rule] :
      ( ( ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra7798328242894269833e_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_562_pos__least,axiom,
    ! [N: nat,Rs: stream8953843411776101167m_rule,R: produc7920154866009513897m_rule] :
      ( ( ( shd_Pr1208753253167450709m_rule @ ( sdrop_746751008794826386m_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra2305546423536801965m_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_563_pos__least,axiom,
    ! [N: nat,Rs: stream8099677779113257519e_rule,R: produc9112364199808626345e_rule] :
      ( ( ( shd_Pr2400962586966563157e_rule @ ( sdrop_1938960342593938834e_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra3497755757335914413e_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_564_pos__least,axiom,
    ! [N: nat,Rs: stream6210534828274662995m_rule,R: produc8828831911945107917m_rule] :
      ( ( ( shd_Pr4461660664618831993m_rule @ ( sdrop_7192298464603511222m_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra1175832030460196561m_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_565_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_566_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_567_sset__fenum,axiom,
    ( ( sset_rule @ ( abstra745658567949189203m_rule @ rules ) )
    = ( sset_rule @ rules ) ) ).

% sset_fenum
thf(fact_568_paramsts__subset,axiom,
    ! [A: list_tm,B2: list_tm] :
      ( ( ord_less_eq_set_tm @ ( set_tm2 @ A ) @ ( set_tm2 @ B2 ) )
     => ( ord_less_eq_set_nat @ ( paramsts @ A ) @ ( paramsts @ B2 ) ) ) ).

% paramsts_subset
thf(fact_569_enabled__R,axiom,
    ! [S: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ? [X3: rule] :
          ( ( member_rule3 @ X3 @ ( sset_rule @ rules ) )
          & ? [X_1: fset_P8989946509869081563ist_fm] : ( eff @ X3 @ S @ X_1 ) ) ) ).

% enabled_R
thf(fact_570_Greatest__equality,axiom,
    ! [P4: set_tm > $o,X: set_tm] :
      ( ( P4 @ X )
     => ( ! [Y4: set_tm] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_set_tm @ Y4 @ X ) )
       => ( ( order_4248741476211579294set_tm @ P4 )
          = X ) ) ) ).

% Greatest_equality
thf(fact_571_Greatest__equality,axiom,
    ! [P4: set_nat > $o,X: set_nat] :
      ( ( P4 @ X )
     => ( ! [Y4: set_nat] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_set_nat @ Y4 @ X ) )
       => ( ( order_5724808138429204845et_nat @ P4 )
          = X ) ) ) ).

% Greatest_equality
thf(fact_572_Greatest__equality,axiom,
    ! [P4: set_fm > $o,X: set_fm] :
      ( ( P4 @ X )
     => ( ! [Y4: set_fm] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_set_fm @ Y4 @ X ) )
       => ( ( order_4244789451125944108set_fm @ P4 )
          = X ) ) ) ).

% Greatest_equality
thf(fact_573_Greatest__equality,axiom,
    ! [P4: extended_enat > $o,X: extended_enat] :
      ( ( P4 @ X )
     => ( ! [Y4: extended_enat] :
            ( ( P4 @ Y4 )
           => ( ord_le2932123472753598470d_enat @ Y4 @ X ) )
       => ( ( order_2428742583041560895d_enat @ P4 )
          = X ) ) ) ).

% Greatest_equality
thf(fact_574_Greatest__equality,axiom,
    ! [P4: nat > $o,X: nat] :
      ( ( P4 @ X )
     => ( ! [Y4: nat] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ X ) )
       => ( ( order_Greatest_nat @ P4 )
          = X ) ) ) ).

% Greatest_equality
thf(fact_575_UNIV__I,axiom,
    ! [X: rule] : ( member_rule3 @ X @ top_top_set_rule ) ).

% UNIV_I
thf(fact_576_UNIV__I,axiom,
    ! [X: list_fm] : ( member_list_fm3 @ X @ top_top_set_list_fm ) ).

% UNIV_I
thf(fact_577_UNIV__I,axiom,
    ! [X: tm] : ( member_tm3 @ X @ top_top_set_tm ) ).

% UNIV_I
thf(fact_578_UNIV__I,axiom,
    ! [X: fm] : ( member_fm3 @ X @ top_top_set_fm ) ).

% UNIV_I
thf(fact_579_UNIV__I,axiom,
    ! [X: nat] : ( member_nat3 @ X @ top_top_set_nat ) ).

% UNIV_I
thf(fact_580_UNIV__I,axiom,
    ! [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ top_to1730629564055774885ist_fm ) ).

% UNIV_I
thf(fact_581_RuleSystem__Defs_Ofenum_Ocong,axiom,
    abstra745658567949189203m_rule = abstra745658567949189203m_rule ).

% RuleSystem_Defs.fenum.cong
thf(fact_582_UNIV__witness,axiom,
    ? [X3: rule] : ( member_rule3 @ X3 @ top_top_set_rule ) ).

% UNIV_witness
thf(fact_583_UNIV__witness,axiom,
    ? [X3: list_fm] : ( member_list_fm3 @ X3 @ top_top_set_list_fm ) ).

% UNIV_witness
thf(fact_584_UNIV__witness,axiom,
    ? [X3: tm] : ( member_tm3 @ X3 @ top_top_set_tm ) ).

% UNIV_witness
thf(fact_585_UNIV__witness,axiom,
    ? [X3: fm] : ( member_fm3 @ X3 @ top_top_set_fm ) ).

% UNIV_witness
thf(fact_586_UNIV__witness,axiom,
    ? [X3: nat] : ( member_nat3 @ X3 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_587_UNIV__witness,axiom,
    ? [X3: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X3 @ top_to1730629564055774885ist_fm ) ).

% UNIV_witness
thf(fact_588_UNIV__eq__I,axiom,
    ! [A: set_rule] :
      ( ! [X3: rule] : ( member_rule3 @ X3 @ A )
     => ( top_top_set_rule = A ) ) ).

% UNIV_eq_I
thf(fact_589_UNIV__eq__I,axiom,
    ! [A: set_list_fm] :
      ( ! [X3: list_fm] : ( member_list_fm3 @ X3 @ A )
     => ( top_top_set_list_fm = A ) ) ).

% UNIV_eq_I
thf(fact_590_UNIV__eq__I,axiom,
    ! [A: set_tm] :
      ( ! [X3: tm] : ( member_tm3 @ X3 @ A )
     => ( top_top_set_tm = A ) ) ).

% UNIV_eq_I
thf(fact_591_UNIV__eq__I,axiom,
    ! [A: set_fm] :
      ( ! [X3: fm] : ( member_fm3 @ X3 @ A )
     => ( top_top_set_fm = A ) ) ).

% UNIV_eq_I
thf(fact_592_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] : ( member_nat3 @ X3 @ A )
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_593_UNIV__eq__I,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] :
      ( ! [X3: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X3 @ A )
     => ( top_to1730629564055774885ist_fm = A ) ) ).

% UNIV_eq_I
thf(fact_594_params_Osimps_I1_J,axiom,
    ! [B: nat,Ts: list_tm] :
      ( ( params @ ( pre @ B @ Ts ) )
      = ( paramsts @ Ts ) ) ).

% params.simps(1)
thf(fact_595_top__greatest,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] : ( ord_le1771420097867575541ist_fm @ A2 @ top_to1730629564055774885ist_fm ) ).

% top_greatest
thf(fact_596_top__greatest,axiom,
    ! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ top_top_set_tm ) ).

% top_greatest
thf(fact_597_top__greatest,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% top_greatest
thf(fact_598_top__greatest,axiom,
    ! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ top_top_set_fm ) ).

% top_greatest
thf(fact_599_top__greatest,axiom,
    ! [A2: extended_enat] : ( ord_le2932123472753598470d_enat @ A2 @ top_to3028658606643905974d_enat ) ).

% top_greatest
thf(fact_600_top_Oextremum__unique,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ top_to1730629564055774885ist_fm @ A2 )
      = ( A2 = top_to1730629564055774885ist_fm ) ) ).

% top.extremum_unique
thf(fact_601_top_Oextremum__unique,axiom,
    ! [A2: set_tm] :
      ( ( ord_less_eq_set_tm @ top_top_set_tm @ A2 )
      = ( A2 = top_top_set_tm ) ) ).

% top.extremum_unique
thf(fact_602_top_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
      = ( A2 = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_603_top_Oextremum__unique,axiom,
    ! [A2: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A2 )
      = ( A2 = top_top_set_fm ) ) ).

% top.extremum_unique
thf(fact_604_top_Oextremum__unique,axiom,
    ! [A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ top_to3028658606643905974d_enat @ A2 )
      = ( A2 = top_to3028658606643905974d_enat ) ) ).

% top.extremum_unique
thf(fact_605_top_Oextremum__uniqueI,axiom,
    ! [A2: set_Pr5202636777678657877ist_fm] :
      ( ( ord_le1771420097867575541ist_fm @ top_to1730629564055774885ist_fm @ A2 )
     => ( A2 = top_to1730629564055774885ist_fm ) ) ).

% top.extremum_uniqueI
thf(fact_606_top_Oextremum__uniqueI,axiom,
    ! [A2: set_tm] :
      ( ( ord_less_eq_set_tm @ top_top_set_tm @ A2 )
     => ( A2 = top_top_set_tm ) ) ).

% top.extremum_uniqueI
thf(fact_607_top_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
     => ( A2 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_608_top_Oextremum__uniqueI,axiom,
    ! [A2: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A2 )
     => ( A2 = top_top_set_fm ) ) ).

% top.extremum_uniqueI
thf(fact_609_top_Oextremum__uniqueI,axiom,
    ! [A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ top_to3028658606643905974d_enat @ A2 )
     => ( A2 = top_to3028658606643905974d_enat ) ) ).

% top.extremum_uniqueI
thf(fact_610_subset__UNIV,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] : ( ord_le1771420097867575541ist_fm @ A @ top_to1730629564055774885ist_fm ) ).

% subset_UNIV
thf(fact_611_subset__UNIV,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ A @ top_top_set_tm ) ).

% subset_UNIV
thf(fact_612_subset__UNIV,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_613_subset__UNIV,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ top_top_set_fm ) ).

% subset_UNIV
thf(fact_614_insert__UNIV,axiom,
    ! [X: rule] :
      ( ( insert_rule2 @ X @ top_top_set_rule )
      = top_top_set_rule ) ).

% insert_UNIV
thf(fact_615_insert__UNIV,axiom,
    ! [X: fm] :
      ( ( insert_fm2 @ X @ top_top_set_fm )
      = top_top_set_fm ) ).

% insert_UNIV
thf(fact_616_insert__UNIV,axiom,
    ! [X: nat] :
      ( ( insert_nat2 @ X @ top_top_set_nat )
      = top_top_set_nat ) ).

% insert_UNIV
thf(fact_617_insert__UNIV,axiom,
    ! [X: produc6018962875968178549ist_fm] :
      ( ( insert4380798862274524997ist_fm @ X @ top_to1730629564055774885ist_fm )
      = top_to1730629564055774885ist_fm ) ).

% insert_UNIV
thf(fact_618_params_Osimps_I7_J,axiom,
    ! [P3: fm] :
      ( ( params @ ( neg @ P3 ) )
      = ( params @ P3 ) ) ).

% params.simps(7)
thf(fact_619_params_Osimps_I6_J,axiom,
    ! [P3: fm] :
      ( ( params @ ( uni @ P3 ) )
      = ( params @ P3 ) ) ).

% params.simps(6)
thf(fact_620_i_Osset__fenum,axiom,
    ! [Rules2: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ ( abstra4452268713091674743m_rule @ Rules2 ) )
      = ( sset_P4484857331586881186m_rule @ Rules2 ) ) ).

% i.sset_fenum
thf(fact_621_i_Osset__fenum,axiom,
    ! [Rules2: stream_rule] :
      ( ( sset_rule @ ( abstra745658567949189203m_rule @ Rules2 ) )
      = ( sset_rule @ Rules2 ) ) ).

% i.sset_fenum
thf(fact_622_RuleSystem__Defs_Osset__fenum,axiom,
    ! [Rules2: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ ( abstra4452268713091674743m_rule @ Rules2 ) )
      = ( sset_P4484857331586881186m_rule @ Rules2 ) ) ).

% RuleSystem_Defs.sset_fenum
thf(fact_623_RuleSystem__Defs_Osset__fenum,axiom,
    ! [Rules2: stream_rule] :
      ( ( sset_rule @ ( abstra745658567949189203m_rule @ Rules2 ) )
      = ( sset_rule @ Rules2 ) ) ).

% RuleSystem_Defs.sset_fenum
thf(fact_624_subtermFm__subset__params,axiom,
    ! [P3: fm,A: list_tm] :
      ( ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermFm @ P3 ) ) @ ( set_tm2 @ A ) )
     => ( ord_less_eq_set_nat @ ( params @ P3 ) @ ( paramsts @ A ) ) ) ).

% subtermFm_subset_params
thf(fact_625_GreatestI2__order,axiom,
    ! [P4: set_tm > $o,X: set_tm,Q2: set_tm > $o] :
      ( ( P4 @ X )
     => ( ! [Y4: set_tm] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_set_tm @ Y4 @ X ) )
       => ( ! [X3: set_tm] :
              ( ( P4 @ X3 )
             => ( ! [Y5: set_tm] :
                    ( ( P4 @ Y5 )
                   => ( ord_less_eq_set_tm @ Y5 @ X3 ) )
               => ( Q2 @ X3 ) ) )
         => ( Q2 @ ( order_4248741476211579294set_tm @ P4 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_626_GreatestI2__order,axiom,
    ! [P4: set_nat > $o,X: set_nat,Q2: set_nat > $o] :
      ( ( P4 @ X )
     => ( ! [Y4: set_nat] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_set_nat @ Y4 @ X ) )
       => ( ! [X3: set_nat] :
              ( ( P4 @ X3 )
             => ( ! [Y5: set_nat] :
                    ( ( P4 @ Y5 )
                   => ( ord_less_eq_set_nat @ Y5 @ X3 ) )
               => ( Q2 @ X3 ) ) )
         => ( Q2 @ ( order_5724808138429204845et_nat @ P4 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_627_GreatestI2__order,axiom,
    ! [P4: set_fm > $o,X: set_fm,Q2: set_fm > $o] :
      ( ( P4 @ X )
     => ( ! [Y4: set_fm] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_set_fm @ Y4 @ X ) )
       => ( ! [X3: set_fm] :
              ( ( P4 @ X3 )
             => ( ! [Y5: set_fm] :
                    ( ( P4 @ Y5 )
                   => ( ord_less_eq_set_fm @ Y5 @ X3 ) )
               => ( Q2 @ X3 ) ) )
         => ( Q2 @ ( order_4244789451125944108set_fm @ P4 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_628_GreatestI2__order,axiom,
    ! [P4: extended_enat > $o,X: extended_enat,Q2: extended_enat > $o] :
      ( ( P4 @ X )
     => ( ! [Y4: extended_enat] :
            ( ( P4 @ Y4 )
           => ( ord_le2932123472753598470d_enat @ Y4 @ X ) )
       => ( ! [X3: extended_enat] :
              ( ( P4 @ X3 )
             => ( ! [Y5: extended_enat] :
                    ( ( P4 @ Y5 )
                   => ( ord_le2932123472753598470d_enat @ Y5 @ X3 ) )
               => ( Q2 @ X3 ) ) )
         => ( Q2 @ ( order_2428742583041560895d_enat @ P4 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_629_GreatestI2__order,axiom,
    ! [P4: nat > $o,X: nat,Q2: nat > $o] :
      ( ( P4 @ X )
     => ( ! [Y4: nat] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ X ) )
       => ( ! [X3: nat] :
              ( ( P4 @ X3 )
             => ( ! [Y5: nat] :
                    ( ( P4 @ Y5 )
                   => ( ord_less_eq_nat @ Y5 @ X3 ) )
               => ( Q2 @ X3 ) ) )
         => ( Q2 @ ( order_Greatest_nat @ P4 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_630_fenum__uu__def,axiom,
    abstra1582897422107675196u_rule = abstra745658567949189203m_rule ).

% fenum_uu_def
thf(fact_631_iso__tuple__UNIV__I,axiom,
    ! [X: rule] : ( member_rule3 @ X @ top_top_set_rule ) ).

% iso_tuple_UNIV_I
thf(fact_632_iso__tuple__UNIV__I,axiom,
    ! [X: list_fm] : ( member_list_fm3 @ X @ top_top_set_list_fm ) ).

% iso_tuple_UNIV_I
thf(fact_633_iso__tuple__UNIV__I,axiom,
    ! [X: tm] : ( member_tm3 @ X @ top_top_set_tm ) ).

% iso_tuple_UNIV_I
thf(fact_634_iso__tuple__UNIV__I,axiom,
    ! [X: fm] : ( member_fm3 @ X @ top_top_set_fm ) ).

% iso_tuple_UNIV_I
thf(fact_635_iso__tuple__UNIV__I,axiom,
    ! [X: nat] : ( member_nat3 @ X @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_636_iso__tuple__UNIV__I,axiom,
    ! [X: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X @ top_to1730629564055774885ist_fm ) ).

% iso_tuple_UNIV_I
thf(fact_637_per,axiom,
    ! [R: rule] :
      ( ( member_rule3 @ R @ ( sset_rule @ rules ) )
     => ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R ) ) ).

% per
thf(fact_638_all__rules__persistent,axiom,
    ! [R4: rule] :
      ( ( member_rule3 @ R4 @ ( sset_rule @ rules ) )
     => ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R4 ) ) ).

% all_rules_persistent
thf(fact_639_s1_I1_J,axiom,
    ( new_term
    = ( ^ [C5: nat,T2: tm] :
          ~ ( member_nat3 @ C5 @ ( paramst @ T2 ) ) ) ) ).

% s1(1)
thf(fact_640_PersistentRuleSystem__axioms,axiom,
    abstra3967921542344661089ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).

% PersistentRuleSystem_axioms
thf(fact_641_p1,axiom,
    paramst2 = paramst ).

% p1
thf(fact_642_RuleSystem_Opos__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream4408948924543953275ist_fm,R: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( ( shd_Pr3211216682057661985ist_fm @ ( sdrop_9176333610110415838ist_fm @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra2497064141961437305ist_fm @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_643_RuleSystem_Opos__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream4385846686851721995e_rule,R: produc7694839378271647877e_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra7798328242894269833e_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_644_RuleSystem_Opos__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream8953843411776101167m_rule,R: produc7920154866009513897m_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( ( shd_Pr1208753253167450709m_rule @ ( sdrop_746751008794826386m_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra2305546423536801965m_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_645_RuleSystem_Opos__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream8099677779113257519e_rule,R: produc9112364199808626345e_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( ( shd_Pr2400962586966563157e_rule @ ( sdrop_1938960342593938834e_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra3497755757335914413e_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_646_RuleSystem_Opos__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream6210534828274662995m_rule,R: produc8828831911945107917m_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( ( shd_Pr4461660664618831993m_rule @ ( sdrop_7192298464603511222m_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra1175832030460196561m_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_647_RuleSystem_Opos__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_648_RuleSystem_Opos__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream_rule,R: rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_649_PersistentRuleSystem_Oaxioms_I1_J,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 ) ) ).

% PersistentRuleSystem.axioms(1)
thf(fact_650_RuleSystem_Oper_Ocong,axiom,
    abstra5255361903751151037ist_fm = abstra5255361903751151037ist_fm ).

% RuleSystem.per.cong
thf(fact_651_top__set__def,axiom,
    ( top_to1730629564055774885ist_fm
    = ( collec5845330005437105888ist_fm @ top_to4151594781713979000t_fm_o ) ) ).

% top_set_def
thf(fact_652_PersistentRuleSystem_Oper,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,R: rule] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ R @ ( sset_rule @ Rules2 ) )
       => ( abstra5255361903751151037ist_fm @ Eff4 @ Rules2 @ S5 @ R ) ) ) ).

% PersistentRuleSystem.per
thf(fact_653_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule,S: rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ? [X3: rule] :
            ( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_rule] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_654_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S5: set_tm,S: tm] :
      ( ( abstra4909020524820736232ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ? [X3: rule] :
            ( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_tm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_655_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm,S: fm] :
      ( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ? [X3: rule] :
            ( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_fm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_656_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S5: set_nat,S: nat] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ? [X3: rule] :
            ( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_nat] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_657_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > list_fm > fset_list_fm > $o,Rules2: stream_rule,S5: set_list_fm,S: list_fm] :
      ( ( abstra9153614075763688444ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ? [X3: rule] :
            ( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_list_fm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_658_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member4699826688122452638ist_fm @ S @ S5 )
       => ? [X3: rule] :
            ( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_1: fset_P8989946509869081563ist_fm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_659_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: produc340336539035504054m_rule > rule > fset_rule > $o,Rules2: stream2709947120125613254m_rule,S5: set_rule,S: rule] :
      ( ( abstra7792261008698710771e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ? [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Rules2 ) )
            & ? [X_1: fset_rule] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_660_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: produc340336539035504054m_rule > tm > fset_tm > $o,Rules2: stream2709947120125613254m_rule,S5: set_tm,S: tm] :
      ( ( abstra6746019703890151108ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ? [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Rules2 ) )
            & ? [X_1: fset_tm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_661_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: produc340336539035504054m_rule > fm > fset_fm > $o,Rules2: stream2709947120125613254m_rule,S5: set_fm,S: fm] :
      ( ( abstra6746019703889232722ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ? [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Rules2 ) )
            & ? [X_1: fset_fm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_662_RuleSystem_Oenabled__R,axiom,
    ! [Eff4: produc340336539035504054m_rule > nat > fset_nat > $o,Rules2: stream2709947120125613254m_rule,S5: set_nat,S: nat] :
      ( ( abstra2021474039004664625le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ? [X3: produc340336539035504054m_rule] :
            ( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Rules2 ) )
            & ? [X_1: fset_nat] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_663_RuleSystem__axioms,axiom,
    abstra5221733350967904376ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).

% RuleSystem_axioms
thf(fact_664_RuleSystem_Opos,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,Rs: stream_rule,R: rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
       => ( ( member_rule3 @ R @ ( sset_rule @ Rules2 ) )
         => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
            = R ) ) ) ) ).

% RuleSystem.pos
thf(fact_665_new__term_Osimps_I2_J,axiom,
    ! [C: nat,I2: nat,L2: list_tm] :
      ( ( new_term @ C @ ( fun @ I2 @ L2 ) )
      = ( ( I2 != C )
        & ( ( I2 != C )
         => ( new_list @ C @ L2 ) ) ) ) ).

% new_term.simps(2)
thf(fact_666_PersistentRuleSystem_Ointro,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( abstra9069655090277518462ist_fm @ Eff4 @ Rules2 @ S5 )
       => ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S5 ) ) ) ).

% PersistentRuleSystem.intro
thf(fact_667_PersistentRuleSystem__def,axiom,
    ( abstra3967921542344661089ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm] :
          ( ( abstra5221733350967904376ist_fm @ Eff @ Rules @ S6 )
          & ( abstra9069655090277518462ist_fm @ Eff @ Rules @ S6 ) ) ) ) ).

% PersistentRuleSystem_def
thf(fact_668_PersistentRuleSystem__axioms__def,axiom,
    ( abstra9069655090277518462ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,S6: set_Pr5202636777678657877ist_fm] :
        ! [R2: rule] :
          ( ( member_rule3 @ R2 @ ( sset_rule @ Rules ) )
         => ( abstra5255361903751151037ist_fm @ Eff @ Rules @ S6 @ R2 ) ) ) ) ).

% PersistentRuleSystem_axioms_def
thf(fact_669_PersistentRuleSystem__axioms_Ointro,axiom,
    ! [Rules2: stream_rule,Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,S5: set_Pr5202636777678657877ist_fm] :
      ( ! [R5: rule] :
          ( ( member_rule3 @ R5 @ ( sset_rule @ Rules2 ) )
         => ( abstra5255361903751151037ist_fm @ Eff4 @ Rules2 @ S5 @ R5 ) )
     => ( abstra9069655090277518462ist_fm @ Eff4 @ Rules2 @ S5 ) ) ).

% PersistentRuleSystem_axioms.intro
thf(fact_670_s1_I2_J,axiom,
    ( new_list
    = ( ^ [C5: nat,L3: list_tm] :
          ~ ( member_nat3 @ C5 @ ( paramsts @ L3 ) ) ) ) ).

% s1(2)
thf(fact_671_paramst__liftt_I2_J,axiom,
    ! [Ts: list_tm] :
      ( ( paramsts @ ( liftts @ Ts ) )
      = ( paramsts @ Ts ) ) ).

% paramst_liftt(2)
thf(fact_672_RuleSystem__Defs_Ofair_Ocong,axiom,
    abstra3799686578551160190r_rule = abstra3799686578551160190r_rule ).

% RuleSystem_Defs.fair.cong
thf(fact_673_fair__rules,axiom,
    abstra3799686578551160190r_rule @ rules @ rules ).

% fair_rules
thf(fact_674_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule,M2: nat] :
      ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
     => ( abstra3665372904625986210m_rule @ Rules2 @ ( sdrop_8169176516188972301m_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_675_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream4408948924543953275ist_fm,Rs: stream4408948924543953275ist_fm,M2: nat] :
      ( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs )
     => ( abstra7670258426472572809ist_fm @ Rules2 @ ( sdrop_9176333610110415838ist_fm @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_676_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream4385846686851721995e_rule,Rs: stream4385846686851721995e_rule,M2: nat] :
      ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
     => ( abstra2528290469512735897e_rule @ Rules2 @ ( sdrop_9113879250048157294e_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_677_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream8953843411776101167m_rule,Rs: stream8953843411776101167m_rule,M2: nat] :
      ( ( abstra4920968922616850365m_rule @ Rules2 @ Rs )
     => ( abstra4920968922616850365m_rule @ Rules2 @ ( sdrop_746751008794826386m_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_678_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream8099677779113257519e_rule,Rs: stream8099677779113257519e_rule,M2: nat] :
      ( ( abstra6113178256415962813e_rule @ Rules2 @ Rs )
     => ( abstra6113178256415962813e_rule @ Rules2 @ ( sdrop_1938960342593938834e_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_679_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream6210534828274662995m_rule,Rs: stream6210534828274662995m_rule,M2: nat] :
      ( ( abstra1378428917839123937m_rule @ Rules2 @ Rs )
     => ( abstra1378428917839123937m_rule @ Rules2 @ ( sdrop_7192298464603511222m_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_680_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream_rule,Rs: stream_rule,M2: nat] :
      ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
     => ( abstra3799686578551160190r_rule @ Rules2 @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_681_i_Osdrop__fair,axiom,
    ! [Rules2: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule,M2: nat] :
      ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
     => ( abstra3665372904625986210m_rule @ Rules2 @ ( sdrop_8169176516188972301m_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_682_i_Osdrop__fair,axiom,
    ! [Rules2: stream4408948924543953275ist_fm,Rs: stream4408948924543953275ist_fm,M2: nat] :
      ( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs )
     => ( abstra7670258426472572809ist_fm @ Rules2 @ ( sdrop_9176333610110415838ist_fm @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_683_i_Osdrop__fair,axiom,
    ! [Rules2: stream4385846686851721995e_rule,Rs: stream4385846686851721995e_rule,M2: nat] :
      ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
     => ( abstra2528290469512735897e_rule @ Rules2 @ ( sdrop_9113879250048157294e_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_684_i_Osdrop__fair,axiom,
    ! [Rules2: stream8953843411776101167m_rule,Rs: stream8953843411776101167m_rule,M2: nat] :
      ( ( abstra4920968922616850365m_rule @ Rules2 @ Rs )
     => ( abstra4920968922616850365m_rule @ Rules2 @ ( sdrop_746751008794826386m_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_685_i_Osdrop__fair,axiom,
    ! [Rules2: stream8099677779113257519e_rule,Rs: stream8099677779113257519e_rule,M2: nat] :
      ( ( abstra6113178256415962813e_rule @ Rules2 @ Rs )
     => ( abstra6113178256415962813e_rule @ Rules2 @ ( sdrop_1938960342593938834e_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_686_i_Osdrop__fair,axiom,
    ! [Rules2: stream6210534828274662995m_rule,Rs: stream6210534828274662995m_rule,M2: nat] :
      ( ( abstra1378428917839123937m_rule @ Rules2 @ Rs )
     => ( abstra1378428917839123937m_rule @ Rules2 @ ( sdrop_7192298464603511222m_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_687_i_Osdrop__fair,axiom,
    ! [Rules2: stream_rule,Rs: stream_rule,M2: nat] :
      ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
     => ( abstra3799686578551160190r_rule @ Rules2 @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_688_RuleSystem__Defs_Ofair__fenum,axiom,
    ! [Rules2: stream_rule] : ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra745658567949189203m_rule @ Rules2 ) ) ).

% RuleSystem_Defs.fair_fenum
thf(fact_689_i_Ofair__fenum,axiom,
    ! [Rules2: stream_rule] : ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra745658567949189203m_rule @ Rules2 ) ) ).

% i.fair_fenum
thf(fact_690_sdrop__fair,axiom,
    ! [Rs: stream_rule,M2: nat] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( abstra3799686578551160190r_rule @ rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% sdrop_fair
thf(fact_691_fair__fenum,axiom,
    abstra3799686578551160190r_rule @ rules @ ( abstra745658567949189203m_rule @ rules ) ).

% fair_fenum
thf(fact_692_PersistentRuleSystem_Oaxioms_I2_J,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( abstra9069655090277518462ist_fm @ Eff4 @ Rules2 @ S5 ) ) ).

% PersistentRuleSystem.axioms(2)
thf(fact_693_pos,axiom,
    ! [Rs: stream_rule,R: rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( ( member_rule3 @ R @ ( sset_rule @ rules ) )
       => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
          = R ) ) ) ).

% pos
thf(fact_694_new__list_Osimps_I2_J,axiom,
    ! [C: nat,T: tm,L2: list_tm] :
      ( ( new_list @ C @ ( cons_tm @ T @ L2 ) )
      = ( ( ( new_term @ C @ T )
         => ( new_list @ C @ L2 ) )
        & ( new_term @ C @ T ) ) ) ).

% new_list.simps(2)
thf(fact_695_fair__uu__def,axiom,
    abstra4598247580560492629u_rule = abstra3799686578551160190r_rule ).

% fair_uu_def
thf(fact_696_top__empty__eq,axiom,
    ( top_top_rule_o
    = ( ^ [X2: rule] : ( member_rule3 @ X2 @ top_top_set_rule ) ) ) ).

% top_empty_eq
thf(fact_697_top__empty__eq,axiom,
    ( top_top_list_fm_o
    = ( ^ [X2: list_fm] : ( member_list_fm3 @ X2 @ top_top_set_list_fm ) ) ) ).

% top_empty_eq
thf(fact_698_top__empty__eq,axiom,
    ( top_top_tm_o
    = ( ^ [X2: tm] : ( member_tm3 @ X2 @ top_top_set_tm ) ) ) ).

% top_empty_eq
thf(fact_699_top__empty__eq,axiom,
    ( top_top_fm_o
    = ( ^ [X2: fm] : ( member_fm3 @ X2 @ top_top_set_fm ) ) ) ).

% top_empty_eq
thf(fact_700_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X2: nat] : ( member_nat3 @ X2 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_701_top__empty__eq,axiom,
    ( top_to4151594781713979000t_fm_o
    = ( ^ [X2: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X2 @ top_to1730629564055774885ist_fm ) ) ) ).

% top_empty_eq
thf(fact_702_s4_I2_J,axiom,
    inc_list = liftts ).

% s4(2)
thf(fact_703_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra7234149737463204544e_rule @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_704_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S5: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra8423890812398148919ule_tm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_705_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra8423890812397230533ule_fm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_706_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S5: set_nat,S: nat,Rs: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra8624044281687814142le_nat @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_707_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: rule > list_fm > fset_list_fm > $o,Rules2: stream_rule,S5: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra5712636991756272843ist_fm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_708_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: produc7694839378271647877e_rule > rule > fset_rule > $o,Rules2: stream4385846686851721995e_rule,S5: set_rule,S: rule,Rs: stream4385846686851721995e_rule] :
      ( ( abstra2905114889036010542e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra6715931297792236797e_rule @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_709_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: produc7694839378271647877e_rule > tm > fset_tm > $o,Rules2: stream4385846686851721995e_rule,S5: set_tm,S: tm,Rs: stream4385846686851721995e_rule] :
      ( ( abstra1261657321278996809ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra5903841454926380346ule_tm @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_710_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: produc7694839378271647877e_rule > fm > fset_fm > $o,Rules2: stream4385846686851721995e_rule,S5: set_fm,S: fm,Rs: stream4385846686851721995e_rule] :
      ( ( abstra1261657321278078423ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra5903841454925461960ule_fm @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_711_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: produc7694839378271647877e_rule > nat > fset_nat > $o,Rules2: stream4385846686851721995e_rule,S5: set_nat,S: nat,Rs: stream4385846686851721995e_rule] :
      ( ( abstra5123640173530818412le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra7127153494424084283le_nat @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_712_RuleSystem_OminWait__ex,axiom,
    ! [Eff4: produc7694839378271647877e_rule > list_fm > fset_list_fm > $o,Rules2: stream4385846686851721995e_rule,S5: set_list_fm,S: list_fm,Rs: stream4385846686851721995e_rule] :
      ( ( abstra3101592633059391453ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra6098267074846246734ist_fm @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_713_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: tm > rule > fset_rule > $o,Rules2: stream_tm,S5: set_rule,S: rule,Rs: stream_tm] :
      ( ( abstra7099963851359259790m_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra6979500642109889529air_tm @ Rules2 @ Rs )
         => ( member_tm3 @ ( shd_tm @ ( abstra7005108030455562926m_rule @ Eff4 @ Rs @ S ) ) @ ( sset_tm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_714_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: fm > rule > fset_rule > $o,Rules2: stream_fm,S5: set_rule,S: rule,Rs: stream_fm] :
      ( ( abstra7659285206637183772m_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra6979500642108971143air_fm @ Rules2 @ Rs )
         => ( member_fm3 @ ( shd_fm @ ( abstra7564429385733486908m_rule @ Eff4 @ Rs @ S ) ) @ ( sset_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_715_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: nat > rule > fset_rule > $o,Rules2: stream_nat,S5: set_rule,S: rule,Rs: stream_nat] :
      ( ( abstra7777025902706499341t_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs )
         => ( member_nat3 @ ( shd_nat @ ( abstra3609009080554578157t_rule @ Eff4 @ Rs @ S ) ) @ ( sset_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_716_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: tm > tm > fset_tm > $o,Rules2: stream_tm,S5: set_tm,S: tm,Rs: stream_tm] :
      ( ( abstra5526243810835382505_tm_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra6979500642109889529air_tm @ Rules2 @ Rs )
         => ( member_tm3 @ ( shd_tm @ ( abstra3994774178628530889_tm_tm @ Eff4 @ Rs @ S ) ) @ ( sset_tm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_717_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: fm > tm > fset_tm > $o,Rules2: stream_fm,S5: set_tm,S: tm,Rs: stream_fm] :
      ( ( abstra966159493039759323_fm_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra6979500642108971143air_fm @ Rules2 @ Rs )
         => ( member_fm3 @ ( shd_fm @ ( abstra8658061897687683515_fm_tm @ Eff4 @ Rs @ S ) ) @ ( sset_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_718_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: nat > tm > fset_tm > $o,Rules2: stream_nat,S5: set_tm,S: tm,Rs: stream_nat] :
      ( ( abstra733049861595049770nat_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs )
         => ( member_nat3 @ ( shd_nat @ ( abstra1615446718767695178nat_tm @ Eff4 @ Rs @ S ) ) @ ( sset_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_719_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: tm > fm > fset_fm > $o,Rules2: stream_tm,S5: set_fm,S: fm,Rs: stream_tm] :
      ( ( abstra5526243810834464119_tm_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra6979500642109889529air_tm @ Rules2 @ Rs )
         => ( member_tm3 @ ( shd_tm @ ( abstra3994774178627612503_tm_fm @ Eff4 @ Rs @ S ) ) @ ( sset_tm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_720_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: fm > fm > fset_fm > $o,Rules2: stream_fm,S5: set_fm,S: fm,Rs: stream_fm] :
      ( ( abstra966159493038840937_fm_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra6979500642108971143air_fm @ Rules2 @ Rs )
         => ( member_fm3 @ ( shd_fm @ ( abstra8658061897686765129_fm_fm @ Eff4 @ Rs @ S ) ) @ ( sset_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_721_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: nat > fm > fset_fm > $o,Rules2: stream_nat,S5: set_fm,S: fm,Rs: stream_nat] :
      ( ( abstra733049861594131384nat_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs )
         => ( member_nat3 @ ( shd_nat @ ( abstra1615446718766776792nat_fm @ Eff4 @ Rs @ S ) ) @ ( sset_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_722_RuleSystem_Otrim__in__R,axiom,
    ! [Eff4: tm > nat > fset_nat > $o,Rules2: stream_tm,S5: set_nat,S: nat,Rs: stream_tm] :
      ( ( abstra8150451472847306700tm_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra6979500642109889529air_tm @ Rules2 @ Rs )
         => ( member_tm3 @ ( shd_tm @ ( abstra9032848330019952108tm_nat @ Eff4 @ Rs @ S ) ) @ ( sset_tm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_723_liftt_Osimps_I2_J,axiom,
    ! [A2: nat,Ts: list_tm] :
      ( ( liftt @ ( fun @ A2 @ Ts ) )
      = ( fun @ A2 @ ( liftts @ Ts ) ) ) ).

% liftt.simps(2)
thf(fact_724_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_725_trim__in__R,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( member_rule3 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ ( sset_rule @ rules ) ) ) ) ).

% trim_in_R
thf(fact_726_paramst__liftt_I1_J,axiom,
    ! [T: tm] :
      ( ( paramst @ ( liftt @ T ) )
      = ( paramst @ T ) ) ).

% paramst_liftt(1)
thf(fact_727_RuleSystem__Defs_Otrim_Ocong,axiom,
    abstra7806236140005899928ist_fm = abstra7806236140005899928ist_fm ).

% RuleSystem_Defs.trim.cong
thf(fact_728_RuleSystem__Defs_Oenabled__def,axiom,
    ( abstra1707737187183579335ist_fm
    = ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,R2: rule,S3: produc6018962875968178549ist_fm] :
        ? [X8: fset_P8989946509869081563ist_fm] : ( Eff @ R2 @ S3 @ X8 ) ) ) ).

% RuleSystem_Defs.enabled_def
thf(fact_729_RuleSystem__Defs_Oenabled_Ocong,axiom,
    abstra1707737187183579335ist_fm = abstra1707737187183579335ist_fm ).

% RuleSystem_Defs.enabled.cong
thf(fact_730_enabled__def,axiom,
    ! [R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
      = ( ? [X8: fset_P8989946509869081563ist_fm] : ( eff @ R @ S @ X8 ) ) ) ).

% enabled_def
thf(fact_731_i_Oenabled__def,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ R @ S )
      = ( ? [X8: fset_P8989946509869081563ist_fm] : ( abstra2682625350522704545ist_fm @ Eff2 @ R @ S @ X8 ) ) ) ).

% i.enabled_def
thf(fact_732_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_733_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra7234149737463204544e_rule @ Eff4 @ ( shd_rule @ ( abstra2680930950814381487e_rule @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_734_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S5: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra8423890812398148919ule_tm @ Eff4 @ ( shd_rule @ ( abstra4814164703917039368ule_tm @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_735_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra8423890812397230533ule_fm @ Eff4 @ ( shd_rule @ ( abstra4814164703916120982ule_fm @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_736_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S5: set_nat,S: nat,Rs: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra8624044281687814142le_nat @ Eff4 @ ( shd_rule @ ( abstra8319304012324366829le_nat @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_737_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > list_fm > fset_list_fm > $o,Rules2: stream_rule,S5: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra5712636991756272843ist_fm @ Eff4 @ ( shd_rule @ ( abstra593219384675117596ist_fm @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_738_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member4699826688122452638ist_fm @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra1707737187183579335ist_fm @ Eff4 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_739_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: produc340336539035504054m_rule > rule > fset_rule > $o,Rules2: stream2709947120125613254m_rule,S5: set_rule,S: rule,Rs: stream2709947120125613254m_rule] :
      ( ( abstra7792261008698710771e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
         => ( abstra1957041411669721828e_rule @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra6343601422196104403e_rule @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_740_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: produc340336539035504054m_rule > tm > fset_tm > $o,Rules2: stream2709947120125613254m_rule,S5: set_tm,S: tm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6746019703890151108ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
         => ( abstra1468671940392962579ule_tm @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra1639899932327861476ule_tm @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_741_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: produc340336539035504054m_rule > fm > fset_fm > $o,Rules2: stream2709947120125613254m_rule,S5: set_fm,S: fm,Rs: stream2709947120125613254m_rule] :
      ( ( abstra6746019703889232722ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
         => ( abstra1468671940392044193ule_fm @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra1639899932326943090ule_fm @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_742_RuleSystem_Otrim__enabled,axiom,
    ! [Eff4: produc340336539035504054m_rule > nat > fset_nat > $o,Rules2: stream2709947120125613254m_rule,S5: set_nat,S: nat,Rs: stream2709947120125613254m_rule] :
      ( ( abstra2021474039004664625le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
         => ( abstra4513569833231518754le_nat @ Eff4 @ ( shd_Pr4562317740776619530m_rule @ ( abstra7926412722278118673le_nat @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_743_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra2680930950814381487e_rule @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_744_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > list_fm > fset_list_fm > $o,Rules2: stream_rule,S5: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra593219384675117596ist_fm @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_745_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S5: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra4814164703917039368ule_tm @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_746_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra4814164703916120982ule_fm @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_747_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S5: set_nat,S: nat,Rs: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra8319304012324366829le_nat @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_748_RuleSystem_Otrim__fair,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member4699826688122452638ist_fm @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_749_subrelI,axiom,
    ! [R: set_Pr5202636777678657877ist_fm,S: set_Pr5202636777678657877ist_fm] :
      ( ! [X3: list_tm,Y4: list_fm] :
          ( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y4 ) @ R )
         => ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y4 ) @ S ) )
     => ( ord_le1771420097867575541ist_fm @ R @ S ) ) ).

% subrelI
thf(fact_750_subrelI,axiom,
    ! [R: set_Pr1822751329126368876m_rule,S: set_Pr1822751329126368876m_rule] :
      ( ! [X3: produc6018962875968178549ist_fm,Y4: rule] :
          ( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y4 ) @ R )
         => ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ X3 @ Y4 ) @ S ) )
     => ( ord_le6390412330253371084m_rule @ R @ S ) ) ).

% subrelI
thf(fact_751_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% subrelI
thf(fact_752_subrelI,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% subrelI
thf(fact_753_liftts_Osimps_I2_J,axiom,
    ! [T: tm,Ts: list_tm] :
      ( ( liftts @ ( cons_tm @ T @ Ts ) )
      = ( cons_tm @ ( liftt @ T ) @ ( liftts @ Ts ) ) ) ).

% liftts.simps(2)
thf(fact_754_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_755_RuleSystem_OminWait__le__pos,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,Rs: stream_rule,R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
       => ( ( member_rule3 @ R @ ( sset_rule @ Rules2 ) )
         => ( ( abstra1707737187183579335ist_fm @ Eff4 @ R @ S )
           => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ Eff4 @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) ) ) ) ) ) ).

% RuleSystem.minWait_le_pos
thf(fact_756_minWait__le__pos,axiom,
    ! [Rs: stream_rule,R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( ( member_rule3 @ 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_757_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_758_RuleSystem_OminWait__least,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( abstra1707737187183579335ist_fm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S )
       => ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ Eff4 @ Rs @ S ) @ N ) ) ) ).

% RuleSystem.minWait_least
thf(fact_759_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_760_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra2680930950814381487e_rule @ Eff4 @ Rs @ S )
            = ( sdrop_rule @ ( abstra538998172947062484e_rule @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_761_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S5: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra4814164703917039368ule_tm @ Eff4 @ Rs @ S )
            = ( sdrop_rule @ ( abstra3486989597109074467ule_tm @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_762_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra4814164703916120982ule_fm @ Eff4 @ Rs @ S )
            = ( sdrop_rule @ ( abstra3486989597108156081ule_fm @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_763_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S5: set_nat,S: nat,Rs: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra8319304012324366829le_nat @ Eff4 @ Rs @ S )
            = ( sdrop_rule @ ( abstra6265646195202037778le_nat @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_764_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: rule > list_fm > fset_list_fm > $o,Rules2: stream_rule,S5: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra593219384675117596ist_fm @ Eff4 @ Rs @ S )
            = ( sdrop_rule @ ( abstra101201851149073335ist_fm @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_765_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: produc7694839378271647877e_rule > rule > fset_rule > $o,Rules2: stream4385846686851721995e_rule,S5: set_rule,S: rule,Rs: stream4385846686851721995e_rule] :
      ( ( abstra2905114889036010542e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ( ( abstra6190269242410416718e_rule @ Eff4 @ Rs @ S )
            = ( sdrop_9113879250048157294e_rule @ ( abstra6973468861704762345e_rule @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_766_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: produc7694839378271647877e_rule > tm > fset_tm > $o,Rules2: stream4385846686851721995e_rule,S5: set_tm,S: tm,Rs: stream4385846686851721995e_rule] :
      ( ( abstra1261657321278996809ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ( ( abstra4260711665088906025ule_tm @ Eff4 @ Rs @ S )
            = ( sdrop_9113879250048157294e_rule @ ( abstra7063818707277690190ule_tm @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_767_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: produc7694839378271647877e_rule > fm > fset_fm > $o,Rules2: stream4385846686851721995e_rule,S5: set_fm,S: fm,Rs: stream4385846686851721995e_rule] :
      ( ( abstra1261657321278078423ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ( ( abstra4260711665087987639ule_fm @ Eff4 @ Rs @ S )
            = ( sdrop_9113879250048157294e_rule @ ( abstra7063818707276771804ule_fm @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_768_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: produc7694839378271647877e_rule > nat > fset_nat > $o,Rules2: stream4385846686851721995e_rule,S5: set_nat,S: nat,Rs: stream4385846686851721995e_rule] :
      ( ( abstra5123640173530818412le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ( ( abstra8638744061939379596le_nat @ Eff4 @ Rs @ S )
            = ( sdrop_9113879250048157294e_rule @ ( abstra8209949696089825831le_nat @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_769_RuleSystem_Otrim__alt,axiom,
    ! [Eff4: produc7694839378271647877e_rule > list_fm > fset_list_fm > $o,Rules2: stream4385846686851721995e_rule,S5: set_list_fm,S: list_fm,Rs: stream4385846686851721995e_rule] :
      ( ( abstra3101592633059391453ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ( ( abstra4302619736147003837ist_fm @ Eff4 @ Rs @ S )
            = ( sdrop_9113879250048157294e_rule @ ( abstra6387871064559822178ist_fm @ Eff4 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_770_inc__list_Osimps_I2_J,axiom,
    ! [T: tm,L2: list_tm] :
      ( ( inc_list @ ( cons_tm @ T @ L2 ) )
      = ( cons_tm @ ( inc_term @ T ) @ ( inc_list @ L2 ) ) ) ).

% inc_list.simps(2)
thf(fact_771_s4_I1_J,axiom,
    inc_term = liftt ).

% s4(1)
thf(fact_772_RuleSystem_OminWait_Ocong,axiom,
    abstra1963046427795717555ist_fm = abstra1963046427795717555ist_fm ).

% RuleSystem.minWait.cong
thf(fact_773_inc__term_Osimps_I2_J,axiom,
    ! [I2: nat,L2: list_tm] :
      ( ( inc_term @ ( fun @ I2 @ L2 ) )
      = ( fun @ I2 @ ( inc_list @ L2 ) ) ) ).

% inc_term.simps(2)
thf(fact_774_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_rule3 @ R @ ( sset_rule @ rules ) )
           => ( abstra2533313685312581075ist_fm @ eff @ R @ Steps ) ) ) ) ) ).

% ipath_mkTree_saturated
thf(fact_775_eff__S,axiom,
    ! [S: produc6018962875968178549ist_fm,R: rule,Sl: fset_P8989946509869081563ist_fm,S4: produc6018962875968178549ist_fm] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( member_rule3 @ R @ ( sset_rule @ rules ) )
       => ( ( eff @ R @ S @ Sl )
         => ( ( fmembe3754813877001230652ist_fm @ S4 @ Sl )
           => ( member4699826688122452638ist_fm @ S4 @ top_to1730629564055774885ist_fm ) ) ) ) ) ).

% eff_S
thf(fact_776_ipath__mkTree__sdrop,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule,M2: nat] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) @ Steps )
         => ? [N2: nat,S7: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S7 @ top_to1730629564055774885ist_fm )
              & ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ eff @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_8169176516188972301m_rule @ M2 @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_sdrop
thf(fact_777_new_Osimps_I1_J,axiom,
    ! [C: nat,I2: nat,L2: list_tm] :
      ( ( new @ C @ ( pre @ I2 @ L2 ) )
      = ( new_list @ C @ L2 ) ) ).

% new.simps(1)
thf(fact_778_listFunTm__paramst_I1_J,axiom,
    ! [T: tm] :
      ( ( set_nat2 @ ( listFunTm @ T ) )
      = ( paramst @ T ) ) ).

% listFunTm_paramst(1)
thf(fact_779_s2,axiom,
    ( new
    = ( ^ [C5: nat,P: fm] :
          ~ ( member_nat3 @ C5 @ ( params @ P ) ) ) ) ).

% s2
thf(fact_780_RuleSystem__Defs_OmkTree_Ocong,axiom,
    abstra7840189589880272446ist_fm = abstra7840189589880272446ist_fm ).

% RuleSystem_Defs.mkTree.cong
thf(fact_781_new_Osimps_I7_J,axiom,
    ! [C: nat,P3: fm] :
      ( ( new @ C @ ( neg @ P3 ) )
      = ( new @ C @ P3 ) ) ).

% new.simps(7)
thf(fact_782_new_Osimps_I6_J,axiom,
    ! [C: nat,P3: fm] :
      ( ( new @ C @ ( uni @ P3 ) )
      = ( new @ C @ P3 ) ) ).

% new.simps(6)
thf(fact_783_RuleSystem_Oeff__S,axiom,
    ! [Eff4: tm > rule > fset_rule > $o,Rules2: stream_tm,S5: set_rule,S: rule,R: tm,Sl: fset_rule,S4: rule] :
      ( ( abstra7099963851359259790m_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( member_tm3 @ R @ ( sset_tm @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_rule @ S4 @ Sl )
             => ( member_rule3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_784_RuleSystem_Oeff__S,axiom,
    ! [Eff4: fm > rule > fset_rule > $o,Rules2: stream_fm,S5: set_rule,S: rule,R: fm,Sl: fset_rule,S4: rule] :
      ( ( abstra7659285206637183772m_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( member_fm3 @ R @ ( sset_fm @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_rule @ S4 @ Sl )
             => ( member_rule3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_785_RuleSystem_Oeff__S,axiom,
    ! [Eff4: nat > rule > fset_rule > $o,Rules2: stream_nat,S5: set_rule,S: rule,R: nat,Sl: fset_rule,S4: rule] :
      ( ( abstra7777025902706499341t_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( member_nat3 @ R @ ( sset_nat @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_rule @ S4 @ Sl )
             => ( member_rule3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_786_RuleSystem_Oeff__S,axiom,
    ! [Eff4: tm > tm > fset_tm > $o,Rules2: stream_tm,S5: set_tm,S: tm,R: tm,Sl: fset_tm,S4: tm] :
      ( ( abstra5526243810835382505_tm_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( member_tm3 @ R @ ( sset_tm @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_tm @ S4 @ Sl )
             => ( member_tm3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_787_RuleSystem_Oeff__S,axiom,
    ! [Eff4: fm > tm > fset_tm > $o,Rules2: stream_fm,S5: set_tm,S: tm,R: fm,Sl: fset_tm,S4: tm] :
      ( ( abstra966159493039759323_fm_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( member_fm3 @ R @ ( sset_fm @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_tm @ S4 @ Sl )
             => ( member_tm3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_788_RuleSystem_Oeff__S,axiom,
    ! [Eff4: nat > tm > fset_tm > $o,Rules2: stream_nat,S5: set_tm,S: tm,R: nat,Sl: fset_tm,S4: tm] :
      ( ( abstra733049861595049770nat_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( member_nat3 @ R @ ( sset_nat @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_tm @ S4 @ Sl )
             => ( member_tm3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_789_RuleSystem_Oeff__S,axiom,
    ! [Eff4: tm > fm > fset_fm > $o,Rules2: stream_tm,S5: set_fm,S: fm,R: tm,Sl: fset_fm,S4: fm] :
      ( ( abstra5526243810834464119_tm_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( member_tm3 @ R @ ( sset_tm @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_fm @ S4 @ Sl )
             => ( member_fm3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_790_RuleSystem_Oeff__S,axiom,
    ! [Eff4: fm > fm > fset_fm > $o,Rules2: stream_fm,S5: set_fm,S: fm,R: fm,Sl: fset_fm,S4: fm] :
      ( ( abstra966159493038840937_fm_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( member_fm3 @ R @ ( sset_fm @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_fm @ S4 @ Sl )
             => ( member_fm3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_791_RuleSystem_Oeff__S,axiom,
    ! [Eff4: nat > fm > fset_fm > $o,Rules2: stream_nat,S5: set_fm,S: fm,R: nat,Sl: fset_fm,S4: fm] :
      ( ( abstra733049861594131384nat_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( member_nat3 @ R @ ( sset_nat @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_fm @ S4 @ Sl )
             => ( member_fm3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_792_RuleSystem_Oeff__S,axiom,
    ! [Eff4: tm > nat > fset_nat > $o,Rules2: stream_tm,S5: set_nat,S: nat,R: tm,Sl: fset_nat,S4: nat] :
      ( ( abstra8150451472847306700tm_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( member_tm3 @ R @ ( sset_tm @ Rules2 ) )
         => ( ( Eff4 @ R @ S @ Sl )
           => ( ( fmember_nat @ S4 @ Sl )
             => ( member_nat3 @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_793_RuleSystem_Ointro,axiom,
    ! [S5: set_rule,Rules2: stream_tm,Eff4: tm > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule3 @ S8 @ S5 )
         => ! [R5: tm] :
              ( ( member_tm3 @ R5 @ ( sset_tm @ Rules2 ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule3 @ S8 @ S5 )
           => ? [X5: tm] :
                ( ( member_tm3 @ X5 @ ( sset_tm @ Rules2 ) )
                & ? [X_12: fset_rule] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra7099963851359259790m_rule @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_794_RuleSystem_Ointro,axiom,
    ! [S5: set_rule,Rules2: stream_fm,Eff4: fm > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule3 @ S8 @ S5 )
         => ! [R5: fm] :
              ( ( member_fm3 @ R5 @ ( sset_fm @ Rules2 ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule3 @ S8 @ S5 )
           => ? [X5: fm] :
                ( ( member_fm3 @ X5 @ ( sset_fm @ Rules2 ) )
                & ? [X_12: fset_rule] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra7659285206637183772m_rule @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_795_RuleSystem_Ointro,axiom,
    ! [S5: set_rule,Rules2: stream_nat,Eff4: nat > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule3 @ S8 @ S5 )
         => ! [R5: nat] :
              ( ( member_nat3 @ R5 @ ( sset_nat @ Rules2 ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule3 @ S8 @ S5 )
           => ? [X5: nat] :
                ( ( member_nat3 @ X5 @ ( sset_nat @ Rules2 ) )
                & ? [X_12: fset_rule] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra7777025902706499341t_rule @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_796_RuleSystem_Ointro,axiom,
    ! [S5: set_tm,Rules2: stream_tm,Eff4: tm > tm > fset_tm > $o] :
      ( ! [S8: tm] :
          ( ( member_tm3 @ S8 @ S5 )
         => ! [R5: tm] :
              ( ( member_tm3 @ R5 @ ( sset_tm @ Rules2 ) )
             => ! [Sl2: fset_tm] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: tm] :
                      ( ( fmember_tm @ S7 @ Sl2 )
                     => ( member_tm3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: tm] :
            ( ( member_tm3 @ S8 @ S5 )
           => ? [X5: tm] :
                ( ( member_tm3 @ X5 @ ( sset_tm @ Rules2 ) )
                & ? [X_12: fset_tm] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra5526243810835382505_tm_tm @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_797_RuleSystem_Ointro,axiom,
    ! [S5: set_tm,Rules2: stream_fm,Eff4: fm > tm > fset_tm > $o] :
      ( ! [S8: tm] :
          ( ( member_tm3 @ S8 @ S5 )
         => ! [R5: fm] :
              ( ( member_fm3 @ R5 @ ( sset_fm @ Rules2 ) )
             => ! [Sl2: fset_tm] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: tm] :
                      ( ( fmember_tm @ S7 @ Sl2 )
                     => ( member_tm3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: tm] :
            ( ( member_tm3 @ S8 @ S5 )
           => ? [X5: fm] :
                ( ( member_fm3 @ X5 @ ( sset_fm @ Rules2 ) )
                & ? [X_12: fset_tm] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra966159493039759323_fm_tm @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_798_RuleSystem_Ointro,axiom,
    ! [S5: set_tm,Rules2: stream_nat,Eff4: nat > tm > fset_tm > $o] :
      ( ! [S8: tm] :
          ( ( member_tm3 @ S8 @ S5 )
         => ! [R5: nat] :
              ( ( member_nat3 @ R5 @ ( sset_nat @ Rules2 ) )
             => ! [Sl2: fset_tm] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: tm] :
                      ( ( fmember_tm @ S7 @ Sl2 )
                     => ( member_tm3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: tm] :
            ( ( member_tm3 @ S8 @ S5 )
           => ? [X5: nat] :
                ( ( member_nat3 @ X5 @ ( sset_nat @ Rules2 ) )
                & ? [X_12: fset_tm] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra733049861595049770nat_tm @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_799_RuleSystem_Ointro,axiom,
    ! [S5: set_fm,Rules2: stream_tm,Eff4: tm > fm > fset_fm > $o] :
      ( ! [S8: fm] :
          ( ( member_fm3 @ S8 @ S5 )
         => ! [R5: tm] :
              ( ( member_tm3 @ R5 @ ( sset_tm @ Rules2 ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: fm] :
            ( ( member_fm3 @ S8 @ S5 )
           => ? [X5: tm] :
                ( ( member_tm3 @ X5 @ ( sset_tm @ Rules2 ) )
                & ? [X_12: fset_fm] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra5526243810834464119_tm_fm @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_800_RuleSystem_Ointro,axiom,
    ! [S5: set_fm,Rules2: stream_fm,Eff4: fm > fm > fset_fm > $o] :
      ( ! [S8: fm] :
          ( ( member_fm3 @ S8 @ S5 )
         => ! [R5: fm] :
              ( ( member_fm3 @ R5 @ ( sset_fm @ Rules2 ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: fm] :
            ( ( member_fm3 @ S8 @ S5 )
           => ? [X5: fm] :
                ( ( member_fm3 @ X5 @ ( sset_fm @ Rules2 ) )
                & ? [X_12: fset_fm] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra966159493038840937_fm_fm @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_801_RuleSystem_Ointro,axiom,
    ! [S5: set_fm,Rules2: stream_nat,Eff4: nat > fm > fset_fm > $o] :
      ( ! [S8: fm] :
          ( ( member_fm3 @ S8 @ S5 )
         => ! [R5: nat] :
              ( ( member_nat3 @ R5 @ ( sset_nat @ Rules2 ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: fm] :
            ( ( member_fm3 @ S8 @ S5 )
           => ? [X5: nat] :
                ( ( member_nat3 @ X5 @ ( sset_nat @ Rules2 ) )
                & ? [X_12: fset_fm] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra733049861594131384nat_fm @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_802_RuleSystem_Ointro,axiom,
    ! [S5: set_nat,Rules2: stream_tm,Eff4: tm > nat > fset_nat > $o] :
      ( ! [S8: nat] :
          ( ( member_nat3 @ S8 @ S5 )
         => ! [R5: tm] :
              ( ( member_tm3 @ R5 @ ( sset_tm @ Rules2 ) )
             => ! [Sl2: fset_nat] :
                  ( ( Eff4 @ R5 @ S8 @ Sl2 )
                 => ! [S7: nat] :
                      ( ( fmember_nat @ S7 @ Sl2 )
                     => ( member_nat3 @ S7 @ S5 ) ) ) ) )
     => ( ! [S8: nat] :
            ( ( member_nat3 @ S8 @ S5 )
           => ? [X5: tm] :
                ( ( member_tm3 @ X5 @ ( sset_tm @ Rules2 ) )
                & ? [X_12: fset_nat] : ( Eff4 @ X5 @ S8 @ X_12 ) ) )
       => ( abstra8150451472847306700tm_nat @ Eff4 @ Rules2 @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_803_RuleSystem__def,axiom,
    ( abstra7099963851359259790m_rule
    = ( ^ [Eff: tm > rule > fset_rule > $o,Rules: stream_tm,S6: set_rule] :
          ( ! [S3: rule] :
              ( ( member_rule3 @ S3 @ S6 )
             => ! [R2: tm] :
                  ( ( member_tm3 @ R2 @ ( sset_tm @ Rules ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl3 )
                         => ( member_rule3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: rule] :
              ( ( member_rule3 @ S3 @ S6 )
             => ? [X2: tm] :
                  ( ( member_tm3 @ X2 @ ( sset_tm @ Rules ) )
                  & ? [X8: fset_rule] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_804_RuleSystem__def,axiom,
    ( abstra7659285206637183772m_rule
    = ( ^ [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S6: set_rule] :
          ( ! [S3: rule] :
              ( ( member_rule3 @ S3 @ S6 )
             => ! [R2: fm] :
                  ( ( member_fm3 @ R2 @ ( sset_fm @ Rules ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl3 )
                         => ( member_rule3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: rule] :
              ( ( member_rule3 @ S3 @ S6 )
             => ? [X2: fm] :
                  ( ( member_fm3 @ X2 @ ( sset_fm @ Rules ) )
                  & ? [X8: fset_rule] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_805_RuleSystem__def,axiom,
    ( abstra7777025902706499341t_rule
    = ( ^ [Eff: nat > rule > fset_rule > $o,Rules: stream_nat,S6: set_rule] :
          ( ! [S3: rule] :
              ( ( member_rule3 @ S3 @ S6 )
             => ! [R2: nat] :
                  ( ( member_nat3 @ R2 @ ( sset_nat @ Rules ) )
                 => ! [Sl3: fset_rule] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl3 )
                         => ( member_rule3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: rule] :
              ( ( member_rule3 @ S3 @ S6 )
             => ? [X2: nat] :
                  ( ( member_nat3 @ X2 @ ( sset_nat @ Rules ) )
                  & ? [X8: fset_rule] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_806_RuleSystem__def,axiom,
    ( abstra5526243810835382505_tm_tm
    = ( ^ [Eff: tm > tm > fset_tm > $o,Rules: stream_tm,S6: set_tm] :
          ( ! [S3: tm] :
              ( ( member_tm3 @ S3 @ S6 )
             => ! [R2: tm] :
                  ( ( member_tm3 @ R2 @ ( sset_tm @ Rules ) )
                 => ! [Sl3: fset_tm] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: tm] :
                          ( ( fmember_tm @ S9 @ Sl3 )
                         => ( member_tm3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: tm] :
              ( ( member_tm3 @ S3 @ S6 )
             => ? [X2: tm] :
                  ( ( member_tm3 @ X2 @ ( sset_tm @ Rules ) )
                  & ? [X8: fset_tm] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_807_RuleSystem__def,axiom,
    ( abstra966159493039759323_fm_tm
    = ( ^ [Eff: fm > tm > fset_tm > $o,Rules: stream_fm,S6: set_tm] :
          ( ! [S3: tm] :
              ( ( member_tm3 @ S3 @ S6 )
             => ! [R2: fm] :
                  ( ( member_fm3 @ R2 @ ( sset_fm @ Rules ) )
                 => ! [Sl3: fset_tm] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: tm] :
                          ( ( fmember_tm @ S9 @ Sl3 )
                         => ( member_tm3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: tm] :
              ( ( member_tm3 @ S3 @ S6 )
             => ? [X2: fm] :
                  ( ( member_fm3 @ X2 @ ( sset_fm @ Rules ) )
                  & ? [X8: fset_tm] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_808_RuleSystem__def,axiom,
    ( abstra733049861595049770nat_tm
    = ( ^ [Eff: nat > tm > fset_tm > $o,Rules: stream_nat,S6: set_tm] :
          ( ! [S3: tm] :
              ( ( member_tm3 @ S3 @ S6 )
             => ! [R2: nat] :
                  ( ( member_nat3 @ R2 @ ( sset_nat @ Rules ) )
                 => ! [Sl3: fset_tm] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: tm] :
                          ( ( fmember_tm @ S9 @ Sl3 )
                         => ( member_tm3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: tm] :
              ( ( member_tm3 @ S3 @ S6 )
             => ? [X2: nat] :
                  ( ( member_nat3 @ X2 @ ( sset_nat @ Rules ) )
                  & ? [X8: fset_tm] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_809_RuleSystem__def,axiom,
    ( abstra5526243810834464119_tm_fm
    = ( ^ [Eff: tm > fm > fset_fm > $o,Rules: stream_tm,S6: set_fm] :
          ( ! [S3: fm] :
              ( ( member_fm3 @ S3 @ S6 )
             => ! [R2: tm] :
                  ( ( member_tm3 @ R2 @ ( sset_tm @ Rules ) )
                 => ! [Sl3: fset_fm] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl3 )
                         => ( member_fm3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: fm] :
              ( ( member_fm3 @ S3 @ S6 )
             => ? [X2: tm] :
                  ( ( member_tm3 @ X2 @ ( sset_tm @ Rules ) )
                  & ? [X8: fset_fm] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_810_RuleSystem__def,axiom,
    ( abstra966159493038840937_fm_fm
    = ( ^ [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S6: set_fm] :
          ( ! [S3: fm] :
              ( ( member_fm3 @ S3 @ S6 )
             => ! [R2: fm] :
                  ( ( member_fm3 @ R2 @ ( sset_fm @ Rules ) )
                 => ! [Sl3: fset_fm] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl3 )
                         => ( member_fm3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: fm] :
              ( ( member_fm3 @ S3 @ S6 )
             => ? [X2: fm] :
                  ( ( member_fm3 @ X2 @ ( sset_fm @ Rules ) )
                  & ? [X8: fset_fm] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_811_RuleSystem__def,axiom,
    ( abstra733049861594131384nat_fm
    = ( ^ [Eff: nat > fm > fset_fm > $o,Rules: stream_nat,S6: set_fm] :
          ( ! [S3: fm] :
              ( ( member_fm3 @ S3 @ S6 )
             => ! [R2: nat] :
                  ( ( member_nat3 @ R2 @ ( sset_nat @ Rules ) )
                 => ! [Sl3: fset_fm] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl3 )
                         => ( member_fm3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: fm] :
              ( ( member_fm3 @ S3 @ S6 )
             => ? [X2: nat] :
                  ( ( member_nat3 @ X2 @ ( sset_nat @ Rules ) )
                  & ? [X8: fset_fm] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_812_RuleSystem__def,axiom,
    ( abstra8150451472847306700tm_nat
    = ( ^ [Eff: tm > nat > fset_nat > $o,Rules: stream_tm,S6: set_nat] :
          ( ! [S3: nat] :
              ( ( member_nat3 @ S3 @ S6 )
             => ! [R2: tm] :
                  ( ( member_tm3 @ R2 @ ( sset_tm @ Rules ) )
                 => ! [Sl3: fset_nat] :
                      ( ( Eff @ R2 @ S3 @ Sl3 )
                     => ! [S9: nat] :
                          ( ( fmember_nat @ S9 @ Sl3 )
                         => ( member_nat3 @ S9 @ S6 ) ) ) ) )
          & ! [S3: nat] :
              ( ( member_nat3 @ S3 @ S6 )
             => ? [X2: tm] :
                  ( ( member_tm3 @ X2 @ ( sset_tm @ Rules ) )
                  & ? [X8: fset_nat] : ( Eff @ X2 @ S3 @ X8 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_813_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule,S: rule,Rs: stream_rule,Steps: stream4385846686851721995e_rule] :
      ( ( abstra2971612018481630886e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff4 @ Rs @ S ) @ Steps )
           => ( abstra8307935604394493588e_rule @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_814_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > list_fm > fset_list_fm > $o,Rules2: stream_rule,S5: set_list_fm,S: list_fm,Rs: stream_rule,Steps: stream1460591043787736522m_rule] :
      ( ( abstra3805608168546402149ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra2882552375471187403m_rule @ ( abstra6869381243922927298ist_fm @ Eff4 @ Rs @ S ) @ Steps )
           => ( abstra3959174058192036855ist_fm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_815_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S5: set_tm,S: tm,Rs: stream_rule,Steps: stream3258364598272819510m_rule] :
      ( ( abstra3960359434151636177ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra3841055140891909431m_rule @ ( abstra3300998584369155758ule_tm @ Eff4 @ Rs @ S ) @ Steps )
           => ( abstra2738778320243819107ule_tm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_816_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm,S: fm,Rs: stream_rule,Steps: stream6755863580823097540m_rule] :
      ( ( abstra3960359434150717791ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra4400376496169833413m_rule @ ( abstra3300998584368237372ule_fm @ Eff4 @ Rs @ S ) @ Steps )
           => ( abstra2738778320242900721ule_fm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_817_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S5: set_nat,S: nat,Rs: stream_rule,Steps: stream8912087984624402761t_rule] :
      ( ( abstra4367803203942586852le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra5135089046934318962t_rule @ ( abstra7662105669874348231le_nat @ Eff4 @ Rs @ S ) @ Steps )
           => ( abstra3870052345063948242le_nat @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_818_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra3967921542344661089ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member4699826688122452638ist_fm @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra8836096083317999815m_rule @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs @ S ) @ Steps )
           => ( abstra6097777249025082867ist_fm @ Eff4 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_819_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S5: set_tm,S: tm,Rs: stream_rule,Steps: stream3258364598272819510m_rule,M2: nat] :
      ( ( abstra4909020524820736232ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra3841055140891909431m_rule @ ( abstra3300998584369155758ule_tm @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: tm] :
                ( ( member_tm3 @ S7 @ S5 )
                & ( abstra3841055140891909431m_rule @ ( abstra3300998584369155758ule_tm @ Eff4 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_3187986897935118717m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_820_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm,S: fm,Rs: stream_rule,Steps: stream6755863580823097540m_rule,M2: nat] :
      ( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra4400376496169833413m_rule @ ( abstra3300998584368237372ule_fm @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm3 @ S7 @ S5 )
                & ( abstra4400376496169833413m_rule @ ( abstra3300998584368237372ule_fm @ Eff4 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_3747308253213042699m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_821_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S5: set_nat,S: nat,Rs: stream_rule,Steps: stream8912087984624402761t_rule,M2: nat] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra5135089046934318962t_rule @ ( abstra7662105669874348231le_nat @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: nat] :
                ( ( member_nat3 @ S7 @ S5 )
                & ( abstra5135089046934318962t_rule @ ( abstra7662105669874348231le_nat @ Eff4 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_3122356783091591596t_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_822_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule,S: rule,Rs: stream_rule,Steps: stream4385846686851721995e_rule,M2: nat] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: rule] :
                ( ( member_rule3 @ S7 @ S5 )
                & ( abstra1660175255847126580e_rule @ ( abstra2300331465871827593e_rule @ Eff4 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_9113879250048157294e_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_823_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: rule > list_fm > fset_list_fm > $o,Rules2: stream_rule,S5: set_list_fm,S: list_fm,Rs: stream_rule,Steps: stream1460591043787736522m_rule,M2: nat] :
      ( ( abstra9153614075763688444ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( ( abstra2882552375471187403m_rule @ ( abstra6869381243922927298ist_fm @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: list_fm] :
                ( ( member_list_fm3 @ S7 @ S5 )
                & ( abstra2882552375471187403m_rule @ ( abstra6869381243922927298ist_fm @ Eff4 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_779662454523830033m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_824_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: list_fm > list_tm > fset_list_tm > $o,Rules2: stream_list_fm,S5: set_list_tm,S: list_tm,Rs: stream_list_fm,Steps: stream4408948924543953275ist_fm,M2: nat] :
      ( ( abstra4205323827771120475ist_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_tm @ S @ S5 )
       => ( ( abstra7078398738616236685ist_fm @ Rules2 @ Rs )
         => ( ( abstra2293663390402184868ist_fm @ ( abstra4388168957404043029ist_tm @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: list_tm] :
                ( ( member_list_tm @ S7 @ S5 )
                & ( abstra2293663390402184868ist_fm @ ( abstra4388168957404043029ist_tm @ Eff4 @ ( sdrop_list_fm @ N2 @ Rs ) @ S7 ) @ ( sdrop_9176333610110415838ist_fm @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_825_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: produc7694839378271647877e_rule > rule > fset_rule > $o,Rules2: stream4385846686851721995e_rule,S5: set_rule,S: rule,Rs: stream4385846686851721995e_rule,Steps: stream6316532330152022520e_rule,M2: nat] :
      ( ( abstra2905114889036010542e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ( ( abstra5549271708691466489e_rule @ ( abstra5501172023691128564e_rule @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: rule] :
                ( ( member_rule3 @ S7 @ S5 )
                & ( abstra5549271708691466489e_rule @ ( abstra5501172023691128564e_rule @ Eff4 @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_4667533458460693055e_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_826_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: produc7694839378271647877e_rule > tm > fset_tm > $o,Rules2: stream4385846686851721995e_rule,S5: set_tm,S: tm,Rs: stream4385846686851721995e_rule,Steps: stream5245948007271892237e_rule,M2: nat] :
      ( ( abstra1261657321278996809ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ( ( abstra6981858451359999286e_rule @ ( abstra8399882063072596483ule_tm @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: tm] :
                ( ( member_tm3 @ S7 @ S5 )
                & ( abstra6981858451359999286e_rule @ ( abstra8399882063072596483ule_tm @ Eff4 @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_1553690009073952624e_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_827_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: produc7694839378271647877e_rule > fm > fset_fm > $o,Rules2: stream4385846686851721995e_rule,S5: set_fm,S: fm,Rs: stream4385846686851721995e_rule,Steps: stream1329043581973813759e_rule,M2: nat] :
      ( ( abstra1261657321278078423ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ( ( abstra4318453995962028840e_rule @ ( abstra8399882063071678097ule_fm @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm3 @ S7 @ S5 )
                & ( abstra4318453995962028840e_rule @ ( abstra8399882063071678097ule_fm @ Eff4 @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_8113657590530757986e_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_828_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff4: produc7694839378271647877e_rule > nat > fset_nat > $o,Rules2: stream4385846686851721995e_rule,S5: set_nat,S: nat,Rs: stream4385846686851721995e_rule,Steps: stream791956382124094522e_rule,M2: nat] :
      ( ( abstra5123640173530818412le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
         => ( ( abstra2368212495854999611e_rule @ ( abstra5317780200598325298le_nat @ Eff4 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S7: nat] :
                ( ( member_nat3 @ S7 @ S5 )
                & ( abstra2368212495854999611e_rule @ ( abstra5317780200598325298le_nat @ Eff4 @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_6661281195147388801e_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_829_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: tm > rule > fset_rule > $o,Rules2: stream_tm,S5: set_rule,S: rule,Rs: stream_tm,Steps: stream8415551234601779672ule_tm,R: tm] :
      ( ( abstra6151302760690159735m_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra6979500642109889529air_tm @ Rules2 @ Rs )
         => ( ( abstra7530814456727015129ule_tm @ ( abstra5491941910907679316m_rule @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_tm3 @ R @ ( sset_tm @ Rules2 ) )
             => ( abstra2914376109873778153m_rule @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_830_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: fm > rule > fset_rule > $o,Rules2: stream_fm,S5: set_rule,S: rule,Rs: stream_fm,Steps: stream8658065904927289446ule_fm,R: fm] :
      ( ( abstra6710624115968083717m_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra6979500642108971143air_fm @ Rules2 @ Rs )
         => ( ( abstra7526862431641379943ule_fm @ ( abstra6051263266185603298m_rule @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_fm3 @ R @ ( sset_fm @ Rules2 ) )
             => ( abstra3473697465151702135m_rule @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_831_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: nat > rule > fset_rule > $o,Rules2: stream_nat,S5: set_rule,S: rule,Rs: stream_nat,Steps: stream8814882096039290697le_nat,R: nat] :
      ( ( abstra8880880309027573988t_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs )
         => ( ( abstra8430046938747295090le_nat @ ( abstra2951810738104559559t_rule @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_nat3 @ R @ ( sset_nat @ Rules2 ) )
             => ( abstra6971170162055632626t_rule @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_832_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: tm > tm > fset_tm > $o,Rules2: stream_tm,S5: set_tm,S: tm,Rs: stream_tm,Steps: stream5748647146893912941_tm_tm,R: tm] :
      ( ( abstra5645505808713051712_tm_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra6979500642109889529air_tm @ Rules2 @ Rs )
         => ( ( abstra2363653106310321558_tm_tm @ ( abstra4750730535116555939_tm_tm @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_tm3 @ R @ ( sset_tm @ Rules2 ) )
             => ( abstra589508014764448590_tm_tm @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_833_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: fm > tm > fset_tm > $o,Rules2: stream_fm,S5: set_tm,S: tm,Rs: stream_fm,Steps: stream5991161817219422715_tm_fm,R: fm] :
      ( ( abstra1085421490917428530_fm_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra6979500642108971143air_fm @ Rules2 @ Rs )
         => ( ( abstra2359701081224686372_tm_fm @ ( abstra190646217320932757_fm_tm @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_fm3 @ R @ ( sset_fm @ Rules2 ) )
             => ( abstra5252795733823601216_fm_tm @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_834_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: nat > tm > fset_tm > $o,Rules2: stream_nat,S5: set_tm,S: tm,Rs: stream_nat,Steps: stream2263335421038561396tm_nat,R: nat] :
      ( ( abstra3975341336036260883nat_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs )
         => ( ( abstra8950899790699253621tm_nat @ ( abstra152912839120034544nat_tm @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_nat3 @ R @ ( sset_nat @ Rules2 ) )
             => ( abstra7351407333249067909nat_tm @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_835_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: tm > fm > fset_fm > $o,Rules2: stream_tm,S5: set_fm,S: fm,Rs: stream_tm,Steps: stream7757019309515125087_fm_tm,R: tm] :
      ( ( abstra5645505808712133326_tm_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra6979500642109889529air_tm @ Rules2 @ Rs )
         => ( ( abstra7026940825369474184_fm_tm @ ( abstra4750730535115637553_tm_fm @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_tm3 @ R @ ( sset_tm @ Rules2 ) )
             => ( abstra589508014763530204_tm_fm @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_836_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: fm > fm > fset_fm > $o,Rules2: stream_fm,S5: set_fm,S: fm,Rs: stream_fm,Steps: stream7999533979840634861_fm_fm,R: fm] :
      ( ( abstra1085421490916510144_fm_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra6979500642108971143air_fm @ Rules2 @ Rs )
         => ( ( abstra7022988800283838998_fm_fm @ ( abstra190646217320014371_fm_fm @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_fm3 @ R @ ( sset_fm @ Rules2 ) )
             => ( abstra5252795733822682830_fm_fm @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_837_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: nat > fm > fset_fm > $o,Rules2: stream_nat,S5: set_fm,S: fm,Rs: stream_nat,Steps: stream6917443656897226242fm_nat,R: nat] :
      ( ( abstra3975341336035342497nat_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra7018062486307922108ir_nat @ Rules2 @ Rs )
         => ( ( abstra7286520533628752899fm_nat @ ( abstra152912839119116158nat_fm @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_nat3 @ R @ ( sset_nat @ Rules2 ) )
             => ( abstra7351407333248149523nat_fm @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_838_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff4: tm > nat > fset_nat > $o,Rules2: stream_tm,S5: set_nat,S: nat,Rs: stream_tm,Steps: stream5482842300757736730nat_tm,R: tm] :
      ( ( abstra2169370910433742005tm_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra6979500642109889529air_tm @ Rules2 @ Rs )
         => ( ( abstra8829537861777438491nat_tm @ ( abstra7570314450372291474tm_nat @ Eff4 @ Rs @ S ) @ Steps )
           => ( ( member_tm3 @ R @ ( sset_tm @ Rules2 ) )
             => ( abstra5545436907646549031tm_nat @ Eff4 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_839_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_840_secavProver__def,axiom,
    ( secavProver
    = ( abstra7840189589880272446ist_fm @ eff @ rules ) ) ).

% secavProver_def
thf(fact_841_mkTree__effG__uu__def,axiom,
    ( abstra9183632535595798085ist_fm
    = ( ^ [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra7840189589880272446ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) ) ) ) ).

% mkTree_effG_uu_def
thf(fact_842_fsubsetI,axiom,
    ! [A: fset_P8989946509869081563ist_fm,B2: fset_P8989946509869081563ist_fm] :
      ( ! [X3: produc6018962875968178549ist_fm] :
          ( ( fmembe3754813877001230652ist_fm @ X3 @ A )
         => ( fmembe3754813877001230652ist_fm @ X3 @ B2 ) )
     => ( ord_le3986950534092794747ist_fm @ A @ B2 ) ) ).

% fsubsetI
thf(fact_843_fsubsetI,axiom,
    ! [A: fset_A8849196716589201442m_rule,B2: fset_A8849196716589201442m_rule] :
      ( ! [X3: abstra8520858721168826610m_rule] :
          ( ( fmembe1099044791951439339m_rule @ X3 @ A )
         => ( fmembe1099044791951439339m_rule @ X3 @ B2 ) )
     => ( ord_le59936386403465346m_rule @ A @ B2 ) ) ).

% fsubsetI
thf(fact_844_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_845_listFunTm_Osimps_I1_J,axiom,
    ! [N: nat,Ts: list_tm] :
      ( ( listFunTm @ ( fun @ N @ Ts ) )
      = ( cons_nat @ N @ ( listFunTms @ Ts ) ) ) ).

% listFunTm.simps(1)
thf(fact_846_listFunTm__paramst_I2_J,axiom,
    ! [Ts: list_tm] :
      ( ( set_nat2 @ ( listFunTms @ Ts ) )
      = ( paramsts @ Ts ) ) ).

% listFunTm_paramst(2)
thf(fact_847_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_848_RuleSystem__Defs_Owf_Ocong,axiom,
    abstra8167393772827826667ist_fm = abstra8167393772827826667ist_fm ).

% RuleSystem_Defs.wf.cong
thf(fact_849_RuleSystem__Defs_Owf__ipath__epath,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,T: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ T )
     => ( ( abstra8836096083317999815m_rule @ T @ Steps )
       => ( abstra6789711989322986974ist_fm @ Eff4 @ Rules2 @ Steps ) ) ) ).

% RuleSystem_Defs.wf_ipath_epath
thf(fact_850_fin__mono,axiom,
    ! [A: fset_P8989946509869081563ist_fm,B2: fset_P8989946509869081563ist_fm,X: produc6018962875968178549ist_fm] :
      ( ( ord_le3986950534092794747ist_fm @ A @ B2 )
     => ( ( fmembe3754813877001230652ist_fm @ X @ A )
       => ( fmembe3754813877001230652ist_fm @ X @ B2 ) ) ) ).

% fin_mono
thf(fact_851_fin__mono,axiom,
    ! [A: fset_A8849196716589201442m_rule,B2: fset_A8849196716589201442m_rule,X: abstra8520858721168826610m_rule] :
      ( ( ord_le59936386403465346m_rule @ A @ B2 )
     => ( ( fmembe1099044791951439339m_rule @ X @ A )
       => ( fmembe1099044791951439339m_rule @ X @ B2 ) ) ) ).

% fin_mono
thf(fact_852_fsubsetD,axiom,
    ! [A: fset_P8989946509869081563ist_fm,B2: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
      ( ( ord_le3986950534092794747ist_fm @ A @ B2 )
     => ( ( fmembe3754813877001230652ist_fm @ C @ A )
       => ( fmembe3754813877001230652ist_fm @ C @ B2 ) ) ) ).

% fsubsetD
thf(fact_853_fsubsetD,axiom,
    ! [A: fset_A8849196716589201442m_rule,B2: fset_A8849196716589201442m_rule,C: abstra8520858721168826610m_rule] :
      ( ( ord_le59936386403465346m_rule @ A @ B2 )
     => ( ( fmembe1099044791951439339m_rule @ C @ A )
       => ( fmembe1099044791951439339m_rule @ C @ B2 ) ) ) ).

% fsubsetD
thf(fact_854_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S5 )
     => ( ( member_rule3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra459280387713764508e_rule @ Eff4 @ Rules2 @ ( abstra2300331465871827593e_rule @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_855_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > list_fm > fset_list_fm > $o,Rules2: stream_rule,S5: set_list_fm,S: list_fm,Rs: stream_rule] :
      ( ( abstra9153614075763688444ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_list_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra2093743074600156143ist_fm @ Eff4 @ Rules2 @ ( abstra6869381243922927298ist_fm @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_856_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S5: set_tm,S: tm,Rs: stream_rule] :
      ( ( abstra4909020524820736232ule_tm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_tm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra1214582413480588891ule_tm @ Eff4 @ Rules2 @ ( abstra3300998584369155758ule_tm @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_857_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member_fm3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra1214582413479670505ule_fm @ Eff4 @ Rules2 @ ( abstra3300998584368237372ule_fm @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_858_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S5: set_nat,S: nat,Rs: stream_rule] :
      ( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S5 )
     => ( ( member_nat3 @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra4678492489212612058le_nat @ Eff4 @ Rules2 @ ( abstra7662105669874348231le_nat @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_859_RuleSystem_Owf__mkTree,axiom,
    ! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S5: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule] :
      ( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S5 )
     => ( ( member4699826688122452638ist_fm @ S @ S5 )
       => ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
         => ( abstra8167393772827826667ist_fm @ Eff4 @ Rules2 @ ( abstra7840189589880272446ist_fm @ Eff4 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_860_i_Owf__ipath__epath,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_rule,T: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule] :
      ( ( abstra8167393772827826667ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ T )
     => ( ( abstra8836096083317999815m_rule @ T @ Steps )
       => ( abstra6789711989322986974ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ Steps ) ) ) ).

% i.wf_ipath_epath
thf(fact_861_fset__of__list__subset,axiom,
    ! [Xs: list_list_fm,Ys: list_list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ Ys ) )
     => ( ord_le3182162295460067583ist_fm @ ( fset_of_list_list_fm @ Xs ) @ ( fset_of_list_list_fm @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_862_fset__of__list__subset,axiom,
    ! [Xs: list_tm,Ys: list_tm] :
      ( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ Ys ) )
     => ( ord_less_eq_fset_tm @ ( fset_of_list_tm @ Xs ) @ ( fset_of_list_tm @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_863_fset__of__list__subset,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
     => ( ord_less_eq_fset_nat @ ( fset_of_list_nat @ Xs ) @ ( fset_of_list_nat @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_864_fset__of__list__subset,axiom,
    ! [Xs: list_fm,Ys: list_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) )
     => ( ord_less_eq_fset_fm @ ( fset_of_list_fm @ Xs ) @ ( fset_of_list_fm @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_865_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: list_fm > list_tm > option_fset_list_tm,Rs: stream_list_fm,S: list_tm] :
      ( ( abstra2656519403522841862ist_fm @ ( abstra4388168957404043029ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rs @ S ) )
      = ( produc1414352766439514085ist_fm @ S @ ( shd_list_fm @ ( abstra1372434334096485691ist_tm @ ( abstra9141620886324064114ist_tm @ Eff2 ) @ Rs @ S ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_866_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: nat > nat > option_fset_nat,Rs: stream_nat,S: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff2 ) @ Rs @ S ) )
      = ( product_Pair_nat_nat @ S @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff2 ) @ Rs @ S ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_867_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff2 ) @ Rs @ S ) )
      = ( produc6161850002892822231at_nat @ S @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff2 ) @ Rs @ S ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_868_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra5180277867158163941m_rule @ ( abstra7840189589880272446ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rs @ S ) )
      = ( produc1733806532565653680m_rule @ S @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rs @ S ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_869_parts__in__effect,axiom,
    ! [P3: fm,Z3: list_fm,B2: list_tm,Z5: list_fm,R: rule,A: list_tm] :
      ( ( member_fm3 @ P3 @ ( set_fm2 @ Z3 ) )
     => ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z5 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z3 ) ) )
       => ? [C6: list_tm,Xs3: list_fm] :
            ( ( ord_less_eq_set_tm @ ( set_tm2 @ A ) @ ( set_tm2 @ C6 ) )
            & ( member_list_fm3 @ Xs3 @ ( set_list_fm2 @ ( parts @ C6 @ R @ P3 ) ) )
            & ( ord_less_eq_set_fm @ ( set_fm2 @ Xs3 ) @ ( set_fm2 @ Z5 ) ) ) ) ) ).

% parts_in_effect
thf(fact_870_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_871_in__cont__mkTree,axiom,
    ! [S: produc6018962875968178549ist_fm,Rs: stream_rule,T3: abstra8520858721168826610m_rule] :
      ( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
     => ( ( abstra3799686578551160190r_rule @ rules @ Rs )
       => ( ( fmembe1099044791951439339m_rule @ T3 @ ( abstra2475333618431710837m_rule @ ( abstra7840189589880272446ist_fm @ eff @ Rs @ S ) ) )
         => ? [Sl4: fset_P8989946509869081563ist_fm,S7: produc6018962875968178549ist_fm] :
              ( ( member4699826688122452638ist_fm @ S7 @ top_to1730629564055774885ist_fm )
              & ( eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S @ Sl4 )
              & ( fmembe3754813877001230652ist_fm @ S7 @ Sl4 )
              & ( T3
                = ( abstra7840189589880272446ist_fm @ eff @ ( stl_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S7 ) ) ) ) ) ) ).

% in_cont_mkTree
thf(fact_872_per__def,axiom,
    ! [R: rule] :
      ( ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R )
      = ( ! [S3: produc6018962875968178549ist_fm,R1: rule,Sl5: fset_P8989946509869081563ist_fm,S9: produc6018962875968178549ist_fm] :
            ( ( ( member4699826688122452638ist_fm @ S3 @ top_to1730629564055774885ist_fm )
              & ( abstra1707737187183579335ist_fm @ eff @ R @ S3 )
              & ( member_rule3 @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ rules ) @ ( insert_rule2 @ R @ bot_bot_set_rule ) ) )
              & ( eff @ R1 @ S3 @ Sl5 )
              & ( fmembe3754813877001230652ist_fm @ S9 @ Sl5 ) )
           => ( abstra1707737187183579335ist_fm @ eff @ R @ S9 ) ) ) ) ).

% per_def
thf(fact_873_listFunTms_Osimps_I2_J,axiom,
    ! [T: tm,Ts: list_tm] :
      ( ( listFunTms @ ( cons_tm @ T @ Ts ) )
      = ( append_nat @ ( listFunTm @ T ) @ ( listFunTms @ Ts ) ) ) ).

% listFunTms.simps(2)
thf(fact_874_empty__iff,axiom,
    ! [C: list_fm] :
      ~ ( member_list_fm3 @ C @ bot_bot_set_list_fm ) ).

% empty_iff
thf(fact_875_empty__iff,axiom,
    ! [C: tm] :
      ~ ( member_tm3 @ C @ bot_bot_set_tm ) ).

% empty_iff
thf(fact_876_empty__iff,axiom,
    ! [C: rule] :
      ~ ( member_rule3 @ C @ bot_bot_set_rule ) ).

% empty_iff
thf(fact_877_empty__iff,axiom,
    ! [C: fm] :
      ~ ( member_fm3 @ C @ bot_bot_set_fm ) ).

% empty_iff
thf(fact_878_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat3 @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_879_all__not__in__conv,axiom,
    ! [A: set_list_fm] :
      ( ( ! [X2: list_fm] :
            ~ ( member_list_fm3 @ X2 @ A ) )
      = ( A = bot_bot_set_list_fm ) ) ).

% all_not_in_conv
thf(fact_880_all__not__in__conv,axiom,
    ! [A: set_tm] :
      ( ( ! [X2: tm] :
            ~ ( member_tm3 @ X2 @ A ) )
      = ( A = bot_bot_set_tm ) ) ).

% all_not_in_conv
thf(fact_881_all__not__in__conv,axiom,
    ! [A: set_rule] :
      ( ( ! [X2: rule] :
            ~ ( member_rule3 @ X2 @ A ) )
      = ( A = bot_bot_set_rule ) ) ).

% all_not_in_conv
thf(fact_882_all__not__in__conv,axiom,
    ! [A: set_fm] :
      ( ( ! [X2: fm] :
            ~ ( member_fm3 @ X2 @ A ) )
      = ( A = bot_bot_set_fm ) ) ).

% all_not_in_conv
thf(fact_883_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X2: nat] :
            ~ ( member_nat3 @ X2 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_884_Collect__empty__eq,axiom,
    ! [P4: rule > $o] :
      ( ( ( collect_rule @ P4 )
        = bot_bot_set_rule )
      = ( ! [X2: rule] :
            ~ ( P4 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_885_Collect__empty__eq,axiom,
    ! [P4: fm > $o] :
      ( ( ( collect_fm @ P4 )
        = bot_bot_set_fm )
      = ( ! [X2: fm] :
            ~ ( P4 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_886_Collect__empty__eq,axiom,
    ! [P4: nat > $o] :
      ( ( ( collect_nat @ P4 )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ~ ( P4 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_887_empty__Collect__eq,axiom,
    ! [P4: rule > $o] :
      ( ( bot_bot_set_rule
        = ( collect_rule @ P4 ) )
      = ( ! [X2: rule] :
            ~ ( P4 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_888_empty__Collect__eq,axiom,
    ! [P4: fm > $o] :
      ( ( bot_bot_set_fm
        = ( collect_fm @ P4 ) )
      = ( ! [X2: fm] :
            ~ ( P4 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_889_empty__Collect__eq,axiom,
    ! [P4: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P4 ) )
      = ( ! [X2: nat] :
            ~ ( P4 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_890_append_Oassoc,axiom,
    ! [A2: list_nat,B: list_nat,C: list_nat] :
      ( ( append_nat @ ( append_nat @ A2 @ B ) @ C )
      = ( append_nat @ A2 @ ( append_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_891_append_Oassoc,axiom,
    ! [A2: list_tm,B: list_tm,C: list_tm] :
      ( ( append_tm @ ( append_tm @ A2 @ B ) @ C )
      = ( append_tm @ A2 @ ( append_tm @ B @ C ) ) ) ).

% append.assoc
thf(fact_892_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_893_append__assoc,axiom,
    ! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
      ( ( append_tm @ ( append_tm @ Xs @ Ys ) @ Zs )
      = ( append_tm @ Xs @ ( append_tm @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_894_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_895_append__same__eq,axiom,
    ! [Ys: list_tm,Xs: list_tm,Zs: list_tm] :
      ( ( ( append_tm @ Ys @ Xs )
        = ( append_tm @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_896_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_897_same__append__eq,axiom,
    ! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
      ( ( ( append_tm @ Xs @ Ys )
        = ( append_tm @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_898_DiffI,axiom,
    ! [C: list_fm,A: set_list_fm,B2: set_list_fm] :
      ( ( member_list_fm3 @ C @ A )
     => ( ~ ( member_list_fm3 @ C @ B2 )
       => ( member_list_fm3 @ C @ ( minus_639611354763871680ist_fm @ A @ B2 ) ) ) ) ).

% DiffI
thf(fact_899_DiffI,axiom,
    ! [C: tm,A: set_tm,B2: set_tm] :
      ( ( member_tm3 @ C @ A )
     => ( ~ ( member_tm3 @ C @ B2 )
       => ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B2 ) ) ) ) ).

% DiffI
thf(fact_900_DiffI,axiom,
    ! [C: fm,A: set_fm,B2: set_fm] :
      ( ( member_fm3 @ C @ A )
     => ( ~ ( member_fm3 @ C @ B2 )
       => ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B2 ) ) ) ) ).

% DiffI
thf(fact_901_DiffI,axiom,
    ! [C: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat3 @ C @ A )
     => ( ~ ( member_nat3 @ C @ B2 )
       => ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).

% DiffI
thf(fact_902_DiffI,axiom,
    ! [C: rule,A: set_rule,B2: set_rule] :
      ( ( member_rule3 @ C @ A )
     => ( ~ ( member_rule3 @ C @ B2 )
       => ( member_rule3 @ C @ ( minus_minus_set_rule @ A @ B2 ) ) ) ) ).

% DiffI
thf(fact_903_Diff__iff,axiom,
    ! [C: list_fm,A: set_list_fm,B2: set_list_fm] :
      ( ( member_list_fm3 @ C @ ( minus_639611354763871680ist_fm @ A @ B2 ) )
      = ( ( member_list_fm3 @ C @ A )
        & ~ ( member_list_fm3 @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_904_Diff__iff,axiom,
    ! [C: tm,A: set_tm,B2: set_tm] :
      ( ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B2 ) )
      = ( ( member_tm3 @ C @ A )
        & ~ ( member_tm3 @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_905_Diff__iff,axiom,
    ! [C: fm,A: set_fm,B2: set_fm] :
      ( ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B2 ) )
      = ( ( member_fm3 @ C @ A )
        & ~ ( member_fm3 @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_906_Diff__iff,axiom,
    ! [C: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B2 ) )
      = ( ( member_nat3 @ C @ A )
        & ~ ( member_nat3 @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_907_Diff__iff,axiom,
    ! [C: rule,A: set_rule,B2: set_rule] :
      ( ( member_rule3 @ C @ ( minus_minus_set_rule @ A @ B2 ) )
      = ( ( member_rule3 @ C @ A )
        & ~ ( member_rule3 @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_908_Diff__idemp,axiom,
    ! [A: set_rule,B2: set_rule] :
      ( ( minus_minus_set_rule @ ( minus_minus_set_rule @ A @ B2 ) @ B2 )
      = ( minus_minus_set_rule @ A @ B2 ) ) ).

% Diff_idemp
thf(fact_909_empty__subsetI,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ bot_bot_set_rule @ A ) ).

% empty_subsetI
thf(fact_910_empty__subsetI,axiom,
    ! [A: set_tm] : ( ord_less_eq_set_tm @ bot_bot_set_tm @ A ) ).

% empty_subsetI
thf(fact_911_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_912_empty__subsetI,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A ) ).

% empty_subsetI
thf(fact_913_subset__empty,axiom,
    ! [A: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ bot_bot_set_rule )
      = ( A = bot_bot_set_rule ) ) ).

% subset_empty
thf(fact_914_subset__empty,axiom,
    ! [A: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ bot_bot_set_tm )
      = ( A = bot_bot_set_tm ) ) ).

% subset_empty
thf(fact_915_subset__empty,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_916_subset__empty,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ bot_bot_set_fm )
      = ( A = bot_bot_set_fm ) ) ).

% subset_empty
thf(fact_917_singletonI,axiom,
    ! [A2: list_fm] : ( member_list_fm3 @ A2 @ ( insert_list_fm2 @ A2 @ bot_bot_set_list_fm ) ) ).

% singletonI
thf(fact_918_singletonI,axiom,
    ! [A2: tm] : ( member_tm3 @ A2 @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) ) ).

% singletonI
thf(fact_919_singletonI,axiom,
    ! [A2: rule] : ( member_rule3 @ A2 @ ( insert_rule2 @ A2 @ bot_bot_set_rule ) ) ).

% singletonI
thf(fact_920_singletonI,axiom,
    ! [A2: fm] : ( member_fm3 @ A2 @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ).

% singletonI
thf(fact_921_singletonI,axiom,
    ! [A2: nat] : ( member_nat3 @ A2 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_922_Diff__empty,axiom,
    ! [A: set_fm] :
      ( ( minus_minus_set_fm @ A @ bot_bot_set_fm )
      = A ) ).

% Diff_empty
thf(fact_923_Diff__empty,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Diff_empty
thf(fact_924_Diff__empty,axiom,
    ! [A: set_rule] :
      ( ( minus_minus_set_rule @ A @ bot_bot_set_rule )
      = A ) ).

% Diff_empty
thf(fact_925_empty__Diff,axiom,
    ! [A: set_fm] :
      ( ( minus_minus_set_fm @ bot_bot_set_fm @ A )
      = bot_bot_set_fm ) ).

% empty_Diff
thf(fact_926_empty__Diff,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_927_empty__Diff,axiom,
    ! [A: set_rule] :
      ( ( minus_minus_set_rule @ bot_bot_set_rule @ A )
      = bot_bot_set_rule ) ).

% empty_Diff
thf(fact_928_Diff__cancel,axiom,
    ! [A: set_fm] :
      ( ( minus_minus_set_fm @ A @ A )
      = bot_bot_set_fm ) ).

% Diff_cancel
thf(fact_929_Diff__cancel,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ A )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_930_Diff__cancel,axiom,
    ! [A: set_rule] :
      ( ( minus_minus_set_rule @ A @ A )
      = bot_bot_set_rule ) ).

% Diff_cancel
thf(fact_931_insert__Diff1,axiom,
    ! [X: list_fm,B2: set_list_fm,A: set_list_fm] :
      ( ( member_list_fm3 @ X @ B2 )
     => ( ( minus_639611354763871680ist_fm @ ( insert_list_fm2 @ X @ A ) @ B2 )
        = ( minus_639611354763871680ist_fm @ A @ B2 ) ) ) ).

% insert_Diff1
thf(fact_932_insert__Diff1,axiom,
    ! [X: tm,B2: set_tm,A: set_tm] :
      ( ( member_tm3 @ X @ B2 )
     => ( ( minus_minus_set_tm @ ( insert_tm2 @ X @ A ) @ B2 )
        = ( minus_minus_set_tm @ A @ B2 ) ) ) ).

% insert_Diff1
thf(fact_933_insert__Diff1,axiom,
    ! [X: fm,B2: set_fm,A: set_fm] :
      ( ( member_fm3 @ X @ B2 )
     => ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ B2 )
        = ( minus_minus_set_fm @ A @ B2 ) ) ) ).

% insert_Diff1
thf(fact_934_insert__Diff1,axiom,
    ! [X: nat,B2: set_nat,A: set_nat] :
      ( ( member_nat3 @ X @ B2 )
     => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B2 )
        = ( minus_minus_set_nat @ A @ B2 ) ) ) ).

% insert_Diff1
thf(fact_935_insert__Diff1,axiom,
    ! [X: rule,B2: set_rule,A: set_rule] :
      ( ( member_rule3 @ X @ B2 )
     => ( ( minus_minus_set_rule @ ( insert_rule2 @ X @ A ) @ B2 )
        = ( minus_minus_set_rule @ A @ B2 ) ) ) ).

% insert_Diff1
thf(fact_936_Diff__insert0,axiom,
    ! [X: list_fm,A: set_list_fm,B2: set_list_fm] :
      ( ~ ( member_list_fm3 @ X @ A )
     => ( ( minus_639611354763871680ist_fm @ A @ ( insert_list_fm2 @ X @ B2 ) )
        = ( minus_639611354763871680ist_fm @ A @ B2 ) ) ) ).

% Diff_insert0
thf(fact_937_Diff__insert0,axiom,
    ! [X: tm,A: set_tm,B2: set_tm] :
      ( ~ ( member_tm3 @ X @ A )
     => ( ( minus_minus_set_tm @ A @ ( insert_tm2 @ X @ B2 ) )
        = ( minus_minus_set_tm @ A @ B2 ) ) ) ).

% Diff_insert0
thf(fact_938_Diff__insert0,axiom,
    ! [X: fm,A: set_fm,B2: set_fm] :
      ( ~ ( member_fm3 @ X @ A )
     => ( ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ B2 ) )
        = ( minus_minus_set_fm @ A @ B2 ) ) ) ).

% Diff_insert0
thf(fact_939_Diff__insert0,axiom,
    ! [X: nat,A: set_nat,B2: set_nat] :
      ( ~ ( member_nat3 @ X @ A )
     => ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ B2 ) )
        = ( minus_minus_set_nat @ A @ B2 ) ) ) ).

% Diff_insert0
thf(fact_940_Diff__insert0,axiom,
    ! [X: rule,A: set_rule,B2: set_rule] :
      ( ~ ( member_rule3 @ X @ A )
     => ( ( minus_minus_set_rule @ A @ ( insert_rule2 @ X @ B2 ) )
        = ( minus_minus_set_rule @ A @ B2 ) ) ) ).

% Diff_insert0
thf(fact_941_singleton__insert__inj__eq,axiom,
    ! [B: rule,A2: rule,A: set_rule] :
      ( ( ( insert_rule2 @ B @ bot_bot_set_rule )
        = ( insert_rule2 @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_rule @ A @ ( insert_rule2 @ B @ bot_bot_set_rule ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_942_singleton__insert__inj__eq,axiom,
    ! [B: tm,A2: tm,A: set_tm] :
      ( ( ( insert_tm2 @ B @ bot_bot_set_tm )
        = ( insert_tm2 @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ B @ bot_bot_set_tm ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_943_singleton__insert__inj__eq,axiom,
    ! [B: nat,A2: nat,A: set_nat] :
      ( ( ( insert_nat2 @ B @ bot_bot_set_nat )
        = ( insert_nat2 @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_944_singleton__insert__inj__eq,axiom,
    ! [B: fm,A2: fm,A: set_fm] :
      ( ( ( insert_fm2 @ B @ bot_bot_set_fm )
        = ( insert_fm2 @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B @ bot_bot_set_fm ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_945_singleton__insert__inj__eq_H,axiom,
    ! [A2: rule,A: set_rule,B: rule] :
      ( ( ( insert_rule2 @ A2 @ A )
        = ( insert_rule2 @ B @ bot_bot_set_rule ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_rule @ A @ ( insert_rule2 @ B @ bot_bot_set_rule ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_946_singleton__insert__inj__eq_H,axiom,
    ! [A2: tm,A: set_tm,B: tm] :
      ( ( ( insert_tm2 @ A2 @ A )
        = ( insert_tm2 @ B @ bot_bot_set_tm ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ B @ bot_bot_set_tm ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_947_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B: nat] :
      ( ( ( insert_nat2 @ A2 @ A )
        = ( insert_nat2 @ B @ bot_bot_set_nat ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_948_singleton__insert__inj__eq_H,axiom,
    ! [A2: fm,A: set_fm,B: fm] :
      ( ( ( insert_fm2 @ A2 @ A )
        = ( insert_fm2 @ B @ bot_bot_set_fm ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ B @ bot_bot_set_fm ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_949_Diff__UNIV,axiom,
    ! [A: set_fm] :
      ( ( minus_minus_set_fm @ A @ top_top_set_fm )
      = bot_bot_set_fm ) ).

% Diff_UNIV
thf(fact_950_Diff__UNIV,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ top_top_set_nat )
      = bot_bot_set_nat ) ).

% Diff_UNIV
thf(fact_951_Diff__UNIV,axiom,
    ! [A: set_Pr5202636777678657877ist_fm] :
      ( ( minus_7584274679010293948ist_fm @ A @ top_to1730629564055774885ist_fm )
      = bot_bo1798318783834493889ist_fm ) ).

% Diff_UNIV
thf(fact_952_Diff__UNIV,axiom,
    ! [A: set_rule] :
      ( ( minus_minus_set_rule @ A @ top_top_set_rule )
      = bot_bot_set_rule ) ).

% Diff_UNIV
thf(fact_953_Diff__eq__empty__iff,axiom,
    ! [A: set_rule,B2: set_rule] :
      ( ( ( minus_minus_set_rule @ A @ B2 )
        = bot_bot_set_rule )
      = ( ord_less_eq_set_rule @ A @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_954_Diff__eq__empty__iff,axiom,
    ! [A: set_tm,B2: set_tm] :
      ( ( ( minus_minus_set_tm @ A @ B2 )
        = bot_bot_set_tm )
      = ( ord_less_eq_set_tm @ A @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_955_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( ( minus_minus_set_nat @ A @ B2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_956_Diff__eq__empty__iff,axiom,
    ! [A: set_fm,B2: set_fm] :
      ( ( ( minus_minus_set_fm @ A @ B2 )
        = bot_bot_set_fm )
      = ( ord_less_eq_set_fm @ A @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_957_insert__Diff__single,axiom,
    ! [A2: fm,A: set_fm] :
      ( ( insert_fm2 @ A2 @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) )
      = ( insert_fm2 @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_958_insert__Diff__single,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
      = ( insert_nat2 @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_959_insert__Diff__single,axiom,
    ! [A2: rule,A: set_rule] :
      ( ( insert_rule2 @ A2 @ ( minus_minus_set_rule @ A @ ( insert_rule2 @ A2 @ bot_bot_set_rule ) ) )
      = ( insert_rule2 @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_960_ipath_Ocases,axiom,
    ! [A1: abstract_tree_rule,A22: stream_rule] :
      ( ( abstract_ipath_rule @ A1 @ A22 )
     => ~ ( ( ( abstract_root_rule @ A1 )
            = ( shd_rule @ A22 ) )
         => ! [T4: abstract_tree_rule] :
              ( ( fmembe4260254584891149511e_rule @ T4 @ ( abstract_cont_rule @ A1 ) )
             => ~ ( abstract_ipath_rule @ T4 @ ( stl_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_961_ipath_Ocases,axiom,
    ! [A1: abstra8520858721168826610m_rule,A22: stream2709947120125613254m_rule] :
      ( ( abstra8836096083317999815m_rule @ A1 @ A22 )
     => ~ ( ( ( abstra5180277867158163941m_rule @ A1 )
            = ( shd_Pr4562317740776619530m_rule @ A22 ) )
         => ! [T4: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ T4 @ ( abstra2475333618431710837m_rule @ A1 ) )
             => ~ ( abstra8836096083317999815m_rule @ T4 @ ( stl_Pr12655793849929990m_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_962_ipath_Osimps,axiom,
    ( abstract_ipath_rule
    = ( ^ [A12: abstract_tree_rule,A23: stream_rule] :
        ? [T2: abstract_tree_rule,Steps2: stream_rule,T5: abstract_tree_rule] :
          ( ( A12 = T2 )
          & ( A23 = Steps2 )
          & ( ( abstract_root_rule @ T2 )
            = ( shd_rule @ Steps2 ) )
          & ( fmembe4260254584891149511e_rule @ T5 @ ( abstract_cont_rule @ T2 ) )
          & ( abstract_ipath_rule @ T5 @ ( stl_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_963_ipath_Osimps,axiom,
    ( abstra8836096083317999815m_rule
    = ( ^ [A12: abstra8520858721168826610m_rule,A23: stream2709947120125613254m_rule] :
        ? [T2: abstra8520858721168826610m_rule,Steps2: stream2709947120125613254m_rule,T5: abstra8520858721168826610m_rule] :
          ( ( A12 = T2 )
          & ( A23 = Steps2 )
          & ( ( abstra5180277867158163941m_rule @ T2 )
            = ( shd_Pr4562317740776619530m_rule @ Steps2 ) )
          & ( fmembe1099044791951439339m_rule @ T5 @ ( abstra2475333618431710837m_rule @ T2 ) )
          & ( abstra8836096083317999815m_rule @ T5 @ ( stl_Pr12655793849929990m_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_964_ipath_Ointros,axiom,
    ! [T: abstract_tree_rule,Steps: stream_rule,T3: abstract_tree_rule] :
      ( ( ( abstract_root_rule @ T )
        = ( shd_rule @ Steps ) )
     => ( ( fmembe4260254584891149511e_rule @ T3 @ ( abstract_cont_rule @ T ) )
       => ( ( abstract_ipath_rule @ T3 @ ( stl_rule @ Steps ) )
         => ( abstract_ipath_rule @ T @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_965_ipath_Ointros,axiom,
    ! [T: abstra8520858721168826610m_rule,Steps: stream2709947120125613254m_rule,T3: abstra8520858721168826610m_rule] :
      ( ( ( abstra5180277867158163941m_rule @ T )
        = ( shd_Pr4562317740776619530m_rule @ Steps ) )
     => ( ( fmembe1099044791951439339m_rule @ T3 @ ( abstra2475333618431710837m_rule @ T ) )
       => ( ( abstra8836096083317999815m_rule @ T3 @ ( stl_Pr12655793849929990m_rule @ Steps ) )
         => ( abstra8836096083317999815m_rule @ T @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_966_ipath_Ocoinduct,axiom,
    ! [X4: abstract_tree_rule > stream_rule > $o,X1: abstract_tree_rule,X23: stream_rule] :
      ( ( X4 @ X1 @ X23 )
     => ( ! [X13: abstract_tree_rule,X24: stream_rule] :
            ( ( X4 @ X13 @ X24 )
           => ? [T6: abstract_tree_rule,Steps3: stream_rule,T7: abstract_tree_rule] :
                ( ( X13 = T6 )
                & ( X24 = Steps3 )
                & ( ( abstract_root_rule @ T6 )
                  = ( shd_rule @ Steps3 ) )
                & ( fmembe4260254584891149511e_rule @ T7 @ ( abstract_cont_rule @ T6 ) )
                & ( ( X4 @ T7 @ ( stl_rule @ Steps3 ) )
                  | ( abstract_ipath_rule @ T7 @ ( stl_rule @ Steps3 ) ) ) ) )
       => ( abstract_ipath_rule @ X1 @ X23 ) ) ) ).

% ipath.coinduct
thf(fact_967_ipath_Ocoinduct,axiom,
    ! [X4: abstra8520858721168826610m_rule > stream2709947120125613254m_rule > $o,X1: abstra8520858721168826610m_rule,X23: stream2709947120125613254m_rule] :
      ( ( X4 @ X1 @ X23 )
     => ( ! [X13: abstra8520858721168826610m_rule,X24: stream2709947120125613254m_rule] :
            ( ( X4 @ X13 @ X24 )
           => ? [T6: abstra8520858721168826610m_rule,Steps3: stream2709947120125613254m_rule,T7: abstra8520858721168826610m_rule] :
                ( ( X13 = T6 )
                & ( X24 = Steps3 )
                & ( ( abstra5180277867158163941m_rule @ T6 )
                  = ( shd_Pr4562317740776619530m_rule @ Steps3 ) )
                & ( fmembe1099044791951439339m_rule @ T7 @ ( abstra2475333618431710837m_rule @ T6 ) )
                & ( ( X4 @ T7 @ ( stl_Pr12655793849929990m_rule @ Steps3 ) )
                  | ( abstra8836096083317999815m_rule @ T7 @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) ) )
       => ( abstra8836096083317999815m_rule @ X1 @ X23 ) ) ) ).

% ipath.coinduct
thf(fact_968_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( stl_Pr2595822214614905141e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_969_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_Pr8529079933065201753m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
      = ( szip_r6533552703345879972m_rule @ ( stl_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_970_szip_Osimps_I2_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream_rule] :
      ( ( stl_Pr497917230009538393e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
      = ( szip_P2044787555563963556e_rule @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_971_szip_Osimps_I2_J,axiom,
    ! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_Pr2679468901532635773m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
      = ( szip_P811719526838699976m_rule @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_972_szip_Osimps_I2_J,axiom,
    ! [S1: stream4408948924543953275ist_fm,S2: stream_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
      = ( szip_P2924820683901490861m_rule @ ( stl_Pr1506262294867171877ist_fm @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_973_DiffE,axiom,
    ! [C: list_fm,A: set_list_fm,B2: set_list_fm] :
      ( ( member_list_fm3 @ C @ ( minus_639611354763871680ist_fm @ A @ B2 ) )
     => ~ ( ( member_list_fm3 @ C @ A )
         => ( member_list_fm3 @ C @ B2 ) ) ) ).

% DiffE
thf(fact_974_DiffE,axiom,
    ! [C: tm,A: set_tm,B2: set_tm] :
      ( ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B2 ) )
     => ~ ( ( member_tm3 @ C @ A )
         => ( member_tm3 @ C @ B2 ) ) ) ).

% DiffE
thf(fact_975_DiffE,axiom,
    ! [C: fm,A: set_fm,B2: set_fm] :
      ( ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B2 ) )
     => ~ ( ( member_fm3 @ C @ A )
         => ( member_fm3 @ C @ B2 ) ) ) ).

% DiffE
thf(fact_976_DiffE,axiom,
    ! [C: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B2 ) )
     => ~ ( ( member_nat3 @ C @ A )
         => ( member_nat3 @ C @ B2 ) ) ) ).

% DiffE
thf(fact_977_DiffE,axiom,
    ! [C: rule,A: set_rule,B2: set_rule] :
      ( ( member_rule3 @ C @ ( minus_minus_set_rule @ A @ B2 ) )
     => ~ ( ( member_rule3 @ C @ A )
         => ( member_rule3 @ C @ B2 ) ) ) ).

% DiffE
thf(fact_978_DiffD1,axiom,
    ! [C: list_fm,A: set_list_fm,B2: set_list_fm] :
      ( ( member_list_fm3 @ C @ ( minus_639611354763871680ist_fm @ A @ B2 ) )
     => ( member_list_fm3 @ C @ A ) ) ).

% DiffD1
thf(fact_979_DiffD1,axiom,
    ! [C: tm,A: set_tm,B2: set_tm] :
      ( ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B2 ) )
     => ( member_tm3 @ C @ A ) ) ).

% DiffD1
thf(fact_980_DiffD1,axiom,
    ! [C: fm,A: set_fm,B2: set_fm] :
      ( ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B2 ) )
     => ( member_fm3 @ C @ A ) ) ).

% DiffD1
thf(fact_981_DiffD1,axiom,
    ! [C: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B2 ) )
     => ( member_nat3 @ C @ A ) ) ).

% DiffD1
thf(fact_982_DiffD1,axiom,
    ! [C: rule,A: set_rule,B2: set_rule] :
      ( ( member_rule3 @ C @ ( minus_minus_set_rule @ A @ B2 ) )
     => ( member_rule3 @ C @ A ) ) ).

% DiffD1
thf(fact_983_DiffD2,axiom,
    ! [C: list_fm,A: set_list_fm,B2: set_list_fm] :
      ( ( member_list_fm3 @ C @ ( minus_639611354763871680ist_fm @ A @ B2 ) )
     => ~ ( member_list_fm3 @ C @ B2 ) ) ).

% DiffD2
thf(fact_984_DiffD2,axiom,
    ! [C: tm,A: set_tm,B2: set_tm] :
      ( ( member_tm3 @ C @ ( minus_minus_set_tm @ A @ B2 ) )
     => ~ ( member_tm3 @ C @ B2 ) ) ).

% DiffD2
thf(fact_985_DiffD2,axiom,
    ! [C: fm,A: set_fm,B2: set_fm] :
      ( ( member_fm3 @ C @ ( minus_minus_set_fm @ A @ B2 ) )
     => ~ ( member_fm3 @ C @ B2 ) ) ).

% DiffD2
thf(fact_986_DiffD2,axiom,
    ! [C: nat,A: set_nat,B2: set_nat] :
      ( ( member_nat3 @ C @ ( minus_minus_set_nat @ A @ B2 ) )
     => ~ ( member_nat3 @ C @ B2 ) ) ).

% DiffD2
thf(fact_987_DiffD2,axiom,
    ! [C: rule,A: set_rule,B2: set_rule] :
      ( ( member_rule3 @ C @ ( minus_minus_set_rule @ A @ B2 ) )
     => ~ ( member_rule3 @ C @ B2 ) ) ).

% DiffD2
thf(fact_988_emptyE,axiom,
    ! [A2: list_fm] :
      ~ ( member_list_fm3 @ A2 @ bot_bot_set_list_fm ) ).

% emptyE
thf(fact_989_emptyE,axiom,
    ! [A2: tm] :
      ~ ( member_tm3 @ A2 @ bot_bot_set_tm ) ).

% emptyE
thf(fact_990_emptyE,axiom,
    ! [A2: rule] :
      ~ ( member_rule3 @ A2 @ bot_bot_set_rule ) ).

% emptyE
thf(fact_991_emptyE,axiom,
    ! [A2: fm] :
      ~ ( member_fm3 @ A2 @ bot_bot_set_fm ) ).

% emptyE
thf(fact_992_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat3 @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_993_equals0D,axiom,
    ! [A: set_list_fm,A2: list_fm] :
      ( ( A = bot_bot_set_list_fm )
     => ~ ( member_list_fm3 @ A2 @ A ) ) ).

% equals0D
thf(fact_994_equals0D,axiom,
    ! [A: set_tm,A2: tm] :
      ( ( A = bot_bot_set_tm )
     => ~ ( member_tm3 @ A2 @ A ) ) ).

% equals0D
thf(fact_995_equals0D,axiom,
    ! [A: set_rule,A2: rule] :
      ( ( A = bot_bot_set_rule )
     => ~ ( member_rule3 @ A2 @ A ) ) ).

% equals0D
thf(fact_996_equals0D,axiom,
    ! [A: set_fm,A2: fm] :
      ( ( A = bot_bot_set_fm )
     => ~ ( member_fm3 @ A2 @ A ) ) ).

% equals0D
thf(fact_997_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat3 @ A2 @ A ) ) ).

% equals0D
thf(fact_998_equals0I,axiom,
    ! [A: set_list_fm] :
      ( ! [Y4: list_fm] :
          ~ ( member_list_fm3 @ Y4 @ A )
     => ( A = bot_bot_set_list_fm ) ) ).

% equals0I
thf(fact_999_equals0I,axiom,
    ! [A: set_tm] :
      ( ! [Y4: tm] :
          ~ ( member_tm3 @ Y4 @ A )
     => ( A = bot_bot_set_tm ) ) ).

% equals0I
thf(fact_1000_equals0I,axiom,
    ! [A: set_rule] :
      ( ! [Y4: rule] :
          ~ ( member_rule3 @ Y4 @ A )
     => ( A = bot_bot_set_rule ) ) ).

% equals0I
thf(fact_1001_equals0I,axiom,
    ! [A: set_fm] :
      ( ! [Y4: fm] :
          ~ ( member_fm3 @ Y4 @ A )
     => ( A = bot_bot_set_fm ) ) ).

% equals0I
thf(fact_1002_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat3 @ Y4 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_1003_ex__in__conv,axiom,
    ! [A: set_list_fm] :
      ( ( ? [X2: list_fm] : ( member_list_fm3 @ X2 @ A ) )
      = ( A != bot_bot_set_list_fm ) ) ).

% ex_in_conv
thf(fact_1004_ex__in__conv,axiom,
    ! [A: set_tm] :
      ( ( ? [X2: tm] : ( member_tm3 @ X2 @ A ) )
      = ( A != bot_bot_set_tm ) ) ).

% ex_in_conv
thf(fact_1005_ex__in__conv,axiom,
    ! [A: set_rule] :
      ( ( ? [X2: rule] : ( member_rule3 @ X2 @ A ) )
      = ( A != bot_bot_set_rule ) ) ).

% ex_in_conv
thf(fact_1006_ex__in__conv,axiom,
    ! [A: set_fm] :
      ( ( ? [X2: fm] : ( member_fm3 @ X2 @ A ) )
      = ( A != bot_bot_set_fm ) ) ).

% ex_in_conv
thf(fact_1007_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X2: nat] : ( member_nat3 @ X2 @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_1008_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_1009_append__eq__appendI,axiom,
    ! [Xs: list_tm,Xs1: list_tm,Zs: list_tm,Ys: list_tm,Us: list_tm] :
      ( ( ( append_tm @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_tm @ Xs1 @ Us ) )
       => ( ( append_tm @ Xs @ Ys )
          = ( append_tm @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_1010_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_1011_append__eq__append__conv2,axiom,
    ! [Xs: list_tm,Ys: list_tm,Zs: list_tm,Ts: list_tm] :
      ( ( ( append_tm @ Xs @ Ys )
        = ( append_tm @ Zs @ Ts ) )
      = ( ? [Us2: list_tm] :
            ( ( ( Xs
                = ( append_tm @ Zs @ Us2 ) )
              & ( ( append_tm @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_tm @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_tm @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_1012_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_1013_Diff__insert__absorb,axiom,
    ! [X: list_fm,A: set_list_fm] :
      ( ~ ( member_list_fm3 @ X @ A )
     => ( ( minus_639611354763871680ist_fm @ ( insert_list_fm2 @ X @ A ) @ ( insert_list_fm2 @ X @ bot_bot_set_list_fm ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1014_Diff__insert__absorb,axiom,
    ! [X: tm,A: set_tm] :
      ( ~ ( member_tm3 @ X @ A )
     => ( ( minus_minus_set_tm @ ( insert_tm2 @ X @ A ) @ ( insert_tm2 @ X @ bot_bot_set_tm ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1015_Diff__insert__absorb,axiom,
    ! [X: fm,A: set_fm] :
      ( ~ ( member_fm3 @ X @ A )
     => ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ ( insert_fm2 @ X @ bot_bot_set_fm ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1016_Diff__insert__absorb,axiom,
    ! [X: nat,A: set_nat] :
      ( ~ ( member_nat3 @ X @ A )
     => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1017_Diff__insert__absorb,axiom,
    ! [X: rule,A: set_rule] :
      ( ~ ( member_rule3 @ X @ A )
     => ( ( minus_minus_set_rule @ ( insert_rule2 @ X @ A ) @ ( insert_rule2 @ X @ bot_bot_set_rule ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1018_Diff__insert2,axiom,
    ! [A: set_fm,A2: fm,B2: set_fm] :
      ( ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ B2 ) )
      = ( minus_minus_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_1019_Diff__insert2,axiom,
    ! [A: set_nat,A2: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_1020_Diff__insert2,axiom,
    ! [A: set_rule,A2: rule,B2: set_rule] :
      ( ( minus_minus_set_rule @ A @ ( insert_rule2 @ A2 @ B2 ) )
      = ( minus_minus_set_rule @ ( minus_minus_set_rule @ A @ ( insert_rule2 @ A2 @ bot_bot_set_rule ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_1021_insert__Diff,axiom,
    ! [A2: list_fm,A: set_list_fm] :
      ( ( member_list_fm3 @ A2 @ A )
     => ( ( insert_list_fm2 @ A2 @ ( minus_639611354763871680ist_fm @ A @ ( insert_list_fm2 @ A2 @ bot_bot_set_list_fm ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1022_insert__Diff,axiom,
    ! [A2: tm,A: set_tm] :
      ( ( member_tm3 @ A2 @ A )
     => ( ( insert_tm2 @ A2 @ ( minus_minus_set_tm @ A @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1023_insert__Diff,axiom,
    ! [A2: fm,A: set_fm] :
      ( ( member_fm3 @ A2 @ A )
     => ( ( insert_fm2 @ A2 @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1024_insert__Diff,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat3 @ A2 @ A )
     => ( ( insert_nat2 @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1025_insert__Diff,axiom,
    ! [A2: rule,A: set_rule] :
      ( ( member_rule3 @ A2 @ A )
     => ( ( insert_rule2 @ A2 @ ( minus_minus_set_rule @ A @ ( insert_rule2 @ A2 @ bot_bot_set_rule ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1026_Diff__insert,axiom,
    ! [A: set_fm,A2: fm,B2: set_fm] :
      ( ( minus_minus_set_fm @ A @ ( insert_fm2 @ A2 @ B2 ) )
      = ( minus_minus_set_fm @ ( minus_minus_set_fm @ A @ B2 ) @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) ) ) ).

% Diff_insert
thf(fact_1027_Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat2 @ A2 @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_1028_Diff__insert,axiom,
    ! [A: set_rule,A2: rule,B2: set_rule] :
      ( ( minus_minus_set_rule @ A @ ( insert_rule2 @ A2 @ B2 ) )
      = ( minus_minus_set_rule @ ( minus_minus_set_rule @ A @ B2 ) @ ( insert_rule2 @ A2 @ bot_bot_set_rule ) ) ) ).

% Diff_insert
thf(fact_1029_subset__insert__iff,axiom,
    ! [A: set_list_fm,X: list_fm,B2: set_list_fm] :
      ( ( ord_le7838213414353715577ist_fm @ A @ ( insert_list_fm2 @ X @ B2 ) )
      = ( ( ( member_list_fm3 @ X @ A )
         => ( ord_le7838213414353715577ist_fm @ ( minus_639611354763871680ist_fm @ A @ ( insert_list_fm2 @ X @ bot_bot_set_list_fm ) ) @ B2 ) )
        & ( ~ ( member_list_fm3 @ X @ A )
         => ( ord_le7838213414353715577ist_fm @ A @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_1030_subset__insert__iff,axiom,
    ! [A: set_rule,X: rule,B2: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ ( insert_rule2 @ X @ B2 ) )
      = ( ( ( member_rule3 @ X @ A )
         => ( ord_less_eq_set_rule @ ( minus_minus_set_rule @ A @ ( insert_rule2 @ X @ bot_bot_set_rule ) ) @ B2 ) )
        & ( ~ ( member_rule3 @ X @ A )
         => ( ord_less_eq_set_rule @ A @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_1031_subset__insert__iff,axiom,
    ! [A: set_tm,X: tm,B2: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ X @ B2 ) )
      = ( ( ( member_tm3 @ X @ A )
         => ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A @ ( insert_tm2 @ X @ bot_bot_set_tm ) ) @ B2 ) )
        & ( ~ ( member_tm3 @ X @ A )
         => ( ord_less_eq_set_tm @ A @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_1032_subset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B2 ) )
      = ( ( ( member_nat3 @ X @ A )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) )
        & ( ~ ( member_nat3 @ X @ A )
         => ( ord_less_eq_set_nat @ A @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_1033_subset__insert__iff,axiom,
    ! [A: set_fm,X: fm,B2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B2 ) )
      = ( ( ( member_fm3 @ X @ A )
         => ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) @ B2 ) )
        & ( ~ ( member_fm3 @ X @ A )
         => ( ord_less_eq_set_fm @ A @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_1034_Diff__single__insert,axiom,
    ! [A: set_rule,X: rule,B2: set_rule] :
      ( ( ord_less_eq_set_rule @ ( minus_minus_set_rule @ A @ ( insert_rule2 @ X @ bot_bot_set_rule ) ) @ B2 )
     => ( ord_less_eq_set_rule @ A @ ( insert_rule2 @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_1035_Diff__single__insert,axiom,
    ! [A: set_tm,X: tm,B2: set_tm] :
      ( ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A @ ( insert_tm2 @ X @ bot_bot_set_tm ) ) @ B2 )
     => ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_1036_Diff__single__insert,axiom,
    ! [A: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_1037_Diff__single__insert,axiom,
    ! [A: set_fm,X: fm,B2: set_fm] :
      ( ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ ( insert_fm2 @ X @ bot_bot_set_fm ) ) @ B2 )
     => ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_1038_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ bot_bot_set_rule )
     => ( A2 = bot_bot_set_rule ) ) ).

% bot.extremum_uniqueI
thf(fact_1039_bot_Oextremum__uniqueI,axiom,
    ! [A2: fset_P8989946509869081563ist_fm] :
      ( ( ord_le3986950534092794747ist_fm @ A2 @ bot_bo6461889142629771335ist_fm )
     => ( A2 = bot_bo6461889142629771335ist_fm ) ) ).

% bot.extremum_uniqueI
thf(fact_1040_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ bot_bot_set_tm )
     => ( A2 = bot_bot_set_tm ) ) ).

% bot.extremum_uniqueI
thf(fact_1041_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1042_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1043_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
     => ( A2 = bot_bot_set_fm ) ) ).

% bot.extremum_uniqueI
thf(fact_1044_bot_Oextremum__uniqueI,axiom,
    ! [A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ bot_bo4199563552545308370d_enat )
     => ( A2 = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_uniqueI
thf(fact_1045_bot_Oextremum__unique,axiom,
    ! [A2: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ bot_bot_set_rule )
      = ( A2 = bot_bot_set_rule ) ) ).

% bot.extremum_unique
thf(fact_1046_bot_Oextremum__unique,axiom,
    ! [A2: fset_P8989946509869081563ist_fm] :
      ( ( ord_le3986950534092794747ist_fm @ A2 @ bot_bo6461889142629771335ist_fm )
      = ( A2 = bot_bo6461889142629771335ist_fm ) ) ).

% bot.extremum_unique
thf(fact_1047_bot_Oextremum__unique,axiom,
    ! [A2: set_tm] :
      ( ( ord_less_eq_set_tm @ A2 @ bot_bot_set_tm )
      = ( A2 = bot_bot_set_tm ) ) ).

% bot.extremum_unique
thf(fact_1048_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_1049_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1050_bot_Oextremum__unique,axiom,
    ! [A2: set_fm] :
      ( ( ord_less_eq_set_fm @ A2 @ bot_bot_set_fm )
      = ( A2 = bot_bot_set_fm ) ) ).

% bot.extremum_unique
thf(fact_1051_bot_Oextremum__unique,axiom,
    ! [A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ bot_bo4199563552545308370d_enat )
      = ( A2 = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_unique
thf(fact_1052_bot_Oextremum,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ bot_bot_set_rule @ A2 ) ).

% bot.extremum
thf(fact_1053_bot_Oextremum,axiom,
    ! [A2: fset_P8989946509869081563ist_fm] : ( ord_le3986950534092794747ist_fm @ bot_bo6461889142629771335ist_fm @ A2 ) ).

% bot.extremum
thf(fact_1054_bot_Oextremum,axiom,
    ! [A2: set_tm] : ( ord_less_eq_set_tm @ bot_bot_set_tm @ A2 ) ).

% bot.extremum
thf(fact_1055_bot_Oextremum,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% bot.extremum
thf(fact_1056_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_1057_bot_Oextremum,axiom,
    ! [A2: set_fm] : ( ord_less_eq_set_fm @ bot_bot_set_fm @ A2 ) ).

% bot.extremum
thf(fact_1058_bot_Oextremum,axiom,
    ! [A2: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A2 ) ).

% bot.extremum
thf(fact_1059_append__Cons,axiom,
    ! [X: tm,Xs: list_tm,Ys: list_tm] :
      ( ( append_tm @ ( cons_tm @ X @ Xs ) @ Ys )
      = ( cons_tm @ X @ ( append_tm @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_1060_append__Cons,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
      = ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_1061_append__Cons,axiom,
    ! [X: list_fm,Xs: list_list_fm,Ys: list_list_fm] :
      ( ( append_list_fm @ ( cons_list_fm @ X @ Xs ) @ Ys )
      = ( cons_list_fm @ X @ ( append_list_fm @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_1062_append__Cons,axiom,
    ! [X: fm,Xs: list_fm,Ys: list_fm] :
      ( ( append_fm @ ( cons_fm @ X @ Xs ) @ Ys )
      = ( cons_fm @ X @ ( append_fm @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_1063_Cons__eq__appendI,axiom,
    ! [X: tm,Xs1: list_tm,Ys: list_tm,Xs: list_tm,Zs: list_tm] :
      ( ( ( cons_tm @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_tm @ Xs1 @ Zs ) )
       => ( ( cons_tm @ X @ Xs )
          = ( append_tm @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_1064_Cons__eq__appendI,axiom,
    ! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs ) )
       => ( ( cons_nat @ X @ Xs )
          = ( append_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_1065_Cons__eq__appendI,axiom,
    ! [X: list_fm,Xs1: list_list_fm,Ys: list_list_fm,Xs: list_list_fm,Zs: list_list_fm] :
      ( ( ( cons_list_fm @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_list_fm @ Xs1 @ Zs ) )
       => ( ( cons_list_fm @ X @ Xs )
          = ( append_list_fm @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_1066_Cons__eq__appendI,axiom,
    ! [X: fm,Xs1: list_fm,Ys: list_fm,Xs: list_fm,Zs: list_fm] :
      ( ( ( cons_fm @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_fm @ Xs1 @ Zs ) )
       => ( ( cons_fm @ X @ Xs )
          = ( append_fm @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_1067_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_1068_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_1069_stream_Ocoinduct,axiom,
    ! [R3: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( R3 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_1070_stream_Ocoinduct,axiom,
    ! [R3: stream2709947120125613254m_rule > stream2709947120125613254m_rule > $o,Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream2709947120125613254m_rule,Stream4: stream2709947120125613254m_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr4562317740776619530m_rule @ Stream3 )
                = ( shd_Pr4562317740776619530m_rule @ Stream4 ) )
              & ( R3 @ ( stl_Pr12655793849929990m_rule @ Stream3 ) @ ( stl_Pr12655793849929990m_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_1071_stream_Ocoinduct__strong,axiom,
    ! [R3: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( ( R3 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) )
                | ( ( stl_rule @ Stream3 )
                  = ( stl_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_1072_stream_Ocoinduct__strong,axiom,
    ! [R3: stream2709947120125613254m_rule > stream2709947120125613254m_rule > $o,Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream2709947120125613254m_rule,Stream4: stream2709947120125613254m_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr4562317740776619530m_rule @ Stream3 )
                = ( shd_Pr4562317740776619530m_rule @ Stream4 ) )
              & ( ( R3 @ ( stl_Pr12655793849929990m_rule @ Stream3 ) @ ( stl_Pr12655793849929990m_rule @ Stream4 ) )
                | ( ( stl_Pr12655793849929990m_rule @ Stream3 )
                  = ( stl_Pr12655793849929990m_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_1073_stl__sset,axiom,
    ! [X: list_fm,A2: stream_list_fm] :
      ( ( member_list_fm3 @ X @ ( sset_list_fm @ ( stl_list_fm @ A2 ) ) )
     => ( member_list_fm3 @ X @ ( sset_list_fm @ A2 ) ) ) ).

% stl_sset
thf(fact_1074_stl__sset,axiom,
    ! [X: tm,A2: stream_tm] :
      ( ( member_tm3 @ X @ ( sset_tm @ ( stl_tm @ A2 ) ) )
     => ( member_tm3 @ X @ ( sset_tm @ A2 ) ) ) ).

% stl_sset
thf(fact_1075_stl__sset,axiom,
    ! [X: fm,A2: stream_fm] :
      ( ( member_fm3 @ X @ ( sset_fm @ ( stl_fm @ A2 ) ) )
     => ( member_fm3 @ X @ ( sset_fm @ A2 ) ) ) ).

% stl_sset
thf(fact_1076_stl__sset,axiom,
    ! [X: nat,A2: stream_nat] :
      ( ( member_nat3 @ X @ ( sset_nat @ ( stl_nat @ A2 ) ) )
     => ( member_nat3 @ X @ ( sset_nat @ A2 ) ) ) ).

% stl_sset
thf(fact_1077_stl__sset,axiom,
    ! [X: produc340336539035504054m_rule,A2: stream2709947120125613254m_rule] :
      ( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ ( stl_Pr12655793849929990m_rule @ A2 ) ) )
     => ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ A2 ) ) ) ).

% stl_sset
thf(fact_1078_stl__sset,axiom,
    ! [X: rule,A2: stream_rule] :
      ( ( member_rule3 @ X @ ( sset_rule @ ( stl_rule @ A2 ) ) )
     => ( member_rule3 @ X @ ( sset_rule @ A2 ) ) ) ).

% stl_sset
thf(fact_1079_double__diff,axiom,
    ! [A: set_rule,B2: set_rule,C2: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B2 )
     => ( ( ord_less_eq_set_rule @ B2 @ C2 )
       => ( ( minus_minus_set_rule @ B2 @ ( minus_minus_set_rule @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_1080_double__diff,axiom,
    ! [A: set_tm,B2: set_tm,C2: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ B2 )
     => ( ( ord_less_eq_set_tm @ B2 @ C2 )
       => ( ( minus_minus_set_tm @ B2 @ ( minus_minus_set_tm @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_1081_double__diff,axiom,
    ! [A: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_1082_double__diff,axiom,
    ! [A: set_fm,B2: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B2 )
     => ( ( ord_less_eq_set_fm @ B2 @ C2 )
       => ( ( minus_minus_set_fm @ B2 @ ( minus_minus_set_fm @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_1083_Diff__subset,axiom,
    ! [A: set_rule,B2: set_rule] : ( ord_less_eq_set_rule @ ( minus_minus_set_rule @ A @ B2 ) @ A ) ).

% Diff_subset
thf(fact_1084_Diff__subset,axiom,
    ! [A: set_tm,B2: set_tm] : ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A @ B2 ) @ A ) ).

% Diff_subset
thf(fact_1085_Diff__subset,axiom,
    ! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ A ) ).

% Diff_subset
thf(fact_1086_Diff__subset,axiom,
    ! [A: set_fm,B2: set_fm] : ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ B2 ) @ A ) ).

% Diff_subset
thf(fact_1087_Diff__mono,axiom,
    ! [A: set_rule,C2: set_rule,D: set_rule,B2: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ C2 )
     => ( ( ord_less_eq_set_rule @ D @ B2 )
       => ( ord_less_eq_set_rule @ ( minus_minus_set_rule @ A @ B2 ) @ ( minus_minus_set_rule @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_1088_Diff__mono,axiom,
    ! [A: set_tm,C2: set_tm,D: set_tm,B2: set_tm] :
      ( ( ord_less_eq_set_tm @ A @ C2 )
     => ( ( ord_less_eq_set_tm @ D @ B2 )
       => ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A @ B2 ) @ ( minus_minus_set_tm @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_1089_Diff__mono,axiom,
    ! [A: set_nat,C2: set_nat,D: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ D @ B2 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_1090_Diff__mono,axiom,
    ! [A: set_fm,C2: set_fm,D: set_fm,B2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ C2 )
     => ( ( ord_less_eq_set_fm @ D @ B2 )
       => ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A @ B2 ) @ ( minus_minus_set_fm @ C2 @ D ) ) ) ) ).

% Diff_mono
thf(fact_1091_sdrop__stl,axiom,
    ! [N: nat,S: stream4408948924543953275ist_fm] :
      ( ( sdrop_9176333610110415838ist_fm @ N @ ( stl_Pr1506262294867171877ist_fm @ S ) )
      = ( stl_Pr1506262294867171877ist_fm @ ( sdrop_9176333610110415838ist_fm @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_1092_sdrop__stl,axiom,
    ! [N: nat,S: stream4385846686851721995e_rule] :
      ( ( sdrop_9113879250048157294e_rule @ N @ ( stl_Pr2595822214614905141e_rule @ S ) )
      = ( stl_Pr2595822214614905141e_rule @ ( sdrop_9113879250048157294e_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_1093_sdrop__stl,axiom,
    ! [N: nat,S: stream8953843411776101167m_rule] :
      ( ( sdrop_746751008794826386m_rule @ N @ ( stl_Pr8529079933065201753m_rule @ S ) )
      = ( stl_Pr8529079933065201753m_rule @ ( sdrop_746751008794826386m_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_1094_sdrop__stl,axiom,
    ! [N: nat,S: stream8099677779113257519e_rule] :
      ( ( sdrop_1938960342593938834e_rule @ N @ ( stl_Pr497917230009538393e_rule @ S ) )
      = ( stl_Pr497917230009538393e_rule @ ( sdrop_1938960342593938834e_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_1095_sdrop__stl,axiom,
    ! [N: nat,S: stream6210534828274662995m_rule] :
      ( ( sdrop_7192298464603511222m_rule @ N @ ( stl_Pr2679468901532635773m_rule @ S ) )
      = ( stl_Pr2679468901532635773m_rule @ ( sdrop_7192298464603511222m_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_1096_sdrop__stl,axiom,
    ! [N: nat,S: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_1097_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_1098_insert__Diff__if,axiom,
    ! [X: list_fm,B2: set_list_fm,A: set_list_fm] :
      ( ( ( member_list_fm3 @ X @ B2 )
       => ( ( minus_639611354763871680ist_fm @ ( insert_list_fm2 @ X @ A ) @ B2 )
          = ( minus_639611354763871680ist_fm @ A @ B2 ) ) )
      & ( ~ ( member_list_fm3 @ X @ B2 )
       => ( ( minus_639611354763871680ist_fm @ ( insert_list_fm2 @ X @ A ) @ B2 )
          = ( insert_list_fm2 @ X @ ( minus_639611354763871680ist_fm @ A @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1099_insert__Diff__if,axiom,
    ! [X: tm,B2: set_tm,A: set_tm] :
      ( ( ( member_tm3 @ X @ B2 )
       => ( ( minus_minus_set_tm @ ( insert_tm2 @ X @ A ) @ B2 )
          = ( minus_minus_set_tm @ A @ B2 ) ) )
      & ( ~ ( member_tm3 @ X @ B2 )
       => ( ( minus_minus_set_tm @ ( insert_tm2 @ X @ A ) @ B2 )
          = ( insert_tm2 @ X @ ( minus_minus_set_tm @ A @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1100_insert__Diff__if,axiom,
    ! [X: fm,B2: set_fm,A: set_fm] :
      ( ( ( member_fm3 @ X @ B2 )
       => ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ B2 )
          = ( minus_minus_set_fm @ A @ B2 ) ) )
      & ( ~ ( member_fm3 @ X @ B2 )
       => ( ( minus_minus_set_fm @ ( insert_fm2 @ X @ A ) @ B2 )
          = ( insert_fm2 @ X @ ( minus_minus_set_fm @ A @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1101_insert__Diff__if,axiom,
    ! [X: nat,B2: set_nat,A: set_nat] :
      ( ( ( member_nat3 @ X @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B2 )
          = ( minus_minus_set_nat @ A @ B2 ) ) )
      & ( ~ ( member_nat3 @ X @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A ) @ B2 )
          = ( insert_nat2 @ X @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1102_insert__Diff__if,axiom,
    ! [X: rule,B2: set_rule,A: set_rule] :
      ( ( ( member_rule3 @ X @ B2 )
       => ( ( minus_minus_set_rule @ ( insert_rule2 @ X @ A ) @ B2 )
          = ( minus_minus_set_rule @ A @ B2 ) ) )
      & ( ~ ( member_rule3 @ X @ B2 )
       => ( ( minus_minus_set_rule @ ( insert_rule2 @ X @ A ) @ B2 )
          = ( insert_rule2 @ X @ ( minus_minus_set_rule @ A @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1103_i_Ofair__stl,axiom,
    ! [Rules2: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule] :
      ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
     => ( abstra3665372904625986210m_rule @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_1104_i_Ofair__stl,axiom,
    ! [Rules2: stream_rule,Rs: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
     => ( abstra3799686578551160190r_rule @ Rules2 @ ( stl_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_1105_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules2: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule] :
      ( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
     => ( abstra3665372904625986210m_rule @ Rules2 @ ( stl_Pr12655793849929990m_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_1106_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules2: stream_rule,Rs: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
     => ( abstra3799686578551160190r_rule @ Rules2 @ ( stl_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_1107_empty__not__UNIV,axiom,
    bot_bot_set_rule != top_top_set_rule ).

% empty_not_UNIV
thf(fact_1108_empty__not__UNIV,axiom,
    bot_bot_set_fm != top_top_set_fm ).

% empty_not_UNIV
thf(fact_1109_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_1110_empty__not__UNIV,axiom,
    bot_bo1798318783834493889ist_fm != top_to1730629564055774885ist_fm ).

% empty_not_UNIV
thf(fact_1111_subset__emptyI,axiom,
    ! [A: set_list_fm] :
      ( ! [X3: list_fm] :
          ~ ( member_list_fm3 @ X3 @ A )
     => ( ord_le7838213414353715577ist_fm @ A @ bot_bot_set_list_fm ) ) ).

% subset_emptyI
thf(fact_1112_subset__emptyI,axiom,
    ! [A: set_rule] :
      ( ! [X3: rule] :
          ~ ( member_rule3 @ X3 @ A )
     => ( ord_less_eq_set_rule @ A @ bot_bot_set_rule ) ) ).

% subset_emptyI
thf(fact_1113_subset__emptyI,axiom,
    ! [A: set_tm] :
      ( ! [X3: tm] :
          ~ ( member_tm3 @ X3 @ A )
     => ( ord_less_eq_set_tm @ A @ bot_bot_set_tm ) ) ).

% subset_emptyI
thf(fact_1114_subset__emptyI,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat3 @ X3 @ A )
     => ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_1115_subset__emptyI,axiom,
    ! [A: set_fm] :
      ( ! [X3: fm] :
          ~ ( member_fm3 @ X3 @ A )
     => ( ord_less_eq_set_fm @ A @ bot_bot_set_fm ) ) ).

% subset_emptyI
thf(fact_1116_singleton__inject,axiom,
    ! [A2: rule,B: rule] :
      ( ( ( insert_rule2 @ A2 @ bot_bot_set_rule )
        = ( insert_rule2 @ B @ bot_bot_set_rule ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_1117_singleton__inject,axiom,
    ! [A2: fm,B: fm] :
      ( ( ( insert_fm2 @ A2 @ bot_bot_set_fm )
        = ( insert_fm2 @ B @ bot_bot_set_fm ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_1118_singleton__inject,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( insert_nat2 @ A2 @ bot_bot_set_nat )
        = ( insert_nat2 @ B @ bot_bot_set_nat ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_1119_insert__not__empty,axiom,
    ! [A2: rule,A: set_rule] :
      ( ( insert_rule2 @ A2 @ A )
     != bot_bot_set_rule ) ).

% insert_not_empty
thf(fact_1120_insert__not__empty,axiom,
    ! [A2: fm,A: set_fm] :
      ( ( insert_fm2 @ A2 @ A )
     != bot_bot_set_fm ) ).

% insert_not_empty
thf(fact_1121_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat2 @ A2 @ A )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_1122_doubleton__eq__iff,axiom,
    ! [A2: rule,B: rule,C: rule,D2: rule] :
      ( ( ( insert_rule2 @ A2 @ ( insert_rule2 @ B @ bot_bot_set_rule ) )
        = ( insert_rule2 @ C @ ( insert_rule2 @ D2 @ bot_bot_set_rule ) ) )
      = ( ( ( A2 = C )
          & ( B = D2 ) )
        | ( ( A2 = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1123_doubleton__eq__iff,axiom,
    ! [A2: fm,B: fm,C: fm,D2: fm] :
      ( ( ( insert_fm2 @ A2 @ ( insert_fm2 @ B @ bot_bot_set_fm ) )
        = ( insert_fm2 @ C @ ( insert_fm2 @ D2 @ bot_bot_set_fm ) ) )
      = ( ( ( A2 = C )
          & ( B = D2 ) )
        | ( ( A2 = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1124_doubleton__eq__iff,axiom,
    ! [A2: nat,B: nat,C: nat,D2: nat] :
      ( ( ( insert_nat2 @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
        = ( insert_nat2 @ C @ ( insert_nat2 @ D2 @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C )
          & ( B = D2 ) )
        | ( ( A2 = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1125_singleton__iff,axiom,
    ! [B: list_fm,A2: list_fm] :
      ( ( member_list_fm3 @ B @ ( insert_list_fm2 @ A2 @ bot_bot_set_list_fm ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_1126_singleton__iff,axiom,
    ! [B: tm,A2: tm] :
      ( ( member_tm3 @ B @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_1127_singleton__iff,axiom,
    ! [B: rule,A2: rule] :
      ( ( member_rule3 @ B @ ( insert_rule2 @ A2 @ bot_bot_set_rule ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_1128_singleton__iff,axiom,
    ! [B: fm,A2: fm] :
      ( ( member_fm3 @ B @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_1129_singleton__iff,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat3 @ B @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_1130_singletonD,axiom,
    ! [B: list_fm,A2: list_fm] :
      ( ( member_list_fm3 @ B @ ( insert_list_fm2 @ A2 @ bot_bot_set_list_fm ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_1131_singletonD,axiom,
    ! [B: tm,A2: tm] :
      ( ( member_tm3 @ B @ ( insert_tm2 @ A2 @ bot_bot_set_tm ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_1132_singletonD,axiom,
    ! [B: rule,A2: rule] :
      ( ( member_rule3 @ B @ ( insert_rule2 @ A2 @ bot_bot_set_rule ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_1133_singletonD,axiom,
    ! [B: fm,A2: fm] :
      ( ( member_fm3 @ B @ ( insert_fm2 @ A2 @ bot_bot_set_fm ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_1134_singletonD,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat3 @ B @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_1135_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules2: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ Rules2 )
     != bot_bo3426410157013319936m_rule ) ).

% RuleSystem_Defs.NE_R
thf(fact_1136_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules2: stream_rule] :
      ( ( sset_rule @ Rules2 )
     != bot_bot_set_rule ) ).

% RuleSystem_Defs.NE_R
thf(fact_1137_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules2: stream_fm] :
      ( ( sset_fm @ Rules2 )
     != bot_bot_set_fm ) ).

% RuleSystem_Defs.NE_R
thf(fact_1138_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules2: stream_nat] :
      ( ( sset_nat @ Rules2 )
     != bot_bot_set_nat ) ).

% RuleSystem_Defs.NE_R
thf(fact_1139_i_ONE__R,axiom,
    ! [Rules2: stream2709947120125613254m_rule] :
      ( ( sset_P4484857331586881186m_rule @ Rules2 )
     != bot_bo3426410157013319936m_rule ) ).

% i.NE_R
thf(fact_1140_i_ONE__R,axiom,
    ! [Rules2: stream_rule] :
      ( ( sset_rule @ Rules2 )
     != bot_bot_set_rule ) ).

% i.NE_R
thf(fact_1141_i_ONE__R,axiom,
    ! [Rules2: stream_fm] :
      ( ( sset_fm @ Rules2 )
     != bot_bot_set_fm ) ).

% i.NE_R
thf(fact_1142_i_ONE__R,axiom,
    ! [Rules2: stream_nat] :
      ( ( sset_nat @ Rules2 )
     != bot_bot_set_nat ) ).

% i.NE_R
thf(fact_1143_smap2_Osimps_I2_J,axiom,
    ! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
      ( ( stl_Pr12655793849929990m_rule @ ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 ) )
      = ( smap2_6945574070536835298m_rule @ F @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_1144_fair__stl,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra3799686578551160190r_rule @ rules @ Rs )
     => ( abstra3799686578551160190r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).

% fair_stl
thf(fact_1145_NE__R,axiom,
    ( ( sset_rule @ rules )
   != bot_bot_set_rule ) ).

% NE_R
thf(fact_1146_parts__in__children,axiom,
    ! [P3: fm,Z3: list_fm,Z5: list_fm,A: list_tm,R: rule] :
      ( ( member_fm3 @ P3 @ ( set_fm2 @ Z3 ) )
     => ( ( member_list_fm3 @ Z5 @ ( set_list_fm2 @ ( children @ A @ R @ Z3 ) ) )
       => ? [B6: list_tm,Xs3: list_fm] :
            ( ( ord_less_eq_set_tm @ ( set_tm2 @ A ) @ ( set_tm2 @ B6 ) )
            & ( member_list_fm3 @ Xs3 @ ( set_list_fm2 @ ( parts @ B6 @ R @ P3 ) ) )
            & ( ord_less_eq_set_fm @ ( set_fm2 @ Xs3 ) @ ( set_fm2 @ Z5 ) ) ) ) ) ).

% parts_in_children
thf(fact_1147_effect__preserves__unaffected,axiom,
    ! [P3: fm,Z3: list_fm,R: rule,B2: list_tm,Z5: list_fm,A: list_tm] :
      ( ( member_fm3 @ P3 @ ( set_fm2 @ Z3 ) )
     => ( ~ ( affects @ R @ P3 )
       => ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z5 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z3 ) ) )
         => ( member_fm3 @ P3 @ ( set_fm2 @ Z5 ) ) ) ) ) ).

% effect_preserves_unaffected
thf(fact_1148_parts__preserves__unaffected,axiom,
    ! [R: rule,P3: fm,Z5: list_fm,A: list_tm] :
      ( ~ ( affects @ R @ P3 )
     => ( ( member_list_fm3 @ Z5 @ ( set_list_fm2 @ ( parts @ A @ R @ P3 ) ) )
       => ( member_fm3 @ P3 @ ( set_fm2 @ Z5 ) ) ) ) ).

% parts_preserves_unaffected
thf(fact_1149_children__preserves__unaffected,axiom,
    ! [P3: fm,Z3: list_fm,R: rule,Z5: list_fm,A: list_tm] :
      ( ( member_fm3 @ P3 @ ( set_fm2 @ Z3 ) )
     => ( ~ ( affects @ R @ P3 )
       => ( ( member_list_fm3 @ Z5 @ ( set_list_fm2 @ ( children @ A @ R @ Z3 ) ) )
         => ( member_fm3 @ P3 @ ( set_fm2 @ Z5 ) ) ) ) ) ).

% children_preserves_unaffected
thf(fact_1150_preds_Osimps_I1_J,axiom,
    ! [N: nat,Ts: list_tm] :
      ( ( preds @ ( pre @ N @ Ts ) )
      = ( insert_fm2 @ ( pre @ N @ Ts ) @ bot_bot_set_fm ) ) ).

% preds.simps(1)
thf(fact_1151_epath__effect,axiom,
    ! [Steps: stream2709947120125613254m_rule,A: list_tm,Z3: list_fm,R: rule] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
     => ( ( ( shd_Pr4562317740776619530m_rule @ Steps )
          = ( produc1733806532565653680m_rule @ ( produc1414352766439514085ist_fm @ A @ Z3 ) @ R ) )
       => ? [B6: list_tm,Z6: list_fm,R6: rule] :
            ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B6 @ Z6 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z3 ) ) )
            & ( ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ Steps ) )
              = ( produc1733806532565653680m_rule @ ( produc1414352766439514085ist_fm @ B6 @ Z6 ) @ R6 ) )
            & ( B6
              = ( remdups_tm @ ( append_tm @ A @ ( append_tm @ ( subterms @ Z3 ) @ ( subterms @ Z6 ) ) ) ) ) ) ) ) ).

% epath_effect
thf(fact_1152_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_1153_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
        = I2 ) ) ).

% diff_diff_cancel
thf(fact_1154_Nat_Oex__has__greatest__nat,axiom,
    ! [P4: nat > $o,K: nat,B: nat] :
      ( ( P4 @ K )
     => ( ! [Y4: nat] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X3: nat] :
            ( ( P4 @ X3 )
            & ! [Y5: nat] :
                ( ( P4 @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1155_GreatestI__ex__nat,axiom,
    ! [P4: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P4 @ X_12 )
     => ( ! [Y4: nat] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( P4 @ ( order_Greatest_nat @ P4 ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_1156_Greatest__le__nat,axiom,
    ! [P4: nat > $o,K: nat,B: nat] :
      ( ( P4 @ K )
     => ( ! [Y4: nat] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P4 ) ) ) ) ).

% Greatest_le_nat
thf(fact_1157_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_1158_GreatestI__nat,axiom,
    ! [P4: nat > $o,K: nat,B: nat] :
      ( ( P4 @ K )
     => ( ! [Y4: nat] :
            ( ( P4 @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( P4 @ ( order_Greatest_nat @ P4 ) ) ) ) ).

% GreatestI_nat
thf(fact_1159_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_1160_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_1161_le__trans,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K )
       => ( ord_less_eq_nat @ I2 @ K ) ) ) ).

% le_trans
thf(fact_1162_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_1163_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_1164_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_1165_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_1166_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).

% diff_le_mono
thf(fact_1167_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1168_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1169_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M2 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1170_effect__tms,axiom,
    ! [B2: list_tm,Z5: list_fm,R: rule,A: list_tm,Z3: list_fm] :
      ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z5 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z3 ) ) )
     => ( B2
        = ( remdups_tm @ ( append_tm @ A @ ( append_tm @ ( subterms @ Z3 ) @ ( subterms @ Z5 ) ) ) ) ) ) ).

% effect_tms
thf(fact_1171_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_rule3 @ 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_1172_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_1173_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% Suc_le_mono
thf(fact_1174_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R3: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ! [X3: nat] : ( R3 @ X3 @ X3 )
       => ( ! [X3: nat,Y4: nat,Z7: nat] :
              ( ( R3 @ X3 @ Y4 )
             => ( ( R3 @ Y4 @ Z7 )
               => ( R3 @ X3 @ Z7 ) ) )
         => ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
           => ( R3 @ M2 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1175_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P4: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( P4 @ M2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ( P4 @ N2 )
               => ( P4 @ ( suc @ N2 ) ) ) )
         => ( P4 @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1176_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1177_full__nat__induct,axiom,
    ! [P4: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
             => ( P4 @ M3 ) )
         => ( P4 @ N2 ) )
     => ( P4 @ N ) ) ).

% full_nat_induct
thf(fact_1178_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_1179_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1180_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_Suc_eq_le
thf(fact_1181_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_1182_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1183_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_lessD
thf(fact_1184_inc__induct,axiom,
    ! [I2: nat,J2: nat,P4: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P4 @ J2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P4 @ ( suc @ N2 ) )
                 => ( P4 @ N2 ) ) ) )
         => ( P4 @ I2 ) ) ) ) ).

% inc_induct
thf(fact_1185_dec__induct,axiom,
    ! [I2: nat,J2: nat,P4: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P4 @ I2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P4 @ N2 )
                 => ( P4 @ ( suc @ N2 ) ) ) ) )
         => ( P4 @ J2 ) ) ) ) ).

% dec_induct
thf(fact_1186_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M2 @ N )
        | ( M2
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1187_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_eq
thf(fact_1188_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
     => ? [M: nat] :
          ( M4
          = ( suc @ M ) ) ) ).

% Suc_le_D
thf(fact_1189_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1190_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( M2
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1191_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_1192_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_leD
thf(fact_1193_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
          & ( M5 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_1194_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_1195_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_nat @ M5 @ N3 )
          | ( M5 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1196_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_1197_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_1198_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I2: nat,J2: nat] :
      ( ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1199_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1200_diff__less__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1201_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1202_rules__repeat,axiom,
    ! [M2: nat,R: rule] :
    ? [N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
      & ( ( snth_rule @ rules @ N2 )
        = R ) ) ).

% rules_repeat
thf(fact_1203_epath_Ocoinduct,axiom,
    ! [X4: stream2709947120125613254m_rule > $o,X: stream2709947120125613254m_rule] :
      ( ( X4 @ X )
     => ( ! [X3: stream2709947120125613254m_rule] :
            ( ( X4 @ X3 )
           => ? [Steps3: stream2709947120125613254m_rule,Sl6: fset_P8989946509869081563ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_rule3 @ ( 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 )
                & ( ( X4 @ ( stl_Pr12655793849929990m_rule @ Steps3 ) )
                  | ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ Steps3 ) ) ) ) )
       => ( abstra6789711989322986974ist_fm @ eff @ rules @ X ) ) ) ).

% epath.coinduct
thf(fact_1204_epath_Ointros,axiom,
    ! [Steps: stream2709947120125613254m_rule,Sl: fset_P8989946509869081563ist_fm] :
      ( ( member_rule3 @ ( 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_1205_epath_Osimps,axiom,
    ! [A2: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ A2 )
      = ( ? [Steps2: stream2709947120125613254m_rule,Sl3: fset_P8989946509869081563ist_fm] :
            ( ( A2 = Steps2 )
            & ( member_rule3 @ ( 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_1206_epath_Ocases,axiom,
    ! [A2: stream2709947120125613254m_rule] :
      ( ( abstra6789711989322986974ist_fm @ eff @ rules @ A2 )
     => ~ ( ( member_rule3 @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ ( sset_rule @ rules ) )
         => ! [Sl2: fset_P8989946509869081563ist_fm] :
              ( ( fmembe3754813877001230652ist_fm @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) @ Sl2 )
             => ( ( eff @ ( produc6514711208036230658m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ ( produc993771546023551940m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) ) @ Sl2 )
               => ~ ( abstra6789711989322986974ist_fm @ eff @ rules @ ( stl_Pr12655793849929990m_rule @ A2 ) ) ) ) ) ) ).

% epath.cases
thf(fact_1207_ptms__def,axiom,
    ( ptms
    = ( ^ [Z: produc340336539035504054m_rule] : ( produc4874987732838623697ist_fm @ ( produc993771546023551940m_rule @ Z ) ) ) ) ).

% ptms_def
thf(fact_1208_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_1209_pseq__def,axiom,
    ( pseq
    = ( ^ [Z: produc340336539035504054m_rule] : ( produc3242453064496107283ist_fm @ ( produc993771546023551940m_rule @ Z ) ) ) ) ).

% pseq_def
thf(fact_1210_nat__descend__induct,axiom,
    ! [N: nat,P4: nat > $o,M2: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P4 @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P4 @ I4 ) )
             => ( P4 @ K2 ) ) )
       => ( P4 @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_1211_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y2: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa )
        = Y2 )
     => ( ( ( ord_less_eq_nat @ Xa @ X )
         => ( Y2
            = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa @ X )
         => ( Y2
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_1212_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M5 @ K3 ) @ ( product_Pair_nat_nat @ M5 @ ( minus_minus_nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_1213_pair__lessI2,axiom,
    ! [A2: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1214_subst_Osimps_I6_J,axiom,
    ! [P3: fm,S: tm,K: nat] :
      ( ( subst @ ( uni @ P3 ) @ S @ K )
      = ( uni @ ( subst @ P3 @ ( liftt @ S ) @ ( suc @ K ) ) ) ) ).

% subst.simps(6)
thf(fact_1215_subst_Osimps_I7_J,axiom,
    ! [P3: fm,S: tm,K: nat] :
      ( ( subst @ ( neg @ P3 ) @ S @ K )
      = ( neg @ ( subst @ P3 @ S @ K ) ) ) ).

% subst.simps(7)
thf(fact_1216_pair__leqI2,axiom,
    ! [A2: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1217_subst_Osimps_I1_J,axiom,
    ! [B: nat,Ts: list_tm,S: tm,K: nat] :
      ( ( subst @ ( pre @ B @ Ts ) @ S @ K )
      = ( pre @ B @ ( substts @ Ts @ S @ K ) ) ) ).

% subst.simps(1)
thf(fact_1218_substts_Osimps_I2_J,axiom,
    ! [T: tm,Ts: list_tm,S: tm,K: nat] :
      ( ( substts @ ( cons_tm @ T @ Ts ) @ S @ K )
      = ( cons_tm @ ( substt @ T @ S @ K ) @ ( substts @ Ts @ S @ K ) ) ) ).

% substts.simps(2)
thf(fact_1219_substt_Osimps_I2_J,axiom,
    ! [A2: nat,Ts: list_tm,S: tm,K: nat] :
      ( ( substt @ ( fun @ A2 @ Ts ) @ S @ K )
      = ( fun @ A2 @ ( substts @ Ts @ S @ K ) ) ) ).

% substt.simps(2)
thf(fact_1220_s5_I1_J,axiom,
    ( sub_term
    = ( ^ [V: nat,S3: tm,T2: tm] : ( substt @ T2 @ S3 @ V ) ) ) ).

% s5(1)
thf(fact_1221_wf_Ocoinduct,axiom,
    ! [X4: abstra8520858721168826610m_rule > $o,X: abstra8520858721168826610m_rule] :
      ( ( X4 @ X )
     => ( ! [X3: abstra8520858721168826610m_rule] :
            ( ( X4 @ X3 )
           => ? [T6: abstra8520858721168826610m_rule] :
                ( ( X3 = T6 )
                & ( member_rule3 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( sset_rule @ rules ) )
                & ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T6 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T6 ) ) )
                & ! [Xa2: abstra8520858721168826610m_rule] :
                    ( ( fmembe1099044791951439339m_rule @ Xa2 @ ( abstra2475333618431710837m_rule @ T6 ) )
                   => ( ( X4 @ Xa2 )
                      | ( abstra8167393772827826667ist_fm @ eff @ rules @ Xa2 ) ) ) ) )
       => ( abstra8167393772827826667ist_fm @ eff @ rules @ X ) ) ) ).

% wf.coinduct
thf(fact_1222_wf_Ointros,axiom,
    ! [T: abstra8520858721168826610m_rule] :
      ( ( member_rule3 @ ( 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 ) ) )
       => ( ! [T4: abstra8520858721168826610m_rule] :
              ( ( fmembe1099044791951439339m_rule @ T4 @ ( abstra2475333618431710837m_rule @ T ) )
             => ( abstra8167393772827826667ist_fm @ eff @ rules @ T4 ) )
         => ( abstra8167393772827826667ist_fm @ eff @ rules @ T ) ) ) ) ).

% wf.intros
thf(fact_1223_wf_Ocases,axiom,
    ! [A2: abstra8520858721168826610m_rule] :
      ( ( abstra8167393772827826667ist_fm @ eff @ rules @ A2 )
     => ~ ( ( member_rule3 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A2 ) ) @ ( sset_rule @ rules ) )
         => ( ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ A2 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ A2 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ A2 ) ) )
           => ~ ! [T7: abstra8520858721168826610m_rule] :
                  ( ( fmembe1099044791951439339m_rule @ T7 @ ( abstra2475333618431710837m_rule @ A2 ) )
                 => ( abstra8167393772827826667ist_fm @ eff @ rules @ T7 ) ) ) ) ) ).

% wf.cases
thf(fact_1224_wf_Osimps,axiom,
    ! [A2: abstra8520858721168826610m_rule] :
      ( ( abstra8167393772827826667ist_fm @ eff @ rules @ A2 )
      = ( ? [T2: abstra8520858721168826610m_rule] :
            ( ( A2 = T2 )
            & ( member_rule3 @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( sset_rule @ rules ) )
            & ( eff @ ( produc6514711208036230658m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( produc993771546023551940m_rule @ ( abstra5180277867158163941m_rule @ T2 ) ) @ ( fimage3741683731962187332ist_fm @ ( comp_P5969275118686715022m_rule @ produc993771546023551940m_rule @ abstra5180277867158163941m_rule ) @ ( abstra2475333618431710837m_rule @ T2 ) ) )
            & ! [X2: abstra8520858721168826610m_rule] :
                ( ( fmembe1099044791951439339m_rule @ X2 @ ( abstra2475333618431710837m_rule @ T2 ) )
               => ( abstra8167393772827826667ist_fm @ eff @ rules @ X2 ) ) ) ) ) ).

% wf.simps
thf(fact_1225_sub__term__const__transfer_I1_J,axiom,
    ! [M2: nat,A2: nat,T: tm,S: tm] :
      ( ( ( sub_term @ M2 @ ( fun @ A2 @ nil_tm ) @ T )
       != ( sub_term @ M2 @ S @ T ) )
     => ( member_tm3 @ ( fun @ A2 @ nil_tm ) @ ( set_tm2 @ ( subtermTm @ ( sub_term @ M2 @ ( fun @ A2 @ nil_tm ) @ T ) ) ) ) ) ).

% sub_term_const_transfer(1)
thf(fact_1226_sub__list_Osimps_I2_J,axiom,
    ! [V2: nat,S: tm,T: tm,L2: list_tm] :
      ( ( sub_list @ V2 @ S @ ( cons_tm @ T @ L2 ) )
      = ( cons_tm @ ( sub_term @ V2 @ S @ T ) @ ( sub_list @ V2 @ S @ L2 ) ) ) ).

% sub_list.simps(2)
thf(fact_1227_s5_I2_J,axiom,
    ( sub_list
    = ( ^ [V: nat,S3: tm,L3: list_tm] : ( substts @ L3 @ S3 @ V ) ) ) ).

% s5(2)
thf(fact_1228_substts_Osimps_I1_J,axiom,
    ! [S: tm,K: nat] :
      ( ( substts @ nil_tm @ S @ K )
      = nil_tm ) ).

% substts.simps(1)
thf(fact_1229_inc__list_Osimps_I1_J,axiom,
    ( ( inc_list @ nil_tm )
    = nil_tm ) ).

% inc_list.simps(1)
thf(fact_1230_liftts_Osimps_I1_J,axiom,
    ( ( liftts @ nil_tm )
    = nil_tm ) ).

% liftts.simps(1)
thf(fact_1231_sub__list_Osimps_I1_J,axiom,
    ! [V2: nat,S: tm] :
      ( ( sub_list @ V2 @ S @ nil_tm )
      = nil_tm ) ).

% sub_list.simps(1)
thf(fact_1232_new__list_Osimps_I1_J,axiom,
    ! [C: nat] : ( new_list @ C @ nil_tm ) ).

% new_list.simps(1)
thf(fact_1233_paramsts_Osimps_I1_J,axiom,
    ( ( paramsts @ nil_tm )
    = bot_bot_set_nat ) ).

% paramsts.simps(1)
thf(fact_1234_sub__term_Osimps_I2_J,axiom,
    ! [V2: nat,S: tm,I2: nat,L2: list_tm] :
      ( ( sub_term @ V2 @ S @ ( fun @ I2 @ L2 ) )
      = ( fun @ I2 @ ( sub_list @ V2 @ S @ L2 ) ) ) ).

% sub_term.simps(2)
thf(fact_1235_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_1236_children_Osimps_I1_J,axiom,
    ! [Uu: list_tm,Uv: rule] :
      ( ( children @ Uu @ Uv @ nil_fm )
      = ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).

% children.simps(1)
thf(fact_1237_listFunTms_Osimps_I1_J,axiom,
    ( ( listFunTms @ nil_tm )
    = nil_nat ) ).

% listFunTms.simps(1)
thf(fact_1238_pickEff,axiom,
    ! [R: rule,S: produc6018962875968178549ist_fm] :
      ( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
     => ( eff @ R @ S @ ( abstra3309574903381802724ist_fm @ eff @ R @ S ) ) ) ).

% pickEff
thf(fact_1239_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_1240_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_1241_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( suc @ J2 ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_1242_remdups__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( remdups_nat @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% remdups_upt
thf(fact_1243_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( upt @ I2 @ J2 )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_1244_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M2 @ Q ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_1245_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ J2 )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) ) ) ).

% upt_conv_Cons
thf(fact_1246_upt__rec,axiom,
    ( upt
    = ( ^ [I: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I @ J ) @ ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_1247_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_1248_enat__ord__number_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_1249_paramst__sub__term_I2_J,axiom,
    ! [M2: nat,S: tm,L2: list_tm] : ( ord_less_eq_set_nat @ ( paramsts @ ( sub_list @ M2 @ S @ L2 ) ) @ ( sup_sup_set_nat @ ( paramst @ S ) @ ( paramsts @ L2 ) ) ) ).

% paramst_sub_term(2)
thf(fact_1250_paramsts_Osimps_I2_J,axiom,
    ! [T: tm,Ts: list_tm] :
      ( ( paramsts @ ( cons_tm @ T @ Ts ) )
      = ( sup_sup_set_nat @ ( paramst @ T ) @ ( paramsts @ Ts ) ) ) ).

% paramsts.simps(2)
thf(fact_1251_paramst__sub__term_I1_J,axiom,
    ! [M2: nat,S: tm,T: tm] : ( ord_less_eq_set_nat @ ( paramst @ ( sub_term @ M2 @ S @ T ) ) @ ( sup_sup_set_nat @ ( paramst @ S ) @ ( paramst @ T ) ) ) ).

% paramst_sub_term(1)
thf(fact_1252_paramst_Osimps_I2_J,axiom,
    ! [A2: nat,Ts: list_tm] :
      ( ( paramst @ ( fun @ A2 @ Ts ) )
      = ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) @ ( paramsts @ Ts ) ) ) ).

% paramst.simps(2)
thf(fact_1253_params__sub,axiom,
    ! [M2: nat,T: tm,P3: fm] : ( ord_less_eq_set_nat @ ( params @ ( sub @ M2 @ T @ P3 ) ) @ ( sup_sup_set_nat @ ( paramst @ T ) @ ( params @ P3 ) ) ) ).

% params_sub
thf(fact_1254_s6,axiom,
    ( sub
    = ( ^ [V: nat,S3: tm,P: fm] : ( subst @ P @ S3 @ V ) ) ) ).

% s6
thf(fact_1255_sub_Osimps_I1_J,axiom,
    ! [V2: nat,S: tm,I2: nat,L2: list_tm] :
      ( ( sub @ V2 @ S @ ( pre @ I2 @ L2 ) )
      = ( pre @ I2 @ ( sub_list @ V2 @ S @ L2 ) ) ) ).

% sub.simps(1)
thf(fact_1256_sub_Osimps_I7_J,axiom,
    ! [V2: nat,S: tm,P3: fm] :
      ( ( sub @ V2 @ S @ ( neg @ P3 ) )
      = ( neg @ ( sub @ V2 @ S @ P3 ) ) ) ).

% sub.simps(7)
thf(fact_1257_sub__const__transfer,axiom,
    ! [M2: nat,A2: nat,P3: fm,T: tm] :
      ( ( ( sub @ M2 @ ( fun @ A2 @ nil_tm ) @ P3 )
       != ( sub @ M2 @ T @ P3 ) )
     => ( member_tm3 @ ( fun @ A2 @ nil_tm ) @ ( set_tm2 @ ( subtermFm @ ( sub @ M2 @ ( fun @ A2 @ nil_tm ) @ P3 ) ) ) ) ) ).

% sub_const_transfer
thf(fact_1258_numeral__le__enat__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ N ) ) ).

% numeral_le_enat_iff
thf(fact_1259_enat__ord__simps_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% enat_ord_simps(1)
thf(fact_1260_Suc__ile__eq,axiom,
    ! [M2: nat,N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M2 ) ) @ N )
      = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M2 ) @ N ) ) ).

% Suc_ile_eq
thf(fact_1261_enat__ile,axiom,
    ! [N: extended_enat,M2: nat] :
      ( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M2 ) )
     => ? [K2: nat] :
          ( N
          = ( extended_enat2 @ K2 ) ) ) ).

% enat_ile
thf(fact_1262_iless__Suc__eq,axiom,
    ! [M2: nat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M2 ) @ ( extended_eSuc @ N ) )
      = ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M2 ) @ N ) ) ).

% iless_Suc_eq
thf(fact_1263_eSuc__ile__mono,axiom,
    ! [N: extended_enat,M2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M2 ) )
      = ( ord_le2932123472753598470d_enat @ N @ M2 ) ) ).

% eSuc_ile_mono
thf(fact_1264_ileI1,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ M2 @ N )
     => ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ M2 ) @ N ) ) ).

% ileI1
thf(fact_1265_ile__eSuc,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ N @ ( extended_eSuc @ N ) ) ).

% ile_eSuc
thf(fact_1266_eff__children,axiom,
    ! [Z3: list_fm,R: rule,A: list_tm,Ss: fset_P8989946509869081563ist_fm] :
      ( ~ ( branchDone @ Z3 )
     => ( ( eff @ R @ ( produc1414352766439514085ist_fm @ A @ Z3 ) @ Ss )
       => ! [X5: list_fm] :
            ( ( member_list_fm3 @ X5 @ ( set_list_fm2 @ ( children @ ( remdups_tm @ ( append_tm @ A @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ Z3 ) ) ) ) @ R @ Z3 ) ) )
           => ? [B6: list_tm] : ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B6 @ X5 ) @ Ss ) ) ) ) ).

% eff_children
thf(fact_1267_branchDone_Osimps_I1_J,axiom,
    ~ ( branchDone @ nil_fm ) ).

% branchDone.simps(1)
thf(fact_1268_branchDone__contradiction,axiom,
    ( branchDone
    = ( ^ [Z: list_fm] :
        ? [P: fm] :
          ( ( member_fm3 @ P @ ( set_fm2 @ Z ) )
          & ( member_fm3 @ ( neg @ P ) @ ( set_fm2 @ Z ) ) ) ) ) ).

% branchDone_contradiction
thf(fact_1269_branchDone_Osimps_I2_J,axiom,
    ! [P3: fm,Z3: list_fm] :
      ( ( branchDone @ ( cons_fm @ ( neg @ P3 ) @ Z3 ) )
      = ( ( member_fm3 @ P3 @ ( set_fm2 @ Z3 ) )
        | ( member_fm3 @ ( neg @ ( neg @ P3 ) ) @ ( set_fm2 @ Z3 ) )
        | ( branchDone @ Z3 ) ) ) ).

% branchDone.simps(2)
thf(fact_1270_branchDone_Osimps_I8_J,axiom,
    ! [V2: fm,Z3: list_fm] :
      ( ( branchDone @ ( cons_fm @ ( uni @ V2 ) @ Z3 ) )
      = ( ( member_fm3 @ ( neg @ ( uni @ V2 ) ) @ ( set_fm2 @ Z3 ) )
        | ( branchDone @ Z3 ) ) ) ).

% branchDone.simps(8)
thf(fact_1271_branchDone_Osimps_I3_J,axiom,
    ! [V2: nat,Va: list_tm,Z3: list_fm] :
      ( ( branchDone @ ( cons_fm @ ( pre @ V2 @ Va ) @ Z3 ) )
      = ( ( member_fm3 @ ( neg @ ( pre @ V2 @ Va ) ) @ ( set_fm2 @ Z3 ) )
        | ( branchDone @ Z3 ) ) ) ).

% branchDone.simps(3)
thf(fact_1272_ne__effect__not__branchDone,axiom,
    ! [B2: list_tm,Z5: list_fm,R: rule,A: list_tm,Z3: list_fm] :
      ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z5 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z3 ) ) )
     => ~ ( branchDone @ Z3 ) ) ).

% ne_effect_not_branchDone
thf(fact_1273_effect_Osimps,axiom,
    ! [Z3: list_fm,R: rule,A: list_tm] :
      ( ( ( branchDone @ Z3 )
       => ( ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z3 ) )
          = bot_bo6461889142629771335ist_fm ) )
      & ( ~ ( branchDone @ Z3 )
       => ( ( effect @ R @ ( produc1414352766439514085ist_fm @ A @ Z3 ) )
          = ( fimage4743371125182381497ist_fm
            @ ^ [Z8: list_fm] : ( produc1414352766439514085ist_fm @ ( remdups_tm @ ( append_tm @ A @ ( append_tm @ ( subterms @ Z3 ) @ ( subterms @ Z8 ) ) ) ) @ Z8 )
            @ ( fset_of_list_list_fm @ ( children @ ( remdups_tm @ ( append_tm @ A @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ Z3 ) ) ) ) @ R @ Z3 ) ) ) ) ) ) ).

% effect.simps
thf(fact_1274_subtermTm_Osimps_I1_J,axiom,
    ! [N: nat,Ts: list_tm] :
      ( ( subtermTm @ ( fun @ N @ Ts ) )
      = ( cons_tm @ ( fun @ N @ Ts ) @ ( remdups_tm @ ( concat_tm @ ( map_tm_list_tm @ subtermTm @ Ts ) ) ) ) ) ).

% subtermTm.simps(1)
thf(fact_1275_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_1276_eff__def,axiom,
    ( eff
    = ( ^ [R2: rule,S3: produc6018962875968178549ist_fm] :
          ( ^ [Y3: fset_P8989946509869081563ist_fm,Z4: fset_P8989946509869081563ist_fm] : ( Y3 = Z4 )
          @ ( effect @ R2 @ S3 ) ) ) ) ).

% eff_def

% Helper facts (13)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y2: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y2: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y2 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
    ! [X: list_fm,Y2: list_fm] :
      ( ( if_list_fm @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
    ! [X: list_fm,Y2: list_fm] :
      ( ( if_list_fm @ $true @ X @ Y2 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
    ! [X: list_tm,Y2: list_tm] :
      ( ( if_list_tm @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
    ! [X: list_tm,Y2: list_tm] :
      ( ( if_list_tm @ $true @ X @ Y2 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Prover__Orule_J_T,axiom,
    ! [X: list_rule,Y2: list_rule] :
      ( ( if_list_rule @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Prover__Orule_J_T,axiom,
    ! [X: list_rule,Y2: list_rule] :
      ( ( if_list_rule @ $true @ X @ Y2 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
    ! [X: list_list_fm,Y2: list_list_fm] :
      ( ( if_list_list_fm @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
    ! [X: list_list_fm,Y2: list_list_fm] :
      ( ( if_list_list_fm @ $true @ X @ Y2 )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P4: $o] :
      ( ( P4 = $true )
      | ( P4 = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y2 )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ? [M3: nat] : ( member_tm3 @ t @ ( set_tm2 @ ( subterms @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ M3 @ steps ) ) ) ) ) ) ).

%------------------------------------------------------------------------------