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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FOL_Seq_Calc3/0011_Completeness/prob_00196_007964__12380958_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2086 ( 661 unt; 788 typ;   0 def)
%            Number of atoms       : 3851 (1255 equ;  15 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 15086 ( 249   ~;  52   |; 429   &;12621   @)
%                                         (   0 <=>;1735  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :  142 ( 141 usr)
%            Number of type conns  : 3301 (3301   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  651 ( 647 usr;  23 con; 0-4 aty)
%            Number of variables   : 4763 ( 817   ^;3664   !; 282   ?;4763   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:28:48.726
%------------------------------------------------------------------------------
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    comp_P2882693189979712244m_rule: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ) > ( abstra4572470701423150929m_rule > produc1996495991257130529ist_fm ) > abstra4572470701423150929m_rule > produc1996495991257130529ist_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P8245093624798776376m_rule: ( produc1996495991257130529ist_fm > produc1996495991257130529ist_fm ) > ( produc164195504107695125m_rule > produc1996495991257130529ist_fm ) > produc164195504107695125m_rule > produc1996495991257130529ist_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P2017078995931194366m_rule: ( produc1996495991257130529ist_fm > produc1754732471204517991ist_fm ) > ( produc164195504107695125m_rule > produc1996495991257130529ist_fm ) > produc164195504107695125m_rule > produc1754732471204517991ist_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_P1567445206330693457at_nat: ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P1085356462900927218m_rule: ( produc1754732471204517991ist_fm > produc1996495991257130529ist_fm ) > ( produc164195504107695125m_rule > produc1754732471204517991ist_fm ) > produc164195504107695125m_rule > produc1996495991257130529ist_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    comp_P4378417847848123130m_rule: ( produc164195504107695125m_rule > produc1996495991257130529ist_fm ) > ( abstra4572470701423150929m_rule > produc164195504107695125m_rule ) > abstra4572470701423150929m_rule > produc1996495991257130529ist_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P7463969011078018366m_rule: ( produc164195504107695125m_rule > produc1996495991257130529ist_fm ) > ( produc164195504107695125m_rule > produc164195504107695125m_rule ) > produc164195504107695125m_rule > produc1996495991257130529ist_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    comp_P2707267495825058490ist_fm: ( produc164195504107695125m_rule > produc1996495991257130529ist_fm ) > ( produc4630866025709511057ist_fm > produc164195504107695125m_rule ) > produc4630866025709511057ist_fm > produc1996495991257130529ist_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    comp_P1643750715206316048m_rule: ( produc164195504107695125m_rule > rule ) > ( abstra4572470701423150929m_rule > produc164195504107695125m_rule ) > abstra4572470701423150929m_rule > rule ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P422132551275928404m_rule: ( produc164195504107695125m_rule > rule ) > ( produc164195504107695125m_rule > produc164195504107695125m_rule ) > produc164195504107695125m_rule > rule ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    comp_P4888803072877744336ist_fm: ( produc164195504107695125m_rule > rule ) > ( produc4630866025709511057ist_fm > produc164195504107695125m_rule ) > produc4630866025709511057ist_fm > rule ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    comp_P4814319827393324394at_nat: ( produc859450856879609959at_nat > product_prod_nat_nat ) > ( produc859450856879609959at_nat > produc859450856879609959at_nat ) > produc859450856879609959at_nat > product_prod_nat_nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P6646430376872094402m_rule: ( produc4630866025709511057ist_fm > produc1996495991257130529ist_fm ) > ( produc164195504107695125m_rule > produc4630866025709511057ist_fm ) > produc164195504107695125m_rule > produc1996495991257130529ist_fm ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P3413117018928051920m_rule: ( produc4630866025709511057ist_fm > rule ) > ( produc164195504107695125m_rule > produc4630866025709511057ist_fm ) > produc164195504107695125m_rule > rule ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_P2548450393677710116m_rule: ( produc780299800278082923e_rule > rule ) > ( produc164195504107695125m_rule > produc780299800278082923e_rule ) > produc164195504107695125m_rule > rule ).

thf(sy_c_Fun_Ocomp_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    comp_r8929483813444710872m_rule: ( rule > produc780299800278082923e_rule ) > ( produc164195504107695125m_rule > rule ) > produc164195504107695125m_rule > produc780299800278082923e_rule ).

thf(sy_c_Fun_Ocomp_001t__Syntax__Orule_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
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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__Syntax__Orule_J,type,
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thf(sy_c_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    if_Abs4566652138351786251m_rule: $o > abstra4572470701423150929m_rule > abstra4572470701423150929m_rule > abstra4572470701423150929m_rule ).

thf(sy_c_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J,type,
    if_fse3370020817956878651m_rule: $o > fset_A5396818216090053761m_rule > fset_A5396818216090053761m_rule > fset_A5396818216090053761m_rule ).

thf(sy_c_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    if_str8948254419368749791m_rule: $o > stream727092118206550309m_rule > stream727092118206550309m_rule > stream727092118206550309m_rule ).

thf(sy_c_If_001t__Stream__Ostream_It__Syntax__Orule_J,type,
    if_stream_rule: $o > stream_rule > stream_rule > stream_rule ).

thf(sy_c_List_OBleast_001t__Nat__Onat,type,
    bleast_nat: set_nat > ( nat > $o ) > nat ).

thf(sy_c_List_Oabort__Bleast_001t__Nat__Onat,type,
    abort_Bleast_nat: set_nat > ( nat > $o ) > nat ).

thf(sy_c_List_Olist_Oset_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).

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

thf(sy_c_List_Olist_Oset_001t__Syntax__Orule,type,
    set_rule2: list_rule > set_rule ).

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

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

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

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

thf(sy_c_Option_Ooption_ONone_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
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thf(sy_c_Option_Ooption_Othe_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    the_fs400605564276472219ist_fm: option6029102822600979917ist_fm > fset_P661503646757059847ist_fm ).

thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Syntax__Orule_J,type,
    bot_bot_set_rule: set_rule ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_It__Syntax__Ofm_J,type,
    ord_Least_set_fm: ( set_fm > $o ) > set_fm ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Syntax__Ofm_J_M_062_It__List__Olist_It__Syntax__Ofm_J_M_Eo_J_J,type,
    ord_le1774903256965159270t_fm_o: ( list_fm > list_fm > $o ) > ( list_fm > list_fm > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_062_It__Syntax__Orule_M_Eo_J_J,type,
    ord_le4891813983447887534rule_o: ( produc1996495991257130529ist_fm > rule > $o ) > ( produc1996495991257130529ist_fm > rule > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_M_Eo_J,type,
    ord_le8518180823940269308t_fm_o: ( produc1996495991257130529ist_fm > $o ) > ( produc1996495991257130529ist_fm > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    ord_le7858099551454983350_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Syntax__Ofm_M_Eo_J,type,
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    ord_le8079975967087607008t_fm_o: ( rule > produc1996495991257130529ist_fm > $o ) > ( rule > produc1996495991257130529ist_fm > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Syntax__Orule_M_Eo_J,type,
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thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    stl_Pr5027827701538482609at_nat: stream6724221391990029191at_nat > stream6724221391990029191at_nat ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    stl_Pr8700568296466786583ist_fm: stream3409308193418444653ist_fm > stream3409308193418444653ist_fm ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    stl_Pr1564049449699658205at_nat: stream1622999302952468147at_nat > stream1622999302952468147at_nat ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Ofm_J,type,
    stl_Pr2715999368752424048_fm_fm: stream5128679079952569136_fm_fm > stream5128679079952569136_fm_fm ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    stl_Pr950425576149878629m_rule: stream727092118206550309m_rule > stream727092118206550309m_rule ).

thf(sy_c_Stream_Ostream_Ostl_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,
    stl_Pr5010465336888994071at_nat: stream8372878641218411373at_nat > stream8372878641218411373at_nat ).

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

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_Mt__Syntax__Orule_J,type,
    stl_Pr830518440239794027e_rule: stream2312013117288958913e_rule > stream2312013117288958913e_rule ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    stl_Pr5425948690297933931m_rule: stream1878305877988316353m_rule > stream1878305877988316353m_rule ).

thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
    stl_Pr4904654673476115995e_rule: stream3588992565182678257e_rule > stream3588992565182678257e_rule ).

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

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

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

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

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

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

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

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

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
    szip_P4443206486938582938e_rule: stream3588992565182678257e_rule > stream3588992565182678257e_rule > stream3281885149994500589e_rule ).

thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J_001t__Syntax__Orule,type,
    szip_P3067794746305893590e_rule: stream3588992565182678257e_rule > stream_rule > stream5285359198808078703e_rule ).

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

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

thf(sy_c_Stream_Oszip_001t__Syntax__Orule_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Syntax__Orule_J,type,
    szip_r730181393530711434e_rule: stream_rule > stream3588992565182678257e_rule > stream4003613590975974891e_rule ).

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

thf(sy_c_Syntax_Ofm_OFalsity,type,
    falsity: fm ).

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

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

thf(sy_c_Syntax_Orule_OAxiom,type,
    axiom: nat > list_tm > rule ).

thf(sy_c_Syntax_Orule_OFlsL,type,
    flsL: rule ).

thf(sy_c_Syntax_Orule_OFlsR,type,
    flsR: rule ).

thf(sy_c_Syntax_Orule_OIdle,type,
    idle: rule ).

thf(sy_c_Syntax_Orule_OUniL,type,
    uniL: tm > fm > rule ).

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

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    accp_P2380921852949180872ist_fm: ( produc4630866025709511057ist_fm > produc4630866025709511057ist_fm > $o ) > produc4630866025709511057ist_fm > $o ).

thf(sy_c_fChoice_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    fChoic7543385803022166781m_rule: ( abstra4572470701423150929m_rule > $o ) > abstra4572470701423150929m_rule ).

thf(sy_c_fChoice_001t__Abstract____Completeness__Otree_It__Syntax__Orule_J,type,
    fChoic8596484284512669613e_rule: ( abstract_tree_rule > $o ) > abstract_tree_rule ).

thf(sy_c_fChoice_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    fChoic3892542562059901019ist_fm: ( fset_P661503646757059847ist_fm > $o ) > fset_P661503646757059847ist_fm ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    fChoic1980167584695909237ist_fm: ( produc1996495991257130529ist_fm > $o ) > produc1996495991257130529ist_fm ).

thf(sy_c_fChoice_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,
    fChoic6673916560228149179at_nat: ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat ).

thf(sy_c_fChoice_001t__Syntax__Ofm,type,
    fChoice_fm: ( fm > $o ) > fm ).

thf(sy_c_fChoice_001t__Syntax__Orule,type,
    fChoice_rule: ( rule > $o ) > rule ).

thf(sy_c_member_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    member1590433669991072232m_rule: abstra4572470701423150929m_rule > set_Ab3744519486624674951m_rule > $o ).

thf(sy_c_member_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    member3493611713653719344ist_fm: fset_P661503646757059847ist_fm > set_fs7534060674568567527ist_fm > $o ).

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

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

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    member8686995742288324264ist_fm: produc4630866025709511057ist_fm > set_Pr2241365836748572487ist_fm > $o ).

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

thf(sy_c_member_001t__Syntax__Orule,type,
    member_rule: rule > set_rule > $o ).

thf(sy_v_p____,type,
    p: fm ).

thf(sy_v_steps,type,
    steps: stream727092118206550309m_rule ).

thf(sy_v_t____,type,
    t: tm ).

% Relevant facts (1275)
thf(fact_0__C_K_C,axiom,
    member_fm @ ( uni @ p ) @ ( treeA @ steps ) ).

% "*"
thf(fact_1_enabled__def,axiom,
    ! [R: rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R
        @ S )
      = ( ? [Sl: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Sl ) ) ) ) ).

% enabled_def
thf(fact_2_rule_Oinject_I4_J,axiom,
    ! [X71: tm,X72: fm,Y71: tm,Y72: fm] :
      ( ( ( uniL @ X71 @ X72 )
        = ( uniL @ Y71 @ Y72 ) )
      = ( ( X71 = Y71 )
        & ( X72 = Y72 ) ) ) ).

% rule.inject(4)
thf(fact_3_assms_I2_J,axiom,
    ( abstra8647779336736900908ist_fm
    @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R2 @ S2 )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ steps ) ).

% assms(2)
thf(fact_4_option_Oinject,axiom,
    ! [X2: fset_P661503646757059847ist_fm,Y2: fset_P661503646757059847ist_fm] :
      ( ( ( some_f2162114139285431020ist_fm @ X2 )
        = ( some_f2162114139285431020ist_fm @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_5_assms_I1_J,axiom,
    ( abstra5415202973855822231ist_fm
    @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R2 @ S2 )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ steps ) ).

% assms(1)
thf(fact_6_RuleSystem__Defs_Oenabled__def,axiom,
    ( abstra3055540779341473792ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R2: rule,S2: produc1996495991257130529ist_fm] :
        ? [X: fset_P661503646757059847ist_fm] : ( Eff @ R2 @ S2 @ X ) ) ) ).

% RuleSystem_Defs.enabled_def
thf(fact_7_RuleSystem__Defs_Oenabled_Ocong,axiom,
    abstra3055540779341473792ist_fm = abstra3055540779341473792ist_fm ).

% RuleSystem_Defs.enabled.cong
thf(fact_8_pickEff,axiom,
    ! [R: rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R
        @ S )
     => ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm
          @ ( abstra458490714785879965ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R
            @ S ) ) ) ) ).

% pickEff
thf(fact_9_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S22: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( szip_rule_rule @ S1 @ S22 ) )
      = ( szip_rule_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_10_sdrop__szip,axiom,
    ! [N: nat,S1: stream8299795917829157543ist_fm,S22: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) )
      = ( szip_P1977448745965526924m_rule @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_11_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( sdrop_6392754500180887844e_rule @ N @ ( szip_P553291425819358518e_rule @ S1 @ S22 ) )
      = ( szip_P553291425819358518e_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_12_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N @ ( szip_r2254114822203693110m_rule @ S1 @ S22 ) )
      = ( szip_r2254114822203693110m_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_13_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( szip_P2499414959592755846m_rule @ S1 @ S22 ) )
      = ( szip_P2499414959592755846m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_14_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S22: stream3588992565182678257e_rule] :
      ( ( sdrop_3185404920971892658e_rule @ N @ ( szip_r730181393530711434e_rule @ S1 @ S22 ) )
      = ( szip_r730181393530711434e_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_15_sdrop__szip,axiom,
    ! [N: nat,S1: stream3588992565182678257e_rule,S22: stream_rule] :
      ( ( sdrop_2433707926279816502e_rule @ N @ ( szip_P3067794746305893590e_rule @ S1 @ S22 ) )
      = ( szip_P3067794746305893590e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_16_sdrop__szip,axiom,
    ! [N: nat,S1: stream_list_fm,S22: stream_list_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( szip_list_fm_list_fm @ S1 @ S22 ) )
      = ( szip_list_fm_list_fm @ ( sdrop_list_fm @ N @ S1 ) @ ( sdrop_list_fm @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_17_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S22: stream8299795917829157543ist_fm] :
      ( ( sdrop_2468034597186633064ist_fm @ N @ ( szip_r8549736553299883584ist_fm @ S1 @ S22 ) )
      = ( szip_r8549736553299883584ist_fm @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_18_sdrop__szip,axiom,
    ! [N: nat,S1: stream3588992565182678257e_rule,S22: stream3588992565182678257e_rule] :
      ( ( sdrop_7075696550712280400e_rule @ N @ ( szip_P4443206486938582938e_rule @ S1 @ S22 ) )
      = ( szip_P4443206486938582938e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_19_Saturated__sdrop,axiom,
    ! [Steps: stream727092118206550309m_rule,N: nat] :
      ( ( abstra8647779336736900908ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( abstra8647779336736900908ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) ) ).

% Saturated_sdrop
thf(fact_20_epath__sdrop,axiom,
    ! [Steps: stream727092118206550309m_rule,N: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) ) ).

% epath_sdrop
thf(fact_21_enabled__ex__taken,axiom,
    ! [Steps: stream727092118206550309m_rule,R: rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( abstra8647779336736900908ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ Steps )
       => ( ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R
            @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
         => ? [K: nat] :
              ( ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ K @ Steps ) ) )
              = R ) ) ) ) ).

% enabled_ex_taken
thf(fact_22_fm_Oinject_I3_J,axiom,
    ! [X4: fm,Y4: fm] :
      ( ( ( uni @ X4 )
        = ( uni @ Y4 ) )
      = ( X4 = Y4 ) ) ).

% fm.inject(3)
thf(fact_23_RuleSystem__Defs_Oepath_Ocong,axiom,
    abstra5415202973855822231ist_fm = abstra5415202973855822231ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_24_RuleSystem__Defs_OpickEff_Ocong,axiom,
    abstra458490714785879965ist_fm = abstra458490714785879965ist_fm ).

% RuleSystem_Defs.pickEff.cong
thf(fact_25_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra8647779336736900908ist_fm = abstra8647779336736900908ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_26_RuleSystem__Defs_OpickEff,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R: rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ Eff2 @ R @ S )
     => ( Eff2 @ R @ S @ ( abstra458490714785879965ist_fm @ Eff2 @ R @ S ) ) ) ).

% RuleSystem_Defs.pickEff
thf(fact_27_prod__eq__iff,axiom,
    ( ( ^ [Y: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y = Z ) )
    = ( ^ [S2: produc164195504107695125m_rule,T: produc164195504107695125m_rule] :
          ( ( ( produc6879501374131015971m_rule @ S2 )
            = ( produc6879501374131015971m_rule @ T ) )
          & ( ( produc7165828336582415457m_rule @ S2 )
            = ( produc7165828336582415457m_rule @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_28_prod__eq__iff,axiom,
    ( ( ^ [Y: produc1996495991257130529ist_fm,Z: produc1996495991257130529ist_fm] : ( Y = Z ) )
    = ( ^ [S2: produc1996495991257130529ist_fm,T: produc1996495991257130529ist_fm] :
          ( ( ( produc1501393135466168645ist_fm @ S2 )
            = ( produc1501393135466168645ist_fm @ T ) )
          & ( ( produc4588648349897876871ist_fm @ S2 )
            = ( produc4588648349897876871ist_fm @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_29_exE__realizer_H,axiom,
    ! [P: rule > produc1996495991257130529ist_fm > $o,P2: produc164195504107695125m_rule] :
      ( ( P @ ( produc7165828336582415457m_rule @ P2 ) @ ( produc6879501374131015971m_rule @ P2 ) )
     => ~ ! [X3: produc1996495991257130529ist_fm,Y3: rule] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_30_exE__realizer_H,axiom,
    ! [P: list_fm > list_fm > $o,P2: produc1996495991257130529ist_fm] :
      ( ( P @ ( produc4588648349897876871ist_fm @ P2 ) @ ( produc1501393135466168645ist_fm @ P2 ) )
     => ~ ! [X3: list_fm,Y3: list_fm] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_31_prod__eqI,axiom,
    ! [P2: produc164195504107695125m_rule,Q: produc164195504107695125m_rule] :
      ( ( ( produc6879501374131015971m_rule @ P2 )
        = ( produc6879501374131015971m_rule @ Q ) )
     => ( ( ( produc7165828336582415457m_rule @ P2 )
          = ( produc7165828336582415457m_rule @ Q ) )
       => ( P2 = Q ) ) ) ).

% prod_eqI
thf(fact_32_prod__eqI,axiom,
    ! [P2: produc1996495991257130529ist_fm,Q: produc1996495991257130529ist_fm] :
      ( ( ( produc1501393135466168645ist_fm @ P2 )
        = ( produc1501393135466168645ist_fm @ Q ) )
     => ( ( ( produc4588648349897876871ist_fm @ P2 )
          = ( produc4588648349897876871ist_fm @ Q ) )
       => ( P2 = Q ) ) ) ).

% prod_eqI
thf(fact_33_prod_Oexpand,axiom,
    ! [Prod: produc164195504107695125m_rule,Prod2: produc164195504107695125m_rule] :
      ( ( ( ( produc6879501374131015971m_rule @ Prod )
          = ( produc6879501374131015971m_rule @ Prod2 ) )
        & ( ( produc7165828336582415457m_rule @ Prod )
          = ( produc7165828336582415457m_rule @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_34_prod_Oexpand,axiom,
    ! [Prod: produc1996495991257130529ist_fm,Prod2: produc1996495991257130529ist_fm] :
      ( ( ( ( produc1501393135466168645ist_fm @ Prod )
          = ( produc1501393135466168645ist_fm @ Prod2 ) )
        & ( ( produc4588648349897876871ist_fm @ Prod )
          = ( produc4588648349897876871ist_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_35_epath__eff,axiom,
    ! [Steps: stream727092118206550309m_rule,Ss2: fset_P661503646757059847ist_fm] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
          = ( some_f2162114139285431020ist_fm @ Ss2 ) )
       => ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Ss2 ) ) ) ).

% epath_eff
thf(fact_36_szip_Osimps_I1_J,axiom,
    ! [S1: stream_list_fm,S22: stream_list_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( szip_list_fm_list_fm @ S1 @ S22 ) )
      = ( produc381145313068854617ist_fm @ ( shd_list_fm @ S1 ) @ ( shd_list_fm @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_37_szip_Osimps_I1_J,axiom,
    ! [S1: stream_nat,S22: stream_nat] :
      ( ( shd_Pr4260400998323988397at_nat @ ( szip_nat_nat @ S1 @ S22 ) )
      = ( product_Pair_nat_nat @ ( shd_nat @ S1 ) @ ( shd_nat @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_38_szip_Osimps_I1_J,axiom,
    ! [S1: stream6724221391990029191at_nat,S22: stream6724221391990029191at_nat] :
      ( ( shd_Pr8412153233960533267at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S22 ) )
      = ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S1 ) @ ( shd_Pr4260400998323988397at_nat @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_39_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S22 ) )
      = ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_40_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( shd_Pr4631506658928855911e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S22 ) )
      = ( produc4383365014245906419e_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_41_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream8299795917829157543ist_fm] :
      ( ( shd_Pr2478396429205514213ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S22 ) )
      = ( produc7063755442767259331ist_fm @ ( shd_rule @ S1 ) @ ( shd_Pr772355297128350925ist_fm @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_42_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr3564872132220007m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S22 ) )
      = ( produc6084188410630241011m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_43_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( shd_Pr4573454438745646103e_rule @ ( szip_rule_rule @ S1 @ S22 ) )
      = ( produc1769597910952761123e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_44_szip_Osimps_I1_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S22: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) )
      = ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_45_prover__def,axiom,
    ( prover
    = ( abstra178918799166095863ist_fm
      @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R2 @ S2 )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ rules ) ) ).

% prover_def
thf(fact_46_treeA__sdrop,axiom,
    ! [N: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeA @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) @ ( treeA @ Steps ) ) ).

% treeA_sdrop
thf(fact_47_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream8299795917829157543ist_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S ) )
      = ( snth_P2624749845746815685ist_fm @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_48_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream3588992565182678257e_rule] :
      ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S ) )
      = ( snth_P8487267234472522255e_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_49_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream1878305877988316353m_rule] :
      ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S ) )
      = ( snth_P7967347330282200671m_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_50_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream2312013117288958913e_rule] :
      ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S ) )
      = ( snth_P3371917080224060767e_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_51_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream6494289010434245521m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S ) )
      = ( snth_P8950227884416379311m_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_52_sdrop__simps_I1_J,axiom,
    ! [N: nat,S: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S ) )
      = ( snth_P1712579846118164081m_rule @ S @ N ) ) ).

% sdrop_simps(1)
thf(fact_53_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_54_Saturated__def,axiom,
    ! [Steps: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
      = ( ! [X5: rule] :
            ( ( member_rule @ X5 @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ X5
              @ Steps ) ) ) ) ).

% Saturated_def
thf(fact_55_wf__ipath__epath,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps )
       => ( abstra5415202973855822231ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ Steps ) ) ) ).

% wf_ipath_epath
thf(fact_56_shd__def,axiom,
    ( shd_Pr7235097944458474089m_rule
    = ( case_s4599053733867249880m_rule
      @ ^ [X1: produc164195504107695125m_rule,X22: stream727092118206550309m_rule] : X1 ) ) ).

% shd_def
thf(fact_57_shd__def,axiom,
    ( shd_rule
    = ( case_s3980788006751379256e_rule
      @ ^ [X1: rule,X22: stream_rule] : X1 ) ) ).

% shd_def
thf(fact_58_old_Oprod_Oinject,axiom,
    ! [A: list_fm,B: list_fm,A2: list_fm,B2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_59_old_Oprod_Oinject,axiom,
    ! [A: produc1996495991257130529ist_fm,B: rule,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( ( produc491467635432902671m_rule @ A @ B )
        = ( produc491467635432902671m_rule @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_60_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_61_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_62_old_Oprod_Oinject,axiom,
    ! [A: rule,B: produc1996495991257130529ist_fm,A2: rule,B2: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B )
        = ( produc7063755442767259331ist_fm @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_63_prod_Oinject,axiom,
    ! [X12: list_fm,X2: list_fm,Y1: list_fm,Y2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ X12 @ X2 )
        = ( produc381145313068854617ist_fm @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_64_prod_Oinject,axiom,
    ! [X12: produc1996495991257130529ist_fm,X2: rule,Y1: produc1996495991257130529ist_fm,Y2: rule] :
      ( ( ( produc491467635432902671m_rule @ X12 @ X2 )
        = ( produc491467635432902671m_rule @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_65_prod_Oinject,axiom,
    ! [X12: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X12 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_66_prod_Oinject,axiom,
    ! [X12: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X12 @ X2 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_67_prod_Oinject,axiom,
    ! [X12: rule,X2: produc1996495991257130529ist_fm,Y1: rule,Y2: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ X12 @ X2 )
        = ( produc7063755442767259331ist_fm @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_68_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_69_prod_Ocollapse,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_70_prod_Ocollapse,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_71_prod_Ocollapse,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_72_prod_Ocollapse,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_73_snth__szip,axiom,
    ! [S1: stream_rule,S22: stream_rule,N: nat] :
      ( ( snth_P8487267234472522255e_rule @ ( szip_rule_rule @ S1 @ S22 ) @ N )
      = ( produc1769597910952761123e_rule @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_74_snth__szip,axiom,
    ! [S1: stream_rule,S22: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P7967347330282200671m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S22 ) @ N )
      = ( produc6084188410630241011m_rule @ ( snth_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_75_snth__szip,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream_rule,N: nat] :
      ( ( snth_P3371917080224060767e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S22 ) @ N )
      = ( produc4383365014245906419e_rule @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_rule @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_76_snth__szip,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P8950227884416379311m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S22 ) @ N )
      = ( produc5927390650430071747m_rule @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_77_snth__szip,axiom,
    ! [S1: stream_list_fm,S22: stream_list_fm,N: nat] :
      ( ( snth_P2624749845746815685ist_fm @ ( szip_list_fm_list_fm @ S1 @ S22 ) @ N )
      = ( produc381145313068854617ist_fm @ ( snth_list_fm @ S1 @ N ) @ ( snth_list_fm @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_78_snth__szip,axiom,
    ! [S1: stream8299795917829157543ist_fm,S22: stream_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) @ N )
      = ( produc491467635432902671m_rule @ ( snth_P2624749845746815685ist_fm @ S1 @ N ) @ ( snth_rule @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_79_snth__szip,axiom,
    ! [S1: stream_nat,S22: stream_nat,N: nat] :
      ( ( snth_P9042041890044241317at_nat @ ( szip_nat_nat @ S1 @ S22 ) @ N )
      = ( product_Pair_nat_nat @ ( snth_nat @ S1 @ N ) @ ( snth_nat @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_80_snth__szip,axiom,
    ! [S1: stream6724221391990029191at_nat,S22: stream6724221391990029191at_nat,N: nat] :
      ( ( snth_P9098630209796499211at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S22 ) @ N )
      = ( produc6161850002892822231at_nat @ ( snth_P9042041890044241317at_nat @ S1 @ N ) @ ( snth_P9042041890044241317at_nat @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_81_snth__szip,axiom,
    ! [S1: stream_rule,S22: stream8299795917829157543ist_fm,N: nat] :
      ( ( snth_P6179250367719980013ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S22 ) @ N )
      = ( produc7063755442767259331ist_fm @ ( snth_rule @ S1 @ N ) @ ( snth_P2624749845746815685ist_fm @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_82_stl__def,axiom,
    ( stl_Pr950425576149878629m_rule
    = ( case_s474603282864825704m_rule
      @ ^ [X1: produc164195504107695125m_rule,X22: stream727092118206550309m_rule] : X22 ) ) ).

% stl_def
thf(fact_83_stl__def,axiom,
    ( stl_rule
    = ( case_s5295311805034529096m_rule
      @ ^ [X1: rule,X22: stream_rule] : X22 ) ) ).

% stl_def
thf(fact_84_RuleSystem__Defs_Owf_Ocong,axiom,
    abstra4476735159789068068ist_fm = abstra4476735159789068068ist_fm ).

% RuleSystem_Defs.wf.cong
thf(fact_85_RuleSystem__Defs_OmkTree_Ocong,axiom,
    abstra178918799166095863ist_fm = abstra178918799166095863ist_fm ).

% RuleSystem_Defs.mkTree.cong
thf(fact_86_stl__sset,axiom,
    ! [X6: fm,A: stream_fm] :
      ( ( member_fm @ X6 @ ( sset_fm @ ( stl_fm @ A ) ) )
     => ( member_fm @ X6 @ ( sset_fm @ A ) ) ) ).

% stl_sset
thf(fact_87_stl__sset,axiom,
    ! [X6: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm] :
      ( ( member8102475879199740618ist_fm @ X6 @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) )
     => ( member8102475879199740618ist_fm @ X6 @ ( sset_P3431374123360452149ist_fm @ A ) ) ) ).

% stl_sset
thf(fact_88_stl__sset,axiom,
    ! [X6: produc859450856879609959at_nat,A: stream8372878641218411373at_nat] :
      ( ( member8206827879206165904at_nat @ X6 @ ( sset_P9164763103296375419at_nat @ ( stl_Pr5010465336888994071at_nat @ A ) ) )
     => ( member8206827879206165904at_nat @ X6 @ ( sset_P9164763103296375419at_nat @ A ) ) ) ).

% stl_sset
thf(fact_89_stl__sset,axiom,
    ! [X6: produc164195504107695125m_rule,A: stream727092118206550309m_rule] :
      ( ( member4220325220686508332m_rule @ X6 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) )
     => ( member4220325220686508332m_rule @ X6 @ ( sset_P4979403222752902913m_rule @ A ) ) ) ).

% stl_sset
thf(fact_90_stl__sset,axiom,
    ! [X6: rule,A: stream_rule] :
      ( ( member_rule @ X6 @ ( sset_rule @ ( stl_rule @ A ) ) )
     => ( member_rule @ X6 @ ( sset_rule @ A ) ) ) ).

% stl_sset
thf(fact_91_snth__sset,axiom,
    ! [S: stream_fm,N: nat] : ( member_fm @ ( snth_fm @ S @ N ) @ ( sset_fm @ S ) ) ).

% snth_sset
thf(fact_92_snth__sset,axiom,
    ! [S: stream8299795917829157543ist_fm,N: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S @ N ) @ ( sset_P3431374123360452149ist_fm @ S ) ) ).

% snth_sset
thf(fact_93_snth__sset,axiom,
    ! [S: stream8372878641218411373at_nat,N: nat] : ( member8206827879206165904at_nat @ ( snth_P9098630209796499211at_nat @ S @ N ) @ ( sset_P9164763103296375419at_nat @ S ) ) ).

% snth_sset
thf(fact_94_snth__sset,axiom,
    ! [S: stream_rule,N: nat] : ( member_rule @ ( snth_rule @ S @ N ) @ ( sset_rule @ S ) ) ).

% snth_sset
thf(fact_95_prod__induct3,axiom,
    ! [P: produc859450856879609959at_nat > $o,X6: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: nat,C: nat] : ( P @ ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) )
     => ( P @ X6 ) ) ).

% prod_induct3
thf(fact_96_prod__induct3,axiom,
    ! [P: produc4630866025709511057ist_fm > $o,X6: produc4630866025709511057ist_fm] :
      ( ! [A3: rule,B3: list_fm,C: list_fm] : ( P @ ( produc7063755442767259331ist_fm @ A3 @ ( produc381145313068854617ist_fm @ B3 @ C ) ) )
     => ( P @ X6 ) ) ).

% prod_induct3
thf(fact_97_mem__Collect__eq,axiom,
    ! [A: fm,P: fm > $o] :
      ( ( member_fm @ A @ ( collect_fm @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
    ! [A: rule,P: rule > $o] :
      ( ( member_rule @ A @ ( collect_rule @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_99_mem__Collect__eq,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ A @ ( collec7231575776666803724ist_fm @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_100_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_101_Collect__mem__eq,axiom,
    ! [A4: set_fm] :
      ( ( collect_fm
        @ ^ [X5: fm] : ( member_fm @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
    ! [A4: set_rule] :
      ( ( collect_rule
        @ ^ [X5: rule] : ( member_rule @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_103_Collect__mem__eq,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm] :
      ( ( collec7231575776666803724ist_fm
        @ ^ [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_104_Collect__mem__eq,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X5: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_105_prod__cases3,axiom,
    ! [Y5: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: nat,C: nat] :
          ( Y5
         != ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_106_prod__cases3,axiom,
    ! [Y5: produc4630866025709511057ist_fm] :
      ~ ! [A3: rule,B3: list_fm,C: list_fm] :
          ( Y5
         != ( produc7063755442767259331ist_fm @ A3 @ ( produc381145313068854617ist_fm @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_107_Pair__inject,axiom,
    ! [A: list_fm,B: list_fm,A2: list_fm,B2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_108_Pair__inject,axiom,
    ! [A: produc1996495991257130529ist_fm,B: rule,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( ( produc491467635432902671m_rule @ A @ B )
        = ( produc491467635432902671m_rule @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_109_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_110_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_111_Pair__inject,axiom,
    ! [A: rule,B: produc1996495991257130529ist_fm,A2: rule,B2: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B )
        = ( produc7063755442767259331ist_fm @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_112_prod__cases,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm] :
      ( ! [A3: list_fm,B3: list_fm] : ( P @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_113_prod__cases,axiom,
    ! [P: produc164195504107695125m_rule > $o,P2: produc164195504107695125m_rule] :
      ( ! [A3: produc1996495991257130529ist_fm,B3: rule] : ( P @ ( produc491467635432902671m_rule @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_114_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_115_prod__cases,axiom,
    ! [P: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_116_prod__cases,axiom,
    ! [P: produc4630866025709511057ist_fm > $o,P2: produc4630866025709511057ist_fm] :
      ( ! [A3: rule,B3: produc1996495991257130529ist_fm] : ( P @ ( produc7063755442767259331ist_fm @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_117_surj__pair,axiom,
    ! [P2: produc1996495991257130529ist_fm] :
    ? [X3: list_fm,Y3: list_fm] :
      ( P2
      = ( produc381145313068854617ist_fm @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_118_surj__pair,axiom,
    ! [P2: produc164195504107695125m_rule] :
    ? [X3: produc1996495991257130529ist_fm,Y3: rule] :
      ( P2
      = ( produc491467635432902671m_rule @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_119_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_120_surj__pair,axiom,
    ! [P2: produc859450856879609959at_nat] :
    ? [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( P2
      = ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_121_surj__pair,axiom,
    ! [P2: produc4630866025709511057ist_fm] :
    ? [X3: rule,Y3: produc1996495991257130529ist_fm] :
      ( P2
      = ( produc7063755442767259331ist_fm @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_122_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc1996495991257130529ist_fm] :
      ~ ! [A3: list_fm,B3: list_fm] :
          ( Y5
         != ( produc381145313068854617ist_fm @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_123_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc164195504107695125m_rule] :
      ~ ! [A3: produc1996495991257130529ist_fm,B3: rule] :
          ( Y5
         != ( produc491467635432902671m_rule @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_124_old_Oprod_Oexhaust,axiom,
    ! [Y5: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y5
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_125_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
          ( Y5
         != ( produc6161850002892822231at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_126_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc4630866025709511057ist_fm] :
      ~ ! [A3: rule,B3: produc1996495991257130529ist_fm] :
          ( Y5
         != ( produc7063755442767259331ist_fm @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_127_sset__induct,axiom,
    ! [Y5: fm,S: stream_fm,P: fm > stream_fm > $o] :
      ( ( member_fm @ Y5 @ ( sset_fm @ S ) )
     => ( ! [S3: stream_fm] : ( P @ ( shd_fm @ S3 ) @ S3 )
       => ( ! [S3: stream_fm,Y3: fm] :
              ( ( member_fm @ Y3 @ ( sset_fm @ ( stl_fm @ S3 ) ) )
             => ( ( P @ Y3 @ ( stl_fm @ S3 ) )
               => ( P @ Y3 @ S3 ) ) )
         => ( P @ Y5 @ S ) ) ) ) ).

% sset_induct
thf(fact_128_sset__induct,axiom,
    ! [Y5: produc1996495991257130529ist_fm,S: stream8299795917829157543ist_fm,P: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Y5 @ ( sset_P3431374123360452149ist_fm @ S ) )
     => ( ! [S3: stream8299795917829157543ist_fm] : ( P @ ( shd_Pr772355297128350925ist_fm @ S3 ) @ S3 )
       => ( ! [S3: stream8299795917829157543ist_fm,Y3: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Y3 @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ S3 ) ) )
             => ( ( P @ Y3 @ ( stl_Pr5027778045650968273ist_fm @ S3 ) )
               => ( P @ Y3 @ S3 ) ) )
         => ( P @ Y5 @ S ) ) ) ) ).

% sset_induct
thf(fact_129_sset__induct,axiom,
    ! [Y5: produc859450856879609959at_nat,S: stream8372878641218411373at_nat,P: produc859450856879609959at_nat > stream8372878641218411373at_nat > $o] :
      ( ( member8206827879206165904at_nat @ Y5 @ ( sset_P9164763103296375419at_nat @ S ) )
     => ( ! [S3: stream8372878641218411373at_nat] : ( P @ ( shd_Pr8412153233960533267at_nat @ S3 ) @ S3 )
       => ( ! [S3: stream8372878641218411373at_nat,Y3: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Y3 @ ( sset_P9164763103296375419at_nat @ ( stl_Pr5010465336888994071at_nat @ S3 ) ) )
             => ( ( P @ Y3 @ ( stl_Pr5010465336888994071at_nat @ S3 ) )
               => ( P @ Y3 @ S3 ) ) )
         => ( P @ Y5 @ S ) ) ) ) ).

% sset_induct
thf(fact_130_sset__induct,axiom,
    ! [Y5: produc164195504107695125m_rule,S: stream727092118206550309m_rule,P: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ Y5 @ ( sset_P4979403222752902913m_rule @ S ) )
     => ( ! [S3: stream727092118206550309m_rule] : ( P @ ( shd_Pr7235097944458474089m_rule @ S3 ) @ S3 )
       => ( ! [S3: stream727092118206550309m_rule,Y3: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ S3 ) ) )
             => ( ( P @ Y3 @ ( stl_Pr950425576149878629m_rule @ S3 ) )
               => ( P @ Y3 @ S3 ) ) )
         => ( P @ Y5 @ S ) ) ) ) ).

% sset_induct
thf(fact_131_sset__induct,axiom,
    ! [Y5: rule,S: stream_rule,P: rule > stream_rule > $o] :
      ( ( member_rule @ Y5 @ ( sset_rule @ S ) )
     => ( ! [S3: stream_rule] : ( P @ ( shd_rule @ S3 ) @ S3 )
       => ( ! [S3: stream_rule,Y3: rule] :
              ( ( member_rule @ Y3 @ ( sset_rule @ ( stl_rule @ S3 ) ) )
             => ( ( P @ Y3 @ ( stl_rule @ S3 ) )
               => ( P @ Y3 @ S3 ) ) )
         => ( P @ Y5 @ S ) ) ) ) ).

% sset_induct
thf(fact_132_RuleSystem__Defs_Osaturated_Ocong,axiom,
    abstra7252176601000738572ist_fm = abstra7252176601000738572ist_fm ).

% RuleSystem_Defs.saturated.cong
thf(fact_133_sset__sdrop,axiom,
    ! [N: nat,S: stream727092118206550309m_rule] : ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S ) ) @ ( sset_P4979403222752902913m_rule @ S ) ) ).

% sset_sdrop
thf(fact_134_sset__sdrop,axiom,
    ! [N: nat,S: stream8299795917829157543ist_fm] : ( ord_le2055072402069232929ist_fm @ ( sset_P3431374123360452149ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S ) ) @ ( sset_P3431374123360452149ist_fm @ S ) ) ).

% sset_sdrop
thf(fact_135_sset__sdrop,axiom,
    ! [N: nat,S: stream3588992565182678257e_rule] : ( ord_le4515388646595166059e_rule @ ( sset_P4203794550272384383e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S ) ) @ ( sset_P4203794550272384383e_rule @ S ) ) ).

% sset_sdrop
thf(fact_136_sset__sdrop,axiom,
    ! [N: nat,S: stream1878305877988316353m_rule] : ( ord_le6218097786609347643m_rule @ ( sset_P8020534884870570447m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S ) ) @ ( sset_P8020534884870570447m_rule @ S ) ) ).

% sset_sdrop
thf(fact_137_sset__sdrop,axiom,
    ! [N: nat,S: stream2312013117288958913e_rule] : ( ord_le6651805025909990203e_rule @ ( sset_P3425104634812430543e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S ) ) @ ( sset_P3425104634812430543e_rule @ S ) ) ).

% sset_sdrop
thf(fact_138_sset__sdrop,axiom,
    ! [N: nat,S: stream6494289010434245521m_rule] : ( ord_le3684904531561281547m_rule @ ( sset_P270791984657859871m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S ) ) @ ( sset_P270791984657859871m_rule @ S ) ) ).

% sset_sdrop
thf(fact_139_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_140_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_141_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S22 ) )
      = ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_142_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( stl_Pr830518440239794027e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S22 ) )
      = ( szip_P553291425819358518e_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_143_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr5425948690297933931m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S22 ) )
      = ( szip_r2254114822203693110m_rule @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_144_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( stl_Pr4904654673476115995e_rule @ ( szip_rule_rule @ S1 @ S22 ) )
      = ( szip_rule_rule @ ( stl_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_145_szip_Osimps_I2_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S22: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) )
      = ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_146_fst__eqD,axiom,
    ! [X6: nat,Y5: nat,A: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) )
        = A )
     => ( X6 = A ) ) ).

% fst_eqD
thf(fact_147_fst__eqD,axiom,
    ! [X6: product_prod_nat_nat,Y5: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) )
        = A )
     => ( X6 = A ) ) ).

% fst_eqD
thf(fact_148_fst__eqD,axiom,
    ! [X6: rule,Y5: produc1996495991257130529ist_fm,A: rule] :
      ( ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) )
        = A )
     => ( X6 = A ) ) ).

% fst_eqD
thf(fact_149_fst__eqD,axiom,
    ! [X6: produc1996495991257130529ist_fm,Y5: rule,A: produc1996495991257130529ist_fm] :
      ( ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) )
        = A )
     => ( X6 = A ) ) ).

% fst_eqD
thf(fact_150_fst__eqD,axiom,
    ! [X6: list_fm,Y5: list_fm,A: list_fm] :
      ( ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) )
        = A )
     => ( X6 = A ) ) ).

% fst_eqD
thf(fact_151_fst__conv,axiom,
    ! [X12: nat,X2: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X12 @ X2 ) )
      = X12 ) ).

% fst_conv
thf(fact_152_fst__conv,axiom,
    ! [X12: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X12 @ X2 ) )
      = X12 ) ).

% fst_conv
thf(fact_153_fst__conv,axiom,
    ! [X12: rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X12 @ X2 ) )
      = X12 ) ).

% fst_conv
thf(fact_154_fst__conv,axiom,
    ! [X12: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X12 @ X2 ) )
      = X12 ) ).

% fst_conv
thf(fact_155_fst__conv,axiom,
    ! [X12: list_fm,X2: list_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X12 @ X2 ) )
      = X12 ) ).

% fst_conv
thf(fact_156_snd__conv,axiom,
    ! [X12: nat,X2: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X12 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_157_snd__conv,axiom,
    ! [X12: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X12 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_158_snd__conv,axiom,
    ! [X12: rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X12 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_159_snd__conv,axiom,
    ! [X12: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X12 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_160_snd__conv,axiom,
    ! [X12: list_fm,X2: list_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X12 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_161_snd__eqD,axiom,
    ! [X6: nat,Y5: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_162_snd__eqD,axiom,
    ! [X6: product_prod_nat_nat,Y5: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_163_snd__eqD,axiom,
    ! [X6: rule,Y5: produc1996495991257130529ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_164_snd__eqD,axiom,
    ! [X6: produc1996495991257130529ist_fm,Y5: rule,A: rule] :
      ( ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_165_snd__eqD,axiom,
    ! [X6: list_fm,Y5: list_fm,A: list_fm] :
      ( ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_166_stream_Ocase__eq__if,axiom,
    ( case_s2494923709252486212m_rule
    = ( ^ [F: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] : ( F @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ) ).

% stream.case_eq_if
thf(fact_167_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra8647779336736900908ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,Steps2: stream727092118206550309m_rule] :
        ! [X5: rule] :
          ( ( member_rule @ X5 @ ( sset_rule @ Rules ) )
         => ( abstra7252176601000738572ist_fm @ Eff @ X5 @ Steps2 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_168_RuleSystem__Defs_Owf__ipath__epath,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps )
       => ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ Steps ) ) ) ).

% RuleSystem_Defs.wf_ipath_epath
thf(fact_169_shd__sset,axiom,
    ! [A: stream_fm] : ( member_fm @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).

% shd_sset
thf(fact_170_shd__sset,axiom,
    ! [A: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) @ ( sset_P3431374123360452149ist_fm @ A ) ) ).

% shd_sset
thf(fact_171_shd__sset,axiom,
    ! [A: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ A ) @ ( sset_P9164763103296375419at_nat @ A ) ) ).

% shd_sset
thf(fact_172_shd__sset,axiom,
    ! [A: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) @ ( sset_P4979403222752902913m_rule @ A ) ) ).

% shd_sset
thf(fact_173_shd__sset,axiom,
    ! [A: stream_rule] : ( member_rule @ ( shd_rule @ A ) @ ( sset_rule @ A ) ) ).

% shd_sset
thf(fact_174_stream_Ocase__distrib,axiom,
    ! [H: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,F2: rule > stream_rule > abstra4572470701423150929m_rule,Stream2: stream_rule] :
      ( ( H @ ( case_s2494923709252486212m_rule @ F2 @ Stream2 ) )
      = ( case_s2494923709252486212m_rule
        @ ^ [X1: rule,X22: stream_rule] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Stream2 ) ) ).

% stream.case_distrib
thf(fact_175_stream_Oexpand,axiom,
    ! [Stream2: stream727092118206550309m_rule,Stream3: stream727092118206550309m_rule] :
      ( ( ( ( shd_Pr7235097944458474089m_rule @ Stream2 )
          = ( shd_Pr7235097944458474089m_rule @ Stream3 ) )
        & ( ( stl_Pr950425576149878629m_rule @ Stream2 )
          = ( stl_Pr950425576149878629m_rule @ Stream3 ) ) )
     => ( Stream2 = Stream3 ) ) ).

% stream.expand
thf(fact_176_stream_Oexpand,axiom,
    ! [Stream2: stream_rule,Stream3: stream_rule] :
      ( ( ( ( shd_rule @ Stream2 )
          = ( shd_rule @ Stream3 ) )
        & ( ( stl_rule @ Stream2 )
          = ( stl_rule @ Stream3 ) ) )
     => ( Stream2 = Stream3 ) ) ).

% stream.expand
thf(fact_177_stream_Ocoinduct,axiom,
    ! [R3: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream2: stream727092118206550309m_rule,Stream3: stream727092118206550309m_rule] :
      ( ( R3 @ Stream2 @ Stream3 )
     => ( ! [Stream4: stream727092118206550309m_rule,Stream5: stream727092118206550309m_rule] :
            ( ( R3 @ Stream4 @ Stream5 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream4 )
                = ( shd_Pr7235097944458474089m_rule @ Stream5 ) )
              & ( R3 @ ( stl_Pr950425576149878629m_rule @ Stream4 ) @ ( stl_Pr950425576149878629m_rule @ Stream5 ) ) ) )
       => ( Stream2 = Stream3 ) ) ) ).

% stream.coinduct
thf(fact_178_stream_Ocoinduct,axiom,
    ! [R3: stream_rule > stream_rule > $o,Stream2: stream_rule,Stream3: stream_rule] :
      ( ( R3 @ Stream2 @ Stream3 )
     => ( ! [Stream4: stream_rule,Stream5: stream_rule] :
            ( ( R3 @ Stream4 @ Stream5 )
           => ( ( ( shd_rule @ Stream4 )
                = ( shd_rule @ Stream5 ) )
              & ( R3 @ ( stl_rule @ Stream4 ) @ ( stl_rule @ Stream5 ) ) ) )
       => ( Stream2 = Stream3 ) ) ) ).

% stream.coinduct
thf(fact_179_stream_Ocoinduct__strong,axiom,
    ! [R3: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream2: stream727092118206550309m_rule,Stream3: stream727092118206550309m_rule] :
      ( ( R3 @ Stream2 @ Stream3 )
     => ( ! [Stream4: stream727092118206550309m_rule,Stream5: stream727092118206550309m_rule] :
            ( ( R3 @ Stream4 @ Stream5 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream4 )
                = ( shd_Pr7235097944458474089m_rule @ Stream5 ) )
              & ( ( R3 @ ( stl_Pr950425576149878629m_rule @ Stream4 ) @ ( stl_Pr950425576149878629m_rule @ Stream5 ) )
                | ( ( stl_Pr950425576149878629m_rule @ Stream4 )
                  = ( stl_Pr950425576149878629m_rule @ Stream5 ) ) ) ) )
       => ( Stream2 = Stream3 ) ) ) ).

% stream.coinduct_strong
thf(fact_180_stream_Ocoinduct__strong,axiom,
    ! [R3: stream_rule > stream_rule > $o,Stream2: stream_rule,Stream3: stream_rule] :
      ( ( R3 @ Stream2 @ Stream3 )
     => ( ! [Stream4: stream_rule,Stream5: stream_rule] :
            ( ( R3 @ Stream4 @ Stream5 )
           => ( ( ( shd_rule @ Stream4 )
                = ( shd_rule @ Stream5 ) )
              & ( ( R3 @ ( stl_rule @ Stream4 ) @ ( stl_rule @ Stream5 ) )
                | ( ( stl_rule @ Stream4 )
                  = ( stl_rule @ Stream5 ) ) ) ) )
       => ( Stream2 = Stream3 ) ) ) ).

% stream.coinduct_strong
thf(fact_181_sdrop__stl,axiom,
    ! [N: nat,S: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( stl_Pr5027778045650968273ist_fm @ S ) )
      = ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_182_sdrop__stl,axiom,
    ! [N: nat,S: stream3588992565182678257e_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( stl_Pr4904654673476115995e_rule @ S ) )
      = ( stl_Pr4904654673476115995e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_183_sdrop__stl,axiom,
    ! [N: nat,S: stream1878305877988316353m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N @ ( stl_Pr5425948690297933931m_rule @ S ) )
      = ( stl_Pr5425948690297933931m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_184_sdrop__stl,axiom,
    ! [N: nat,S: stream2312013117288958913e_rule] :
      ( ( sdrop_6392754500180887844e_rule @ N @ ( stl_Pr830518440239794027e_rule @ S ) )
      = ( stl_Pr830518440239794027e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_185_sdrop__stl,axiom,
    ! [N: nat,S: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( stl_Pr2200982372225425851m_rule @ S ) )
      = ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_186_sdrop__stl,axiom,
    ! [N: nat,S: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( stl_Pr950425576149878629m_rule @ S ) )
      = ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_187_sdrop__stl,axiom,
    ! [N: nat,S: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S ) ) ) ).

% sdrop_stl
thf(fact_188_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream5128679079952569136_fm_fm > $o,X6: stream5128679079952569136_fm_fm,Rules2: stream_fm,Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream5128679079952569136_fm_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream5128679079952569136_fm_fm,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules2 ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) )
                  | ( abstra9155489748554225612ist_fm @ Eff2 @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_189_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream3409308193418444653ist_fm > $o,X6: stream3409308193418444653ist_fm,Rules2: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream3409308193418444653ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream3409308193418444653ist_fm,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) )
                  | ( abstra5061764961602482947ist_fm @ Eff2 @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_190_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream1622999302952468147at_nat > $o,X6: stream1622999302952468147at_nat,Rules2: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream1622999302952468147at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream1622999302952468147at_nat,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) )
                  | ( abstra1311857326046938941ist_fm @ Eff2 @ Rules2 @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) ) )
       => ( abstra1311857326046938941ist_fm @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_191_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream812778442145099016ule_fm > $o,X6: stream812778442145099016ule_fm,Rules2: stream_fm,Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream812778442145099016ule_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream812778442145099016ule_fm,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules2 ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) )
                  | ( abstra1364558218721404774m_rule @ Eff2 @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_192_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream7989562319868115861ist_fm > $o,X6: stream7989562319868115861ist_fm,Rules2: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream7989562319868115861ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7989562319868115861ist_fm,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) )
                  | ( abstra8066765647771630831m_rule @ Eff2 @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_193_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream1819640557982756571at_nat > $o,X6: stream1819640557982756571at_nat,Rules2: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream1819640557982756571at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream1819640557982756571at_nat,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) )
                  | ( abstra3157339599784851765m_rule @ Eff2 @ Rules2 @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) ) )
       => ( abstra3157339599784851765m_rule @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_194_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream7755889244956737277e_rule > $o,X6: stream7755889244956737277e_rule,Rules2: stream_rule,Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream7755889244956737277e_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7755889244956737277e_rule,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules2 ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) )
                  | ( abstra5823804033175701339m_rule @ Eff2 @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_195_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream8299795917829157543ist_fm > $o,X6: stream8299795917829157543ist_fm,Rules2: stream_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream8299795917829157543ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream8299795917829157543ist_fm,Sl2: fset_list_fm] :
                ( ( X3 = Steps3 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules2 ) )
                & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) )
                  | ( abstra6579460235205289861ist_fm @ Eff2 @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) ) )
       => ( abstra6579460235205289861ist_fm @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_196_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X6: stream727092118206550309m_rule,Rules2: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream727092118206550309m_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream727092118206550309m_rule,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules2 ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl2 )
                & ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_197_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream5128679079952569136_fm_fm,Rules2: stream_fm,Sl3: fset_P661503646757059847ist_fm,Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( sset_fm @ Rules2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra9155489748554225612ist_fm @ Eff2 @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) )
           => ( abstra9155489748554225612ist_fm @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_198_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream3409308193418444653ist_fm,Rules2: stream8299795917829157543ist_fm,Sl3: fset_P661503646757059847ist_fm,Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra5061764961602482947ist_fm @ Eff2 @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ Steps ) )
           => ( abstra5061764961602482947ist_fm @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_199_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream1622999302952468147at_nat,Rules2: stream8372878641218411373at_nat,Sl3: fset_P661503646757059847ist_fm,Eff2: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra1311857326046938941ist_fm @ Eff2 @ Rules2 @ ( stl_Pr1564049449699658205at_nat @ Steps ) )
           => ( abstra1311857326046938941ist_fm @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_200_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream812778442145099016ule_fm,Rules2: stream_fm,Sl3: fset_A5396818216090053761m_rule,Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( sset_fm @ Rules2 ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra1364558218721404774m_rule @ Eff2 @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ Steps ) )
           => ( abstra1364558218721404774m_rule @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_201_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7989562319868115861ist_fm,Rules2: stream8299795917829157543ist_fm,Sl3: fset_A5396818216090053761m_rule,Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra8066765647771630831m_rule @ Eff2 @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ Steps ) )
           => ( abstra8066765647771630831m_rule @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_202_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream1819640557982756571at_nat,Rules2: stream8372878641218411373at_nat,Sl3: fset_A5396818216090053761m_rule,Eff2: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra3157339599784851765m_rule @ Eff2 @ Rules2 @ ( stl_Pr2369401607900929691at_nat @ Steps ) )
           => ( abstra3157339599784851765m_rule @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_203_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7755889244956737277e_rule,Rules2: stream_rule,Sl3: fset_A5396818216090053761m_rule,Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5823804033175701339m_rule @ Eff2 @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ Steps ) )
           => ( abstra5823804033175701339m_rule @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_204_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream8299795917829157543ist_fm,Rules2: stream_list_fm,Sl3: fset_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules2 ) )
     => ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra6579460235205289861ist_fm @ Eff2 @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ Steps ) )
           => ( abstra6579460235205289861ist_fm @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_205_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Rules2: stream_rule,Sl3: fset_P661503646757059847ist_fm,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_206_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra9155489748554225612ist_fm
    = ( ^ [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,A5: stream5128679079952569136_fm_fm] :
        ? [Steps2: stream5128679079952569136_fm_fm,Sl: fset_P661503646757059847ist_fm] :
          ( ( A5 = Steps2 )
          & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ Sl )
          & ( abstra9155489748554225612ist_fm @ Eff @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_207_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5061764961602482947ist_fm
    = ( ^ [Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8299795917829157543ist_fm,A5: stream3409308193418444653ist_fm] :
        ? [Steps2: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
          ( ( A5 = Steps2 )
          & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ Sl )
          & ( abstra5061764961602482947ist_fm @ Eff @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_208_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1311857326046938941ist_fm
    = ( ^ [Eff: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8372878641218411373at_nat,A5: stream1622999302952468147at_nat] :
        ? [Steps2: stream1622999302952468147at_nat,Sl: fset_P661503646757059847ist_fm] :
          ( ( A5 = Steps2 )
          & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ Sl )
          & ( abstra1311857326046938941ist_fm @ Eff @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_209_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1364558218721404774m_rule
    = ( ^ [Eff: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_fm,A5: stream812778442145099016ule_fm] :
        ? [Steps2: stream812778442145099016ule_fm,Sl: fset_A5396818216090053761m_rule] :
          ( ( A5 = Steps2 )
          & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
          & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ Sl )
          & ( abstra1364558218721404774m_rule @ Eff @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_210_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra8066765647771630831m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream8299795917829157543ist_fm,A5: stream7989562319868115861ist_fm] :
        ? [Steps2: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
          ( ( A5 = Steps2 )
          & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
          & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ Sl )
          & ( abstra8066765647771630831m_rule @ Eff @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_211_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra3157339599784851765m_rule
    = ( ^ [Eff: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream8372878641218411373at_nat,A5: stream1819640557982756571at_nat] :
        ? [Steps2: stream1819640557982756571at_nat,Sl: fset_A5396818216090053761m_rule] :
          ( ( A5 = Steps2 )
          & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
          & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ Sl )
          & ( abstra3157339599784851765m_rule @ Eff @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_212_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5823804033175701339m_rule
    = ( ^ [Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_rule,A5: stream7755889244956737277e_rule] :
        ? [Steps2: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
          ( ( A5 = Steps2 )
          & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
          & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ Sl )
          & ( abstra5823804033175701339m_rule @ Eff @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_213_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6579460235205289861ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,A5: stream8299795917829157543ist_fm] :
        ? [Steps2: stream8299795917829157543ist_fm,Sl: fset_list_fm] :
          ( ( A5 = Steps2 )
          & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules ) )
          & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ Sl )
          & ( abstra6579460235205289861ist_fm @ Eff @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_214_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5415202973855822231ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A5: stream727092118206550309m_rule] :
        ? [Steps2: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
          ( ( A5 = Steps2 )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ Sl )
          & ( abstra5415202973855822231ist_fm @ Eff @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_215_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ Eff2 @ Rules2 @ A )
     => ~ ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( sset_fm @ Rules2 ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ Sl4 )
               => ~ ( abstra9155489748554225612ist_fm @ Eff2 @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_216_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ Eff2 @ Rules2 @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra5061764961602482947ist_fm @ Eff2 @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_217_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8372878641218411373at_nat,A: stream1622999302952468147at_nat] :
      ( ( abstra1311857326046938941ist_fm @ Eff2 @ Rules2 @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ Sl4 )
               => ~ ( abstra1311857326046938941ist_fm @ Eff2 @ Rules2 @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_218_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ Eff2 @ Rules2 @ A )
     => ~ ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( sset_fm @ Rules2 ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ Sl4 )
               => ~ ( abstra1364558218721404774m_rule @ Eff2 @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_219_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ Eff2 @ Rules2 @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra8066765647771630831m_rule @ Eff2 @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_220_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8372878641218411373at_nat,A: stream1819640557982756571at_nat] :
      ( ( abstra3157339599784851765m_rule @ Eff2 @ Rules2 @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ Sl4 )
               => ~ ( abstra3157339599784851765m_rule @ Eff2 @ Rules2 @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_221_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ Eff2 @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ Sl4 )
               => ~ ( abstra5823804033175701339m_rule @ Eff2 @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_222_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ Eff2 @ Rules2 @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( sset_list_fm @ Rules2 ) )
         => ! [Sl4: fset_list_fm] :
              ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra6579460235205289861ist_fm @ Eff2 @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_223_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) ) @ Sl4 )
             => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ Sl4 )
               => ~ ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_224_surjective__pairing,axiom,
    ! [T2: product_prod_nat_nat] :
      ( T2
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T2 ) @ ( product_snd_nat_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_225_surjective__pairing,axiom,
    ! [T2: produc859450856879609959at_nat] :
      ( T2
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ T2 ) @ ( produc6408287024330202629at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_226_surjective__pairing,axiom,
    ! [T2: produc4630866025709511057ist_fm] :
      ( T2
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ T2 ) @ ( produc4514744107061996309ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_227_surjective__pairing,axiom,
    ! [T2: produc164195504107695125m_rule] :
      ( T2
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ T2 ) @ ( produc7165828336582415457m_rule @ T2 ) ) ) ).

% surjective_pairing
thf(fact_228_surjective__pairing,axiom,
    ! [T2: produc1996495991257130529ist_fm] :
      ( T2
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ T2 ) @ ( produc4588648349897876871ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_229_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q2: nat > $o,Q: nat] :
      ( ( P @ P2 )
     => ( ( Q2 @ Q )
       => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q ) ) )
          & ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_230_conjI__realizer,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat,Q2: product_prod_nat_nat > $o,Q: product_prod_nat_nat] :
      ( ( P @ P2 )
     => ( ( Q2 @ Q )
       => ( ( P @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ P2 @ Q ) ) )
          & ( Q2 @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ P2 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_231_conjI__realizer,axiom,
    ! [P: rule > $o,P2: rule,Q2: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm] :
      ( ( P @ P2 )
     => ( ( Q2 @ Q )
       => ( ( P @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ P2 @ Q ) ) )
          & ( Q2 @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ P2 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_232_conjI__realizer,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm,Q2: rule > $o,Q: rule] :
      ( ( P @ P2 )
     => ( ( Q2 @ Q )
       => ( ( P @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ P2 @ Q ) ) )
          & ( Q2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ P2 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_233_conjI__realizer,axiom,
    ! [P: list_fm > $o,P2: list_fm,Q2: list_fm > $o,Q: list_fm] :
      ( ( P @ P2 )
     => ( ( Q2 @ Q )
       => ( ( P @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ P2 @ Q ) ) )
          & ( Q2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ P2 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_234_exI__realizer,axiom,
    ! [P: nat > nat > $o,Y5: nat,X6: nat] :
      ( ( P @ Y5 @ X6 )
     => ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) ) ) ) ).

% exI_realizer
thf(fact_235_exI__realizer,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o,Y5: product_prod_nat_nat,X6: product_prod_nat_nat] :
      ( ( P @ Y5 @ X6 )
     => ( P @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) ) @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) ) ) ) ).

% exI_realizer
thf(fact_236_exI__realizer,axiom,
    ! [P: produc1996495991257130529ist_fm > rule > $o,Y5: produc1996495991257130529ist_fm,X6: rule] :
      ( ( P @ Y5 @ X6 )
     => ( P @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) ) @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) ) ) ) ).

% exI_realizer
thf(fact_237_exI__realizer,axiom,
    ! [P: rule > produc1996495991257130529ist_fm > $o,Y5: rule,X6: produc1996495991257130529ist_fm] :
      ( ( P @ Y5 @ X6 )
     => ( P @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) ) ) ) ).

% exI_realizer
thf(fact_238_exI__realizer,axiom,
    ! [P: list_fm > list_fm > $o,Y5: list_fm,X6: list_fm] :
      ( ( P @ Y5 @ X6 )
     => ( P @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) ) @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) ) ) ) ).

% exI_realizer
thf(fact_239_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( Prod
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_240_prod_Oexhaust__sel,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( Prod
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_241_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( Prod
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_242_prod_Oexhaust__sel,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( Prod
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_243_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( Prod
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_244_per__rules_H,axiom,
    ! [R: rule,A4: list_fm,B4: list_fm,A6: list_fm,B5: list_fm,R4: rule,Ss3: fset_P661503646757059847ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R
        @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
     => ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R
            @ ( produc381145313068854617ist_fm @ A6 @ B5 ) )
       => ( ( ( eff @ R4 @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
            = ( some_f2162114139285431020ist_fm @ Ss3 ) )
         => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B5 ) @ Ss3 )
           => ( R4 = R ) ) ) ) ) ).

% per_rules'
thf(fact_245_epath_Ocases,axiom,
    ! [A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) ) @ Sl4 )
             => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) )
                  = ( some_f2162114139285431020ist_fm @ Sl4 ) )
               => ~ ( abstra5415202973855822231ist_fm
                    @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S2 )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% epath.cases
thf(fact_246_epath_Osimps,axiom,
    ! [A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
      = ( ? [Steps2: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl )
            & ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) )
              = ( some_f2162114139285431020ist_fm @ Sl ) )
            & ( abstra5415202973855822231ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% epath.simps
thf(fact_247_epath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Sl3: fset_P661503646757059847ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
            = ( some_f2162114139285431020ist_fm @ Sl3 ) )
         => ( ( abstra5415202973855822231ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps ) ) ) ) ) ).

% epath.intros
thf(fact_248_epath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X6: stream727092118206550309m_rule] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream727092118206550309m_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream727092118206550309m_rule,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl2 )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) )
                  = ( some_f2162114139285431020ist_fm @ Sl2 ) )
                & ( ( X7 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra5415202973855822231ist_fm
                    @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S2 )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ X6 ) ) ) ).

% epath.coinduct
thf(fact_249_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: nat > nat > $o,X6: nat,Y5: nat,A: product_prod_nat_nat] :
      ( ( P @ X6 @ Y5 )
     => ( ( A
          = ( product_Pair_nat_nat @ X6 @ Y5 ) )
       => ( P @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_250_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o,X6: product_prod_nat_nat,Y5: product_prod_nat_nat,A: produc859450856879609959at_nat] :
      ( ( P @ X6 @ Y5 )
     => ( ( A
          = ( produc6161850002892822231at_nat @ X6 @ Y5 ) )
       => ( P @ ( produc3213797794245857475at_nat @ A ) @ ( produc6408287024330202629at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_251_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: rule > produc1996495991257130529ist_fm > $o,X6: rule,Y5: produc1996495991257130529ist_fm,A: produc4630866025709511057ist_fm] :
      ( ( P @ X6 @ Y5 )
     => ( ( A
          = ( produc7063755442767259331ist_fm @ X6 @ Y5 ) )
       => ( P @ ( produc4228417144610596823ist_fm @ A ) @ ( produc4514744107061996309ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_252_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: produc1996495991257130529ist_fm > rule > $o,X6: produc1996495991257130529ist_fm,Y5: rule,A: produc164195504107695125m_rule] :
      ( ( P @ X6 @ Y5 )
     => ( ( A
          = ( produc491467635432902671m_rule @ X6 @ Y5 ) )
       => ( P @ ( produc6879501374131015971m_rule @ A ) @ ( produc7165828336582415457m_rule @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_253_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: list_fm > list_fm > $o,X6: list_fm,Y5: list_fm,A: produc1996495991257130529ist_fm] :
      ( ( P @ X6 @ Y5 )
     => ( ( A
          = ( produc381145313068854617ist_fm @ X6 @ Y5 ) )
       => ( P @ ( produc1501393135466168645ist_fm @ A ) @ ( produc4588648349897876871ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_254_treeB__sdrop,axiom,
    ! [N: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeB @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) @ ( treeB @ Steps ) ) ).

% treeB_sdrop
thf(fact_255_subset__antisym,axiom,
    ! [A4: set_fm,B4: set_fm] :
      ( ( ord_less_eq_set_fm @ A4 @ B4 )
     => ( ( ord_less_eq_set_fm @ B4 @ A4 )
       => ( A4 = B4 ) ) ) ).

% subset_antisym
thf(fact_256_subsetI,axiom,
    ! [A4: set_rule,B4: set_rule] :
      ( ! [X3: rule] :
          ( ( member_rule @ X3 @ A4 )
         => ( member_rule @ X3 @ B4 ) )
     => ( ord_less_eq_set_rule @ A4 @ B4 ) ) ).

% subsetI
thf(fact_257_subsetI,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,B4: set_Pr7058068377845519745ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X3 @ A4 )
         => ( member8102475879199740618ist_fm @ X3 @ B4 ) )
     => ( ord_le2055072402069232929ist_fm @ A4 @ B4 ) ) ).

% subsetI
thf(fact_258_subsetI,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ! [X3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A4 )
         => ( member8206827879206165904at_nat @ X3 @ B4 ) )
     => ( ord_le3000389064537975527at_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_259_subsetI,axiom,
    ! [A4: set_fm,B4: set_fm] :
      ( ! [X3: fm] :
          ( ( member_fm @ X3 @ A4 )
         => ( member_fm @ X3 @ B4 ) )
     => ( ord_less_eq_set_fm @ A4 @ B4 ) ) ).

% subsetI
thf(fact_260_dual__order_Orefl,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).

% dual_order.refl
thf(fact_261_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_262_order__refl,axiom,
    ! [X6: set_fm] : ( ord_less_eq_set_fm @ X6 @ X6 ) ).

% order_refl
thf(fact_263_order__refl,axiom,
    ! [X6: nat] : ( ord_less_eq_nat @ X6 @ X6 ) ).

% order_refl
thf(fact_264_snth__sset__smerge,axiom,
    ! [Ss2: stream_stream_fm,N: nat,M: nat] : ( member_fm @ ( snth_fm @ ( snth_stream_fm @ Ss2 @ N ) @ M ) @ ( sset_fm @ ( smerge_fm @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_265_snth__sset__smerge,axiom,
    ! [Ss2: stream8183306005803988909ist_fm,N: nat,M: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ ( snth_s4276058539312847179ist_fm @ Ss2 @ N ) @ M ) @ ( sset_P3431374123360452149ist_fm @ ( smerge8067294617398470191ist_fm @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_266_snth__sset__smerge,axiom,
    ! [Ss2: stream7445237232777090803at_nat,N: nat,M: nat] : ( member8206827879206165904at_nat @ ( snth_P9098630209796499211at_nat @ ( snth_s7872458624752591889at_nat @ Ss2 @ N ) @ M ) @ ( sset_P9164763103296375419at_nat @ ( smerge8841642037904234357at_nat @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_267_snth__sset__smerge,axiom,
    ! [Ss2: stream_stream_rule,N: nat,M: nat] : ( member_rule @ ( snth_rule @ ( snth_stream_rule @ Ss2 @ N ) @ M ) @ ( sset_rule @ ( smerge_rule @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_268_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream5128679079952569136_fm_fm > $o,X6: stream5128679079952569136_fm_fm,Rules2: stream_fm,Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream5128679079952569136_fm_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream5128679079952569136_fm_fm,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules2 ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) @ Sl2 )
                & ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) )
                  | ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.epath.coinduct
thf(fact_269_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream3409308193418444653ist_fm > $o,X6: stream3409308193418444653ist_fm,Rules2: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream3409308193418444653ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream3409308193418444653ist_fm,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl2 )
                & ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) )
                  | ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.epath.coinduct
thf(fact_270_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream1622999302952468147at_nat > $o,X6: stream1622999302952468147at_nat,Rules2: stream8372878641218411373at_nat,Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream1622999302952468147at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream1622999302952468147at_nat,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) @ Sl2 )
                & ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) )
                  | ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) ) )
       => ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.epath.coinduct
thf(fact_271_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream812778442145099016ule_fm > $o,X6: stream812778442145099016ule_fm,Rules2: stream_fm,Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream812778442145099016ule_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream812778442145099016ule_fm,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules2 ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl2 )
                & ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) )
                  | ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.epath.coinduct
thf(fact_272_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7989562319868115861ist_fm > $o,X6: stream7989562319868115861ist_fm,Rules2: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream7989562319868115861ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7989562319868115861ist_fm,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl2 )
                & ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) )
                  | ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.epath.coinduct
thf(fact_273_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream1819640557982756571at_nat > $o,X6: stream1819640557982756571at_nat,Rules2: stream8372878641218411373at_nat,Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream1819640557982756571at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream1819640557982756571at_nat,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) @ Sl2 )
                & ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) )
                  | ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) ) )
       => ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.epath.coinduct
thf(fact_274_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7755889244956737277e_rule > $o,X6: stream7755889244956737277e_rule,Rules2: stream_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream7755889244956737277e_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7755889244956737277e_rule,Sl2: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules2 ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl2 )
                & ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) )
                  | ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.epath.coinduct
thf(fact_275_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream8299795917829157543ist_fm > $o,X6: stream8299795917829157543ist_fm,Rules2: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream8299795917829157543ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream8299795917829157543ist_fm,Sl2: fset_list_fm] :
                ( ( X3 = Steps3 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules2 ) )
                & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) @ Sl2 )
                & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) )
                  | ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) ) )
       => ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.epath.coinduct
thf(fact_276_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X6: stream727092118206550309m_rule,Rules2: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X6 )
     => ( ! [X3: stream727092118206550309m_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream727092118206550309m_rule,Sl2: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules2 ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl2 )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl2 )
                & ( ( X7 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.epath.coinduct
thf(fact_277_i_Oepath_Ointros,axiom,
    ! [Steps: stream5128679079952569136_fm_fm,Rules2: stream_fm,Sl3: fset_P661503646757059847ist_fm,Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( sset_fm @ Rules2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) )
           => ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_278_i_Oepath_Ointros,axiom,
    ! [Steps: stream3409308193418444653ist_fm,Rules2: stream8299795917829157543ist_fm,Sl3: fset_P661503646757059847ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ Steps ) )
           => ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_279_i_Oepath_Ointros,axiom,
    ! [Steps: stream1622999302952468147at_nat,Rules2: stream8372878641218411373at_nat,Sl3: fset_P661503646757059847ist_fm,Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr1564049449699658205at_nat @ Steps ) )
           => ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_280_i_Oepath_Ointros,axiom,
    ! [Steps: stream812778442145099016ule_fm,Rules2: stream_fm,Sl3: fset_A5396818216090053761m_rule,Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( sset_fm @ Rules2 ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ Steps ) )
           => ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_281_i_Oepath_Ointros,axiom,
    ! [Steps: stream7989562319868115861ist_fm,Rules2: stream8299795917829157543ist_fm,Sl3: fset_A5396818216090053761m_rule,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ Steps ) )
           => ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_282_i_Oepath_Ointros,axiom,
    ! [Steps: stream1819640557982756571at_nat,Rules2: stream8372878641218411373at_nat,Sl3: fset_A5396818216090053761m_rule,Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr2369401607900929691at_nat @ Steps ) )
           => ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_283_i_Oepath_Ointros,axiom,
    ! [Steps: stream7755889244956737277e_rule,Rules2: stream_rule,Sl3: fset_A5396818216090053761m_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) @ Sl3 )
       => ( ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ Steps ) )
           => ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_284_i_Oepath_Ointros,axiom,
    ! [Steps: stream8299795917829157543ist_fm,Rules2: stream_list_fm,Sl3: fset_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules2 ) )
     => ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ Steps ) )
           => ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_285_i_Oepath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Rules2: stream_rule,Sl3: fset_P661503646757059847ist_fm,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_286_i_Oepath_Osimps,axiom,
    ! [Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules2 @ A )
      = ( ? [Steps2: stream5128679079952569136_fm_fm,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( sset_fm @ Rules2 ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ Sl )
            & ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_287_i_Oepath_Osimps,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules2 @ A )
      = ( ? [Steps2: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ Sl )
            & ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_288_i_Oepath_Osimps,axiom,
    ! [Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream8372878641218411373at_nat,A: stream1622999302952468147at_nat] :
      ( ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules2 @ A )
      = ( ? [Steps2: stream1622999302952468147at_nat,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) @ Sl )
            & ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ Sl )
            & ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_289_i_Oepath_Osimps,axiom,
    ! [Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules2: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules2 @ A )
      = ( ? [Steps2: stream812778442145099016ule_fm,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules2 ) )
            & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ Sl )
            & ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_290_i_Oepath_Osimps,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules2: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules2 @ A )
      = ( ? [Steps2: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
            & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ Sl )
            & ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_291_i_Oepath_Osimps,axiom,
    ! [Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules2: stream8372878641218411373at_nat,A: stream1819640557982756571at_nat] :
      ( ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules2 @ A )
      = ( ? [Steps2: stream1819640557982756571at_nat,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
            & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) @ Sl )
            & ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ Sl )
            & ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_292_i_Oepath_Osimps,axiom,
    ! [Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules2: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules2 @ A )
      = ( ? [Steps2: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
            & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) @ Sl )
            & ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ Sl )
            & ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_293_i_Oepath_Osimps,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules2: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ A )
      = ( ? [Steps2: stream8299795917829157543ist_fm,Sl: fset_list_fm] :
            ( ( A = Steps2 )
            & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules2 ) )
            & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ Sl )
            & ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_294_i_Oepath_Osimps,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ A )
      = ( ? [Steps2: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl )
            & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ Sl )
            & ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_295_less__eq__set__def,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B6: set_rule] :
          ( ord_less_eq_rule_o
          @ ^ [X5: rule] : ( member_rule @ X5 @ A7 )
          @ ^ [X5: rule] : ( member_rule @ X5 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_296_less__eq__set__def,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B6: set_Pr7058068377845519745ist_fm] :
          ( ord_le8518180823940269308t_fm_o
          @ ^ [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ A7 )
          @ ^ [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_297_less__eq__set__def,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( ord_le7858099551454983350_nat_o
          @ ^ [X5: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X5 @ A7 )
          @ ^ [X5: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X5 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_298_less__eq__set__def,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B6: set_fm] :
          ( ord_less_eq_fm_o
          @ ^ [X5: fm] : ( member_fm @ X5 @ A7 )
          @ ^ [X5: fm] : ( member_fm @ X5 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_299_i_Oenabled__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R: rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S )
      = ( ? [X: fset_P661503646757059847ist_fm] : ( abstra5459341389059251418ist_fm @ Eff3 @ R @ S @ X ) ) ) ).

% i.enabled_def
thf(fact_300_i_OpickEff,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R: rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S )
     => ( abstra5459341389059251418ist_fm @ Eff3 @ R @ S @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S ) ) ) ).

% i.pickEff
thf(fact_301_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_302_le__cases3,axiom,
    ! [X6: nat,Y5: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X6 @ Y5 )
       => ~ ( ord_less_eq_nat @ Y5 @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y5 @ X6 )
         => ~ ( ord_less_eq_nat @ X6 @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X6 @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y5 ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y5 )
             => ~ ( ord_less_eq_nat @ Y5 @ X6 ) )
           => ( ( ( ord_less_eq_nat @ Y5 @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X6 ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X6 )
                 => ~ ( ord_less_eq_nat @ X6 @ Y5 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_303_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: set_fm,Z: set_fm] : ( Y = Z ) )
    = ( ^ [X5: set_fm,Y6: set_fm] :
          ( ( ord_less_eq_set_fm @ X5 @ Y6 )
          & ( ord_less_eq_set_fm @ Y6 @ X5 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_304_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [X5: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y6 )
          & ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_305_ord__eq__le__trans,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( A = B )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ord_less_eq_set_fm @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_306_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_307_ord__le__eq__trans,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_set_fm @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_308_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_309_order__antisym,axiom,
    ! [X6: set_fm,Y5: set_fm] :
      ( ( ord_less_eq_set_fm @ X6 @ Y5 )
     => ( ( ord_less_eq_set_fm @ Y5 @ X6 )
       => ( X6 = Y5 ) ) ) ).

% order_antisym
thf(fact_310_order__antisym,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X6 @ Y5 )
     => ( ( ord_less_eq_nat @ Y5 @ X6 )
       => ( X6 = Y5 ) ) ) ).

% order_antisym
thf(fact_311_order_Otrans,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ord_less_eq_set_fm @ A @ C2 ) ) ) ).

% order.trans
thf(fact_312_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_313_order__trans,axiom,
    ! [X6: set_fm,Y5: set_fm,Z2: set_fm] :
      ( ( ord_less_eq_set_fm @ X6 @ Y5 )
     => ( ( ord_less_eq_set_fm @ Y5 @ Z2 )
       => ( ord_less_eq_set_fm @ X6 @ Z2 ) ) ) ).

% order_trans
thf(fact_314_order__trans,axiom,
    ! [X6: nat,Y5: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X6 @ Y5 )
     => ( ( ord_less_eq_nat @ Y5 @ Z2 )
       => ( ord_less_eq_nat @ X6 @ Z2 ) ) ) ).

% order_trans
thf(fact_315_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_316_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: set_fm,Z: set_fm] : ( Y = Z ) )
    = ( ^ [A5: set_fm,B7: set_fm] :
          ( ( ord_less_eq_set_fm @ B7 @ A5 )
          & ( ord_less_eq_set_fm @ A5 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_317_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A5: nat,B7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_318_dual__order_Oantisym,axiom,
    ! [B: set_fm,A: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_319_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_320_dual__order_Otrans,axiom,
    ! [B: set_fm,A: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ C2 @ B )
       => ( ord_less_eq_set_fm @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_321_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_322_antisym,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_323_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_324_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: set_fm,Z: set_fm] : ( Y = Z ) )
    = ( ^ [A5: set_fm,B7: set_fm] :
          ( ( ord_less_eq_set_fm @ A5 @ B7 )
          & ( ord_less_eq_set_fm @ B7 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_325_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A5: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A5 @ B7 )
          & ( ord_less_eq_nat @ B7 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_326_order__subst1,axiom,
    ! [A: set_fm,F2: set_fm > set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_327_order__subst1,axiom,
    ! [A: set_fm,F2: nat > set_fm,B: nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_328_order__subst1,axiom,
    ! [A: nat,F2: set_fm > nat,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_329_order__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_330_order__subst2,axiom,
    ! [A: set_fm,B: set_fm,F2: set_fm > set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F2 @ B ) @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_fm @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_331_order__subst2,axiom,
    ! [A: set_fm,B: set_fm,F2: set_fm > nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_332_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > set_fm,C2: set_fm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_fm @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_333_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_334_order__eq__refl,axiom,
    ! [X6: set_fm,Y5: set_fm] :
      ( ( X6 = Y5 )
     => ( ord_less_eq_set_fm @ X6 @ Y5 ) ) ).

% order_eq_refl
thf(fact_335_order__eq__refl,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( X6 = Y5 )
     => ( ord_less_eq_nat @ X6 @ Y5 ) ) ).

% order_eq_refl
thf(fact_336_linorder__linear,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X6 @ Y5 )
      | ( ord_less_eq_nat @ Y5 @ X6 ) ) ).

% linorder_linear
thf(fact_337_ord__eq__le__subst,axiom,
    ! [A: set_fm,F2: set_fm > set_fm,B: set_fm,C2: set_fm] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_338_ord__eq__le__subst,axiom,
    ! [A: nat,F2: set_fm > nat,B: set_fm,C2: set_fm] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_339_ord__eq__le__subst,axiom,
    ! [A: set_fm,F2: nat > set_fm,B: nat,C2: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_fm @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_340_ord__eq__le__subst,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_341_ord__le__eq__subst,axiom,
    ! [A: set_fm,B: set_fm,F2: set_fm > set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_fm @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_342_ord__le__eq__subst,axiom,
    ! [A: set_fm,B: set_fm,F2: set_fm > nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_343_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > set_fm,C2: set_fm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_fm @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_344_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_345_linorder__le__cases,axiom,
    ! [X6: nat,Y5: nat] :
      ( ~ ( ord_less_eq_nat @ X6 @ Y5 )
     => ( ord_less_eq_nat @ Y5 @ X6 ) ) ).

% linorder_le_cases
thf(fact_346_order__antisym__conv,axiom,
    ! [Y5: set_fm,X6: set_fm] :
      ( ( ord_less_eq_set_fm @ Y5 @ X6 )
     => ( ( ord_less_eq_set_fm @ X6 @ Y5 )
        = ( X6 = Y5 ) ) ) ).

% order_antisym_conv
thf(fact_347_order__antisym__conv,axiom,
    ! [Y5: nat,X6: nat] :
      ( ( ord_less_eq_nat @ Y5 @ X6 )
     => ( ( ord_less_eq_nat @ X6 @ Y5 )
        = ( X6 = Y5 ) ) ) ).

% order_antisym_conv
thf(fact_348_in__mono,axiom,
    ! [A4: set_rule,B4: set_rule,X6: rule] :
      ( ( ord_less_eq_set_rule @ A4 @ B4 )
     => ( ( member_rule @ X6 @ A4 )
       => ( member_rule @ X6 @ B4 ) ) ) ).

% in_mono
thf(fact_349_in__mono,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,B4: set_Pr7058068377845519745ist_fm,X6: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A4 @ B4 )
     => ( ( member8102475879199740618ist_fm @ X6 @ A4 )
       => ( member8102475879199740618ist_fm @ X6 @ B4 ) ) ) ).

% in_mono
thf(fact_350_in__mono,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,X6: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A4 @ B4 )
     => ( ( member8206827879206165904at_nat @ X6 @ A4 )
       => ( member8206827879206165904at_nat @ X6 @ B4 ) ) ) ).

% in_mono
thf(fact_351_in__mono,axiom,
    ! [A4: set_fm,B4: set_fm,X6: fm] :
      ( ( ord_less_eq_set_fm @ A4 @ B4 )
     => ( ( member_fm @ X6 @ A4 )
       => ( member_fm @ X6 @ B4 ) ) ) ).

% in_mono
thf(fact_352_subsetD,axiom,
    ! [A4: set_rule,B4: set_rule,C2: rule] :
      ( ( ord_less_eq_set_rule @ A4 @ B4 )
     => ( ( member_rule @ C2 @ A4 )
       => ( member_rule @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_353_subsetD,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,B4: set_Pr7058068377845519745ist_fm,C2: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A4 @ B4 )
     => ( ( member8102475879199740618ist_fm @ C2 @ A4 )
       => ( member8102475879199740618ist_fm @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_354_subsetD,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,C2: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A4 @ B4 )
     => ( ( member8206827879206165904at_nat @ C2 @ A4 )
       => ( member8206827879206165904at_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_355_subsetD,axiom,
    ! [A4: set_fm,B4: set_fm,C2: fm] :
      ( ( ord_less_eq_set_fm @ A4 @ B4 )
     => ( ( member_fm @ C2 @ A4 )
       => ( member_fm @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_356_equalityE,axiom,
    ! [A4: set_fm,B4: set_fm] :
      ( ( A4 = B4 )
     => ~ ( ( ord_less_eq_set_fm @ A4 @ B4 )
         => ~ ( ord_less_eq_set_fm @ B4 @ A4 ) ) ) ).

% equalityE
thf(fact_357_subset__eq,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B6: set_rule] :
        ! [X5: rule] :
          ( ( member_rule @ X5 @ A7 )
         => ( member_rule @ X5 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_358_subset__eq,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B6: set_Pr7058068377845519745ist_fm] :
        ! [X5: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X5 @ A7 )
         => ( member8102475879199740618ist_fm @ X5 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_359_subset__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
        ! [X5: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X5 @ A7 )
         => ( member8206827879206165904at_nat @ X5 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_360_subset__eq,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B6: set_fm] :
        ! [X5: fm] :
          ( ( member_fm @ X5 @ A7 )
         => ( member_fm @ X5 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_361_equalityD1,axiom,
    ! [A4: set_fm,B4: set_fm] :
      ( ( A4 = B4 )
     => ( ord_less_eq_set_fm @ A4 @ B4 ) ) ).

% equalityD1
thf(fact_362_equalityD2,axiom,
    ! [A4: set_fm,B4: set_fm] :
      ( ( A4 = B4 )
     => ( ord_less_eq_set_fm @ B4 @ A4 ) ) ).

% equalityD2
thf(fact_363_subset__iff,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B6: set_rule] :
        ! [T: rule] :
          ( ( member_rule @ T @ A7 )
         => ( member_rule @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_364_subset__iff,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B6: set_Pr7058068377845519745ist_fm] :
        ! [T: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ T @ A7 )
         => ( member8102475879199740618ist_fm @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_365_subset__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
        ! [T: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ T @ A7 )
         => ( member8206827879206165904at_nat @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_366_subset__iff,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B6: set_fm] :
        ! [T: fm] :
          ( ( member_fm @ T @ A7 )
         => ( member_fm @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_367_subset__refl,axiom,
    ! [A4: set_fm] : ( ord_less_eq_set_fm @ A4 @ A4 ) ).

% subset_refl
thf(fact_368_Collect__mono,axiom,
    ! [P: fm > $o,Q2: fm > $o] :
      ( ! [X3: fm] :
          ( ( P @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_less_eq_set_fm @ ( collect_fm @ P ) @ ( collect_fm @ Q2 ) ) ) ).

% Collect_mono
thf(fact_369_subset__trans,axiom,
    ! [A4: set_fm,B4: set_fm,C3: set_fm] :
      ( ( ord_less_eq_set_fm @ A4 @ B4 )
     => ( ( ord_less_eq_set_fm @ B4 @ C3 )
       => ( ord_less_eq_set_fm @ A4 @ C3 ) ) ) ).

% subset_trans
thf(fact_370_set__eq__subset,axiom,
    ( ( ^ [Y: set_fm,Z: set_fm] : ( Y = Z ) )
    = ( ^ [A7: set_fm,B6: set_fm] :
          ( ( ord_less_eq_set_fm @ A7 @ B6 )
          & ( ord_less_eq_set_fm @ B6 @ A7 ) ) ) ) ).

% set_eq_subset
thf(fact_371_Collect__mono__iff,axiom,
    ! [P: fm > $o,Q2: fm > $o] :
      ( ( ord_less_eq_set_fm @ ( collect_fm @ P ) @ ( collect_fm @ Q2 ) )
      = ( ! [X5: fm] :
            ( ( P @ X5 )
           => ( Q2 @ X5 ) ) ) ) ).

% Collect_mono_iff
thf(fact_372_i_Owf__ipath__epath,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream_rule,T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ Steps ) ) ) ).

% i.wf_ipath_epath
thf(fact_373_i_OSaturated__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ Steps )
      = ( ! [X5: rule] :
            ( ( member_rule @ X5 @ ( sset_rule @ Rules2 ) )
           => ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ X5 @ Steps ) ) ) ) ).

% i.Saturated_def
thf(fact_374_Collect__subset,axiom,
    ! [A4: set_rule,P: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X5: rule] :
            ( ( member_rule @ X5 @ A4 )
            & ( P @ X5 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_375_Collect__subset,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X5: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X5 @ A4 )
            & ( P @ X5 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_376_Collect__subset,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X5: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X5 @ A4 )
            & ( P @ X5 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_377_Collect__subset,axiom,
    ! [A4: set_fm,P: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X5: fm] :
            ( ( member_fm @ X5 @ A4 )
            & ( P @ X5 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_378_subset__CollectI,axiom,
    ! [B4: set_rule,A4: set_rule,Q2: rule > $o,P: rule > $o] :
      ( ( ord_less_eq_set_rule @ B4 @ A4 )
     => ( ! [X3: rule] :
            ( ( member_rule @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_rule
          @ ( collect_rule
            @ ^ [X5: rule] :
                ( ( member_rule @ X5 @ B4 )
                & ( Q2 @ X5 ) ) )
          @ ( collect_rule
            @ ^ [X5: rule] :
                ( ( member_rule @ X5 @ A4 )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_379_subset__CollectI,axiom,
    ! [B4: set_Pr7058068377845519745ist_fm,A4: set_Pr7058068377845519745ist_fm,Q2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B4 @ A4 )
     => ( ! [X3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le2055072402069232929ist_fm
          @ ( collec7231575776666803724ist_fm
            @ ^ [X5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X5 @ B4 )
                & ( Q2 @ X5 ) ) )
          @ ( collec7231575776666803724ist_fm
            @ ^ [X5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X5 @ A4 )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_380_subset__CollectI,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,Q2: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B4 @ A4 )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3000389064537975527at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X5 @ B4 )
                & ( Q2 @ X5 ) ) )
          @ ( collec7088162979684241874at_nat
            @ ^ [X5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X5 @ A4 )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_381_subset__CollectI,axiom,
    ! [B4: set_fm,A4: set_fm,Q2: fm > $o,P: fm > $o] :
      ( ( ord_less_eq_set_fm @ B4 @ A4 )
     => ( ! [X3: fm] :
            ( ( member_fm @ X3 @ B4 )
           => ( ( Q2 @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_fm
          @ ( collect_fm
            @ ^ [X5: fm] :
                ( ( member_fm @ X5 @ B4 )
                & ( Q2 @ X5 ) ) )
          @ ( collect_fm
            @ ^ [X5: fm] :
                ( ( member_fm @ X5 @ A4 )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_382_subset__Collect__iff,axiom,
    ! [B4: set_rule,A4: set_rule,P: rule > $o] :
      ( ( ord_less_eq_set_rule @ B4 @ A4 )
     => ( ( ord_less_eq_set_rule @ B4
          @ ( collect_rule
            @ ^ [X5: rule] :
                ( ( member_rule @ X5 @ A4 )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: rule] :
              ( ( member_rule @ X5 @ B4 )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_383_subset__Collect__iff,axiom,
    ! [B4: set_Pr7058068377845519745ist_fm,A4: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B4 @ A4 )
     => ( ( ord_le2055072402069232929ist_fm @ B4
          @ ( collec7231575776666803724ist_fm
            @ ^ [X5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X5 @ A4 )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ X5 @ B4 )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_384_subset__Collect__iff,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B4 @ A4 )
     => ( ( ord_le3000389064537975527at_nat @ B4
          @ ( collec7088162979684241874at_nat
            @ ^ [X5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X5 @ A4 )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X5 @ B4 )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_385_subset__Collect__iff,axiom,
    ! [B4: set_fm,A4: set_fm,P: fm > $o] :
      ( ( ord_less_eq_set_fm @ B4 @ A4 )
     => ( ( ord_less_eq_set_fm @ B4
          @ ( collect_fm
            @ ^ [X5: fm] :
                ( ( member_fm @ X5 @ A4 )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: fm] :
              ( ( member_fm @ X5 @ B4 )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_386_i_Oepath_Ocases,axiom,
    ! [Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( sset_fm @ Rules2 ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ Sl4 )
               => ~ ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_387_i_Oepath_Ocases,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_388_i_Oepath_Ocases,axiom,
    ! [Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream8372878641218411373at_nat,A: stream1622999302952468147at_nat] :
      ( ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) @ Sl4 )
             => ( ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ Sl4 )
               => ~ ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_389_i_Oepath_Ocases,axiom,
    ! [Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules2: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( sset_fm @ Rules2 ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ Sl4 )
               => ~ ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_390_i_Oepath_Ocases,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules2: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_391_i_Oepath_Ocases,axiom,
    ! [Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules2: stream8372878641218411373at_nat,A: stream1819640557982756571at_nat] :
      ( ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) @ Sl4 )
             => ( ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ Sl4 )
               => ~ ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_392_i_Oepath_Ocases,axiom,
    ! [Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules2: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ! [Sl4: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) @ Sl4 )
             => ( ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ Sl4 )
               => ~ ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_393_i_Oepath_Ocases,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules2: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( sset_list_fm @ Rules2 ) )
         => ! [Sl4: fset_list_fm] :
              ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) @ Sl4 )
             => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ Sl4 )
               => ~ ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_394_i_Oepath_Ocases,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ! [Sl4: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) ) @ Sl4 )
             => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ Sl4 )
               => ~ ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_395_saturated__effG__uu__def,axiom,
    ( abstra7873564507807625473ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% saturated_effG_uu_def
thf(fact_396_Saturated__effG__uu__uu__def,axiom,
    ( abstra5023634976456092961ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra8647779336736900908ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% Saturated_effG_uu_uu_def
thf(fact_397_mkTree__effG__uu__def,axiom,
    ( abstra3917079402916513406ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% mkTree_effG_uu_def
thf(fact_398_fsubsetI,axiom,
    ! [A4: fset_P661503646757059847ist_fm,B4: fset_P661503646757059847ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X3 @ A4 )
         => ( fmembe3381613331217039976ist_fm @ X3 @ B4 ) )
     => ( ord_le2064643713053750439ist_fm @ A4 @ B4 ) ) ).

% fsubsetI
thf(fact_399_fsubsetI,axiom,
    ! [A4: fset_A5396818216090053761m_rule,B4: fset_A5396818216090053761m_rule] :
      ( ! [X3: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X3 @ A4 )
         => ( fmembe1635983029874454474m_rule @ X3 @ B4 ) )
     => ( ord_le6951374404268211425m_rule @ A4 @ B4 ) ) ).

% fsubsetI
thf(fact_400_sndI,axiom,
    ! [X6: product_prod_nat_nat,Y5: nat,Z2: nat] :
      ( ( X6
        = ( product_Pair_nat_nat @ Y5 @ Z2 ) )
     => ( ( product_snd_nat_nat @ X6 )
        = Z2 ) ) ).

% sndI
thf(fact_401_sndI,axiom,
    ! [X6: produc859450856879609959at_nat,Y5: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( X6
        = ( produc6161850002892822231at_nat @ Y5 @ Z2 ) )
     => ( ( produc6408287024330202629at_nat @ X6 )
        = Z2 ) ) ).

% sndI
thf(fact_402_sndI,axiom,
    ! [X6: produc4630866025709511057ist_fm,Y5: rule,Z2: produc1996495991257130529ist_fm] :
      ( ( X6
        = ( produc7063755442767259331ist_fm @ Y5 @ Z2 ) )
     => ( ( produc4514744107061996309ist_fm @ X6 )
        = Z2 ) ) ).

% sndI
thf(fact_403_sndI,axiom,
    ! [X6: produc164195504107695125m_rule,Y5: produc1996495991257130529ist_fm,Z2: rule] :
      ( ( X6
        = ( produc491467635432902671m_rule @ Y5 @ Z2 ) )
     => ( ( produc7165828336582415457m_rule @ X6 )
        = Z2 ) ) ).

% sndI
thf(fact_404_sndI,axiom,
    ! [X6: produc1996495991257130529ist_fm,Y5: list_fm,Z2: list_fm] :
      ( ( X6
        = ( produc381145313068854617ist_fm @ Y5 @ Z2 ) )
     => ( ( produc4588648349897876871ist_fm @ X6 )
        = Z2 ) ) ).

% sndI
thf(fact_405_eq__snd__iff,axiom,
    ! [B: nat,P2: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P2 ) )
      = ( ? [A5: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_406_eq__snd__iff,axiom,
    ! [B: product_prod_nat_nat,P2: produc859450856879609959at_nat] :
      ( ( B
        = ( produc6408287024330202629at_nat @ P2 ) )
      = ( ? [A5: product_prod_nat_nat] :
            ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_407_eq__snd__iff,axiom,
    ! [B: produc1996495991257130529ist_fm,P2: produc4630866025709511057ist_fm] :
      ( ( B
        = ( produc4514744107061996309ist_fm @ P2 ) )
      = ( ? [A5: rule] :
            ( P2
            = ( produc7063755442767259331ist_fm @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_408_eq__snd__iff,axiom,
    ! [B: rule,P2: produc164195504107695125m_rule] :
      ( ( B
        = ( produc7165828336582415457m_rule @ P2 ) )
      = ( ? [A5: produc1996495991257130529ist_fm] :
            ( P2
            = ( produc491467635432902671m_rule @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_409_eq__snd__iff,axiom,
    ! [B: list_fm,P2: produc1996495991257130529ist_fm] :
      ( ( B
        = ( produc4588648349897876871ist_fm @ P2 ) )
      = ( ? [A5: list_fm] :
            ( P2
            = ( produc381145313068854617ist_fm @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_410_eq__fst__iff,axiom,
    ! [A: nat,P2: product_prod_nat_nat] :
      ( ( A
        = ( product_fst_nat_nat @ P2 ) )
      = ( ? [B7: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A @ B7 ) ) ) ) ).

% eq_fst_iff
thf(fact_411_eq__fst__iff,axiom,
    ! [A: product_prod_nat_nat,P2: produc859450856879609959at_nat] :
      ( ( A
        = ( produc3213797794245857475at_nat @ P2 ) )
      = ( ? [B7: product_prod_nat_nat] :
            ( P2
            = ( produc6161850002892822231at_nat @ A @ B7 ) ) ) ) ).

% eq_fst_iff
thf(fact_412_eq__fst__iff,axiom,
    ! [A: rule,P2: produc4630866025709511057ist_fm] :
      ( ( A
        = ( produc4228417144610596823ist_fm @ P2 ) )
      = ( ? [B7: produc1996495991257130529ist_fm] :
            ( P2
            = ( produc7063755442767259331ist_fm @ A @ B7 ) ) ) ) ).

% eq_fst_iff
thf(fact_413_eq__fst__iff,axiom,
    ! [A: produc1996495991257130529ist_fm,P2: produc164195504107695125m_rule] :
      ( ( A
        = ( produc6879501374131015971m_rule @ P2 ) )
      = ( ? [B7: rule] :
            ( P2
            = ( produc491467635432902671m_rule @ A @ B7 ) ) ) ) ).

% eq_fst_iff
thf(fact_414_eq__fst__iff,axiom,
    ! [A: list_fm,P2: produc1996495991257130529ist_fm] :
      ( ( A
        = ( produc1501393135466168645ist_fm @ P2 ) )
      = ( ? [B7: list_fm] :
            ( P2
            = ( produc381145313068854617ist_fm @ A @ B7 ) ) ) ) ).

% eq_fst_iff
thf(fact_415_fstI,axiom,
    ! [X6: product_prod_nat_nat,Y5: nat,Z2: nat] :
      ( ( X6
        = ( product_Pair_nat_nat @ Y5 @ Z2 ) )
     => ( ( product_fst_nat_nat @ X6 )
        = Y5 ) ) ).

% fstI
thf(fact_416_fstI,axiom,
    ! [X6: produc859450856879609959at_nat,Y5: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( X6
        = ( produc6161850002892822231at_nat @ Y5 @ Z2 ) )
     => ( ( produc3213797794245857475at_nat @ X6 )
        = Y5 ) ) ).

% fstI
thf(fact_417_fstI,axiom,
    ! [X6: produc4630866025709511057ist_fm,Y5: rule,Z2: produc1996495991257130529ist_fm] :
      ( ( X6
        = ( produc7063755442767259331ist_fm @ Y5 @ Z2 ) )
     => ( ( produc4228417144610596823ist_fm @ X6 )
        = Y5 ) ) ).

% fstI
thf(fact_418_fstI,axiom,
    ! [X6: produc164195504107695125m_rule,Y5: produc1996495991257130529ist_fm,Z2: rule] :
      ( ( X6
        = ( produc491467635432902671m_rule @ Y5 @ Z2 ) )
     => ( ( produc6879501374131015971m_rule @ X6 )
        = Y5 ) ) ).

% fstI
thf(fact_419_fstI,axiom,
    ! [X6: produc1996495991257130529ist_fm,Y5: list_fm,Z2: list_fm] :
      ( ( X6
        = ( produc381145313068854617ist_fm @ Y5 @ Z2 ) )
     => ( ( produc1501393135466168645ist_fm @ X6 )
        = Y5 ) ) ).

% fstI
thf(fact_420_eff__S,axiom,
    ! [S: produc1996495991257130529ist_fm,R: rule,Sl3: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( member_rule @ R @ ( sset_rule @ rules ) )
       => ( ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Sl3 ) )
         => ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl3 )
           => ( member8102475879199740618ist_fm @ S4 @ top_to728902761071950545ist_fm ) ) ) ) ) ).

% eff_S
thf(fact_421_minWait__least,axiom,
    ! [N: nat,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
        @ S )
     => ( ord_less_eq_nat
        @ ( abstra738319790293016300ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S )
        @ N ) ) ).

% minWait_least
thf(fact_422_UNIV__I,axiom,
    ! [X6: fm] : ( member_fm @ X6 @ top_top_set_fm ) ).

% UNIV_I
thf(fact_423_UNIV__I,axiom,
    ! [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ top_to8454300755403694231at_nat ) ).

% UNIV_I
thf(fact_424_UNIV__I,axiom,
    ! [X6: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X6 @ top_to728902761071950545ist_fm ) ).

% UNIV_I
thf(fact_425_UNIV__I,axiom,
    ! [X6: rule] : ( member_rule @ X6 @ top_top_set_rule ) ).

% UNIV_I
thf(fact_426_UNIV__eq__I,axiom,
    ! [A4: set_fm] :
      ( ! [X3: fm] : ( member_fm @ X3 @ A4 )
     => ( top_top_set_fm = A4 ) ) ).

% UNIV_eq_I
thf(fact_427_UNIV__eq__I,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ! [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ A4 )
     => ( top_to8454300755403694231at_nat = A4 ) ) ).

% UNIV_eq_I
thf(fact_428_UNIV__eq__I,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ A4 )
     => ( top_to728902761071950545ist_fm = A4 ) ) ).

% UNIV_eq_I
thf(fact_429_UNIV__eq__I,axiom,
    ! [A4: set_rule] :
      ( ! [X3: rule] : ( member_rule @ X3 @ A4 )
     => ( top_top_set_rule = A4 ) ) ).

% UNIV_eq_I
thf(fact_430_UNIV__witness,axiom,
    ? [X3: fm] : ( member_fm @ X3 @ top_top_set_fm ) ).

% UNIV_witness
thf(fact_431_UNIV__witness,axiom,
    ? [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ top_to8454300755403694231at_nat ) ).

% UNIV_witness
thf(fact_432_UNIV__witness,axiom,
    ? [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ top_to728902761071950545ist_fm ) ).

% UNIV_witness
thf(fact_433_UNIV__witness,axiom,
    ? [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ).

% UNIV_witness
thf(fact_434_RuleSystem_OminWait_Ocong,axiom,
    abstra738319790293016300ist_fm = abstra738319790293016300ist_fm ).

% RuleSystem.minWait.cong
thf(fact_435_UNIV__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm
      @ ^ [X5: produc1996495991257130529ist_fm] : $true ) ) ).

% UNIV_def
thf(fact_436_UNIV__def,axiom,
    ( top_top_set_rule
    = ( collect_rule
      @ ^ [X5: rule] : $true ) ) ).

% UNIV_def
thf(fact_437_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
     => ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_uniqueI
thf(fact_438_top_Oextremum__uniqueI,axiom,
    ! [A: set_rule] :
      ( ( ord_less_eq_set_rule @ top_top_set_rule @ A )
     => ( A = top_top_set_rule ) ) ).

% top.extremum_uniqueI
thf(fact_439_top_Oextremum__uniqueI,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A )
     => ( A = top_top_set_fm ) ) ).

% top.extremum_uniqueI
thf(fact_440_top_Oextremum__unique,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
      = ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_unique
thf(fact_441_top_Oextremum__unique,axiom,
    ! [A: set_rule] :
      ( ( ord_less_eq_set_rule @ top_top_set_rule @ A )
      = ( A = top_top_set_rule ) ) ).

% top.extremum_unique
thf(fact_442_top_Oextremum__unique,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A )
      = ( A = top_top_set_fm ) ) ).

% top.extremum_unique
thf(fact_443_top__greatest,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A @ top_to728902761071950545ist_fm ) ).

% top_greatest
thf(fact_444_top__greatest,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ top_top_set_rule ) ).

% top_greatest
thf(fact_445_top__greatest,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ top_top_set_fm ) ).

% top_greatest
thf(fact_446_subset__UNIV,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A4 @ top_to728902761071950545ist_fm ) ).

% subset_UNIV
thf(fact_447_subset__UNIV,axiom,
    ! [A4: set_rule] : ( ord_less_eq_set_rule @ A4 @ top_top_set_rule ) ).

% subset_UNIV
thf(fact_448_subset__UNIV,axiom,
    ! [A4: set_fm] : ( ord_less_eq_set_fm @ A4 @ top_top_set_fm ) ).

% subset_UNIV
thf(fact_449_eqfelem__imp__iff,axiom,
    ! [X6: produc1996495991257130529ist_fm,Y5: produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm] :
      ( ( X6 = Y5 )
     => ( ( fmembe3381613331217039976ist_fm @ X6 @ A4 )
        = ( fmembe3381613331217039976ist_fm @ Y5 @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_450_eqfelem__imp__iff,axiom,
    ! [X6: abstra4572470701423150929m_rule,Y5: abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule] :
      ( ( X6 = Y5 )
     => ( ( fmembe1635983029874454474m_rule @ X6 @ A4 )
        = ( fmembe1635983029874454474m_rule @ Y5 @ A4 ) ) ) ).

% eqfelem_imp_iff
thf(fact_451_if__split__fmem2,axiom,
    ! [A: produc1996495991257130529ist_fm,Q2: $o,X6: fset_P661503646757059847ist_fm,Y5: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A @ ( if_fse3714977293194272717ist_fm @ Q2 @ X6 @ Y5 ) )
      = ( ( Q2
         => ( fmembe3381613331217039976ist_fm @ A @ X6 ) )
        & ( ~ Q2
         => ( fmembe3381613331217039976ist_fm @ A @ Y5 ) ) ) ) ).

% if_split_fmem2
thf(fact_452_if__split__fmem2,axiom,
    ! [A: abstra4572470701423150929m_rule,Q2: $o,X6: fset_A5396818216090053761m_rule,Y5: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ A @ ( if_fse3370020817956878651m_rule @ Q2 @ X6 @ Y5 ) )
      = ( ( Q2
         => ( fmembe1635983029874454474m_rule @ A @ X6 ) )
        & ( ~ Q2
         => ( fmembe1635983029874454474m_rule @ A @ Y5 ) ) ) ) ).

% if_split_fmem2
thf(fact_453_if__split__fmem1,axiom,
    ! [Q2: $o,X6: produc1996495991257130529ist_fm,Y5: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ ( if_Pro3930376587665744871ist_fm @ Q2 @ X6 @ Y5 ) @ B )
      = ( ( Q2
         => ( fmembe3381613331217039976ist_fm @ X6 @ B ) )
        & ( ~ Q2
         => ( fmembe3381613331217039976ist_fm @ Y5 @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_454_if__split__fmem1,axiom,
    ! [Q2: $o,X6: abstra4572470701423150929m_rule,Y5: abstra4572470701423150929m_rule,B: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ ( if_Abs4566652138351786251m_rule @ Q2 @ X6 @ Y5 ) @ B )
      = ( ( Q2
         => ( fmembe1635983029874454474m_rule @ X6 @ B ) )
        & ( ~ Q2
         => ( fmembe1635983029874454474m_rule @ Y5 @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_455_eqfset__imp__iff,axiom,
    ! [A4: fset_P661503646757059847ist_fm,B4: fset_P661503646757059847ist_fm,X6: produc1996495991257130529ist_fm] :
      ( ( A4 = B4 )
     => ( ( fmembe3381613331217039976ist_fm @ X6 @ A4 )
        = ( fmembe3381613331217039976ist_fm @ X6 @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_456_eqfset__imp__iff,axiom,
    ! [A4: fset_A5396818216090053761m_rule,B4: fset_A5396818216090053761m_rule,X6: abstra4572470701423150929m_rule] :
      ( ( A4 = B4 )
     => ( ( fmembe1635983029874454474m_rule @ X6 @ A4 )
        = ( fmembe1635983029874454474m_rule @ X6 @ B4 ) ) ) ).

% eqfset_imp_iff
thf(fact_457_eq__fmem__trans,axiom,
    ! [A: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm] :
      ( ( A = B )
     => ( ( fmembe3381613331217039976ist_fm @ B @ A4 )
       => ( fmembe3381613331217039976ist_fm @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_458_eq__fmem__trans,axiom,
    ! [A: abstra4572470701423150929m_rule,B: abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule] :
      ( ( A = B )
     => ( ( fmembe1635983029874454474m_rule @ B @ A4 )
       => ( fmembe1635983029874454474m_rule @ A @ A4 ) ) ) ).

% eq_fmem_trans
thf(fact_459_fequalityCE,axiom,
    ! [A4: fset_P661503646757059847ist_fm,B4: fset_P661503646757059847ist_fm,C2: produc1996495991257130529ist_fm] :
      ( ( A4 = B4 )
     => ( ( ( fmembe3381613331217039976ist_fm @ C2 @ A4 )
         => ~ ( fmembe3381613331217039976ist_fm @ C2 @ B4 ) )
       => ~ ( ~ ( fmembe3381613331217039976ist_fm @ C2 @ A4 )
           => ( fmembe3381613331217039976ist_fm @ C2 @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_460_fequalityCE,axiom,
    ! [A4: fset_A5396818216090053761m_rule,B4: fset_A5396818216090053761m_rule,C2: abstra4572470701423150929m_rule] :
      ( ( A4 = B4 )
     => ( ( ( fmembe1635983029874454474m_rule @ C2 @ A4 )
         => ~ ( fmembe1635983029874454474m_rule @ C2 @ B4 ) )
       => ~ ( ~ ( fmembe1635983029874454474m_rule @ C2 @ A4 )
           => ( fmembe1635983029874454474m_rule @ C2 @ B4 ) ) ) ) ).

% fequalityCE
thf(fact_461_fset__eqI,axiom,
    ! [A4: fset_P661503646757059847ist_fm,B4: fset_P661503646757059847ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X3 @ A4 )
          = ( fmembe3381613331217039976ist_fm @ X3 @ B4 ) )
     => ( A4 = B4 ) ) ).

% fset_eqI
thf(fact_462_fset__eqI,axiom,
    ! [A4: fset_A5396818216090053761m_rule,B4: fset_A5396818216090053761m_rule] :
      ( ! [X3: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X3 @ A4 )
          = ( fmembe1635983029874454474m_rule @ X3 @ B4 ) )
     => ( A4 = B4 ) ) ).

% fset_eqI
thf(fact_463_fsubsetD,axiom,
    ! [A4: fset_P661503646757059847ist_fm,B4: fset_P661503646757059847ist_fm,C2: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A4 @ B4 )
     => ( ( fmembe3381613331217039976ist_fm @ C2 @ A4 )
       => ( fmembe3381613331217039976ist_fm @ C2 @ B4 ) ) ) ).

% fsubsetD
thf(fact_464_fsubsetD,axiom,
    ! [A4: fset_A5396818216090053761m_rule,B4: fset_A5396818216090053761m_rule,C2: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A4 @ B4 )
     => ( ( fmembe1635983029874454474m_rule @ C2 @ A4 )
       => ( fmembe1635983029874454474m_rule @ C2 @ B4 ) ) ) ).

% fsubsetD
thf(fact_465_fin__mono,axiom,
    ! [A4: fset_P661503646757059847ist_fm,B4: fset_P661503646757059847ist_fm,X6: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A4 @ B4 )
     => ( ( fmembe3381613331217039976ist_fm @ X6 @ A4 )
       => ( fmembe3381613331217039976ist_fm @ X6 @ B4 ) ) ) ).

% fin_mono
thf(fact_466_fin__mono,axiom,
    ! [A4: fset_A5396818216090053761m_rule,B4: fset_A5396818216090053761m_rule,X6: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A4 @ B4 )
     => ( ( fmembe1635983029874454474m_rule @ X6 @ A4 )
       => ( fmembe1635983029874454474m_rule @ X6 @ B4 ) ) ) ).

% fin_mono
thf(fact_467_enabled__R,axiom,
    ! [S: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ? [X3: rule] :
          ( ( member_rule @ X3 @ ( sset_rule @ rules ) )
          & ? [Sl4: fset_P661503646757059847ist_fm] :
              ( ( eff @ X3 @ S )
              = ( some_f2162114139285431020ist_fm @ Sl4 ) ) ) ) ).

% enabled_R
thf(fact_468_ipath__mkTree__saturated,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R: rule] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S )
            @ Steps )
         => ( ( member_rule @ R @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R
              @ Steps ) ) ) ) ) ).

% ipath_mkTree_saturated
thf(fact_469_ipath__mkTree__sdrop,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,M: nat] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S )
            @ Steps )
         => ? [N2: nat,S5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S5 @ top_to728902761071950545ist_fm )
              & ( abstra2285467793620430502m_rule
                @ ( abstra178918799166095863ist_fm
                  @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R2 @ S2 )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ ( sdrop_rule @ N2 @ Rs )
                  @ S5 )
                @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_sdrop
thf(fact_470_per,axiom,
    ! [R: rule] :
      ( ( member_rule @ R @ ( sset_rule @ rules ) )
     => ( abstra4153889242452539894ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R ) ) ).

% per
thf(fact_471_ipath__mkTree__Saturated,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S )
            @ Steps )
         => ( abstra8647779336736900908ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ Steps ) ) ) ) ).

% ipath_mkTree_Saturated
thf(fact_472_minWait__ex,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ? [N2: nat] :
            ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) )
            @ S ) ) ) ).

% minWait_ex
thf(fact_473_minWait__def,axiom,
    ! [Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra738319790293016300ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S )
      = ( ord_Least_nat
        @ ^ [N3: nat] :
            ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
            @ S ) ) ) ).

% minWait_def
thf(fact_474_pred__subset__eq,axiom,
    ! [R3: set_rule,S6: set_rule] :
      ( ( ord_less_eq_rule_o
        @ ^ [X5: rule] : ( member_rule @ X5 @ R3 )
        @ ^ [X5: rule] : ( member_rule @ X5 @ S6 ) )
      = ( ord_less_eq_set_rule @ R3 @ S6 ) ) ).

% pred_subset_eq
thf(fact_475_pred__subset__eq,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S6: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le8518180823940269308t_fm_o
        @ ^ [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ R3 )
        @ ^ [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ S6 ) )
      = ( ord_le2055072402069232929ist_fm @ R3 @ S6 ) ) ).

% pred_subset_eq
thf(fact_476_pred__subset__eq,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S6: set_Pr8693737435421807431at_nat] :
      ( ( ord_le7858099551454983350_nat_o
        @ ^ [X5: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X5 @ R3 )
        @ ^ [X5: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X5 @ S6 ) )
      = ( ord_le3000389064537975527at_nat @ R3 @ S6 ) ) ).

% pred_subset_eq
thf(fact_477_pred__subset__eq,axiom,
    ! [R3: set_fm,S6: set_fm] :
      ( ( ord_less_eq_fm_o
        @ ^ [X5: fm] : ( member_fm @ X5 @ R3 )
        @ ^ [X5: fm] : ( member_fm @ X5 @ S6 ) )
      = ( ord_less_eq_set_fm @ R3 @ S6 ) ) ).

% pred_subset_eq
thf(fact_478_PersistentRuleSystem__axioms,axiom,
    ( abstra7887161495092839066ist_fm
    @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R2 @ S2 )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ top_to728902761071950545ist_fm ) ).

% PersistentRuleSystem_axioms
thf(fact_479_iso__tuple__UNIV__I,axiom,
    ! [X6: fm] : ( member_fm @ X6 @ top_top_set_fm ) ).

% iso_tuple_UNIV_I
thf(fact_480_iso__tuple__UNIV__I,axiom,
    ! [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ top_to8454300755403694231at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_481_iso__tuple__UNIV__I,axiom,
    ! [X6: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X6 @ top_to728902761071950545ist_fm ) ).

% iso_tuple_UNIV_I
thf(fact_482_iso__tuple__UNIV__I,axiom,
    ! [X6: rule] : ( member_rule @ X6 @ top_top_set_rule ) ).

% iso_tuple_UNIV_I
thf(fact_483_top__empty__eq,axiom,
    ( top_top_fm_o
    = ( ^ [X5: fm] : ( member_fm @ X5 @ top_top_set_fm ) ) ) ).

% top_empty_eq
thf(fact_484_top__empty__eq,axiom,
    ( top_to1313005431924226822_nat_o
    = ( ^ [X5: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X5 @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq
thf(fact_485_top__empty__eq,axiom,
    ( top_to6798613745154499404t_fm_o
    = ( ^ [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq
thf(fact_486_top__empty__eq,axiom,
    ( top_top_rule_o
    = ( ^ [X5: rule] : ( member_rule @ X5 @ top_top_set_rule ) ) ) ).

% top_empty_eq
thf(fact_487_RuleSystem_Oper_Ocong,axiom,
    abstra4153889242452539894ist_fm = abstra4153889242452539894ist_fm ).

% RuleSystem.per.cong
thf(fact_488_RuleSystem__Defs_Ofair_Ocong,axiom,
    abstra1114877437032970289r_rule = abstra1114877437032970289r_rule ).

% RuleSystem_Defs.fair.cong
thf(fact_489_LeastI2__ex,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_ex
thf(fact_490_LeastI__ex,axiom,
    ! [P: nat > $o] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI_ex
thf(fact_491_LeastI2,axiom,
    ! [P: nat > $o,A: nat,Q2: nat > $o] :
      ( ( P @ A )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2
thf(fact_492_LeastI,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( P @ K2 )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI
thf(fact_493_top__set__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm @ top_to6798613745154499404t_fm_o ) ) ).

% top_set_def
thf(fact_494_top__set__def,axiom,
    ( top_top_set_rule
    = ( collect_rule @ top_top_rule_o ) ) ).

% top_set_def
thf(fact_495_PersistentRuleSystem_Oper,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,R: rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
       => ( abstra4153889242452539894ist_fm @ Eff2 @ Rules2 @ S6 @ R ) ) ) ).

% PersistentRuleSystem.per
thf(fact_496_fair__stl,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).

% fair_stl
thf(fact_497_Least1I,axiom,
    ! [P: set_fm > $o] :
      ( ? [X8: set_fm] :
          ( ( P @ X8 )
          & ! [Y3: set_fm] :
              ( ( P @ Y3 )
             => ( ord_less_eq_set_fm @ X8 @ Y3 ) )
          & ! [Y3: set_fm] :
              ( ( ( P @ Y3 )
                & ! [Ya: set_fm] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_set_fm @ Y3 @ Ya ) ) )
             => ( Y3 = X8 ) ) )
     => ( P @ ( ord_Least_set_fm @ P ) ) ) ).

% Least1I
thf(fact_498_Least1I,axiom,
    ! [P: nat > $o] :
      ( ? [X8: nat] :
          ( ( P @ X8 )
          & ! [Y3: nat] :
              ( ( P @ Y3 )
             => ( ord_less_eq_nat @ X8 @ Y3 ) )
          & ! [Y3: nat] :
              ( ( ( P @ Y3 )
                & ! [Ya: nat] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_nat @ Y3 @ Ya ) ) )
             => ( Y3 = X8 ) ) )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% Least1I
thf(fact_499_Least1__le,axiom,
    ! [P: set_fm > $o,Z2: set_fm] :
      ( ? [X8: set_fm] :
          ( ( P @ X8 )
          & ! [Y3: set_fm] :
              ( ( P @ Y3 )
             => ( ord_less_eq_set_fm @ X8 @ Y3 ) )
          & ! [Y3: set_fm] :
              ( ( ( P @ Y3 )
                & ! [Ya: set_fm] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_set_fm @ Y3 @ Ya ) ) )
             => ( Y3 = X8 ) ) )
     => ( ( P @ Z2 )
       => ( ord_less_eq_set_fm @ ( ord_Least_set_fm @ P ) @ Z2 ) ) ) ).

% Least1_le
thf(fact_500_Least1__le,axiom,
    ! [P: nat > $o,Z2: nat] :
      ( ? [X8: nat] :
          ( ( P @ X8 )
          & ! [Y3: nat] :
              ( ( P @ Y3 )
             => ( ord_less_eq_nat @ X8 @ Y3 ) )
          & ! [Y3: nat] :
              ( ( ( P @ Y3 )
                & ! [Ya: nat] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_nat @ Y3 @ Ya ) ) )
             => ( Y3 = X8 ) ) )
     => ( ( P @ Z2 )
       => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ Z2 ) ) ) ).

% Least1_le
thf(fact_501_LeastI2__order,axiom,
    ! [P: set_fm > $o,X6: set_fm,Q2: set_fm > $o] :
      ( ( P @ X6 )
     => ( ! [Y3: set_fm] :
            ( ( P @ Y3 )
           => ( ord_less_eq_set_fm @ X6 @ Y3 ) )
       => ( ! [X3: set_fm] :
              ( ( P @ X3 )
             => ( ! [Y7: set_fm] :
                    ( ( P @ Y7 )
                   => ( ord_less_eq_set_fm @ X3 @ Y7 ) )
               => ( Q2 @ X3 ) ) )
         => ( Q2 @ ( ord_Least_set_fm @ P ) ) ) ) ) ).

% LeastI2_order
thf(fact_502_LeastI2__order,axiom,
    ! [P: nat > $o,X6: nat,Q2: nat > $o] :
      ( ( P @ X6 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ X6 @ Y3 ) )
       => ( ! [X3: nat] :
              ( ( P @ X3 )
             => ( ! [Y7: nat] :
                    ( ( P @ Y7 )
                   => ( ord_less_eq_nat @ X3 @ Y7 ) )
               => ( Q2 @ X3 ) ) )
         => ( Q2 @ ( ord_Least_nat @ P ) ) ) ) ) ).

% LeastI2_order
thf(fact_503_Least__equality,axiom,
    ! [P: set_fm > $o,X6: set_fm] :
      ( ( P @ X6 )
     => ( ! [Y3: set_fm] :
            ( ( P @ Y3 )
           => ( ord_less_eq_set_fm @ X6 @ Y3 ) )
       => ( ( ord_Least_set_fm @ P )
          = X6 ) ) ) ).

% Least_equality
thf(fact_504_Least__equality,axiom,
    ! [P: nat > $o,X6: nat] :
      ( ( P @ X6 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ X6 @ Y3 ) )
       => ( ( ord_Least_nat @ P )
          = X6 ) ) ) ).

% Least_equality
thf(fact_505_LeastI2__wellorder,axiom,
    ! [P: nat > $o,A: nat,Q2: nat > $o] :
      ( ( P @ A )
     => ( ! [A3: nat] :
            ( ( P @ A3 )
           => ( ! [B8: nat] :
                  ( ( P @ B8 )
                 => ( ord_less_eq_nat @ A3 @ B8 ) )
             => ( Q2 @ A3 ) ) )
       => ( Q2 @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_wellorder
thf(fact_506_LeastI2__wellorder__ex,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [A3: nat] :
            ( ( P @ A3 )
           => ( ! [B8: nat] :
                  ( ( P @ B8 )
                 => ( ord_less_eq_nat @ A3 @ B8 ) )
             => ( Q2 @ A3 ) ) )
       => ( Q2 @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_wellorder_ex
thf(fact_507_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules2: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_508_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules2: stream_rule,Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules2 @ ( stl_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_509_i_Ofair__stl,axiom,
    ! [Rules2: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_510_i_Ofair__stl,axiom,
    ! [Rules2: stream_rule,Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules2 @ ( stl_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_511_top__empty__eq2,axiom,
    ( top_to1955928418919630078rule_o
    = ( ^ [X5: produc1996495991257130529ist_fm,Y6: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X5 @ Y6 ) @ top_to1603951219661559419m_rule ) ) ) ).

% top_empty_eq2
thf(fact_512_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X5: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y6 ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_513_top__empty__eq2,axiom,
    ( top_to2659251575160290430_nat_o
    = ( ^ [X5: product_prod_nat_nat,Y6: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y6 ) @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq2
thf(fact_514_top__empty__eq2,axiom,
    ( top_to5144090402559349552t_fm_o
    = ( ^ [X5: rule,Y6: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y6 ) @ top_to2837172092223966711ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_515_top__empty__eq2,axiom,
    ( top_to7326020031456794390t_fm_o
    = ( ^ [X5: list_fm,Y6: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y6 ) @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_516_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules2 @ ( sdrop_7224736112439592940m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_517_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream8299795917829157543ist_fm,Rs: stream8299795917829157543ist_fm,M: nat] :
      ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
     => ( abstra7454370078233524789ist_fm @ Rules2 @ ( sdrop_4442373711808556042ist_fm @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_518_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream3588992565182678257e_rule,Rs: stream3588992565182678257e_rule,M: nat] :
      ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
     => ( abstra4837122928373946751e_rule @ Rules2 @ ( sdrop_2199339672054592340e_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_519_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream1878305877988316353m_rule,Rs: stream1878305877988316353m_rule,M: nat] :
      ( ( abstra5219219999209536975m_rule @ Rules2 @ Rs )
     => ( abstra5219219999209536975m_rule @ Rules2 @ ( sdrop_1764812713384251940m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_520_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream2312013117288958913e_rule,Rs: stream2312013117288958913e_rule,M: nat] :
      ( ( abstra623789749151397071e_rule @ Rules2 @ Rs )
     => ( abstra623789749151397071e_rule @ Rules2 @ ( sdrop_6392754500180887844e_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_521_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream6494289010434245521m_rule,Rs: stream6494289010434245521m_rule,M: nat] :
      ( ( abstra7961590535338393887m_rule @ Rules2 @ Rs )
     => ( abstra7961590535338393887m_rule @ Rules2 @ ( sdrop_7373388980841566196m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_522_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules2: stream_rule,Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules2 @ ( sdrop_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_523_i_Osdrop__fair,axiom,
    ! [Rules2: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules2 @ ( sdrop_7224736112439592940m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_524_i_Osdrop__fair,axiom,
    ! [Rules2: stream8299795917829157543ist_fm,Rs: stream8299795917829157543ist_fm,M: nat] :
      ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
     => ( abstra7454370078233524789ist_fm @ Rules2 @ ( sdrop_4442373711808556042ist_fm @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_525_i_Osdrop__fair,axiom,
    ! [Rules2: stream3588992565182678257e_rule,Rs: stream3588992565182678257e_rule,M: nat] :
      ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
     => ( abstra4837122928373946751e_rule @ Rules2 @ ( sdrop_2199339672054592340e_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_526_i_Osdrop__fair,axiom,
    ! [Rules2: stream1878305877988316353m_rule,Rs: stream1878305877988316353m_rule,M: nat] :
      ( ( abstra5219219999209536975m_rule @ Rules2 @ Rs )
     => ( abstra5219219999209536975m_rule @ Rules2 @ ( sdrop_1764812713384251940m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_527_i_Osdrop__fair,axiom,
    ! [Rules2: stream2312013117288958913e_rule,Rs: stream2312013117288958913e_rule,M: nat] :
      ( ( abstra623789749151397071e_rule @ Rules2 @ Rs )
     => ( abstra623789749151397071e_rule @ Rules2 @ ( sdrop_6392754500180887844e_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_528_i_Osdrop__fair,axiom,
    ! [Rules2: stream6494289010434245521m_rule,Rs: stream6494289010434245521m_rule,M: nat] :
      ( ( abstra7961590535338393887m_rule @ Rules2 @ Rs )
     => ( abstra7961590535338393887m_rule @ Rules2 @ ( sdrop_7373388980841566196m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_529_i_Osdrop__fair,axiom,
    ! [Rules2: stream_rule,Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules2 @ ( sdrop_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_530_Least__le,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( P @ K2 )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ K2 ) ) ).

% Least_le
thf(fact_531_UNIV__rules,axiom,
    ( ( sset_rule @ rules )
    = top_top_set_rule ) ).

% UNIV_rules
thf(fact_532_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ( abstra7168897066996117381ule_fm @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_533_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule] :
      ( ( abstra8115150628584006924e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S ) @ Steps )
           => ( abstra4228102177642093818e_rule @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_534_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S6: set_Pr8693737435421807431at_nat,S: produc859450856879609959at_nat,Rs: stream_rule,Steps: stream1370203160359889003t_rule] :
      ( ( abstra1526291686923892064at_nat @ Eff2 @ Rules2 @ S6 )
     => ( ( member8206827879206165904at_nat @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra4530403282039525100t_rule @ ( abstra193881638956283709at_nat @ Eff2 @ Rs @ S ) @ Steps )
           => ( abstra7052252741274667762at_nat @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_535_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ( abstra8647779336736900908ist_fm @ Eff2 @ Rules2 @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_536_sdrop__fair,axiom,
    ! [Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% sdrop_fair
thf(fact_537_subrelI,axiom,
    ! [R: set_Pr7058068377845519745ist_fm,S: set_Pr7058068377845519745ist_fm] :
      ( ! [X3: list_fm,Y3: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ R )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ S ) )
     => ( ord_le2055072402069232929ist_fm @ R @ S ) ) ).

% subrelI
thf(fact_538_subrelI,axiom,
    ! [R: set_Pr1008144964186165195m_rule,S: set_Pr1008144964186165195m_rule] :
      ( ! [X3: produc1996495991257130529ist_fm,Y3: rule] :
          ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) @ R )
         => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X3 @ Y3 ) @ S ) )
     => ( ord_le5052251716133888043m_rule @ R @ S ) ) ).

% subrelI
thf(fact_539_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% subrelI
thf(fact_540_subrelI,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% subrelI
thf(fact_541_subrelI,axiom,
    ! [R: set_Pr2241365836748572487ist_fm,S: set_Pr2241365836748572487ist_fm] :
      ( ! [X3: rule,Y3: produc1996495991257130529ist_fm] :
          ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) @ R )
         => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) @ S ) )
     => ( ord_le6285472588696295335ist_fm @ R @ S ) ) ).

% subrelI
thf(fact_542_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules2: stream_fm,S6: set_fm,S: fm,Rs: stream_fm,Steps: stream1375025385734440967_fm_fm,R: fm] :
      ( ( abstra1990789280842237858_fm_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules2 @ Rs )
         => ( ( abstra724614664783131952_fm_fm @ ( abstra5285091746773999237_fm_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
             => ( abstra81079133870296496_fm_fm @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_543_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules2: stream_fm,S6: set_rule,S: rule,Rs: stream_fm,Steps: stream1800043145262542076ule_fm,R: fm] :
      ( ( abstra4039608574777915031m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules2 @ Rs )
         => ( ( abstra8780395760585530021ule_fm @ ( abstra8242076542743448570m_rule @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
             => ( abstra5359274141611169189m_rule @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_544_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule,R: rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
             => ( abstra1108425641574350757ule_fm @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_545_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule,R: rule] :
      ( ( abstra8115150628584006924e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
             => ( abstra6372423936129899802e_rule @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_546_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules2: stream8299795917829157543ist_fm,S6: set_fm,S: fm,Rs: stream8299795917829157543ist_fm,Steps: stream5068232101399153750ist_fm,R: produc1996495991257130529ist_fm] :
      ( ( abstra1665324712063988849_fm_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ( ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
             => ( abstra230716855031908835_fm_fm @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_547_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S6: set_rule,S: rule,Rs: stream8299795917829157543ist_fm,Steps: stream1960312990768957601ist_fm,R: produc1996495991257130529ist_fm] :
      ( ( abstra1314873687758482406m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
             => ( abstra679888793666381912m_rule @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_548_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_fm,Steps: stream5128679079952569136_fm_fm,R: fm] :
      ( ( abstra6333699301575899727ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules2 @ Rs )
         => ( ( abstra2316067621365718065_fm_fm @ ( abstra982595312256735276ist_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
             => ( abstra4899091444543819713ist_fm @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_549_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R: rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
             => ( abstra7252176601000738572ist_fm @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_550_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc859450856879609959at_nat > fm > fset_fm > $o,Rules2: stream8372878641218411373at_nat,S6: set_fm,S: fm,Rs: stream8372878641218411373at_nat,Steps: stream8224063793758848028at_nat,R: produc859450856879609959at_nat] :
      ( ( abstra8110488454511597367nat_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra5618069685180253819at_nat @ Rules2 @ Rs )
         => ( ( abstra8590151062464399645at_nat @ ( abstra8755028071118092180nat_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
             => ( abstra7502804401140805033nat_fm @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_551_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc859450856879609959at_nat > rule > fset_rule > $o,Rules2: stream8372878641218411373at_nat,S6: set_rule,S: rule,Rs: stream8372878641218411373at_nat,Steps: stream7497242749156405735at_nat,R: produc859450856879609959at_nat] :
      ( ( abstra7673653355469228972t_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra5618069685180253819at_nat @ Rules2 @ Rs )
         => ( ( abstra3018042991430833768at_nat @ ( abstra6341243307501620617t_rule @ Eff2 @ Rs @ S ) @ Steps )
           => ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
             => ( abstra436705010757945630t_rule @ Eff2 @ R @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_552_pred__subset__eq2,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S6: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le1774903256965159270t_fm_o
        @ ^ [X5: list_fm,Y6: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y6 ) @ R3 )
        @ ^ [X5: list_fm,Y6: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y6 ) @ S6 ) )
      = ( ord_le2055072402069232929ist_fm @ R3 @ S6 ) ) ).

% pred_subset_eq2
thf(fact_553_pred__subset__eq2,axiom,
    ! [R3: set_Pr1008144964186165195m_rule,S6: set_Pr1008144964186165195m_rule] :
      ( ( ord_le4891813983447887534rule_o
        @ ^ [X5: produc1996495991257130529ist_fm,Y6: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X5 @ Y6 ) @ R3 )
        @ ^ [X5: produc1996495991257130529ist_fm,Y6: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X5 @ Y6 ) @ S6 ) )
      = ( ord_le5052251716133888043m_rule @ R3 @ S6 ) ) ).

% pred_subset_eq2
thf(fact_554_pred__subset__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S6: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X5: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y6 ) @ R3 )
        @ ^ [X5: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y6 ) @ S6 ) )
      = ( ord_le3146513528884898305at_nat @ R3 @ S6 ) ) ).

% pred_subset_eq2
thf(fact_555_pred__subset__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S6: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X5: product_prod_nat_nat,Y6: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y6 ) @ R3 )
        @ ^ [X5: product_prod_nat_nat,Y6: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y6 ) @ S6 ) )
      = ( ord_le3000389064537975527at_nat @ R3 @ S6 ) ) ).

% pred_subset_eq2
thf(fact_556_pred__subset__eq2,axiom,
    ! [R3: set_Pr2241365836748572487ist_fm,S6: set_Pr2241365836748572487ist_fm] :
      ( ( ord_le8079975967087607008t_fm_o
        @ ^ [X5: rule,Y6: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y6 ) @ R3 )
        @ ^ [X5: rule,Y6: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y6 ) @ S6 ) )
      = ( ord_le6285472588696295335ist_fm @ R3 @ S6 ) ) ).

% pred_subset_eq2
thf(fact_557_pred__equals__eq2,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S6: set_Pr7058068377845519745ist_fm] :
      ( ( ( ^ [X5: list_fm,Y6: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y6 ) @ R3 ) )
        = ( ^ [X5: list_fm,Y6: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y6 ) @ S6 ) ) )
      = ( R3 = S6 ) ) ).

% pred_equals_eq2
thf(fact_558_pred__equals__eq2,axiom,
    ! [R3: set_Pr1008144964186165195m_rule,S6: set_Pr1008144964186165195m_rule] :
      ( ( ( ^ [X5: produc1996495991257130529ist_fm,Y6: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X5 @ Y6 ) @ R3 ) )
        = ( ^ [X5: produc1996495991257130529ist_fm,Y6: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X5 @ Y6 ) @ S6 ) ) )
      = ( R3 = S6 ) ) ).

% pred_equals_eq2
thf(fact_559_pred__equals__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S6: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X5: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y6 ) @ R3 ) )
        = ( ^ [X5: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y6 ) @ S6 ) ) )
      = ( R3 = S6 ) ) ).

% pred_equals_eq2
thf(fact_560_pred__equals__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S6: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X5: product_prod_nat_nat,Y6: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y6 ) @ R3 ) )
        = ( ^ [X5: product_prod_nat_nat,Y6: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y6 ) @ S6 ) ) )
      = ( R3 = S6 ) ) ).

% pred_equals_eq2
thf(fact_561_pred__equals__eq2,axiom,
    ! [R3: set_Pr2241365836748572487ist_fm,S6: set_Pr2241365836748572487ist_fm] :
      ( ( ( ^ [X5: rule,Y6: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y6 ) @ R3 ) )
        = ( ^ [X5: rule,Y6: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y6 ) @ S6 ) ) )
      = ( R3 = S6 ) ) ).

% pred_equals_eq2
thf(fact_562_wf__mkTree,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra4476735159789068068ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ ( abstra178918799166095863ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S ) ) ) ) ).

% wf_mkTree
thf(fact_563_per__rules,axiom,
    ! [R: rule] :
      ( abstra4153889242452539894ist_fm
      @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R2 @ S2 )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ rules
      @ top_to728902761071950545ist_fm
      @ R ) ).

% per_rules
thf(fact_564_minWait__le__pos,axiom,
    ! [Rs: stream_rule,R: rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( member_rule @ R @ ( sset_rule @ rules ) )
       => ( ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R
            @ S )
         => ( ord_less_eq_nat
            @ ( abstra738319790293016300ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S )
            @ ( abstract_pos_rule @ Rs @ R ) ) ) ) ) ).

% minWait_le_pos
thf(fact_565_pos,axiom,
    ! [Rs: stream_rule,R: rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( member_rule @ R @ ( sset_rule @ rules ) )
       => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
          = R ) ) ) ).

% pos
thf(fact_566_trim__alt,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2363873759530810705ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S )
          = ( sdrop_rule
            @ ( abstra738319790293016300ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S )
            @ Rs ) ) ) ) ).

% trim_alt
thf(fact_567_trim__enabled,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra3055540779341473792ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S ) )
          @ S ) ) ) ).

% trim_enabled
thf(fact_568_trim__in__R,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( member_rule
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S ) )
          @ ( sset_rule @ rules ) ) ) ) ).

% trim_in_R
thf(fact_569_fair__uu__def,axiom,
    abstra1913438439042302728u_rule = abstra1114877437032970289r_rule ).

% fair_uu_def
thf(fact_570_RuleSystem_OminWait__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S )
        = ( ord_Least_nat
          @ ^ [N3: nat] : ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_def
thf(fact_571_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule,M: nat] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: fm] :
                ( ( member_fm @ S5 @ S6 )
                & ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ ( sdrop_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_432990894411838815m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_572_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule,M: nat] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: rule] :
                ( ( member_rule @ S5 @ S6 )
                & ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ ( sdrop_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_2199339672054592340e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_573_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,S6: set_list_fm,S: list_fm,Rs: stream_list_fm,Steps: stream8299795917829157543ist_fm,M: nat] :
      ( ( abstra12045985430712427ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_list_fm @ S @ S6 )
       => ( ( abstra274702546776099020ist_fm @ Rules2 @ Rs )
         => ( ( abstra5863983821914536912ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: list_fm] :
                ( ( member_list_fm @ S5 @ S6 )
                & ( abstra5863983821914536912ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ ( sdrop_list_fm @ N2 @ Rs ) @ S5 ) @ ( sdrop_4442373711808556042ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_574_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc780299800278082923e_rule > fm > fset_fm > $o,Rules2: stream3588992565182678257e_rule,S6: set_fm,S: fm,Rs: stream3588992565182678257e_rule,Steps: stream3505881084499982752e_rule,M: nat] :
      ( ( abstra6843014240698492626ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
         => ( ( abstra8067268766995864225e_rule @ ( abstra5872912078930511000ule_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: fm] :
                ( ( member_fm @ S5 @ S6 )
                & ( abstra8067268766995864225e_rule @ ( abstra5872912078930511000ule_fm @ Eff2 @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_3146478060095727335e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_575_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc780299800278082923e_rule > rule > fset_rule > $o,Rules2: stream3588992565182678257e_rule,S6: set_rule,S: rule,Rs: stream3588992565182678257e_rule,Steps: stream4003613590975974891e_rule,M: nat] :
      ( ( abstra1742398642700549575e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
         => ( ( abstra4067143171202666092e_rule @ ( abstra4338455777355667597e_rule @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: rule] :
                ( ( member_rule @ S5 @ S6 )
                & ( abstra4067143171202666092e_rule @ ( abstra4338455777355667597e_rule @ Eff2 @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_3185404920971892658e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_576_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules2: stream8299795917829157543ist_fm,S6: set_fm,S: fm,Rs: stream8299795917829157543ist_fm,Steps: stream5068232101399153750ist_fm,M: nat] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ( ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: fm] :
                ( ( member_fm @ S5 @ S6 )
                & ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) @ S5 ) @ ( sdrop_8120954597595522205ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_577_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S6: set_rule,S: rule,Rs: stream8299795917829157543ist_fm,Steps: stream1960312990768957601ist_fm,M: nat] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: rule] :
                ( ( member_rule @ S5 @ S6 )
                & ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) @ S5 ) @ ( sdrop_2468034597186633064ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_578_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,M: nat] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S5 @ S6 )
                & ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ ( sdrop_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_579_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules2: stream727092118206550309m_rule,S6: set_fm,S: fm,Rs: stream727092118206550309m_rule,Steps: stream5001952251264143308m_rule,M: nat] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: fm] :
                ( ( member_fm @ S5 @ S6 )
                & ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_6203755225909610927m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_580_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules2: stream727092118206550309m_rule,S6: set_rule,S: rule,Rs: stream727092118206550309m_rule,Steps: stream1878305877988316353m_rule,M: nat] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ Rs @ S ) @ Steps )
           => ? [N2: nat,S5: rule] :
                ( ( member_rule @ S5 @ S6 )
                & ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_1764812713384251940m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_581_RuleSystem__Defs_Otrim_Ocong,axiom,
    abstra2363873759530810705ist_fm = abstra2363873759530810705ist_fm ).

% RuleSystem_Defs.trim.cong
thf(fact_582_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules2 @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_583_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules2 @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_584_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S6: set_Pr8693737435421807431at_nat,S: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules2 @ S6 )
     => ( ( member8206827879206165904at_nat @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules2 @ ( abstra3437498576158148503at_nat @ Eff2 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_585_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules2 @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_586_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_12: fset_fm] : ( Eff2 @ X3 @ S @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_587_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_12: fset_rule] : ( Eff2 @ X3 @ S @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_588_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S6: set_Pr8693737435421807431at_nat,S: produc859450856879609959at_nat] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules2 @ S6 )
     => ( ( member8206827879206165904at_nat @ S @ S6 )
       => ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_12: fset_P3619101297660450253at_nat] : ( Eff2 @ X3 @ S @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_589_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
            & ? [X_12: fset_P661503646757059847ist_fm] : ( Eff2 @ X3 @ S @ X_12 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_590_PersistentRuleSystem_Oaxioms_I1_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 ) ) ).

% PersistentRuleSystem.axioms(1)
thf(fact_591_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules2: stream_fm,S6: set_fm,S: fm,Rs: stream_fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules2 @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra5942290089224017835_fm_fm @ Eff2 @ Rs @ S ) ) @ ( sset_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_592_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules2: stream_fm,S6: set_rule,S: rule,Rs: stream_fm] :
      ( ( abstra5167362717900046400m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules2 @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra8496492133263135264m_rule @ Eff2 @ Rs @ S ) ) @ ( sset_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_593_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S ) ) @ ( sset_rule @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_594_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S ) ) @ ( sset_rule @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_595_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules2: stream8299795917829157543ist_fm,S6: set_fm,S: fm,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( abstra5503639309725227688_fm_fm @ Eff2 @ Rs @ S ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_596_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S6: set_rule,S: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs @ S ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_597_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_fm] :
      ( ( abstra7587511110199143014ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules2 @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra948641862382362758ist_fm @ Eff2 @ Rs @ S ) ) @ ( sset_fm @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_598_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S ) ) @ ( sset_rule @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_599_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc859450856879609959at_nat > fm > fset_fm > $o,Rules2: stream8372878641218411373at_nat,S6: set_fm,S: fm,Rs: stream8372878641218411373at_nat] :
      ( ( abstra849841822169861838nat_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra5618069685180253819at_nat @ Rules2 @ Rs )
         => ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ ( abstra2782398720704632046nat_fm @ Eff2 @ Rs @ S ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_600_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc859450856879609959at_nat > rule > fset_rule > $o,Rules2: stream8372878641218411373at_nat,S6: set_rule,S: rule,Rs: stream8372878641218411373at_nat] :
      ( ( abstra2579607477568694211t_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra5618069685180253819at_nat @ Rules2 @ Rs )
         => ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ ( abstra361488207848709603t_rule @ Eff2 @ Rs @ S ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_601_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules2: stream727092118206550309m_rule,S6: set_fm,S: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( abstra7152516257876272129ule_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_602_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules2: stream727092118206550309m_rule,S6: set_rule,S: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( abstra2524698237922071414e_rule @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_603_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream727092118206550309m_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( abstra7200325324992684464ist_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_604_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream727092118206550309m_rule,S6: set_Pr8693737435421807431at_nat,S: produc859450856879609959at_nat,Rs: stream727092118206550309m_rule] :
      ( ( abstra8123243527896255399at_nat @ Eff2 @ Rules2 @ S6 )
     => ( ( member8206827879206165904at_nat @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( abstra1800717694132746102at_nat @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra276577985667043783at_nat @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_605_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra7038331712872434097ule_fm @ Eff2 @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_606_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra3154316310710804774e_rule @ Eff2 @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_607_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S6: set_Pr8693737435421807431at_nat,S: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules2 @ S6 )
     => ( ( member8206827879206165904at_nat @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra248173851283269062at_nat @ Eff2 @ ( shd_rule @ ( abstra3437498576158148503at_nat @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_608_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_609_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S )
            = ( sdrop_rule @ ( abstra2294468009765627845ule_fm @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_610_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S )
            = ( sdrop_rule @ ( abstra5682536783049438522e_rule @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_611_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc780299800278082923e_rule > fm > fset_fm > $o,Rules2: stream3588992565182678257e_rule,S6: set_fm,S: fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra6843014240698492626ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
         => ( ( abstra7283367714565168370ule_fm @ Eff2 @ Rs @ S )
            = ( sdrop_2199339672054592340e_rule @ ( abstra1933929482952324109ule_fm @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_612_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc780299800278082923e_rule > rule > fset_rule > $o,Rules2: stream3588992565182678257e_rule,S6: set_rule,S: rule,Rs: stream3588992565182678257e_rule] :
      ( ( abstra1742398642700549575e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
         => ( ( abstra5027552996074955751e_rule @ Eff2 @ Rs @ S )
            = ( sdrop_2199339672054592340e_rule @ ( abstra5810752615369301378e_rule @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_613_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules2: stream8299795917829157543ist_fm,S6: set_fm,S: fm,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ( ( abstra5503639309725227688_fm_fm @ Eff2 @ Rs @ S )
            = ( sdrop_4442373711808556042ist_fm @ ( abstra8883821634369821123_fm_fm @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_614_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S6: set_rule,S: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ( ( abstra5014957989051229853m_rule @ Eff2 @ Rs @ S )
            = ( sdrop_4442373711808556042ist_fm @ ( abstra3389404019813435448m_rule @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_615_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S )
            = ( sdrop_rule @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_616_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules2: stream727092118206550309m_rule,S6: set_fm,S: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S )
            = ( sdrop_7224736112439592940m_rule @ ( abstra6093700014593756181ule_fm @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_617_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules2: stream727092118206550309m_rule,S6: set_rule,S: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S )
            = ( sdrop_7224736112439592940m_rule @ ( abstra6412047235177443722e_rule @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_618_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc780299800278082923e_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream3588992565182678257e_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra2302509236375889695ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
         => ( ( abstra3813788296048736511ist_fm @ Eff2 @ Rs @ S )
            = ( sdrop_2199339672054592340e_rule @ ( abstra4936422202967311524ist_fm @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_619_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream8299795917829157543ist_fm,R: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra3513618393749879589ist_fm @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_620_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream3588992565182678257e_rule,R: produc780299800278082923e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra883788664900704879e_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_621_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream1878305877988316353m_rule,R: produc1660549859252256699m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra7431954940529686207m_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_622_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream2312013117288958913e_rule,R: produc6288491646048892603e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra2836524690471546303e_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_623_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream6494289010434245521m_rule,R: produc3793571666044074251m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra3193765552906301967m_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_624_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream727092118206550309m_rule,R: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstra1882453022147397137m_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_625_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,R: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
          = R )
       => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_626_RuleSystem_Oeff__S,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules2: stream_fm,S6: set_fm,S: fm,R: fm,Sl3: fset_fm,S4: fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmember_fm @ S4 @ Sl3 )
             => ( member_fm @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_627_RuleSystem_Oeff__S,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules2: stream_fm,S6: set_rule,S: rule,R: fm,Sl3: fset_rule,S4: rule] :
      ( ( abstra5167362717900046400m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmember_rule @ S4 @ Sl3 )
             => ( member_rule @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_628_RuleSystem_Oeff__S,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,R: rule,Sl3: fset_fm,S4: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmember_fm @ S4 @ Sl3 )
             => ( member_fm @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_629_RuleSystem_Oeff__S,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,R: rule,Sl3: fset_rule,S4: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmember_rule @ S4 @ Sl3 )
             => ( member_rule @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_630_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules2: stream8299795917829157543ist_fm,S6: set_fm,S: fm,R: produc1996495991257130529ist_fm,Sl3: fset_fm,S4: fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmember_fm @ S4 @ Sl3 )
             => ( member_fm @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_631_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S6: set_rule,S: rule,R: produc1996495991257130529ist_fm,Sl3: fset_rule,S4: rule] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmember_rule @ S4 @ Sl3 )
             => ( member_rule @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_632_RuleSystem_Oeff__S,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,R: fm,Sl3: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
      ( ( abstra7587511110199143014ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl3 )
             => ( member8102475879199740618ist_fm @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_633_RuleSystem_Oeff__S,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,R: rule,Sl3: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl3 )
             => ( member8102475879199740618ist_fm @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_634_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc859450856879609959at_nat > fm > fset_fm > $o,Rules2: stream8372878641218411373at_nat,S6: set_fm,S: fm,R: produc859450856879609959at_nat,Sl3: fset_fm,S4: fm] :
      ( ( abstra849841822169861838nat_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmember_fm @ S4 @ Sl3 )
             => ( member_fm @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_635_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc859450856879609959at_nat > rule > fset_rule > $o,Rules2: stream8372878641218411373at_nat,S6: set_rule,S: rule,R: produc859450856879609959at_nat,Sl3: fset_rule,S4: rule] :
      ( ( abstra2579607477568694211t_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
         => ( ( Eff2 @ R @ S @ Sl3 )
           => ( ( fmember_rule @ S4 @ Sl3 )
             => ( member_rule @ S4 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_636_RuleSystem_Ointro,axiom,
    ! [S6: set_fm,Rules2: stream_fm,Eff2: fm > fm > fset_fm > $o] :
      ( ! [S3: fm] :
          ( ( member_fm @ S3 @ S6 )
         => ! [R5: fm] :
              ( ( member_fm @ R5 @ ( sset_fm @ Rules2 ) )
             => ! [Sl4: fset_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: fm] :
                      ( ( fmember_fm @ S5 @ Sl4 )
                     => ( member_fm @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: fm] :
            ( ( member_fm @ S3 @ S6 )
           => ? [X8: fm] :
                ( ( member_fm @ X8 @ ( sset_fm @ Rules2 ) )
                & ? [X_1: fset_fm] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra886934874521163211_fm_fm @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_637_RuleSystem_Ointro,axiom,
    ! [S6: set_rule,Rules2: stream_fm,Eff2: fm > rule > fset_rule > $o] :
      ( ! [S3: rule] :
          ( ( member_rule @ S3 @ S6 )
         => ! [R5: fm] :
              ( ( member_fm @ R5 @ ( sset_fm @ Rules2 ) )
             => ! [Sl4: fset_rule] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: rule] :
                      ( ( fmember_rule @ S5 @ Sl4 )
                     => ( member_rule @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: rule] :
            ( ( member_rule @ S3 @ S6 )
           => ? [X8: fm] :
                ( ( member_fm @ X8 @ ( sset_fm @ Rules2 ) )
                & ? [X_1: fset_rule] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra5167362717900046400m_rule @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_638_RuleSystem_Ointro,axiom,
    ! [S6: set_fm,Rules2: stream_rule,Eff2: rule > fm > fset_fm > $o] :
      ( ! [S3: fm] :
          ( ( member_fm @ S3 @ S6 )
         => ! [R5: rule] :
              ( ( member_rule @ R5 @ ( sset_rule @ Rules2 ) )
             => ! [Sl4: fset_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: fm] :
                      ( ( fmember_fm @ S5 @ Sl4 )
                     => ( member_fm @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: fm] :
            ( ( member_fm @ S3 @ S6 )
           => ? [X8: rule] :
                ( ( member_rule @ X8 @ ( sset_rule @ Rules2 ) )
                & ? [X_1: fset_fm] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_639_RuleSystem_Ointro,axiom,
    ! [S6: set_rule,Rules2: stream_rule,Eff2: rule > rule > fset_rule > $o] :
      ( ! [S3: rule] :
          ( ( member_rule @ S3 @ S6 )
         => ! [R5: rule] :
              ( ( member_rule @ R5 @ ( sset_rule @ Rules2 ) )
             => ! [Sl4: fset_rule] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: rule] :
                      ( ( fmember_rule @ S5 @ Sl4 )
                     => ( member_rule @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: rule] :
            ( ( member_rule @ S3 @ S6 )
           => ? [X8: rule] :
                ( ( member_rule @ X8 @ ( sset_rule @ Rules2 ) )
                & ? [X_1: fset_rule] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_640_RuleSystem_Ointro,axiom,
    ! [S6: set_fm,Rules2: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o] :
      ( ! [S3: fm] :
          ( ( member_fm @ S3 @ S6 )
         => ! [R5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R5 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
             => ! [Sl4: fset_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: fm] :
                      ( ( fmember_fm @ S5 @ Sl4 )
                     => ( member_fm @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: fm] :
            ( ( member_fm @ S3 @ S6 )
           => ? [X8: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X8 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                & ? [X_1: fset_fm] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_641_RuleSystem_Ointro,axiom,
    ! [S6: set_rule,Rules2: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o] :
      ( ! [S3: rule] :
          ( ( member_rule @ S3 @ S6 )
         => ! [R5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R5 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
             => ! [Sl4: fset_rule] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: rule] :
                      ( ( fmember_rule @ S5 @ Sl4 )
                     => ( member_rule @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: rule] :
            ( ( member_rule @ S3 @ S6 )
           => ? [X8: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X8 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                & ? [X_1: fset_rule] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra9106359784214577277m_rule @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_642_RuleSystem_Ointro,axiom,
    ! [S6: set_Pr7058068377845519745ist_fm,Rules2: stream_fm,Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ! [S3: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ S3 @ S6 )
         => ! [R5: fm] :
              ( ( member_fm @ R5 @ ( sset_fm @ Rules2 ) )
             => ! [Sl4: fset_P661503646757059847ist_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: produc1996495991257130529ist_fm] :
                      ( ( fmembe3381613331217039976ist_fm @ S5 @ Sl4 )
                     => ( member8102475879199740618ist_fm @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ S3 @ S6 )
           => ? [X8: fm] :
                ( ( member_fm @ X8 @ ( sset_fm @ Rules2 ) )
                & ? [X_1: fset_P661503646757059847ist_fm] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra7587511110199143014ist_fm @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_643_RuleSystem_Ointro,axiom,
    ! [S6: set_Pr7058068377845519745ist_fm,Rules2: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ! [S3: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ S3 @ S6 )
         => ! [R5: rule] :
              ( ( member_rule @ R5 @ ( sset_rule @ Rules2 ) )
             => ! [Sl4: fset_P661503646757059847ist_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: produc1996495991257130529ist_fm] :
                      ( ( fmembe3381613331217039976ist_fm @ S5 @ Sl4 )
                     => ( member8102475879199740618ist_fm @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ S3 @ S6 )
           => ? [X8: rule] :
                ( ( member_rule @ X8 @ ( sset_rule @ Rules2 ) )
                & ? [X_1: fset_P661503646757059847ist_fm] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_644_RuleSystem_Ointro,axiom,
    ! [S6: set_fm,Rules2: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > fm > fset_fm > $o] :
      ( ! [S3: fm] :
          ( ( member_fm @ S3 @ S6 )
         => ! [R5: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ R5 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
             => ! [Sl4: fset_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: fm] :
                      ( ( fmember_fm @ S5 @ Sl4 )
                     => ( member_fm @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: fm] :
            ( ( member_fm @ S3 @ S6 )
           => ? [X8: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X8 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ? [X_1: fset_fm] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra849841822169861838nat_fm @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_645_RuleSystem_Ointro,axiom,
    ! [S6: set_rule,Rules2: stream8372878641218411373at_nat,Eff2: produc859450856879609959at_nat > rule > fset_rule > $o] :
      ( ! [S3: rule] :
          ( ( member_rule @ S3 @ S6 )
         => ! [R5: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ R5 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
             => ! [Sl4: fset_rule] :
                  ( ( Eff2 @ R5 @ S3 @ Sl4 )
                 => ! [S5: rule] :
                      ( ( fmember_rule @ S5 @ Sl4 )
                     => ( member_rule @ S5 @ S6 ) ) ) ) )
     => ( ! [S3: rule] :
            ( ( member_rule @ S3 @ S6 )
           => ? [X8: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X8 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                & ? [X_1: fset_rule] : ( Eff2 @ X8 @ S3 @ X_1 ) ) )
       => ( abstra2579607477568694211t_rule @ Eff2 @ Rules2 @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_646_RuleSystem__def,axiom,
    ( abstra886934874521163211_fm_fm
    = ( ^ [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S7: set_fm] :
          ( ! [S2: fm] :
              ( ( member_fm @ S2 @ S7 )
             => ! [R2: fm] :
                  ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S7 ) ) ) ) )
          & ! [S2: fm] :
              ( ( member_fm @ S2 @ S7 )
             => ? [X5: fm] :
                  ( ( member_fm @ X5 @ ( sset_fm @ Rules ) )
                  & ? [X: fset_fm] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_647_RuleSystem__def,axiom,
    ( abstra5167362717900046400m_rule
    = ( ^ [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S7: set_rule] :
          ( ! [S2: rule] :
              ( ( member_rule @ S2 @ S7 )
             => ! [R2: fm] :
                  ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S7 ) ) ) ) )
          & ! [S2: rule] :
              ( ( member_rule @ S2 @ S7 )
             => ? [X5: fm] :
                  ( ( member_fm @ X5 @ ( sset_fm @ Rules ) )
                  & ? [X: fset_rule] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_648_RuleSystem__def,axiom,
    ( abstra916514217863227968ule_fm
    = ( ^ [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S7: set_fm] :
          ( ! [S2: fm] :
              ( ( member_fm @ S2 @ S7 )
             => ! [R2: rule] :
                  ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S7 ) ) ) ) )
          & ! [S2: fm] :
              ( ( member_fm @ S2 @ S7 )
             => ? [X5: rule] :
                  ( ( member_rule @ X5 @ ( sset_rule @ Rules ) )
                  & ? [X: fset_fm] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_649_RuleSystem__def,axiom,
    ( abstra2725590884229999157e_rule
    = ( ^ [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S7: set_rule] :
          ( ! [S2: rule] :
              ( ( member_rule @ S2 @ S7 )
             => ! [R2: rule] :
                  ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S7 ) ) ) ) )
          & ! [S2: rule] :
              ( ( member_rule @ S2 @ S7 )
             => ? [X5: rule] :
                  ( ( member_rule @ X5 @ ( sset_rule @ Rules ) )
                  & ? [X: fset_rule] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_650_RuleSystem__def,axiom,
    ( abstra2919136520687232136_fm_fm
    = ( ^ [Eff: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S7: set_fm] :
          ( ! [S2: fm] :
              ( ( member_fm @ S2 @ S7 )
             => ! [R2: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S7 ) ) ) ) )
          & ! [S2: fm] :
              ( ( member_fm @ S2 @ S7 )
             => ? [X5: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X5 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                  & ? [X: fset_fm] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_651_RuleSystem__def,axiom,
    ( abstra9106359784214577277m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S7: set_rule] :
          ( ! [S2: rule] :
              ( ( member_rule @ S2 @ S7 )
             => ! [R2: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S7 ) ) ) ) )
          & ! [S2: rule] :
              ( ( member_rule @ S2 @ S7 )
             => ? [X5: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X5 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                  & ? [X: fset_rule] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_652_RuleSystem__def,axiom,
    ( abstra7587511110199143014ist_fm
    = ( ^ [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S7: set_Pr7058068377845519745ist_fm] :
          ( ! [S2: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S2 @ S7 )
             => ! [R2: fm] :
                  ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
                 => ! [Sl: fset_P661503646757059847ist_fm] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: produc1996495991257130529ist_fm] :
                          ( ( fmembe3381613331217039976ist_fm @ S8 @ Sl )
                         => ( member8102475879199740618ist_fm @ S8 @ S7 ) ) ) ) )
          & ! [S2: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S2 @ S7 )
             => ? [X5: fm] :
                  ( ( member_fm @ X5 @ ( sset_fm @ Rules ) )
                  & ? [X: fset_P661503646757059847ist_fm] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_653_RuleSystem__def,axiom,
    ( abstra6455275554694158129ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S7: set_Pr7058068377845519745ist_fm] :
          ( ! [S2: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S2 @ S7 )
             => ! [R2: rule] :
                  ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
                 => ! [Sl: fset_P661503646757059847ist_fm] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: produc1996495991257130529ist_fm] :
                          ( ( fmembe3381613331217039976ist_fm @ S8 @ Sl )
                         => ( member8102475879199740618ist_fm @ S8 @ S7 ) ) ) ) )
          & ! [S2: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S2 @ S7 )
             => ? [X5: rule] :
                  ( ( member_rule @ X5 @ ( sset_rule @ Rules ) )
                  & ? [X: fset_P661503646757059847ist_fm] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_654_RuleSystem__def,axiom,
    ( abstra849841822169861838nat_fm
    = ( ^ [Eff: produc859450856879609959at_nat > fm > fset_fm > $o,Rules: stream8372878641218411373at_nat,S7: set_fm] :
          ( ! [S2: fm] :
              ( ( member_fm @ S2 @ S7 )
             => ! [R2: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ R2 @ ( sset_P9164763103296375419at_nat @ Rules ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S7 ) ) ) ) )
          & ! [S2: fm] :
              ( ( member_fm @ S2 @ S7 )
             => ? [X5: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X5 @ ( sset_P9164763103296375419at_nat @ Rules ) )
                  & ? [X: fset_fm] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_655_RuleSystem__def,axiom,
    ( abstra2579607477568694211t_rule
    = ( ^ [Eff: produc859450856879609959at_nat > rule > fset_rule > $o,Rules: stream8372878641218411373at_nat,S7: set_rule] :
          ( ! [S2: rule] :
              ( ( member_rule @ S2 @ S7 )
             => ! [R2: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ R2 @ ( sset_P9164763103296375419at_nat @ Rules ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R2 @ S2 @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S7 ) ) ) ) )
          & ! [S2: rule] :
              ( ( member_rule @ S2 @ S7 )
             => ? [X5: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X5 @ ( sset_P9164763103296375419at_nat @ Rules ) )
                  & ? [X: fset_rule] : ( Eff @ X5 @ S2 @ X ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_656_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream8299795917829157543ist_fm,R: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra3513618393749879589ist_fm @ Rs @ R )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N3 @ Rs ) )
              = R ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_657_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream3588992565182678257e_rule,R: produc780299800278082923e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra883788664900704879e_rule @ Rs @ R )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N3 @ Rs ) )
              = R ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_658_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream1878305877988316353m_rule,R: produc1660549859252256699m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra7431954940529686207m_rule @ Rs @ R )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N3 @ Rs ) )
              = R ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_659_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream2312013117288958913e_rule,R: produc6288491646048892603e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra2836524690471546303e_rule @ Rs @ R )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N3 @ Rs ) )
              = R ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_660_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream6494289010434245521m_rule,R: produc3793571666044074251m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra3193765552906301967m_rule @ Rs @ R )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N3 @ Rs ) )
              = R ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_661_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream727092118206550309m_rule,R: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra1882453022147397137m_rule @ Rs @ R )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N3 @ Rs ) )
              = R ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_662_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstract_pos_rule @ Rs @ R )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
              = R ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_663_RuleSystem_Opos,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
            = R ) ) ) ) ).

% RuleSystem.pos
thf(fact_664_RuleSystem_OminWait__le__pos,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R: rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
       => ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( ( abstra3055540779341473792ist_fm @ Eff2 @ R @ S )
           => ( ord_less_eq_nat @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) ) ) ) ) ) ).

% RuleSystem.minWait_le_pos
thf(fact_665_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra2794488811626936717ule_fm @ Eff2 @ Rules2 @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_666_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra5602818997816140546e_rule @ Eff2 @ Rules2 @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_667_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S6: set_Pr8693737435421807431at_nat,S: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules2 @ S6 )
     => ( ( member8206827879206165904at_nat @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra1446413340331638506at_nat @ Eff2 @ Rules2 @ ( abstra193881638956283709at_nat @ Eff2 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_668_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_669_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra7038331712872434097ule_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_670_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra3154316310710804774e_rule @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_671_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc780299800278082923e_rule > fm > fset_fm > $o,Rules2: stream3588992565182678257e_rule,S6: set_fm,S: fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra6843014240698492626ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra1405456541278283553ule_fm @ Eff2 @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_672_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc780299800278082923e_rule > rule > fset_rule > $o,Rules2: stream3588992565182678257e_rule,S6: set_rule,S: rule,Rs: stream3588992565182678257e_rule] :
      ( ( abstra1742398642700549575e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra5553215051456775830e_rule @ Eff2 @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_673_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules2: stream8299795917829157543ist_fm,S6: set_fm,S: fm,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra8628512393757682903_fm_fm @ Eff2 @ ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_674_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S6: set_rule,S: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra5706625008861892940m_rule @ Eff2 @ ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_675_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_676_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc780299800278082923e_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream3588992565182678257e_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra2302509236375889695ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra4837122928373946751e_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra1049073205711939216ist_fm @ Eff2 @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_677_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules2: stream727092118206550309m_rule,S6: set_fm,S: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra7152516257876272129ule_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_678_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules2: stream727092118206550309m_rule,S6: set_rule,S: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ? [N2: nat] : ( abstra2524698237922071414e_rule @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_679_RuleSystem__axioms,axiom,
    ( abstra6455275554694158129ist_fm
    @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R2 @ S2 )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ top_to728902761071950545ist_fm ) ).

% RuleSystem_axioms
thf(fact_680_pos__least,axiom,
    ! [N: nat,Rs: stream8299795917829157543ist_fm,R: produc1996495991257130529ist_fm] :
      ( ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra3513618393749879589ist_fm @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_681_pos__least,axiom,
    ! [N: nat,Rs: stream3588992565182678257e_rule,R: produc780299800278082923e_rule] :
      ( ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra883788664900704879e_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_682_pos__least,axiom,
    ! [N: nat,Rs: stream1878305877988316353m_rule,R: produc1660549859252256699m_rule] :
      ( ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra7431954940529686207m_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_683_pos__least,axiom,
    ! [N: nat,Rs: stream2312013117288958913e_rule,R: produc6288491646048892603e_rule] :
      ( ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra2836524690471546303e_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_684_pos__least,axiom,
    ! [N: nat,Rs: stream6494289010434245521m_rule,R: produc3793571666044074251m_rule] :
      ( ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra3193765552906301967m_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_685_pos__least,axiom,
    ! [N: nat,Rs: stream727092118206550309m_rule,R: produc164195504107695125m_rule] :
      ( ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Rs ) )
        = R )
     => ( ord_less_eq_nat @ ( abstra1882453022147397137m_rule @ Rs @ R ) @ N ) ) ).

% pos_least
thf(fact_686_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_687_trim__fair,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra1114877437032970289r_rule @ rules
          @ ( abstra2363873759530810705ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S ) ) ) ) ).

% trim_fair
thf(fact_688_RuleSystem_OminWait__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S )
       => ( ord_less_eq_nat @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S ) @ N ) ) ) ).

% RuleSystem.minWait_least
thf(fact_689_pos__def,axiom,
    ( abstra3513618393749879589ist_fm
    = ( ^ [Rs2: stream8299795917829157543ist_fm,R2: produc1996495991257130529ist_fm] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N3 @ Rs2 ) )
              = R2 ) ) ) ) ).

% pos_def
thf(fact_690_pos__def,axiom,
    ( abstra883788664900704879e_rule
    = ( ^ [Rs2: stream3588992565182678257e_rule,R2: produc780299800278082923e_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N3 @ Rs2 ) )
              = R2 ) ) ) ) ).

% pos_def
thf(fact_691_pos__def,axiom,
    ( abstra7431954940529686207m_rule
    = ( ^ [Rs2: stream1878305877988316353m_rule,R2: produc1660549859252256699m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N3 @ Rs2 ) )
              = R2 ) ) ) ) ).

% pos_def
thf(fact_692_pos__def,axiom,
    ( abstra2836524690471546303e_rule
    = ( ^ [Rs2: stream2312013117288958913e_rule,R2: produc6288491646048892603e_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N3 @ Rs2 ) )
              = R2 ) ) ) ) ).

% pos_def
thf(fact_693_pos__def,axiom,
    ( abstra3193765552906301967m_rule
    = ( ^ [Rs2: stream6494289010434245521m_rule,R2: produc3793571666044074251m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N3 @ Rs2 ) )
              = R2 ) ) ) ) ).

% pos_def
thf(fact_694_pos__def,axiom,
    ( abstra1882453022147397137m_rule
    = ( ^ [Rs2: stream727092118206550309m_rule,R2: produc164195504107695125m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N3 @ Rs2 ) )
              = R2 ) ) ) ) ).

% pos_def
thf(fact_695_pos__def,axiom,
    ( abstract_pos_rule
    = ( ^ [Rs2: stream_rule,R2: rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs2 ) )
              = R2 ) ) ) ) ).

% pos_def
thf(fact_696_in__cont__mkTree,axiom,
    ! [S: produc1996495991257130529ist_fm,Rs: stream_rule,T3: abstra4572470701423150929m_rule] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( fmembe1635983029874454474m_rule @ T3
            @ ( abstra4962253831105676628m_rule
              @ ( abstra178918799166095863ist_fm
                @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ S2 )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S ) ) )
         => ? [Sl5: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S5 @ top_to728902761071950545ist_fm )
              & ( ( eff
                  @ ( shd_rule
                    @ ( abstra2363873759530810705ist_fm
                      @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R2 @ S2 )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ Rs
                      @ S ) )
                  @ S )
                = ( some_f2162114139285431020ist_fm @ Sl5 ) )
              & ( fmembe3381613331217039976ist_fm @ S5 @ Sl5 )
              & ( T3
                = ( abstra178918799166095863ist_fm
                  @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R2 @ S2 )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ ( stl_rule
                    @ ( abstra2363873759530810705ist_fm
                      @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R2 @ S2 )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ Rs
                      @ S ) )
                  @ S5 ) ) ) ) ) ) ).

% in_cont_mkTree
thf(fact_697_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules2: stream727092118206550309m_rule,S6: set_fm,S: fm,Rs: stream727092118206550309m_rule,T3: abstra4575282202256230560m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( ( fmembe8691679578407899495m_rule @ T3 @ ( abstra2433431901973683783m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_fm,S5: fm] :
                ( ( member_fm @ S5 @ S6 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmember_fm @ S5 @ Sl5 )
                & ( T3
                  = ( abstra1207708450097661130ule_fm @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_698_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules2: stream727092118206550309m_rule,S6: set_rule,S: rule,Rs: stream727092118206550309m_rule,T3: abstra4791331342562632597m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( ( fmembe1981849019698968796m_rule @ T3 @ ( abstra5403976286072314556m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_rule,S5: rule] :
                ( ( member_rule @ S5 @ S6 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmember_rule @ S5 @ Sl5 )
                & ( T3
                  = ( abstra1231468025953361855e_rule @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_699_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream727092118206550309m_rule,S6: set_Pr8693737435421807431at_nat,S: produc859450856879609959at_nat,Rs: stream727092118206550309m_rule,T3: abstra8173387307635926887m_rule] :
      ( ( abstra8123243527896255399at_nat @ Eff2 @ Rules2 @ S6 )
     => ( ( member8206827879206165904at_nat @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( ( fmembe2908244859456314592m_rule @ T3 @ ( abstra8750048091207030378m_rule @ ( abstra8186850217458087021at_nat @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_P3619101297660450253at_nat,S5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S5 @ S6 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra276577985667043783at_nat @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmembe6431293532144391214at_nat @ S5 @ Sl5 )
                & ( T3
                  = ( abstra8186850217458087021at_nat @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra276577985667043783at_nat @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_700_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream727092118206550309m_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule,T3: abstra4658698708444432161m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( ( fmembe2925761000581315610m_rule @ T3 @ ( abstra5924671108449608740m_rule @ ( abstra2161496378874750759ist_fm @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S5 @ S6 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmembe3381613331217039976ist_fm @ S5 @ Sl5 )
                & ( T3
                  = ( abstra2161496378874750759ist_fm @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_701_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream727092118206550309m_rule,S6: set_Ab3744519486624674951m_rule,S: abstra4572470701423150929m_rule,Rs: stream727092118206550309m_rule,T3: abstra5426939140630736801m_rule] :
      ( ( abstra6898608545798587409m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member1590433669991072232m_rule @ S @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules2 @ Rs )
         => ( ( fmembe8791921695509808616m_rule @ T3 @ ( abstra3071294317365906120m_rule @ ( abstra1631510761171731915m_rule @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_A5396818216090053761m_rule,S5: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S5 @ S6 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra7609338345072424433m_rule @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmembe1635983029874454474m_rule @ S5 @ Sl5 )
                & ( T3
                  = ( abstra1631510761171731915m_rule @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra7609338345072424433m_rule @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_702_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm,S: fm,Rs: stream_rule,T3: abstra4524330545568143696m_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( fmembe2914889872492170903m_rule @ T3 @ ( abstra5134701107988495863m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_fm,S5: fm] :
                ( ( member_fm @ S5 @ S6 )
                & ( Eff2 @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmember_fm @ S5 @ Sl5 )
                & ( T3
                  = ( abstra3991228042706630138ule_fm @ Eff2 @ ( stl_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_703_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule,S: rule,Rs: stream_rule,T3: abstra1071693965251717573e_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( fmembe5852619498312149900e_rule @ T3 @ ( abstra2651384284180525036e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_rule,S5: rule] :
                ( ( member_rule @ S5 @ S6 )
                & ( Eff2 @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmember_rule @ S5 @ Sl5 )
                & ( T3
                  = ( abstra7443870075974203631e_rule @ Eff2 @ ( stl_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_704_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S6: set_Pr8693737435421807431at_nat,S: produc859450856879609959at_nat,Rs: stream_rule,T3: abstra5175790144536110231t_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff2 @ Rules2 @ S6 )
     => ( ( member8206827879206165904at_nat @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( fmembe4064145320595246480t_rule @ T3 @ ( abstra553533796049368218t_rule @ ( abstra193881638956283709at_nat @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_P3619101297660450253at_nat,S5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S5 @ S6 )
                & ( Eff2 @ ( shd_rule @ ( abstra3437498576158148503at_nat @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmembe6431293532144391214at_nat @ S5 @ Sl5 )
                & ( T3
                  = ( abstra193881638956283709at_nat @ Eff2 @ ( stl_rule @ ( abstra3437498576158148503at_nat @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_705_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_rule,S6: set_Ab3744519486624674951m_rule,S: abstra4572470701423150929m_rule,Rs: stream_rule,T3: abstra4125161011840266193e_rule] :
      ( ( abstra6244190358723783233m_rule @ Eff2 @ Rules2 @ S6 )
     => ( ( member1590433669991072232m_rule @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( fmembe8010993199532995224e_rule @ T3 @ ( abstra6162512231279892984e_rule @ ( abstra2861652164185304827m_rule @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_A5396818216090053761m_rule,S5: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S5 @ S6 )
                & ( Eff2 @ ( shd_rule @ ( abstra1865002795007470625m_rule @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmembe1635983029874454474m_rule @ S5 @ Sl5 )
                & ( T3
                  = ( abstra2861652164185304827m_rule @ Eff2 @ ( stl_rule @ ( abstra1865002795007470625m_rule @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_706_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm,Rs: stream_rule,T3: abstra4572470701423150929m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules2 @ Rs )
         => ( ( fmembe1635983029874454474m_rule @ T3 @ ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S ) ) )
           => ? [Sl5: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S5 @ S6 )
                & ( Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
                & ( fmembe3381613331217039976ist_fm @ S5 @ Sl5 )
                & ( T3
                  = ( abstra178918799166095863ist_fm @ Eff2 @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_707_trim__def,axiom,
    ! [Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra2363873759530810705ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S )
      = ( sdrop_while_rule
        @ ^ [R2: rule] :
            ~ ( abstra3055540779341473792ist_fm
              @ ^ [S2: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ S2 @ T )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R2
              @ S )
        @ Rs ) ) ).

% trim_def
thf(fact_708_epath__completeness__Saturated,axiom,
    ! [S: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
     => ( ? [T4: abstra4572470701423150929m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T4 ) )
              = S )
            & ( abstra4476735159789068068ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ T4 )
            & ( abstra3507476289341504315m_rule @ T4 ) )
        | ? [Steps4: stream727092118206550309m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps4 ) )
              = S )
            & ( abstra5415202973855822231ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps4 )
            & ( abstra8647779336736900908ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps4 ) ) ) ) ).

% epath_completeness_Saturated
thf(fact_709_PersistentRuleSystem_Ointro,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( abstra2778581409224112183ist_fm @ Eff2 @ Rules2 @ S6 )
       => ( abstra7887161495092839066ist_fm @ Eff2 @ Rules2 @ S6 ) ) ) ).

% PersistentRuleSystem.intro
thf(fact_710_PersistentRuleSystem__def,axiom,
    ( abstra7887161495092839066ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S7: set_Pr7058068377845519745ist_fm] :
          ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S7 )
          & ( abstra2778581409224112183ist_fm @ Eff @ Rules @ S7 ) ) ) ) ).

% PersistentRuleSystem_def
thf(fact_711_mkTree_Osimps_I1_J,axiom,
    ! [Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S ) )
      = ( produc491467635432902671m_rule @ S
        @ ( shd_rule
          @ ( abstra2363873759530810705ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S ) ) ) ) ).

% mkTree.simps(1)
thf(fact_712_tree_Oexpand,axiom,
    ! [Tree: abstra4572470701423150929m_rule,Tree2: abstra4572470701423150929m_rule] :
      ( ( ( ( abstra3503897793822247108m_rule @ Tree )
          = ( abstra3503897793822247108m_rule @ Tree2 ) )
        & ( ( abstra4962253831105676628m_rule @ Tree )
          = ( abstra4962253831105676628m_rule @ Tree2 ) ) )
     => ( Tree = Tree2 ) ) ).

% tree.expand
thf(fact_713_tfinite,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ! [T5: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T2 ) )
         => ( abstra3507476289341504315m_rule @ T5 ) )
     => ( abstra3507476289341504315m_rule @ T2 ) ) ).

% tfinite
thf(fact_714_tfinite_Osimps,axiom,
    ( abstra3507476289341504315m_rule
    = ( ^ [A5: abstra4572470701423150929m_rule] :
        ? [T: abstra4572470701423150929m_rule] :
          ( ( A5 = T )
          & ! [X5: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X5 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra3507476289341504315m_rule @ X5 ) ) ) ) ) ).

% tfinite.simps
thf(fact_715_tfinite_Ocases,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra3507476289341504315m_rule @ A )
     => ! [T6: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ A ) )
         => ( abstra3507476289341504315m_rule @ T6 ) ) ) ).

% tfinite.cases
thf(fact_716_ipath_Ocoinduct,axiom,
    ! [X7: abstract_tree_rule > stream_rule > $o,X12: abstract_tree_rule,X2: stream_rule] :
      ( ( X7 @ X12 @ X2 )
     => ( ! [X13: abstract_tree_rule,X23: stream_rule] :
            ( ( X7 @ X13 @ X23 )
           => ? [T7: abstract_tree_rule,Steps3: stream_rule,T6: abstract_tree_rule] :
                ( ( X13 = T7 )
                & ( X23 = Steps3 )
                & ( ( abstract_root_rule @ T7 )
                  = ( shd_rule @ Steps3 ) )
                & ( fmembe7964111328986570234e_rule @ T6 @ ( abstract_cont_rule @ T7 ) )
                & ( ( X7 @ T6 @ ( stl_rule @ Steps3 ) )
                  | ( abstract_ipath_rule @ T6 @ ( stl_rule @ Steps3 ) ) ) ) )
       => ( abstract_ipath_rule @ X12 @ X2 ) ) ) ).

% ipath.coinduct
thf(fact_717_ipath_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > stream727092118206550309m_rule > $o,X12: abstra4572470701423150929m_rule,X2: stream727092118206550309m_rule] :
      ( ( X7 @ X12 @ X2 )
     => ( ! [X13: abstra4572470701423150929m_rule,X23: stream727092118206550309m_rule] :
            ( ( X7 @ X13 @ X23 )
           => ? [T7: abstra4572470701423150929m_rule,Steps3: stream727092118206550309m_rule,T6: abstra4572470701423150929m_rule] :
                ( ( X13 = T7 )
                & ( X23 = Steps3 )
                & ( ( abstra3503897793822247108m_rule @ T7 )
                  = ( shd_Pr7235097944458474089m_rule @ Steps3 ) )
                & ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ T7 ) )
                & ( ( X7 @ T6 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra2285467793620430502m_rule @ T6 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra2285467793620430502m_rule @ X12 @ X2 ) ) ) ).

% ipath.coinduct
thf(fact_718_ipath_Ointros,axiom,
    ! [T2: abstract_tree_rule,Steps: stream_rule,T3: abstract_tree_rule] :
      ( ( ( abstract_root_rule @ T2 )
        = ( shd_rule @ Steps ) )
     => ( ( fmembe7964111328986570234e_rule @ T3 @ ( abstract_cont_rule @ T2 ) )
       => ( ( abstract_ipath_rule @ T3 @ ( stl_rule @ Steps ) )
         => ( abstract_ipath_rule @ T2 @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_719_ipath_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule,T3: abstra4572470701423150929m_rule] :
      ( ( ( abstra3503897793822247108m_rule @ T2 )
        = ( shd_Pr7235097944458474089m_rule @ Steps ) )
     => ( ( fmembe1635983029874454474m_rule @ T3 @ ( abstra4962253831105676628m_rule @ T2 ) )
       => ( ( abstra2285467793620430502m_rule @ T3 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
         => ( abstra2285467793620430502m_rule @ T2 @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_720_ipath_Osimps,axiom,
    ( abstract_ipath_rule
    = ( ^ [A1: abstract_tree_rule,A22: stream_rule] :
        ? [T: abstract_tree_rule,Steps2: stream_rule,T8: abstract_tree_rule] :
          ( ( A1 = T )
          & ( A22 = Steps2 )
          & ( ( abstract_root_rule @ T )
            = ( shd_rule @ Steps2 ) )
          & ( fmembe7964111328986570234e_rule @ T8 @ ( abstract_cont_rule @ T ) )
          & ( abstract_ipath_rule @ T8 @ ( stl_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_721_ipath_Osimps,axiom,
    ( abstra2285467793620430502m_rule
    = ( ^ [A1: abstra4572470701423150929m_rule,A22: stream727092118206550309m_rule] :
        ? [T: abstra4572470701423150929m_rule,Steps2: stream727092118206550309m_rule,T8: abstra4572470701423150929m_rule] :
          ( ( A1 = T )
          & ( A22 = Steps2 )
          & ( ( abstra3503897793822247108m_rule @ T )
            = ( shd_Pr7235097944458474089m_rule @ Steps2 ) )
          & ( fmembe1635983029874454474m_rule @ T8 @ ( abstra4962253831105676628m_rule @ T ) )
          & ( abstra2285467793620430502m_rule @ T8 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_722_ipath_Ocases,axiom,
    ! [A12: abstract_tree_rule,A23: stream_rule] :
      ( ( abstract_ipath_rule @ A12 @ A23 )
     => ~ ( ( ( abstract_root_rule @ A12 )
            = ( shd_rule @ A23 ) )
         => ! [T5: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T5 @ ( abstract_cont_rule @ A12 ) )
             => ~ ( abstract_ipath_rule @ T5 @ ( stl_rule @ A23 ) ) ) ) ) ).

% ipath.cases
thf(fact_723_ipath_Ocases,axiom,
    ! [A12: abstra4572470701423150929m_rule,A23: stream727092118206550309m_rule] :
      ( ( abstra2285467793620430502m_rule @ A12 @ A23 )
     => ~ ( ( ( abstra3503897793822247108m_rule @ A12 )
            = ( shd_Pr7235097944458474089m_rule @ A23 ) )
         => ! [T5: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ A12 ) )
             => ~ ( abstra2285467793620430502m_rule @ T5 @ ( stl_Pr950425576149878629m_rule @ A23 ) ) ) ) ) ).

% ipath.cases
thf(fact_724_ftree__no__ipath,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra3507476289341504315m_rule @ T2 )
     => ~ ( abstra2285467793620430502m_rule @ T2 @ Steps ) ) ).

% ftree_no_ipath
thf(fact_725_sdrop__while_Osimps,axiom,
    ( sdrop_5541336384294084785m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o,S2: stream727092118206550309m_rule] : ( if_str8948254419368749791m_rule @ ( P3 @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) @ ( sdrop_5541336384294084785m_rule @ P3 @ ( stl_Pr950425576149878629m_rule @ S2 ) ) @ S2 ) ) ) ).

% sdrop_while.simps
thf(fact_726_sdrop__while_Osimps,axiom,
    ( sdrop_while_rule
    = ( ^ [P3: rule > $o,S2: stream_rule] : ( if_stream_rule @ ( P3 @ ( shd_rule @ S2 ) ) @ ( sdrop_while_rule @ P3 @ ( stl_rule @ S2 ) ) @ S2 ) ) ) ).

% sdrop_while.simps
thf(fact_727_PersistentRuleSystem_Oaxioms_I2_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( abstra2778581409224112183ist_fm @ Eff2 @ Rules2 @ S6 ) ) ).

% PersistentRuleSystem.axioms(2)
thf(fact_728_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,S6: set_list_fm,S: list_fm] :
      ( ( abstra1616580364776158018ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member_list_fm @ S @ S6 )
       => ( ? [T4: abstra5042247676209264507ist_fm] :
              ( ( ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T4 ) )
                = S )
              & ( abstra8964572462466433336ist_fm @ Eff2 @ Rules2 @ T4 )
              & ( abstra5066959810000083067ist_fm @ T4 ) )
          | ? [Steps4: stream8299795917829157543ist_fm] :
              ( ( ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps4 ) )
                = S )
              & ( abstra6579460235205289861ist_fm @ Eff2 @ Rules2 @ Steps4 )
              & ( abstra8946172763262761776ist_fm @ Eff2 @ Rules2 @ Steps4 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_729_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm,S: produc1996495991257130529ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules2 @ S6 )
     => ( ( member8102475879199740618ist_fm @ S @ S6 )
       => ( ? [T4: abstra4572470701423150929m_rule] :
              ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T4 ) )
                = S )
              & ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ T4 )
              & ( abstra3507476289341504315m_rule @ T4 ) )
          | ? [Steps4: stream727092118206550309m_rule] :
              ( ( ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps4 ) )
                = S )
              & ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ Steps4 )
              & ( abstra8647779336736900908ist_fm @ Eff2 @ Rules2 @ Steps4 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_730_RuleSystem__Defs_Otrim__def,axiom,
    ( abstra2363873759530810705ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S2: produc1996495991257130529ist_fm] :
          ( sdrop_while_rule
          @ ^ [R2: rule] :
              ~ ( abstra3055540779341473792ist_fm @ Eff @ R2 @ S2 )
          @ Rs2 ) ) ) ).

% RuleSystem_Defs.trim_def
thf(fact_731_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rs: stream_list_fm,S: list_fm] :
      ( ( abstra778364809706815794ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ Rs @ S ) )
      = ( produc381145313068854617ist_fm @ S @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff2 @ Rs @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_732_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: nat > nat > fset_nat > $o,Rs: stream_nat,S: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ Eff2 @ Rs @ S ) )
      = ( product_Pair_nat_nat @ S @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff2 @ Rs @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_733_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs: stream6724221391990029191at_nat,S: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ Eff2 @ Rs @ S ) )
      = ( produc6161850002892822231at_nat @ S @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff2 @ Rs @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_734_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs: stream8299795917829157543ist_fm,S: rule] :
      ( ( abstra7970568315424063040ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ Rs @ S ) )
      = ( produc7063755442767259331ist_fm @ S @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_735_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S ) )
      = ( produc491467635432902671m_rule @ S @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_736_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S: list_fm] :
      ( ( abstra778364809706815794ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S ) )
      = ( produc381145313068854617ist_fm @ S @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_737_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: nat > nat > option_fset_nat,Rs: stream_nat,S: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S ) )
      = ( product_Pair_nat_nat @ S @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_738_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S ) )
      = ( produc6161850002892822231at_nat @ S @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_739_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S: rule] :
      ( ( abstra7970568315424063040ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S ) )
      = ( produc7063755442767259331ist_fm @ S @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_740_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S ) )
      = ( produc491467635432902671m_rule @ S @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_741_PersistentRuleSystem__axioms__def,axiom,
    ( abstra2778581409224112183ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S7: set_Pr7058068377845519745ist_fm] :
        ! [R2: rule] :
          ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( abstra4153889242452539894ist_fm @ Eff @ Rules @ S7 @ R2 ) ) ) ) ).

% PersistentRuleSystem_axioms_def
thf(fact_742_PersistentRuleSystem__axioms_Ointro,axiom,
    ! [Rules2: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,S6: set_Pr7058068377845519745ist_fm] :
      ( ! [R5: rule] :
          ( ( member_rule @ R5 @ ( sset_rule @ Rules2 ) )
         => ( abstra4153889242452539894ist_fm @ Eff2 @ Rules2 @ S6 @ R5 ) )
     => ( abstra2778581409224112183ist_fm @ Eff2 @ Rules2 @ S6 ) ) ).

% PersistentRuleSystem_axioms.intro
thf(fact_743_i_Otrim__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S )
      = ( sdrop_while_rule
        @ ^ [R2: rule] :
            ~ ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S )
        @ Rs ) ) ).

% i.trim_def
thf(fact_744_mkTree_Osimps_I2_J,axiom,
    ! [Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S ) )
      = ( fimage748043113761278389m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( stl_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S ) ) )
        @ ( abstra458490714785879965ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S ) )
          @ S ) ) ) ).

% mkTree.simps(2)
thf(fact_745_i_OmkTree_Osimps_I2_J,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S ) )
      = ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S ) ) ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S ) ) @ S ) ) ) ).

% i.mkTree.simps(2)
thf(fact_746_RuleSystem__Defs_OmkTree_Osimps_I2_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S ) )
      = ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S ) ) ) @ ( abstra458490714785879965ist_fm @ Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S ) ) @ S ) ) ) ).

% RuleSystem_Defs.mkTree.simps(2)
thf(fact_747_Konig,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ~ ( abstra3507476289341504315m_rule @ T2 )
     => ( abstra2285467793620430502m_rule @ T2 @ ( abstra3079604473823497664m_rule @ T2 ) ) ) ).

% Konig
thf(fact_748_wf_Ocases,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) )
              = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) ) )
           => ~ ! [T6: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm
                    @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S2 )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ T6 ) ) ) ) ) ).

% wf.cases
thf(fact_749_wf_Osimps,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
      = ( ? [T: abstra4572470701423150929m_rule] :
            ( ( A = T )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ rules ) )
            & ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) )
              = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) ) )
            & ! [X5: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X5 @ ( abstra4962253831105676628m_rule @ T ) )
               => ( abstra4476735159789068068ist_fm
                  @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R2 @ S2 )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ rules
                  @ X5 ) ) ) ) ) ).

% wf.simps
thf(fact_750_wf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ rules ) )
     => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) )
          = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) ) )
       => ( ! [T5: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm
                @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ S2 )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ rules
                @ T5 ) )
         => ( abstra4476735159789068068ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ T2 ) ) ) ) ).

% wf.intros
thf(fact_751_fset_Omap__comp,axiom,
    ! [G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc164195504107695125m_rule,V: fset_A5396818216090053761m_rule] :
      ( ( fimage6650965236902554765ist_fm @ G @ ( fimage6814378747598375653m_rule @ F2 @ V ) )
      = ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ G @ F2 ) @ V ) ) ).

% fset.map_comp
thf(fact_752_fset_Omap__comp,axiom,
    ! [G: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,V: fset_P661503646757059847ist_fm] :
      ( ( fimage3349360726776475169m_rule @ G @ ( fimage748043113761278389m_rule @ F2 @ V ) )
      = ( fimage748043113761278389m_rule @ ( comp_A7346078547394085594ist_fm @ G @ F2 ) @ V ) ) ).

% fset.map_comp
thf(fact_753_fset_Omap__comp,axiom,
    ! [G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,V: fset_A5396818216090053761m_rule] :
      ( ( fimage8674584286907012797ist_fm @ G @ ( fimage9164954772359936849ist_fm @ F2 @ V ) )
      = ( fimage9164954772359936849ist_fm @ ( comp_P2882693189979712244m_rule @ G @ F2 ) @ V ) ) ).

% fset.map_comp
thf(fact_754_fset_Omap__comp,axiom,
    ! [G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,V: fset_P661503646757059847ist_fm] :
      ( ( fimage748043113761278389m_rule @ G @ ( fimage8674584286907012797ist_fm @ F2 @ V ) )
      = ( fimage748043113761278389m_rule @ ( comp_P2076232811723594896ist_fm @ G @ F2 ) @ V ) ) ).

% fset.map_comp
thf(fact_755_fset_Omap__comp,axiom,
    ! [G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,V: fset_A5396818216090053761m_rule] :
      ( ( fimage748043113761278389m_rule @ G @ ( fimage9164954772359936849ist_fm @ F2 @ V ) )
      = ( fimage3349360726776475169m_rule @ ( comp_P4439951101548160930m_rule @ G @ F2 ) @ V ) ) ).

% fset.map_comp
thf(fact_756_fset_Omap__comp,axiom,
    ! [G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,V: fset_A5396818216090053761m_rule] :
      ( ( fimage9164954772359936849ist_fm @ G @ ( fimage3349360726776475169m_rule @ F2 @ V ) )
      = ( fimage9164954772359936849ist_fm @ ( comp_A8152538925650202942m_rule @ G @ F2 ) @ V ) ) ).

% fset.map_comp
thf(fact_757_fset_Omap__comp,axiom,
    ! [G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,V: fset_P661503646757059847ist_fm] :
      ( ( fimage9164954772359936849ist_fm @ G @ ( fimage748043113761278389m_rule @ F2 @ V ) )
      = ( fimage8674584286907012797ist_fm @ ( comp_A2205678127158149108ist_fm @ G @ F2 ) @ V ) ) ).

% fset.map_comp
thf(fact_758_fimage__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X6: produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm] :
      ( ( B
        = ( F2 @ X6 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X6 @ A4 )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F2 @ A4 ) ) ) ) ).

% fimage_eqI
thf(fact_759_fimage__eqI,axiom,
    ! [B: abstra4572470701423150929m_rule,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,X6: produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm] :
      ( ( B
        = ( F2 @ X6 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X6 @ A4 )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F2 @ A4 ) ) ) ) ).

% fimage_eqI
thf(fact_760_fimage__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,X6: abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule] :
      ( ( B
        = ( F2 @ X6 ) )
     => ( ( fmembe1635983029874454474m_rule @ X6 @ A4 )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F2 @ A4 ) ) ) ) ).

% fimage_eqI
thf(fact_761_fimage__eqI,axiom,
    ! [B: abstra4572470701423150929m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,X6: abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule] :
      ( ( B
        = ( F2 @ X6 ) )
     => ( ( fmembe1635983029874454474m_rule @ X6 @ A4 )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F2 @ A4 ) ) ) ) ).

% fimage_eqI
thf(fact_762_K__record__comp,axiom,
    ! [C2: produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P4378417847848123130m_rule
        @ ^ [X5: produc164195504107695125m_rule] : C2
        @ F2 )
      = ( ^ [X5: abstra4572470701423150929m_rule] : C2 ) ) ).

% K_record_comp
thf(fact_763_fimage__fimage,axiom,
    ! [F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A4: fset_P661503646757059847ist_fm] :
      ( ( fimage3349360726776475169m_rule @ F2 @ ( fimage748043113761278389m_rule @ G @ A4 ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X5: produc1996495991257130529ist_fm] : ( F2 @ ( G @ X5 ) )
        @ A4 ) ) ).

% fimage_fimage
thf(fact_764_fimage__fimage,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A4: fset_A5396818216090053761m_rule] :
      ( ( fimage8674584286907012797ist_fm @ F2 @ ( fimage9164954772359936849ist_fm @ G @ A4 ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X5: abstra4572470701423150929m_rule] : ( F2 @ ( G @ X5 ) )
        @ A4 ) ) ).

% fimage_fimage
thf(fact_765_fimage__fimage,axiom,
    ! [F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm] :
      ( ( fimage748043113761278389m_rule @ F2 @ ( fimage8674584286907012797ist_fm @ G @ A4 ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X5: produc1996495991257130529ist_fm] : ( F2 @ ( G @ X5 ) )
        @ A4 ) ) ).

% fimage_fimage
thf(fact_766_fimage__fimage,axiom,
    ! [F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A4: fset_A5396818216090053761m_rule] :
      ( ( fimage748043113761278389m_rule @ F2 @ ( fimage9164954772359936849ist_fm @ G @ A4 ) )
      = ( fimage3349360726776475169m_rule
        @ ^ [X5: abstra4572470701423150929m_rule] : ( F2 @ ( G @ X5 ) )
        @ A4 ) ) ).

% fimage_fimage
thf(fact_767_fimage__fimage,axiom,
    ! [F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule] :
      ( ( fimage9164954772359936849ist_fm @ F2 @ ( fimage3349360726776475169m_rule @ G @ A4 ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X5: abstra4572470701423150929m_rule] : ( F2 @ ( G @ X5 ) )
        @ A4 ) ) ).

% fimage_fimage
thf(fact_768_fimage__fimage,axiom,
    ! [F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A4: fset_P661503646757059847ist_fm] :
      ( ( fimage9164954772359936849ist_fm @ F2 @ ( fimage748043113761278389m_rule @ G @ A4 ) )
      = ( fimage8674584286907012797ist_fm
        @ ^ [X5: produc1996495991257130529ist_fm] : ( F2 @ ( G @ X5 ) )
        @ A4 ) ) ).

% fimage_fimage
thf(fact_769_comp__apply__eq,axiom,
    ! [F2: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc164195504107695125m_rule,X6: abstra4572470701423150929m_rule,H: produc164195504107695125m_rule > produc1996495991257130529ist_fm,K2: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( ( F2 @ ( G @ X6 ) )
        = ( H @ ( K2 @ X6 ) ) )
     => ( ( comp_P4378417847848123130m_rule @ F2 @ G @ X6 )
        = ( comp_P4378417847848123130m_rule @ H @ K2 @ X6 ) ) ) ).

% comp_apply_eq
thf(fact_770_rev__fimage__eqI,axiom,
    ! [X6: produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm,B: produc1996495991257130529ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X6 @ A4 )
     => ( ( B
          = ( F2 @ X6 ) )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F2 @ A4 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_771_rev__fimage__eqI,axiom,
    ! [X6: produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm,B: abstra4572470701423150929m_rule,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X6 @ A4 )
     => ( ( B
          = ( F2 @ X6 ) )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F2 @ A4 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_772_rev__fimage__eqI,axiom,
    ! [X6: abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule,B: produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X6 @ A4 )
     => ( ( B
          = ( F2 @ X6 ) )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F2 @ A4 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_773_rev__fimage__eqI,axiom,
    ! [X6: abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule,B: abstra4572470701423150929m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X6 @ A4 )
     => ( ( B
          = ( F2 @ X6 ) )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F2 @ A4 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_774_fimage__cong,axiom,
    ! [M2: fset_P661503646757059847ist_fm,N4: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( M2 = N4 )
     => ( ! [X3: produc1996495991257130529ist_fm] :
            ( ( fmembe3381613331217039976ist_fm @ X3 @ N4 )
           => ( ( F2 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( fimage748043113761278389m_rule @ F2 @ M2 )
          = ( fimage748043113761278389m_rule @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_775_fimage__cong,axiom,
    ! [M2: fset_A5396818216090053761m_rule,N4: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( M2 = N4 )
     => ( ! [X3: abstra4572470701423150929m_rule] :
            ( ( fmembe1635983029874454474m_rule @ X3 @ N4 )
           => ( ( F2 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( fimage9164954772359936849ist_fm @ F2 @ M2 )
          = ( fimage9164954772359936849ist_fm @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_776_fimageI,axiom,
    ! [X6: produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X6 @ A4 )
     => ( fmembe3381613331217039976ist_fm @ ( F2 @ X6 ) @ ( fimage8674584286907012797ist_fm @ F2 @ A4 ) ) ) ).

% fimageI
thf(fact_777_fimageI,axiom,
    ! [X6: produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X6 @ A4 )
     => ( fmembe1635983029874454474m_rule @ ( F2 @ X6 ) @ ( fimage748043113761278389m_rule @ F2 @ A4 ) ) ) ).

% fimageI
thf(fact_778_fimageI,axiom,
    ! [X6: abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X6 @ A4 )
     => ( fmembe3381613331217039976ist_fm @ ( F2 @ X6 ) @ ( fimage9164954772359936849ist_fm @ F2 @ A4 ) ) ) ).

% fimageI
thf(fact_779_fimageI,axiom,
    ! [X6: abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X6 @ A4 )
     => ( fmembe1635983029874454474m_rule @ ( F2 @ X6 ) @ ( fimage3349360726776475169m_rule @ F2 @ A4 ) ) ) ).

% fimageI
thf(fact_780_fimageE,axiom,
    ! [B: produc1996495991257130529ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A4: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F2 @ A4 ) )
     => ~ ! [X3: produc1996495991257130529ist_fm] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X3 @ A4 ) ) ) ).

% fimageE
thf(fact_781_fimageE,axiom,
    ! [B: produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A4: fset_A5396818216090053761m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F2 @ A4 ) )
     => ~ ! [X3: abstra4572470701423150929m_rule] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe1635983029874454474m_rule @ X3 @ A4 ) ) ) ).

% fimageE
thf(fact_782_fimageE,axiom,
    ! [B: abstra4572470701423150929m_rule,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A4: fset_P661503646757059847ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F2 @ A4 ) )
     => ~ ! [X3: produc1996495991257130529ist_fm] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X3 @ A4 ) ) ) ).

% fimageE
thf(fact_783_fimageE,axiom,
    ! [B: abstra4572470701423150929m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A4: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F2 @ A4 ) )
     => ~ ! [X3: abstra4572470701423150929m_rule] :
            ( ( B
              = ( F2 @ X3 ) )
           => ~ ( fmembe1635983029874454474m_rule @ X3 @ A4 ) ) ) ).

% fimageE
thf(fact_784_subset__fimage__iff,axiom,
    ! [B4: fset_A5396818216090053761m_rule,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A4: fset_P661503646757059847ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ B4 @ ( fimage748043113761278389m_rule @ F2 @ A4 ) )
      = ( ? [AA: fset_P661503646757059847ist_fm] :
            ( ( ord_le2064643713053750439ist_fm @ AA @ A4 )
            & ( B4
              = ( fimage748043113761278389m_rule @ F2 @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_785_subset__fimage__iff,axiom,
    ! [B4: fset_P661503646757059847ist_fm,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A4: fset_A5396818216090053761m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ B4 @ ( fimage9164954772359936849ist_fm @ F2 @ A4 ) )
      = ( ? [AA: fset_A5396818216090053761m_rule] :
            ( ( ord_le6951374404268211425m_rule @ AA @ A4 )
            & ( B4
              = ( fimage9164954772359936849ist_fm @ F2 @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_786_fimage__mono,axiom,
    ! [A4: fset_P661503646757059847ist_fm,B4: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ A4 @ B4 )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F2 @ A4 ) @ ( fimage748043113761278389m_rule @ F2 @ B4 ) ) ) ).

% fimage_mono
thf(fact_787_fimage__mono,axiom,
    ! [A4: fset_A5396818216090053761m_rule,B4: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ A4 @ B4 )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F2 @ A4 ) @ ( fimage9164954772359936849ist_fm @ F2 @ B4 ) ) ) ).

% fimage_mono
thf(fact_788_fimage__fsubsetI,axiom,
    ! [A4: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,B4: fset_P661503646757059847ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X3 @ A4 )
         => ( fmembe3381613331217039976ist_fm @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage8674584286907012797ist_fm @ F2 @ A4 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_789_fimage__fsubsetI,axiom,
    ! [A4: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,B4: fset_A5396818216090053761m_rule] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X3 @ A4 )
         => ( fmembe1635983029874454474m_rule @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F2 @ A4 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_790_fimage__fsubsetI,axiom,
    ! [A4: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,B4: fset_P661503646757059847ist_fm] :
      ( ! [X3: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X3 @ A4 )
         => ( fmembe3381613331217039976ist_fm @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F2 @ A4 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_791_fimage__fsubsetI,axiom,
    ! [A4: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,B4: fset_A5396818216090053761m_rule] :
      ( ! [X3: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X3 @ A4 )
         => ( fmembe1635983029874454474m_rule @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage3349360726776475169m_rule @ F2 @ A4 ) @ B4 ) ) ).

% fimage_fsubsetI
thf(fact_792_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ Eff2 @ Rules2 @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( sset_list_fm @ Rules2 ) )
         => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ A ) ) )
           => ~ ! [T6: abstra5042247676209264507ist_fm] :
                  ( ( fmembe913231786053615938ist_fm @ T6 @ ( abstra1905332785217431714ist_fm @ A ) )
                 => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules2 @ T6 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_793_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) )
           => ~ ! [T6: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ T6 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_794_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra8964572462466433336ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,A5: abstra5042247676209264507ist_fm] :
        ? [T: abstra5042247676209264507ist_fm] :
          ( ( A5 = T )
          & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( sset_list_fm @ Rules ) )
          & ( Eff @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T ) ) )
          & ! [X5: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ X5 @ ( abstra1905332785217431714ist_fm @ T ) )
             => ( abstra8964572462466433336ist_fm @ Eff @ Rules @ X5 ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_795_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra4476735159789068068ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A5: abstra4572470701423150929m_rule] :
        ? [T: abstra4572470701423150929m_rule] :
          ( ( A5 = T )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ Rules ) )
          & ( Eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) )
          & ! [X5: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X5 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ X5 ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_796_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T2: abstra5042247676209264507ist_fm,Rules2: stream_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( sset_list_fm @ Rules2 ) )
     => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T2 ) ) )
       => ( ! [T5: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ T5 @ ( abstra1905332785217431714ist_fm @ T2 ) )
             => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules2 @ T5 ) )
         => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules2 @ T2 ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_797_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Rules2: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) )
       => ( ! [T5: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ T5 ) )
         => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ T2 ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_798_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X7: abstra5042247676209264507ist_fm > $o,X6: abstra5042247676209264507ist_fm,Rules2: stream_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: abstra5042247676209264507ist_fm] :
            ( ( X7 @ X3 )
           => ? [T7: abstra5042247676209264507ist_fm] :
                ( ( X3 = T7 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T7 ) ) @ ( sset_list_fm @ Rules2 ) )
                & ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T7 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T7 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T7 ) ) )
                & ! [Xa: abstra5042247676209264507ist_fm] :
                    ( ( fmembe913231786053615938ist_fm @ Xa @ ( abstra1905332785217431714ist_fm @ T7 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra8964572462466433336ist_fm @ Eff2 @ Rules2 @ Xa ) ) ) ) )
       => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_799_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X6: abstra4572470701423150929m_rule,Rules2: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X6 )
     => ( ! [X3: abstra4572470701423150929m_rule] :
            ( ( X7 @ X3 )
           => ? [T7: abstra4572470701423150929m_rule] :
                ( ( X3 = T7 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T7 ) ) @ ( sset_rule @ Rules2 ) )
                & ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T7 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T7 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T7 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T7 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ X6 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_800_i_Owf_Ocases,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules2: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( sset_list_fm @ Rules2 ) )
         => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ A ) ) )
           => ~ ! [T6: abstra5042247676209264507ist_fm] :
                  ( ( fmembe913231786053615938ist_fm @ T6 @ ( abstra1905332785217431714ist_fm @ A ) )
                 => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ T6 ) ) ) ) ) ).

% i.wf.cases
thf(fact_801_i_Owf_Ocases,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ Rules2 ) )
         => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) )
           => ~ ! [T6: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ T6 ) ) ) ) ) ).

% i.wf.cases
thf(fact_802_i_Owf_Osimps,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules2: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ A )
      = ( ? [T: abstra5042247676209264507ist_fm] :
            ( ( A = T )
            & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( sset_list_fm @ Rules2 ) )
            & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T ) ) )
            & ! [X5: abstra5042247676209264507ist_fm] :
                ( ( fmembe913231786053615938ist_fm @ X5 @ ( abstra1905332785217431714ist_fm @ T ) )
               => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ X5 ) ) ) ) ) ).

% i.wf.simps
thf(fact_803_i_Owf_Osimps,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules2: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ A )
      = ( ? [T: abstra4572470701423150929m_rule] :
            ( ( A = T )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ Rules2 ) )
            & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) )
            & ! [X5: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X5 @ ( abstra4962253831105676628m_rule @ T ) )
               => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ X5 ) ) ) ) ) ).

% i.wf.simps
thf(fact_804_i_Owf_Ointros,axiom,
    ! [T2: abstra5042247676209264507ist_fm,Rules2: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( sset_list_fm @ Rules2 ) )
     => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T2 ) ) )
       => ( ! [T5: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ T5 @ ( abstra1905332785217431714ist_fm @ T2 ) )
             => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ T5 ) )
         => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ T2 ) ) ) ) ).

% i.wf.intros
thf(fact_805_i_Owf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Rules2: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ Rules2 ) )
     => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) )
       => ( ! [T5: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ T5 ) )
         => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ T2 ) ) ) ) ).

% i.wf.intros
thf(fact_806_i_Owf_Ocoinduct,axiom,
    ! [X7: abstra5042247676209264507ist_fm > $o,X6: abstra5042247676209264507ist_fm,Rules2: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( X7 @ X6 )
     => ( ! [X3: abstra5042247676209264507ist_fm] :
            ( ( X7 @ X3 )
           => ? [T7: abstra5042247676209264507ist_fm] :
                ( ( X3 = T7 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T7 ) ) @ ( sset_list_fm @ Rules2 ) )
                & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T7 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T7 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T7 ) ) )
                & ! [Xa: abstra5042247676209264507ist_fm] :
                    ( ( fmembe913231786053615938ist_fm @ Xa @ ( abstra1905332785217431714ist_fm @ T7 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ Xa ) ) ) ) )
       => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.wf.coinduct
thf(fact_807_i_Owf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X6: abstra4572470701423150929m_rule,Rules2: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X6 )
     => ( ! [X3: abstra4572470701423150929m_rule] :
            ( ( X7 @ X3 )
           => ? [T7: abstra4572470701423150929m_rule] :
                ( ( X3 = T7 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T7 ) ) @ ( sset_rule @ Rules2 ) )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T7 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T7 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T7 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T7 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules2 @ X6 ) ) ) ).

% i.wf.coinduct
thf(fact_808_konig_Osimps_I1_J,axiom,
    ! [T2: abstract_tree_rule] :
      ( ( shd_rule @ ( abstract_konig_rule @ T2 ) )
      = ( abstract_root_rule @ T2 ) ) ).

% konig.simps(1)
thf(fact_809_konig_Osimps_I1_J,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( abstra3079604473823497664m_rule @ T2 ) )
      = ( abstra3503897793822247108m_rule @ T2 ) ) ).

% konig.simps(1)
thf(fact_810_wf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X6: abstra4572470701423150929m_rule] :
      ( ( X7 @ X6 )
     => ( ! [X3: abstra4572470701423150929m_rule] :
            ( ( X7 @ X3 )
           => ? [T7: abstra4572470701423150929m_rule] :
                ( ( X3 = T7 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T7 ) ) @ ( sset_rule @ rules ) )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T7 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T7 ) ) )
                  = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T7 ) ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T7 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra4476735159789068068ist_fm
                        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                            ( ( eff @ R2 @ S2 )
                            = ( some_f2162114139285431020ist_fm @ Ss ) )
                        @ rules
                        @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ X6 ) ) ) ).

% wf.coinduct
thf(fact_811_mkTree_Ocode,axiom,
    ! [Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S )
      = ( abstra7467780926338743076m_rule
        @ ( produc491467635432902671m_rule @ S
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S ) ) )
        @ ( fimage748043113761278389m_rule
          @ ( abstra178918799166095863ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( stl_rule
              @ ( abstra2363873759530810705ist_fm
                @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ S2 )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S ) ) )
          @ ( abstra458490714785879965ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule
              @ ( abstra2363873759530810705ist_fm
                @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ S2 )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S ) )
            @ S ) ) ) ) ).

% mkTree.code
thf(fact_812_mkTree__unfold,axiom,
    ! [Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S )
      = ( case_s2494923709252486212m_rule
        @ ^ [R2: rule,S8: stream_rule] :
            ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ R2 )
            @ ( fimage748043113761278389m_rule
              @ ( abstra178918799166095863ist_fm
                @ ^ [S2: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S2 @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ S8 )
              @ ( abstra458490714785879965ist_fm
                @ ^ [S2: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S2 @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R2
                @ S ) ) )
        @ ( abstra2363873759530810705ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S ) ) ) ).

% mkTree_unfold
thf(fact_813_i_OmkTree_Ocode,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S: list_fm] :
      ( ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S )
      = ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S ) ) ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ ( stl_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S ) ) ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S ) ) @ S ) ) ) ) ).

% i.mkTree.code
thf(fact_814_i_OmkTree_Ocode,axiom,
    ! [Eff3: nat > nat > option_fset_nat,Rs: stream_nat,S: nat] :
      ( ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S )
      = ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S ) ) ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ ( stl_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S ) ) ) @ ( abstra969719415724737503at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S ) ) @ S ) ) ) ) ).

% i.mkTree.code
thf(fact_815_i_OmkTree_Ocode,axiom,
    ! [Eff3: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S: product_prod_nat_nat] :
      ( ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S )
      = ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S ) ) ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ ( stl_Pr5027827701538482609at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S ) ) ) @ ( abstra673899893360562045at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S ) ) @ S ) ) ) ) ).

% i.mkTree.code
thf(fact_816_i_OmkTree_Ocode,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S: rule] :
      ( ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S )
      = ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S ) ) ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S ) ) ) @ ( abstra3109574944306299113m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S ) ) @ S ) ) ) ) ).

% i.mkTree.code
thf(fact_817_i_OmkTree_Ocode,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S )
      = ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S ) ) ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S ) ) ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S ) ) @ S ) ) ) ) ).

% i.mkTree.code
thf(fact_818_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra3846921640779263781ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rs2: stream_list_fm,S2: list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S2 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S2 ) ) ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ Eff @ ( stl_list_fm @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S2 ) ) ) @ ( abstra1097029126060538623ist_fm @ Eff @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S2 ) ) @ S2 ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_819_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra1192804020863382533at_nat
    = ( ^ [Eff: nat > nat > fset_nat > $o,Rs2: stream_nat,S2: nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S2 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff @ Rs2 @ S2 ) ) ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ Eff @ ( stl_nat @ ( abstra7573781713853518635at_nat @ Eff @ Rs2 @ S2 ) ) ) @ ( abstra969719415724737503at_nat @ Eff @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff @ Rs2 @ S2 ) ) @ S2 ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_820_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra8972947971889770147at_nat
    = ( ^ [Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs2: stream6724221391990029191at_nat,S2: product_prod_nat_nat] : ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S2 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff @ Rs2 @ S2 ) ) ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ Eff @ ( stl_Pr5027827701538482609at_nat @ ( abstra4493250544927931593at_nat @ Eff @ Rs2 @ S2 ) ) ) @ ( abstra673899893360562045at_nat @ Eff @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff @ Rs2 @ S2 ) ) @ S2 ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_821_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra2830003028686515011m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs2: stream8299795917829157543ist_fm,S2: rule] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S2 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S2 ) ) ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S2 ) ) ) @ ( abstra3109574944306299113m_rule @ Eff @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S2 ) ) @ S2 ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_822_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra178918799166095863ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S2: produc1996495991257130529ist_fm] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S2 ) ) ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S2 ) ) ) @ ( abstra458490714785879965ist_fm @ Eff @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S2 ) ) @ S2 ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_823_i_OmkTree__unfold,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S: list_fm] :
      ( ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S )
      = ( case_s5690291548269216621ist_fm
        @ ^ [R2: list_fm,S8: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S @ R2 ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ S8 ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ R2 @ S ) ) )
        @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S ) ) ) ).

% i.mkTree_unfold
thf(fact_824_i_OmkTree__unfold,axiom,
    ! [Eff3: nat > nat > option_fset_nat,Rs: stream_nat,S: nat] :
      ( ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S )
      = ( case_s2280698752336340829at_nat
        @ ^ [R2: nat,S8: stream_nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S @ R2 ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ S8 ) @ ( abstra969719415724737503at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ R2 @ S ) ) )
        @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S ) ) ) ).

% i.mkTree_unfold
thf(fact_825_i_OmkTree__unfold,axiom,
    ! [Eff3: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S: product_prod_nat_nat] :
      ( ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S )
      = ( case_s6993487711257930156at_nat
        @ ^ [R2: product_prod_nat_nat,S8: stream6724221391990029191at_nat] : ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S @ R2 ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ S8 ) @ ( abstra673899893360562045at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ R2 @ S ) ) )
        @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S ) ) ) ).

% i.mkTree_unfold
thf(fact_826_i_OmkTree__unfold,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S: rule] :
      ( ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S )
      = ( case_s482378023405642050ist_fm
        @ ^ [R2: produc1996495991257130529ist_fm,S8: stream8299795917829157543ist_fm] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S @ R2 ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ S8 ) @ ( abstra3109574944306299113m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ R2 @ S ) ) )
        @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S ) ) ) ).

% i.mkTree_unfold
thf(fact_827_i_OmkTree__unfold,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S )
      = ( case_s2494923709252486212m_rule
        @ ^ [R2: rule,S8: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ R2 ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ S8 ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S ) ) )
        @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S ) ) ) ).

% i.mkTree_unfold
thf(fact_828_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra3846921640779263781ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rs2: stream_list_fm,S2: list_fm] :
          ( case_s5690291548269216621ist_fm
          @ ^ [R2: list_fm,S8: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S2 @ R2 ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ Eff @ S8 ) @ ( abstra1097029126060538623ist_fm @ Eff @ R2 @ S2 ) ) )
          @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S2 ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_829_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra1192804020863382533at_nat
    = ( ^ [Eff: nat > nat > fset_nat > $o,Rs2: stream_nat,S2: nat] :
          ( case_s2280698752336340829at_nat
          @ ^ [R2: nat,S8: stream_nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S2 @ R2 ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ Eff @ S8 ) @ ( abstra969719415724737503at_nat @ Eff @ R2 @ S2 ) ) )
          @ ( abstra7573781713853518635at_nat @ Eff @ Rs2 @ S2 ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_830_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra8972947971889770147at_nat
    = ( ^ [Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs2: stream6724221391990029191at_nat,S2: product_prod_nat_nat] :
          ( case_s6993487711257930156at_nat
          @ ^ [R2: product_prod_nat_nat,S8: stream6724221391990029191at_nat] : ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S2 @ R2 ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ Eff @ S8 ) @ ( abstra673899893360562045at_nat @ Eff @ R2 @ S2 ) ) )
          @ ( abstra4493250544927931593at_nat @ Eff @ Rs2 @ S2 ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_831_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra2830003028686515011m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs2: stream8299795917829157543ist_fm,S2: rule] :
          ( case_s482378023405642050ist_fm
          @ ^ [R2: produc1996495991257130529ist_fm,S8: stream8299795917829157543ist_fm] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S2 @ R2 ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ S8 ) @ ( abstra3109574944306299113m_rule @ Eff @ R2 @ S2 ) ) )
          @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S2 ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_832_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra178918799166095863ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S2: produc1996495991257130529ist_fm] :
          ( case_s2494923709252486212m_rule
          @ ^ [R2: rule,S8: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S2 @ R2 ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff @ S8 ) @ ( abstra458490714785879965ist_fm @ Eff @ R2 @ S2 ) ) )
          @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S2 ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_833_tree_Oinject,axiom,
    ! [X12: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule,Y1: produc164195504107695125m_rule,Y2: fset_A5396818216090053761m_rule] :
      ( ( ( abstra7467780926338743076m_rule @ X12 @ X2 )
        = ( abstra7467780926338743076m_rule @ Y1 @ Y2 ) )
      = ( ( X12 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% tree.inject
thf(fact_834_tree_Ocollapse,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) )
      = Tree ) ).

% tree.collapse
thf(fact_835_tree_Oexhaust,axiom,
    ! [Y5: abstra4572470701423150929m_rule] :
      ~ ! [X13: produc164195504107695125m_rule,X23: fset_A5396818216090053761m_rule] :
          ( Y5
         != ( abstra7467780926338743076m_rule @ X13 @ X23 ) ) ).

% tree.exhaust
thf(fact_836_fst__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc5894184038225292483ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X5: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X5 @ X5 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% fst_diag_fst
thf(fact_837_fst__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X5: list_fm] : ( produc381145313068854617ist_fm @ X5 @ X5 )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% fst_diag_fst
thf(fact_838_snd__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc6725306161561501521e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X5: rule] : ( produc1769597910952761123e_rule @ X5 @ X5 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% snd_diag_snd
thf(fact_839_snd__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X5: list_fm] : ( produc381145313068854617ist_fm @ X5 @ X5 )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% snd_diag_snd
thf(fact_840_fst__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc7264528503596239631e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X5: rule] : ( produc1769597910952761123e_rule @ X5 @ X5 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% fst_diag_snd
thf(fact_841_fst__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X5: list_fm] : ( produc381145313068854617ist_fm @ X5 @ X5 )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% fst_diag_snd
thf(fact_842_snd__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc1639249265332130309ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X5: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X5 @ X5 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% snd_diag_fst
thf(fact_843_snd__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X5: list_fm] : ( produc381145313068854617ist_fm @ X5 @ X5 )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% snd_diag_fst
thf(fact_844_tree_Osel_I1_J,axiom,
    ! [X12: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra3503897793822247108m_rule @ ( abstra7467780926338743076m_rule @ X12 @ X2 ) )
      = X12 ) ).

% tree.sel(1)
thf(fact_845_tree_Osel_I2_J,axiom,
    ! [X12: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra4962253831105676628m_rule @ ( abstra7467780926338743076m_rule @ X12 @ X2 ) )
      = X2 ) ).

% tree.sel(2)
thf(fact_846_tree_Oexhaust__sel,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( Tree
      = ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) ) ) ).

% tree.exhaust_sel
thf(fact_847_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P1712579846118164081m_rule @ S @ N5 ) )
     => ( ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P ) @ S )
        = ( sdrop_7224736112439592940m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P1712579846118164081m_rule @ S @ N3 ) ) )
          @ S ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_848_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,S: stream8299795917829157543ist_fm] :
      ( ? [N5: nat] : ( P @ ( snth_P2624749845746815685ist_fm @ S @ N5 ) )
     => ( ( sdrop_7013889347380152453ist_fm @ ( comp_o6353595986216377982ist_fm @ (~) @ P ) @ S )
        = ( sdrop_4442373711808556042ist_fm
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P2624749845746815685ist_fm @ S @ N3 ) ) )
          @ S ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_849_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc780299800278082923e_rule > $o,S: stream3588992565182678257e_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P8487267234472522255e_rule @ S @ N5 ) )
     => ( ( sdrop_4395986648051443663e_rule @ ( comp_o7949119091651446216e_rule @ (~) @ P ) @ S )
        = ( sdrop_2199339672054592340e_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P8487267234472522255e_rule @ S @ N3 ) ) )
          @ S ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_850_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc1660549859252256699m_rule > $o,S: stream1878305877988316353m_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P7967347330282200671m_rule @ S @ N5 ) )
     => ( ( sdrop_6423752018218906655m_rule @ ( comp_o6304513743058443928m_rule @ (~) @ P ) @ S )
        = ( sdrop_1764812713384251940m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P7967347330282200671m_rule @ S @ N3 ) ) )
          @ S ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_851_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc6288491646048892603e_rule > $o,S: stream2312013117288958913e_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P3371917080224060767e_rule @ S @ N5 ) )
     => ( ( sdrop_1828321768160766751e_rule @ ( comp_o1709083493000304024e_rule @ (~) @ P ) @ S )
        = ( sdrop_6392754500180887844e_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P3371917080224060767e_rule @ S @ N3 ) ) )
          @ S ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_852_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc3793571666044074251m_rule > $o,S: stream6494289010434245521m_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P8950227884416379311m_rule @ S @ N5 ) )
     => ( ( sdrop_7771878303739699055m_rule @ ( comp_o8292784708554675816m_rule @ (~) @ P ) @ S )
        = ( sdrop_7373388980841566196m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P8950227884416379311m_rule @ S @ N3 ) ) )
          @ S ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_853_sdrop__while__sdrop__LEAST,axiom,
    ! [P: rule > $o,S: stream_rule] :
      ( ? [N5: nat] : ( P @ ( snth_rule @ S @ N5 ) )
     => ( ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P ) @ S )
        = ( sdrop_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_rule @ S @ N3 ) ) )
          @ S ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_854_konig_Osimps_I2_J,axiom,
    ! [T2: abstract_tree_rule] :
      ( ( stl_rule @ ( abstract_konig_rule @ T2 ) )
      = ( abstract_konig_rule
        @ ( fChoic8596484284512669613e_rule
          @ ^ [T8: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T8 @ ( abstract_cont_rule @ T2 ) )
              & ~ ( abstra6797871900832723819e_rule @ T8 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_855_konig_Osimps_I2_J,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( abstra3079604473823497664m_rule @ T2 ) )
      = ( abstra3079604473823497664m_rule
        @ ( fChoic7543385803022166781m_rule
          @ ^ [T8: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T8 @ ( abstra4962253831105676628m_rule @ T2 ) )
              & ~ ( abstra3507476289341504315m_rule @ T8 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_856_treeB__snth,axiom,
    ! [P2: fm,Steps: stream727092118206550309m_rule] :
      ( ( member_fm @ P2 @ ( treeB @ Steps ) )
     => ? [N2: nat] : ( member_fm @ P2 @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) ) ) ) ) ) ).

% treeB_snth
thf(fact_857_treeA__snth,axiom,
    ! [P2: fm,Steps: stream727092118206550309m_rule] :
      ( ( member_fm @ P2 @ ( treeA @ Steps ) )
     => ? [N2: nat] : ( member_fm @ P2 @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) ) ) ) ) ) ).

% treeA_snth
thf(fact_858_Fair__Stream_Ofair__def,axiom,
    ( fair_fair_rule
    = ( ^ [S2: stream_rule] :
        ! [X5: rule] :
          ( ( member_rule @ X5 @ ( sset_rule @ S2 ) )
         => ! [M3: nat] :
            ? [N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
              & ( ( snth_rule @ S2 @ N3 )
                = X5 ) ) ) ) ) ).

% Fair_Stream.fair_def
thf(fact_859_cont__def,axiom,
    ( abstra4962253831105676628m_rule
    = ( abstra1216198984563783442m_rule
      @ ^ [X1: produc164195504107695125m_rule,X22: fset_A5396818216090053761m_rule] : X22 ) ) ).

% cont_def
thf(fact_860_tree_Oroot__def,axiom,
    ( abstra3503897793822247108m_rule
    = ( abstra6581154509029876902m_rule
      @ ^ [X1: produc164195504107695125m_rule,X22: fset_A5396818216090053761m_rule] : X1 ) ) ).

% tree.root_def
thf(fact_861_some__sym__eq__trivial,axiom,
    ! [X6: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ( ^ [Y: fset_P661503646757059847ist_fm,Z: fset_P661503646757059847ist_fm] : ( Y = Z )
          @ X6 ) )
      = X6 ) ).

% some_sym_eq_trivial
thf(fact_862_some__eq__trivial,axiom,
    ! [X6: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ^ [Y6: fset_P661503646757059847ist_fm] : ( Y6 = X6 ) )
      = X6 ) ).

% some_eq_trivial
thf(fact_863_some__equality,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ( P @ A )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ( fChoic3892542562059901019ist_fm @ P )
          = A ) ) ) ).

% some_equality
thf(fact_864_epath__Pre__sdrop_I2_J,axiom,
    ! [Steps: stream727092118206550309m_rule,P: nat,Ts: list_tm,M: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) ) ) )
       => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ) ).

% epath_Pre_sdrop(2)
thf(fact_865_epath__Pre__sdrop_I1_J,axiom,
    ! [Steps: stream727092118206550309m_rule,P: nat,Ts: list_tm,M: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) ) ) )
       => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ) ).

% epath_Pre_sdrop(1)
thf(fact_866_eff__preserves__Pre_I2_J,axiom,
    ! [A4: list_fm,B4: list_fm,R: rule,Ss2: fset_P661503646757059847ist_fm,A6: list_fm,B5: list_fm,P: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A4 @ B4 ) @ R ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A4 @ B4 ) @ R ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B5 ) @ Ss2 )
       => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B4 ) )
         => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B5 ) ) ) ) ) ).

% eff_preserves_Pre(2)
thf(fact_867_fm_Oinject_I1_J,axiom,
    ! [X21: nat,X222: list_tm,Y21: nat,Y22: list_tm] :
      ( ( ( pre @ X21 @ X222 )
        = ( pre @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% fm.inject(1)
thf(fact_868_fm_Odistinct_I9_J,axiom,
    ! [X21: nat,X222: list_tm,X4: fm] :
      ( ( pre @ X21 @ X222 )
     != ( uni @ X4 ) ) ).

% fm.distinct(9)
thf(fact_869_someI2,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm,Q2: fset_P661503646757059847ist_fm > $o] :
      ( ( P @ A )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( fChoic3892542562059901019ist_fm @ P ) ) ) ) ).

% someI2
thf(fact_870_someI__ex,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_1: fset_P661503646757059847ist_fm] : ( P @ X_1 )
     => ( P @ ( fChoic3892542562059901019ist_fm @ P ) ) ) ).

% someI_ex
thf(fact_871_someI2__ex,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,Q2: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_1: fset_P661503646757059847ist_fm] : ( P @ X_1 )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( fChoic3892542562059901019ist_fm @ P ) ) ) ) ).

% someI2_ex
thf(fact_872_someI2__bex,axiom,
    ! [A4: set_fm,P: fm > $o,Q2: fm > $o] :
      ( ? [X8: fm] :
          ( ( member_fm @ X8 @ A4 )
          & ( P @ X8 ) )
     => ( ! [X3: fm] :
            ( ( ( member_fm @ X3 @ A4 )
              & ( P @ X3 ) )
           => ( Q2 @ X3 ) )
       => ( Q2
          @ ( fChoice_fm
            @ ^ [X5: fm] :
                ( ( member_fm @ X5 @ A4 )
                & ( P @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_873_someI2__bex,axiom,
    ! [A4: set_rule,P: rule > $o,Q2: rule > $o] :
      ( ? [X8: rule] :
          ( ( member_rule @ X8 @ A4 )
          & ( P @ X8 ) )
     => ( ! [X3: rule] :
            ( ( ( member_rule @ X3 @ A4 )
              & ( P @ X3 ) )
           => ( Q2 @ X3 ) )
       => ( Q2
          @ ( fChoice_rule
            @ ^ [X5: rule] :
                ( ( member_rule @ X5 @ A4 )
                & ( P @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_874_someI2__bex,axiom,
    ! [A4: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > $o] :
      ( ? [X8: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X8 @ A4 )
          & ( P @ X8 ) )
     => ( ! [X3: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ X3 @ A4 )
              & ( P @ X3 ) )
           => ( Q2 @ X3 ) )
       => ( Q2
          @ ( fChoic1980167584695909237ist_fm
            @ ^ [X5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X5 @ A4 )
                & ( P @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_875_someI2__bex,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o,Q2: produc859450856879609959at_nat > $o] :
      ( ? [X8: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X8 @ A4 )
          & ( P @ X8 ) )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( ( member8206827879206165904at_nat @ X3 @ A4 )
              & ( P @ X3 ) )
           => ( Q2 @ X3 ) )
       => ( Q2
          @ ( fChoic6673916560228149179at_nat
            @ ^ [X5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X5 @ A4 )
                & ( P @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_876_someI2__bex,axiom,
    ! [A4: set_fs7534060674568567527ist_fm,P: fset_P661503646757059847ist_fm > $o,Q2: fset_P661503646757059847ist_fm > $o] :
      ( ? [X8: fset_P661503646757059847ist_fm] :
          ( ( member3493611713653719344ist_fm @ X8 @ A4 )
          & ( P @ X8 ) )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( ( member3493611713653719344ist_fm @ X3 @ A4 )
              & ( P @ X3 ) )
           => ( Q2 @ X3 ) )
       => ( Q2
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X5: fset_P661503646757059847ist_fm] :
                ( ( member3493611713653719344ist_fm @ X5 @ A4 )
                & ( P @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_877_some__eq__ex,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o] :
      ( ( P @ ( fChoic3892542562059901019ist_fm @ P ) )
      = ( ? [X: fset_P661503646757059847ist_fm] : ( P @ X ) ) ) ).

% some_eq_ex
thf(fact_878_some1__equality,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ? [X8: fset_P661503646757059847ist_fm] :
          ( ( P @ X8 )
          & ! [Y3: fset_P661503646757059847ist_fm] :
              ( ( P @ Y3 )
             => ( Y3 = X8 ) ) )
     => ( ( P @ A )
       => ( ( fChoic3892542562059901019ist_fm @ P )
          = A ) ) ) ).

% some1_equality
thf(fact_879_eff__preserves__Pre_I1_J,axiom,
    ! [A4: list_fm,B4: list_fm,R: rule,Ss2: fset_P661503646757059847ist_fm,A6: list_fm,B5: list_fm,P: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A4 @ B4 ) @ R ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A4 @ B4 ) @ R ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B5 ) @ Ss2 )
       => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A4 ) )
         => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A6 ) ) ) ) ) ).

% eff_preserves_Pre(1)
thf(fact_880_konig_Ocode,axiom,
    ( abstra3079604473823497664m_rule
    = ( ^ [T: abstra4572470701423150929m_rule] :
          ( sCons_5731567480799343806m_rule @ ( abstra3503897793822247108m_rule @ T )
          @ ( abstra3079604473823497664m_rule
            @ ( fChoic7543385803022166781m_rule
              @ ^ [T8: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T8 @ ( abstra4962253831105676628m_rule @ T ) )
                  & ~ ( abstra3507476289341504315m_rule @ T8 ) ) ) ) ) ) ) ).

% konig.code
thf(fact_881_subset__code_I1_J,axiom,
    ! [Xs: list_rule,B4: set_rule] :
      ( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ B4 )
      = ( ! [X5: rule] :
            ( ( member_rule @ X5 @ ( set_rule2 @ Xs ) )
           => ( member_rule @ X5 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_882_subset__code_I1_J,axiom,
    ! [Xs: list_P5616295576739893671ist_fm,B4: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ ( set_Pr8767716839810916150ist_fm @ Xs ) @ B4 )
      = ( ! [X5: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X5 @ ( set_Pr8767716839810916150ist_fm @ Xs ) )
           => ( member8102475879199740618ist_fm @ X5 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_883_subset__code_I1_J,axiom,
    ! [Xs: list_P8469869581646625389at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs ) @ B4 )
      = ( ! [X5: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X5 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ X5 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_884_subset__code_I1_J,axiom,
    ! [Xs: list_fm,B4: set_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B4 )
      = ( ! [X5: fm] :
            ( ( member_fm @ X5 @ ( set_fm2 @ Xs ) )
           => ( member_fm @ X5 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_885_snd__sndOp,axiom,
    ! [P: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > rule > $o] :
      ( produc7165828336582415457m_rule
      = ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( bNF_sn6617447237779055481m_rule @ P @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_886_snd__sndOp,axiom,
    ! [P: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] :
      ( produc4588648349897876871ist_fm
      = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( bNF_sn5292590578203848342ist_fm @ P @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_887_fst__fstOp,axiom,
    ! [P: produc1996495991257130529ist_fm > rule > $o,Q2: rule > rule > $o] :
      ( produc6879501374131015971m_rule
      = ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( bNF_fs2143400664727635963e_rule @ P @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_888_fst__fstOp,axiom,
    ! [P: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] :
      ( produc1501393135466168645ist_fm
      = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( bNF_fs6361025107161210196ist_fm @ P @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_889_snd__comp__apsnd,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F2 ) )
      = ( comp_r641804198332660996m_rule @ F2 @ produc7165828336582415457m_rule ) ) ).

% snd_comp_apsnd
thf(fact_890_snd__comp__apsnd,axiom,
    ! [F2: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc9047611296996456864ist_fm @ F2 ) )
      = ( comp_l1868218281415653180ist_fm @ F2 @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_apsnd
thf(fact_891_fst__comp__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F2 ) )
      = ( comp_P8245093624798776376m_rule @ F2 @ produc6879501374131015971m_rule ) ) ).

% fst_comp_apfst
thf(fact_892_fst__comp__apfst,axiom,
    ! [F2: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc1130293176790411102ist_fm @ F2 ) )
      = ( comp_l1868218281415653180ist_fm @ F2 @ produc1501393135466168645ist_fm ) ) ).

% fst_comp_apfst
thf(fact_893_apfst__conv,axiom,
    ! [F2: list_fm > list_fm,X6: list_fm,Y5: list_fm] :
      ( ( produc1130293176790411102ist_fm @ F2 @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) )
      = ( produc381145313068854617ist_fm @ ( F2 @ X6 ) @ Y5 ) ) ).

% apfst_conv
thf(fact_894_apfst__conv,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X6: produc1996495991257130529ist_fm,Y5: rule] :
      ( ( produc3731403639160201281m_rule @ F2 @ ( produc491467635432902671m_rule @ X6 @ Y5 ) )
      = ( produc491467635432902671m_rule @ ( F2 @ X6 ) @ Y5 ) ) ).

% apfst_conv
thf(fact_895_apfst__conv,axiom,
    ! [F2: nat > nat,X6: nat,Y5: nat] :
      ( ( produc986720760941809198at_nat @ F2 @ ( product_Pair_nat_nat @ X6 @ Y5 ) )
      = ( product_Pair_nat_nat @ ( F2 @ X6 ) @ Y5 ) ) ).

% apfst_conv
thf(fact_896_apfst__conv,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,X6: product_prod_nat_nat,Y5: product_prod_nat_nat] :
      ( ( produc3057951030792887557at_nat @ F2 @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) )
      = ( produc6161850002892822231at_nat @ ( F2 @ X6 ) @ Y5 ) ) ).

% apfst_conv
thf(fact_897_apfst__conv,axiom,
    ! [F2: rule > rule,X6: rule,Y5: produc1996495991257130529ist_fm] :
      ( ( produc5534506254562563289ist_fm @ F2 @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) )
      = ( produc7063755442767259331ist_fm @ ( F2 @ X6 ) @ Y5 ) ) ).

% apfst_conv
thf(fact_898_apsnd__conv,axiom,
    ! [F2: list_fm > list_fm,X6: list_fm,Y5: list_fm] :
      ( ( produc9047611296996456864ist_fm @ F2 @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) )
      = ( produc381145313068854617ist_fm @ X6 @ ( F2 @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_899_apsnd__conv,axiom,
    ! [F2: rule > rule,X6: produc1996495991257130529ist_fm,Y5: rule] :
      ( ( produc3063135074001183255ist_fm @ F2 @ ( produc491467635432902671m_rule @ X6 @ Y5 ) )
      = ( produc491467635432902671m_rule @ X6 @ ( F2 @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_900_apsnd__conv,axiom,
    ! [F2: nat > nat,X6: nat,Y5: nat] :
      ( ( produc3094765310956027504at_nat @ F2 @ ( product_Pair_nat_nat @ X6 @ Y5 ) )
      = ( product_Pair_nat_nat @ X6 @ ( F2 @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_901_apsnd__conv,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,X6: product_prod_nat_nat,Y5: product_prod_nat_nat] :
      ( ( produc8987972918798826819at_nat @ F2 @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) )
      = ( produc6161850002892822231at_nat @ X6 @ ( F2 @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_902_apsnd__conv,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X6: rule,Y5: produc1996495991257130529ist_fm] :
      ( ( produc4332487942798490499m_rule @ F2 @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) )
      = ( produc7063755442767259331ist_fm @ X6 @ ( F2 @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_903_fst__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X6: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F2 @ X6 ) )
      = ( F2 @ ( produc6879501374131015971m_rule @ X6 ) ) ) ).

% fst_apfst
thf(fact_904_fst__apfst,axiom,
    ! [F2: list_fm > list_fm,X6: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc1130293176790411102ist_fm @ F2 @ X6 ) )
      = ( F2 @ ( produc1501393135466168645ist_fm @ X6 ) ) ) ).

% fst_apfst
thf(fact_905_snd__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X6: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F2 @ X6 ) )
      = ( produc7165828336582415457m_rule @ X6 ) ) ).

% snd_apfst
thf(fact_906_snd__apfst,axiom,
    ! [F2: list_fm > list_fm,X6: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F2 @ X6 ) )
      = ( produc4588648349897876871ist_fm @ X6 ) ) ).

% snd_apfst
thf(fact_907_fst__apsnd,axiom,
    ! [F2: rule > rule,X6: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F2 @ X6 ) )
      = ( produc6879501374131015971m_rule @ X6 ) ) ).

% fst_apsnd
thf(fact_908_fst__apsnd,axiom,
    ! [F2: list_fm > list_fm,X6: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F2 @ X6 ) )
      = ( produc1501393135466168645ist_fm @ X6 ) ) ).

% fst_apsnd
thf(fact_909_snd__apsnd,axiom,
    ! [F2: rule > rule,X6: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F2 @ X6 ) )
      = ( F2 @ ( produc7165828336582415457m_rule @ X6 ) ) ) ).

% snd_apsnd
thf(fact_910_snd__apsnd,axiom,
    ! [F2: list_fm > list_fm,X6: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc9047611296996456864ist_fm @ F2 @ X6 ) )
      = ( F2 @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ).

% snd_apsnd
thf(fact_911_stream_Ocollapse,axiom,
    ! [Stream2: stream727092118206550309m_rule] :
      ( ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream2 ) @ ( stl_Pr950425576149878629m_rule @ Stream2 ) )
      = Stream2 ) ).

% stream.collapse
thf(fact_912_stream_Ocollapse,axiom,
    ! [Stream2: stream_rule] :
      ( ( sCons_rule @ ( shd_rule @ Stream2 ) @ ( stl_rule @ Stream2 ) )
      = Stream2 ) ).

% stream.collapse
thf(fact_913_snd__comp__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F2 ) )
      = produc7165828336582415457m_rule ) ).

% snd_comp_apfst
thf(fact_914_snd__comp__apfst,axiom,
    ! [F2: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F2 ) )
      = produc4588648349897876871ist_fm ) ).

% snd_comp_apfst
thf(fact_915_fst__comp__apsnd,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F2 ) )
      = produc6879501374131015971m_rule ) ).

% fst_comp_apsnd
thf(fact_916_fst__comp__apsnd,axiom,
    ! [F2: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F2 ) )
      = produc1501393135466168645ist_fm ) ).

% fst_comp_apsnd
thf(fact_917_apsnd__apfst,axiom,
    ! [F2: rule > list_fm,G: produc1996495991257130529ist_fm > list_fm,X6: produc164195504107695125m_rule] :
      ( ( produc7464737497110386437ist_fm @ F2 @ ( produc7276395001879883020m_rule @ G @ X6 ) )
      = ( produc381145313068854617ist_fm @ ( G @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( F2 @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_918_apsnd__apfst,axiom,
    ! [F2: rule > rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X6: produc164195504107695125m_rule] :
      ( ( produc3063135074001183255ist_fm @ F2 @ ( produc3731403639160201281m_rule @ G @ X6 ) )
      = ( produc491467635432902671m_rule @ ( G @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( F2 @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_919_apsnd__apfst,axiom,
    ! [F2: rule > nat,G: produc1996495991257130529ist_fm > nat,X6: produc164195504107695125m_rule] :
      ( ( produc7007998740204313573at_nat @ F2 @ ( produc6002204213176801020t_rule @ G @ X6 ) )
      = ( product_Pair_nat_nat @ ( G @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( F2 @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_920_apsnd__apfst,axiom,
    ! [F2: rule > product_prod_nat_nat,G: produc1996495991257130529ist_fm > product_prod_nat_nat,X6: produc164195504107695125m_rule] :
      ( ( produc5248576904221492867at_nat @ F2 @ ( produc7833360833687662369t_rule @ G @ X6 ) )
      = ( produc6161850002892822231at_nat @ ( G @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( F2 @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_921_apsnd__apfst,axiom,
    ! [F2: rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > rule,X6: produc164195504107695125m_rule] :
      ( ( produc5714219303521602403m_rule @ F2 @ ( produc1151914582733876081e_rule @ G @ X6 ) )
      = ( produc7063755442767259331ist_fm @ ( G @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( F2 @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_922_apsnd__apfst,axiom,
    ! [F2: list_fm > list_fm,G: list_fm > list_fm,X6: produc1996495991257130529ist_fm] :
      ( ( produc9047611296996456864ist_fm @ F2 @ ( produc1130293176790411102ist_fm @ G @ X6 ) )
      = ( produc381145313068854617ist_fm @ ( G @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( F2 @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_923_apsnd__apfst,axiom,
    ! [F2: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,X6: produc1996495991257130529ist_fm] :
      ( ( produc4802975340153367740ist_fm @ F2 @ ( produc147150953317395813ist_fm @ G @ X6 ) )
      = ( produc491467635432902671m_rule @ ( G @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( F2 @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_924_apsnd__apfst,axiom,
    ! [F2: list_fm > nat,G: list_fm > nat,X6: produc1996495991257130529ist_fm] :
      ( ( produc8916614678339685504at_nat @ F2 @ ( produc4198509902575289422ist_fm @ G @ X6 ) )
      = ( product_Pair_nat_nat @ ( G @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( F2 @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_925_apsnd__apfst,axiom,
    ! [F2: list_fm > product_prod_nat_nat,G: list_fm > product_prod_nat_nat,X6: produc1996495991257130529ist_fm] :
      ( ( produc1584647392062276254at_nat @ F2 @ ( produc8169395892699911749ist_fm @ G @ X6 ) )
      = ( produc6161850002892822231at_nat @ ( G @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( F2 @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_926_apsnd__apfst,axiom,
    ! [F2: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X6: produc1996495991257130529ist_fm] :
      ( ( produc7454059569673786888m_rule @ F2 @ ( produc8579845201498058179ist_fm @ G @ X6 ) )
      = ( produc7063755442767259331ist_fm @ ( G @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( F2 @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apsnd_apfst
thf(fact_927_apfst__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,X6: produc164195504107695125m_rule] :
      ( ( produc6153989603652719015ist_fm @ F2 @ ( produc2413407552729308604ist_fm @ G @ X6 ) )
      = ( produc381145313068854617ist_fm @ ( F2 @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_928_apfst__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X6: produc164195504107695125m_rule] :
      ( ( produc3731403639160201281m_rule @ F2 @ ( produc3063135074001183255ist_fm @ G @ X6 ) )
      = ( produc491467635432902671m_rule @ ( F2 @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_929_apfst__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > nat,G: rule > nat,X6: produc164195504107695125m_rule] :
      ( ( produc8027278558657866375at_nat @ F2 @ ( produc5400006205771932108ist_fm @ G @ X6 ) )
      = ( product_Pair_nat_nat @ ( F2 @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_930_apfst__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > product_prod_nat_nat,G: rule > product_prod_nat_nat,X6: produc164195504107695125m_rule] :
      ( ( produc3405531377997666853at_nat @ F2 @ ( produc5369481475936028579ist_fm @ G @ X6 ) )
      = ( produc6161850002892822231at_nat @ ( F2 @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_931_apfst__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule,G: rule > produc1996495991257130529ist_fm,X6: produc164195504107695125m_rule] :
      ( ( produc1080319409639782133ist_fm @ F2 @ ( produc4729144337346781827ist_fm @ G @ X6 ) )
      = ( produc7063755442767259331ist_fm @ ( F2 @ ( produc6879501374131015971m_rule @ X6 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_932_apfst__apsnd,axiom,
    ! [F2: list_fm > list_fm,G: list_fm > list_fm,X6: produc1996495991257130529ist_fm] :
      ( ( produc1130293176790411102ist_fm @ F2 @ ( produc9047611296996456864ist_fm @ G @ X6 ) )
      = ( produc381145313068854617ist_fm @ ( F2 @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_933_apfst__apsnd,axiom,
    ! [F2: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X6: produc1996495991257130529ist_fm] :
      ( ( produc7806619372436685002m_rule @ F2 @ ( produc4688919482198968325ist_fm @ G @ X6 ) )
      = ( produc491467635432902671m_rule @ ( F2 @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_934_apfst__apsnd,axiom,
    ! [F2: list_fm > nat,G: list_fm > nat,X6: produc1996495991257130529ist_fm] :
      ( ( produc4433738330514286142at_nat @ F2 @ ( produc5009473360894124176ist_fm @ G @ X6 ) )
      = ( product_Pair_nat_nat @ ( F2 @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_935_apfst__apsnd,axiom,
    ! [F2: list_fm > product_prod_nat_nat,G: list_fm > product_prod_nat_nat,X6: produc1996495991257130529ist_fm] :
      ( ( produc2364714475711006556at_nat @ F2 @ ( produc9169599327578890115ist_fm @ G @ X6 ) )
      = ( produc6161850002892822231at_nat @ ( F2 @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_936_apfst__apsnd,axiom,
    ! [F2: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,X6: produc1996495991257130529ist_fm] :
      ( ( produc5155535142916265854ist_fm @ F2 @ ( produc2501461190424677027ist_fm @ G @ X6 ) )
      = ( produc7063755442767259331ist_fm @ ( F2 @ ( produc1501393135466168645ist_fm @ X6 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X6 ) ) ) ) ).

% apfst_apsnd
thf(fact_937_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B: rule,S22: stream_rule] :
      ( ( szip_rule_rule @ ( sCons_rule @ A @ S1 ) @ ( sCons_rule @ B @ S22 ) )
      = ( sCons_7940479310257918722e_rule @ ( produc1769597910952761123e_rule @ A @ B ) @ ( szip_rule_rule @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_938_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B: produc164195504107695125m_rule,S22: stream727092118206550309m_rule] :
      ( ( szip_r2254114822203693110m_rule @ ( sCons_rule @ A @ S1 ) @ ( sCons_5731567480799343806m_rule @ B @ S22 ) )
      = ( sCons_2207919253591517394m_rule @ ( produc6084188410630241011m_rule @ A @ B ) @ ( szip_r2254114822203693110m_rule @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_939_szip__unfold,axiom,
    ! [A: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B: rule,S22: stream_rule] :
      ( ( szip_P553291425819358518e_rule @ ( sCons_5731567480799343806m_rule @ A @ S1 ) @ ( sCons_rule @ B @ S22 ) )
      = ( sCons_6835861040388153298e_rule @ ( produc4383365014245906419e_rule @ A @ B ) @ ( szip_P553291425819358518e_rule @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_940_szip__unfold,axiom,
    ! [A: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B: produc164195504107695125m_rule,S22: stream727092118206550309m_rule] :
      ( ( szip_P2499414959592755846m_rule @ ( sCons_5731567480799343806m_rule @ A @ S1 ) @ ( sCons_5731567480799343806m_rule @ B @ S22 ) )
      = ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ A @ B ) @ ( szip_P2499414959592755846m_rule @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_941_szip__unfold,axiom,
    ! [A: list_fm,S1: stream_list_fm,B: list_fm,S22: stream_list_fm] :
      ( ( szip_list_fm_list_fm @ ( sCons_list_fm @ A @ S1 ) @ ( sCons_list_fm @ B @ S22 ) )
      = ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ A @ B ) @ ( szip_list_fm_list_fm @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_942_szip__unfold,axiom,
    ! [A: produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,B: rule,S22: stream_rule] :
      ( ( szip_P1977448745965526924m_rule @ ( sCons_307821682964077496ist_fm @ A @ S1 ) @ ( sCons_rule @ B @ S22 ) )
      = ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ A @ B ) @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_943_szip__unfold,axiom,
    ! [A: nat,S1: stream_nat,B: nat,S22: stream_nat] :
      ( ( szip_nat_nat @ ( sCons_nat @ A @ S1 ) @ ( sCons_nat @ B @ S22 ) )
      = ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( szip_nat_nat @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_944_szip__unfold,axiom,
    ! [A: product_prod_nat_nat,S1: stream6724221391990029191at_nat,B: product_prod_nat_nat,S22: stream6724221391990029191at_nat] :
      ( ( szip_P4314639285670189082at_nat @ ( sCons_2147322719763279000at_nat @ A @ S1 ) @ ( sCons_2147322719763279000at_nat @ B @ S22 ) )
      = ( sCons_5297226365640217982at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( szip_P4314639285670189082at_nat @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_945_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B: produc1996495991257130529ist_fm,S22: stream8299795917829157543ist_fm] :
      ( ( szip_r8549736553299883584ist_fm @ ( sCons_rule @ A @ S1 ) @ ( sCons_307821682964077496ist_fm @ B @ S22 ) )
      = ( sCons_974865965546383930ist_fm @ ( produc7063755442767259331ist_fm @ A @ B ) @ ( szip_r8549736553299883584ist_fm @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_946_stream_Osel_I1_J,axiom,
    ! [X12: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sCons_5731567480799343806m_rule @ X12 @ X2 ) )
      = X12 ) ).

% stream.sel(1)
thf(fact_947_stream_Osel_I1_J,axiom,
    ! [X12: rule,X2: stream_rule] :
      ( ( shd_rule @ ( sCons_rule @ X12 @ X2 ) )
      = X12 ) ).

% stream.sel(1)
thf(fact_948_stream_Osel_I2_J,axiom,
    ! [X12: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sCons_5731567480799343806m_rule @ X12 @ X2 ) )
      = X2 ) ).

% stream.sel(2)
thf(fact_949_stream_Osel_I2_J,axiom,
    ! [X12: rule,X2: stream_rule] :
      ( ( stl_rule @ ( sCons_rule @ X12 @ X2 ) )
      = X2 ) ).

% stream.sel(2)
thf(fact_950_stream_Oset__intros_I2_J,axiom,
    ! [Y5: fm,X2: stream_fm,X12: fm] :
      ( ( member_fm @ Y5 @ ( sset_fm @ X2 ) )
     => ( member_fm @ Y5 @ ( sset_fm @ ( sCons_fm @ X12 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_951_stream_Oset__intros_I2_J,axiom,
    ! [Y5: produc1996495991257130529ist_fm,X2: stream8299795917829157543ist_fm,X12: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ Y5 @ ( sset_P3431374123360452149ist_fm @ X2 ) )
     => ( member8102475879199740618ist_fm @ Y5 @ ( sset_P3431374123360452149ist_fm @ ( sCons_307821682964077496ist_fm @ X12 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_952_stream_Oset__intros_I2_J,axiom,
    ! [Y5: produc859450856879609959at_nat,X2: stream8372878641218411373at_nat,X12: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Y5 @ ( sset_P9164763103296375419at_nat @ X2 ) )
     => ( member8206827879206165904at_nat @ Y5 @ ( sset_P9164763103296375419at_nat @ ( sCons_5297226365640217982at_nat @ X12 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_953_stream_Oset__intros_I2_J,axiom,
    ! [Y5: rule,X2: stream_rule,X12: rule] :
      ( ( member_rule @ Y5 @ ( sset_rule @ X2 ) )
     => ( member_rule @ Y5 @ ( sset_rule @ ( sCons_rule @ X12 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_954_stream_Oset__intros_I1_J,axiom,
    ! [X12: fm,X2: stream_fm] : ( member_fm @ X12 @ ( sset_fm @ ( sCons_fm @ X12 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_955_stream_Oset__intros_I1_J,axiom,
    ! [X12: produc1996495991257130529ist_fm,X2: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X12 @ ( sset_P3431374123360452149ist_fm @ ( sCons_307821682964077496ist_fm @ X12 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_956_stream_Oset__intros_I1_J,axiom,
    ! [X12: produc859450856879609959at_nat,X2: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ X12 @ ( sset_P9164763103296375419at_nat @ ( sCons_5297226365640217982at_nat @ X12 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_957_stream_Oset__intros_I1_J,axiom,
    ! [X12: rule,X2: stream_rule] : ( member_rule @ X12 @ ( sset_rule @ ( sCons_rule @ X12 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_958_stream_Oset__cases,axiom,
    ! [E: fm,A: stream_fm] :
      ( ( member_fm @ E @ ( sset_fm @ A ) )
     => ( ! [Z22: stream_fm] :
            ( A
           != ( sCons_fm @ E @ Z22 ) )
       => ~ ! [Z1: fm,Z22: stream_fm] :
              ( ( A
                = ( sCons_fm @ Z1 @ Z22 ) )
             => ~ ( member_fm @ E @ ( sset_fm @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_959_stream_Oset__cases,axiom,
    ! [E: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm] :
      ( ( member8102475879199740618ist_fm @ E @ ( sset_P3431374123360452149ist_fm @ A ) )
     => ( ! [Z22: stream8299795917829157543ist_fm] :
            ( A
           != ( sCons_307821682964077496ist_fm @ E @ Z22 ) )
       => ~ ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm] :
              ( ( A
                = ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) )
             => ~ ( member8102475879199740618ist_fm @ E @ ( sset_P3431374123360452149ist_fm @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_960_stream_Oset__cases,axiom,
    ! [E: produc859450856879609959at_nat,A: stream8372878641218411373at_nat] :
      ( ( member8206827879206165904at_nat @ E @ ( sset_P9164763103296375419at_nat @ A ) )
     => ( ! [Z22: stream8372878641218411373at_nat] :
            ( A
           != ( sCons_5297226365640217982at_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc859450856879609959at_nat,Z22: stream8372878641218411373at_nat] :
              ( ( A
                = ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) )
             => ~ ( member8206827879206165904at_nat @ E @ ( sset_P9164763103296375419at_nat @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_961_stream_Oset__cases,axiom,
    ! [E: rule,A: stream_rule] :
      ( ( member_rule @ E @ ( sset_rule @ A ) )
     => ( ! [Z22: stream_rule] :
            ( A
           != ( sCons_rule @ E @ Z22 ) )
       => ~ ! [Z1: rule,Z22: stream_rule] :
              ( ( A
                = ( sCons_rule @ Z1 @ Z22 ) )
             => ~ ( member_rule @ E @ ( sset_rule @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_962_stream_Oset__induct,axiom,
    ! [X6: fm,A: stream_fm,P: fm > stream_fm > $o] :
      ( ( member_fm @ X6 @ ( sset_fm @ A ) )
     => ( ! [Z1: fm,Z22: stream_fm] : ( P @ Z1 @ ( sCons_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: fm,Z22: stream_fm,Xa: fm] :
              ( ( member_fm @ Xa @ ( sset_fm @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_fm @ Z1 @ Z22 ) ) ) )
         => ( P @ X6 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_963_stream_Oset__induct,axiom,
    ! [X6: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm,P: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X6 @ ( sset_P3431374123360452149ist_fm @ A ) )
     => ( ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm] : ( P @ Z1 @ ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm,Xa: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Xa @ ( sset_P3431374123360452149ist_fm @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) ) ) )
         => ( P @ X6 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_964_stream_Oset__induct,axiom,
    ! [X6: produc859450856879609959at_nat,A: stream8372878641218411373at_nat,P: produc859450856879609959at_nat > stream8372878641218411373at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X6 @ ( sset_P9164763103296375419at_nat @ A ) )
     => ( ! [Z1: produc859450856879609959at_nat,Z22: stream8372878641218411373at_nat] : ( P @ Z1 @ ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) )
       => ( ! [Z1: produc859450856879609959at_nat,Z22: stream8372878641218411373at_nat,Xa: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Xa @ ( sset_P9164763103296375419at_nat @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) ) ) )
         => ( P @ X6 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_965_stream_Oset__induct,axiom,
    ! [X6: rule,A: stream_rule,P: rule > stream_rule > $o] :
      ( ( member_rule @ X6 @ ( sset_rule @ A ) )
     => ( ! [Z1: rule,Z22: stream_rule] : ( P @ Z1 @ ( sCons_rule @ Z1 @ Z22 ) )
       => ( ! [Z1: rule,Z22: stream_rule,Xa: rule] :
              ( ( member_rule @ Xa @ ( sset_rule @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_rule @ Z1 @ Z22 ) ) ) )
         => ( P @ X6 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_966_stream_Ocase,axiom,
    ! [F2: rule > stream_rule > abstra4572470701423150929m_rule,X12: rule,X2: stream_rule] :
      ( ( case_s2494923709252486212m_rule @ F2 @ ( sCons_rule @ X12 @ X2 ) )
      = ( F2 @ X12 @ X2 ) ) ).

% stream.case
thf(fact_967_sdrop__while__SCons,axiom,
    ! [P: rule > $o,A: rule,S: stream_rule] :
      ( ( ( P @ A )
       => ( ( sdrop_while_rule @ P @ ( sCons_rule @ A @ S ) )
          = ( sdrop_while_rule @ P @ S ) ) )
      & ( ~ ( P @ A )
       => ( ( sdrop_while_rule @ P @ ( sCons_rule @ A @ S ) )
          = ( sCons_rule @ A @ S ) ) ) ) ).

% sdrop_while_SCons
thf(fact_968_stream_Oexhaust__sel,axiom,
    ! [Stream2: stream727092118206550309m_rule] :
      ( Stream2
      = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream2 ) @ ( stl_Pr950425576149878629m_rule @ Stream2 ) ) ) ).

% stream.exhaust_sel
thf(fact_969_stream_Oexhaust__sel,axiom,
    ! [Stream2: stream_rule] :
      ( Stream2
      = ( sCons_rule @ ( shd_rule @ Stream2 ) @ ( stl_rule @ Stream2 ) ) ) ).

% stream.exhaust_sel
thf(fact_970_stream_Osplit__sel,axiom,
    ! [P: abstra4572470701423150929m_rule > $o,F2: rule > stream_rule > abstra4572470701423150929m_rule,Stream2: stream_rule] :
      ( ( P @ ( case_s2494923709252486212m_rule @ F2 @ Stream2 ) )
      = ( ( Stream2
          = ( sCons_rule @ ( shd_rule @ Stream2 ) @ ( stl_rule @ Stream2 ) ) )
       => ( P @ ( F2 @ ( shd_rule @ Stream2 ) @ ( stl_rule @ Stream2 ) ) ) ) ) ).

% stream.split_sel
thf(fact_971_stream_Osplit__sel__asm,axiom,
    ! [P: abstra4572470701423150929m_rule > $o,F2: rule > stream_rule > abstra4572470701423150929m_rule,Stream2: stream_rule] :
      ( ( P @ ( case_s2494923709252486212m_rule @ F2 @ Stream2 ) )
      = ( ~ ( ( Stream2
              = ( sCons_rule @ ( shd_rule @ Stream2 ) @ ( stl_rule @ Stream2 ) ) )
            & ~ ( P @ ( F2 @ ( shd_rule @ Stream2 ) @ ( stl_rule @ Stream2 ) ) ) ) ) ) ).

% stream.split_sel_asm
thf(fact_972_szip_Ocode,axiom,
    ( szip_list_fm_list_fm
    = ( ^ [S12: stream_list_fm,S23: stream_list_fm] : ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ ( shd_list_fm @ S12 ) @ ( shd_list_fm @ S23 ) ) @ ( szip_list_fm_list_fm @ ( stl_list_fm @ S12 ) @ ( stl_list_fm @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_973_szip_Ocode,axiom,
    ( szip_nat_nat
    = ( ^ [S12: stream_nat,S23: stream_nat] : ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ ( shd_nat @ S12 ) @ ( shd_nat @ S23 ) ) @ ( szip_nat_nat @ ( stl_nat @ S12 ) @ ( stl_nat @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_974_szip_Ocode,axiom,
    ( szip_P4314639285670189082at_nat
    = ( ^ [S12: stream6724221391990029191at_nat,S23: stream6724221391990029191at_nat] : ( sCons_5297226365640217982at_nat @ ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S12 ) @ ( shd_Pr4260400998323988397at_nat @ S23 ) ) @ ( szip_P4314639285670189082at_nat @ ( stl_Pr5027827701538482609at_nat @ S12 ) @ ( stl_Pr5027827701538482609at_nat @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_975_szip_Ocode,axiom,
    ( szip_P1977448745965526924m_rule
    = ( ^ [S12: stream8299795917829157543ist_fm,S23: stream_rule] : ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S12 ) @ ( shd_rule @ S23 ) ) @ ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S12 ) @ ( stl_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_976_szip_Ocode,axiom,
    ( szip_P2499414959592755846m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S23: stream727092118206550309m_rule] : ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S23 ) ) @ ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_977_szip_Ocode,axiom,
    ( szip_P553291425819358518e_rule
    = ( ^ [S12: stream727092118206550309m_rule,S23: stream_rule] : ( sCons_6835861040388153298e_rule @ ( produc4383365014245906419e_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_rule @ S23 ) ) @ ( szip_P553291425819358518e_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_978_szip_Ocode,axiom,
    ( szip_r8549736553299883584ist_fm
    = ( ^ [S12: stream_rule,S23: stream8299795917829157543ist_fm] : ( sCons_974865965546383930ist_fm @ ( produc7063755442767259331ist_fm @ ( shd_rule @ S12 ) @ ( shd_Pr772355297128350925ist_fm @ S23 ) ) @ ( szip_r8549736553299883584ist_fm @ ( stl_rule @ S12 ) @ ( stl_Pr5027778045650968273ist_fm @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_979_szip_Ocode,axiom,
    ( szip_r2254114822203693110m_rule
    = ( ^ [S12: stream_rule,S23: stream727092118206550309m_rule] : ( sCons_2207919253591517394m_rule @ ( produc6084188410630241011m_rule @ ( shd_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S23 ) ) @ ( szip_r2254114822203693110m_rule @ ( stl_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_980_szip_Ocode,axiom,
    ( szip_rule_rule
    = ( ^ [S12: stream_rule,S23: stream_rule] : ( sCons_7940479310257918722e_rule @ ( produc1769597910952761123e_rule @ ( shd_rule @ S12 ) @ ( shd_rule @ S23 ) ) @ ( szip_rule_rule @ ( stl_rule @ S12 ) @ ( stl_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_981_sndOp__def,axiom,
    ( bNF_sn6617447237779055481m_rule
    = ( ^ [P3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( bNF_pi3438825987790746729m_rule @ P3 @ Q3 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_982_sndOp__def,axiom,
    ( bNF_sn5292590578203848342ist_fm
    = ( ^ [P3: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( bNF_pi3539088406881798022ist_fm @ P3 @ Q3 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_983_fstOp__def,axiom,
    ( bNF_fs2143400664727635963e_rule
    = ( ^ [P3: produc1996495991257130529ist_fm > rule > $o,Q3: rule > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Ac ) @ ( bNF_pi580457772512444745e_rule @ P3 @ Q3 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_984_fstOp__def,axiom,
    ( bNF_fs6361025107161210196ist_fm
    = ( ^ [P3: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Ac ) @ ( bNF_pi3539088406881798022ist_fm @ P3 @ Q3 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_985_sfilter_Ocode,axiom,
    ( sfilte7579662470208054517m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o,S2: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P3 ) @ S2 ) ) @ ( sfilte7579662470208054517m_rule @ P3 @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P3 ) @ S2 ) ) ) ) ) ) ).

% sfilter.code
thf(fact_986_sfilter_Ocode,axiom,
    ( sfilter_rule
    = ( ^ [P3: rule > $o,S2: stream_rule] : ( sCons_rule @ ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P3 ) @ S2 ) ) @ ( sfilter_rule @ P3 @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P3 ) @ S2 ) ) ) ) ) ) ).

% sfilter.code
thf(fact_987_apfst__convE,axiom,
    ! [Q: produc1996495991257130529ist_fm,F2: list_fm > list_fm,P2: produc1996495991257130529ist_fm] :
      ( ( Q
        = ( produc1130293176790411102ist_fm @ F2 @ P2 ) )
     => ~ ! [X3: list_fm,Y3: list_fm] :
            ( ( P2
              = ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
           => ( Q
             != ( produc381145313068854617ist_fm @ ( F2 @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_988_apfst__convE,axiom,
    ! [Q: produc164195504107695125m_rule,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,P2: produc164195504107695125m_rule] :
      ( ( Q
        = ( produc3731403639160201281m_rule @ F2 @ P2 ) )
     => ~ ! [X3: produc1996495991257130529ist_fm,Y3: rule] :
            ( ( P2
              = ( produc491467635432902671m_rule @ X3 @ Y3 ) )
           => ( Q
             != ( produc491467635432902671m_rule @ ( F2 @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_989_apfst__convE,axiom,
    ! [Q: product_prod_nat_nat,F2: nat > nat,P2: product_prod_nat_nat] :
      ( ( Q
        = ( produc986720760941809198at_nat @ F2 @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ( Q
             != ( product_Pair_nat_nat @ ( F2 @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_990_apfst__convE,axiom,
    ! [Q: produc859450856879609959at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,P2: produc859450856879609959at_nat] :
      ( ( Q
        = ( produc3057951030792887557at_nat @ F2 @ P2 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
           => ( Q
             != ( produc6161850002892822231at_nat @ ( F2 @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_991_apfst__convE,axiom,
    ! [Q: produc4630866025709511057ist_fm,F2: rule > rule,P2: produc4630866025709511057ist_fm] :
      ( ( Q
        = ( produc5534506254562563289ist_fm @ F2 @ P2 ) )
     => ~ ! [X3: rule,Y3: produc1996495991257130529ist_fm] :
            ( ( P2
              = ( produc7063755442767259331ist_fm @ X3 @ Y3 ) )
           => ( Q
             != ( produc7063755442767259331ist_fm @ ( F2 @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_992_Stream_Osmember__def,axiom,
    ( smember_fm
    = ( ^ [X5: fm,S2: stream_fm] : ( member_fm @ X5 @ ( sset_fm @ S2 ) ) ) ) ).

% Stream.smember_def
thf(fact_993_Stream_Osmember__def,axiom,
    ( smembe2449974647026456671ist_fm
    = ( ^ [X5: produc1996495991257130529ist_fm,S2: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X5 @ ( sset_P3431374123360452149ist_fm @ S2 ) ) ) ) ).

% Stream.smember_def
thf(fact_994_Stream_Osmember__def,axiom,
    ( smembe1741262129464730021at_nat
    = ( ^ [X5: produc859450856879609959at_nat,S2: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ X5 @ ( sset_P9164763103296375419at_nat @ S2 ) ) ) ) ).

% Stream.smember_def
thf(fact_995_Stream_Osmember__def,axiom,
    ( smember_rule
    = ( ^ [X5: rule,S2: stream_rule] : ( member_rule @ X5 @ ( sset_rule @ S2 ) ) ) ) ).

% Stream.smember_def
thf(fact_996_sfilter_Osimps_I1_J,axiom,
    ! [P: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sfilte7579662470208054517m_rule @ P @ S ) )
      = ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P ) @ S ) ) ) ).

% sfilter.simps(1)
thf(fact_997_sfilter_Osimps_I1_J,axiom,
    ! [P: rule > $o,S: stream_rule] :
      ( ( shd_rule @ ( sfilter_rule @ P @ S ) )
      = ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P ) @ S ) ) ) ).

% sfilter.simps(1)
thf(fact_998_sfilter_Osimps_I2_J,axiom,
    ! [P: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sfilte7579662470208054517m_rule @ P @ S ) )
      = ( sfilte7579662470208054517m_rule @ P @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P ) @ S ) ) ) ) ).

% sfilter.simps(2)
thf(fact_999_sfilter_Osimps_I2_J,axiom,
    ! [P: rule > $o,S: stream_rule] :
      ( ( stl_rule @ ( sfilter_rule @ P @ S ) )
      = ( sfilter_rule @ P @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P ) @ S ) ) ) ) ).

% sfilter.simps(2)
thf(fact_1000_sfilter__P,axiom,
    ! [P: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] :
      ( ( P @ ( shd_Pr7235097944458474089m_rule @ S ) )
     => ( ( sfilte7579662470208054517m_rule @ P @ S )
        = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S ) @ ( sfilte7579662470208054517m_rule @ P @ ( stl_Pr950425576149878629m_rule @ S ) ) ) ) ) ).

% sfilter_P
thf(fact_1001_sfilter__P,axiom,
    ! [P: rule > $o,S: stream_rule] :
      ( ( P @ ( shd_rule @ S ) )
     => ( ( sfilter_rule @ P @ S )
        = ( sCons_rule @ ( shd_rule @ S ) @ ( sfilter_rule @ P @ ( stl_rule @ S ) ) ) ) ) ).

% sfilter_P
thf(fact_1002_sfilter__not__P,axiom,
    ! [P: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] :
      ( ~ ( P @ ( shd_Pr7235097944458474089m_rule @ S ) )
     => ( ( sfilte7579662470208054517m_rule @ P @ S )
        = ( sfilte7579662470208054517m_rule @ P @ ( stl_Pr950425576149878629m_rule @ S ) ) ) ) ).

% sfilter_not_P
thf(fact_1003_sfilter__not__P,axiom,
    ! [P: rule > $o,S: stream_rule] :
      ( ~ ( P @ ( shd_rule @ S ) )
     => ( ( sfilter_rule @ P @ S )
        = ( sfilter_rule @ P @ ( stl_rule @ S ) ) ) ) ).

% sfilter_not_P
thf(fact_1004_Bleast__def,axiom,
    ( bleast_nat
    = ( ^ [S7: set_nat,P3: nat > $o] :
          ( ord_Least_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ S7 )
              & ( P3 @ X5 ) ) ) ) ) ).

% Bleast_def
thf(fact_1005_abort__Bleast__def,axiom,
    ( abort_Bleast_nat
    = ( ^ [S7: set_nat,P3: nat > $o] :
          ( ord_Least_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ S7 )
              & ( P3 @ X5 ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_1006_sinterleave_Ocode,axiom,
    ( sinter8084399408526207096m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S23: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( sinter8084399408526207096m_rule @ S23 @ ( stl_Pr950425576149878629m_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_1007_sinterleave_Ocode,axiom,
    ( sinterleave_rule
    = ( ^ [S12: stream_rule,S23: stream_rule] : ( sCons_rule @ ( shd_rule @ S12 ) @ ( sinterleave_rule @ S23 @ ( stl_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_1008_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S22 ) )
      = ( shd_Pr7235097944458474089m_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_1009_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( shd_rule @ ( sinterleave_rule @ S1 @ S22 ) )
      = ( shd_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_1010_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S22 ) )
      = ( sinter8084399408526207096m_rule @ S22 @ ( stl_Pr950425576149878629m_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_1011_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( stl_rule @ ( sinterleave_rule @ S1 @ S22 ) )
      = ( sinterleave_rule @ S22 @ ( stl_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_1012_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_1013_verit__sko__forall__indirect2,axiom,
    ! [X6: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o,P4: fset_P661503646757059847ist_fm > $o] :
      ( ( X6
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X5: fset_P661503646757059847ist_fm] :
              ~ ( P @ X5 ) ) )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
            = ( P4 @ X3 ) )
       => ( ( ! [X: fset_P661503646757059847ist_fm] : ( P4 @ X ) )
          = ( P @ X6 ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_1014_verit__sko__forall__indirect,axiom,
    ! [X6: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o] :
      ( ( X6
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X5: fset_P661503646757059847ist_fm] :
              ~ ( P @ X5 ) ) )
     => ( ( ! [X: fset_P661503646757059847ist_fm] : ( P @ X ) )
        = ( P @ X6 ) ) ) ).

% verit_sko_forall_indirect
thf(fact_1015_verit__sko__ex__indirect2,axiom,
    ! [X6: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o,P4: fset_P661503646757059847ist_fm > $o] :
      ( ( X6
        = ( fChoic3892542562059901019ist_fm @ P ) )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
            = ( P4 @ X3 ) )
       => ( ( ? [X: fset_P661503646757059847ist_fm] : ( P4 @ X ) )
          = ( P @ X6 ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_1016_fset__of__list__elem,axiom,
    ! [X6: rule,Xs: list_rule] :
      ( ( fmember_rule @ X6 @ ( fset_of_list_rule @ Xs ) )
      = ( member_rule @ X6 @ ( set_rule2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1017_fset__of__list__elem,axiom,
    ! [X6: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( fmembe6431293532144391214at_nat @ X6 @ ( fset_o2072219007702160685at_nat @ Xs ) )
      = ( member8206827879206165904at_nat @ X6 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1018_fset__of__list__elem,axiom,
    ! [X6: produc1996495991257130529ist_fm,Xs: list_P5616295576739893671ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X6 @ ( fset_o3706400737857578983ist_fm @ Xs ) )
      = ( member8102475879199740618ist_fm @ X6 @ ( set_Pr8767716839810916150ist_fm @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1019_fset__of__list__elem,axiom,
    ! [X6: abstra4572470701423150929m_rule,Xs: list_A1849579841884561377m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X6 @ ( fset_o8294491781805274379m_rule @ Xs ) )
      = ( member1590433669991072232m_rule @ X6 @ ( set_Ab5312821370972875900m_rule @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1020_fset__of__list__elem,axiom,
    ! [X6: fm,Xs: list_fm] :
      ( ( fmember_fm @ X6 @ ( fset_of_list_fm @ Xs ) )
      = ( member_fm @ X6 @ ( set_fm2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1021_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1022_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1023_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1024_conj__subset__def,axiom,
    ! [A4: set_fm,P: fm > $o,Q2: fm > $o] :
      ( ( ord_less_eq_set_fm @ A4
        @ ( collect_fm
          @ ^ [X5: fm] :
              ( ( P @ X5 )
              & ( Q2 @ X5 ) ) ) )
      = ( ( ord_less_eq_set_fm @ A4 @ ( collect_fm @ P ) )
        & ( ord_less_eq_set_fm @ A4 @ ( collect_fm @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_1025_verit__sko__ex_H,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A4: $o] :
      ( ( ( P @ ( fChoic3892542562059901019ist_fm @ P ) )
        = A4 )
     => ( ( ? [X: fset_P661503646757059847ist_fm] : ( P @ X ) )
        = A4 ) ) ).

% verit_sko_ex'
thf(fact_1026_verit__sko__forall,axiom,
    ( ( ^ [P5: fset_P661503646757059847ist_fm > $o] :
        ! [X9: fset_P661503646757059847ist_fm] : ( P5 @ X9 ) )
    = ( ^ [P3: fset_P661503646757059847ist_fm > $o] :
          ( P3
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X5: fset_P661503646757059847ist_fm] :
                ~ ( P3 @ X5 ) ) ) ) ) ).

% verit_sko_forall
thf(fact_1027_verit__sko__forall_H,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A4: $o] :
      ( ( ( P
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X5: fset_P661503646757059847ist_fm] :
                ~ ( P @ X5 ) ) )
        = A4 )
     => ( ( ! [X: fset_P661503646757059847ist_fm] : ( P @ X ) )
        = A4 ) ) ).

% verit_sko_forall'
thf(fact_1028_verit__sko__forall_H_H,axiom,
    ! [B4: fset_P661503646757059847ist_fm,A4: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o] :
      ( ( B4 = A4 )
     => ( ( ( fChoic3892542562059901019ist_fm @ P )
          = A4 )
        = ( ( fChoic3892542562059901019ist_fm @ P )
          = B4 ) ) ) ).

% verit_sko_forall''
thf(fact_1029_verit__sko__ex__indirect,axiom,
    ! [X6: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o] :
      ( ( X6
        = ( fChoic3892542562059901019ist_fm @ P ) )
     => ( ( ? [X: fset_P661503646757059847ist_fm] : ( P @ X ) )
        = ( P @ X6 ) ) ) ).

% verit_sko_ex_indirect
thf(fact_1030_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > rule > rule,S1: stream_rule,S22: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_rule_rule_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_rule_rule_rule @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1031_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > produc780299800278082923e_rule > rule,S1: stream_rule,S22: stream3588992565182678257e_rule] :
      ( ( sdrop_rule @ N @ ( smap2_4638655784531377133e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_4638655784531377133e_rule @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1032_sdrop__smap2,axiom,
    ! [N: nat,F2: produc780299800278082923e_rule > rule > rule,S1: stream3588992565182678257e_rule,S22: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_4273976090187245369e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_4273976090187245369e_rule @ F2 @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1033_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > rule > produc780299800278082923e_rule,S1: stream_rule,S22: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( smap2_2301042431756194977e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_2301042431756194977e_rule @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1034_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > produc1996495991257130529ist_fm > rule,S1: stream_rule,S22: stream8299795917829157543ist_fm] :
      ( ( sdrop_rule @ N @ ( smap2_3943848181979139747m_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_3943848181979139747m_rule @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1035_sdrop__smap2,axiom,
    ! [N: nat,F2: produc1996495991257130529ist_fm > rule > rule,S1: stream8299795917829157543ist_fm,S22: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_6133544317484809199e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_6133544317484809199e_rule @ F2 @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1036_sdrop__smap2,axiom,
    ! [N: nat,F2: produc780299800278082923e_rule > produc780299800278082923e_rule > rule,S1: stream3588992565182678257e_rule,S22: stream3588992565182678257e_rule] :
      ( ( sdrop_rule @ N @ ( smap2_6314367768709263299e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_6314367768709263299e_rule @ F2 @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1037_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > rule > produc1996495991257130529ist_fm,S1: stream_rule,S22: stream_rule] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( smap2_1292763952458720599ist_fm @ F2 @ S1 @ S22 ) )
      = ( smap2_1292763952458720599ist_fm @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1038_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > produc780299800278082923e_rule > produc780299800278082923e_rule,S1: stream_rule,S22: stream3588992565182678257e_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( smap2_644143445760767683e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_644143445760767683e_rule @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1039_sdrop__smap2,axiom,
    ! [N: nat,F2: produc780299800278082923e_rule > rule > produc780299800278082923e_rule,S1: stream3588992565182678257e_rule,S22: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( smap2_3976754415934081143e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_3976754415934081143e_rule @ F2 @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_1040_smap2_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3865418324013074943m_rule @ F2 @ S1 @ S22 ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_1041_smap2_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3092065398501458735m_rule @ F2 @ S1 @ S22 ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_1042_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_7722561687528711727m_rule @ F2 @ S1 @ S22 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_1043_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_4438185555641665375m_rule @ F2 @ S1 @ S22 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_1044_smap2_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_1391242002117124143e_rule @ F2 @ S1 @ S22 ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_1045_smap2_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( shd_rule @ ( smap2_190827247425732959e_rule @ F2 @ S1 @ S22 ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_1046_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_2737362159257330783e_rule @ F2 @ S1 @ S22 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_1047_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > rule > rule,S1: stream_rule,S22: stream_rule] :
      ( ( shd_rule @ ( smap2_rule_rule_rule @ F2 @ S1 @ S22 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_1048_smap2_Osimps_I2_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3865418324013074943m_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_3865418324013074943m_rule @ F2 @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_1049_smap2_Osimps_I2_J,axiom,
    ! [F2: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3092065398501458735m_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_3092065398501458735m_rule @ F2 @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_1050_smap2_Osimps_I2_J,axiom,
    ! [F2: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_7722561687528711727m_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_7722561687528711727m_rule @ F2 @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_1051_smap2_Osimps_I2_J,axiom,
    ! [F2: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_4438185555641665375m_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_4438185555641665375m_rule @ F2 @ ( stl_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_1052_smap2_Osimps_I2_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_1391242002117124143e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_1391242002117124143e_rule @ F2 @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_1053_smap2_Osimps_I2_J,axiom,
    ! [F2: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( stl_rule @ ( smap2_190827247425732959e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_190827247425732959e_rule @ F2 @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_1054_smap2_Osimps_I2_J,axiom,
    ! [F2: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_2737362159257330783e_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_2737362159257330783e_rule @ F2 @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_1055_smap2_Osimps_I2_J,axiom,
    ! [F2: rule > rule > rule,S1: stream_rule,S22: stream_rule] :
      ( ( stl_rule @ ( smap2_rule_rule_rule @ F2 @ S1 @ S22 ) )
      = ( smap2_rule_rule_rule @ F2 @ ( stl_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_1056_convol__expand__snd,axiom,
    ! [F2: abstra4572470701423150929m_rule > produc164195504107695125m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ F2 )
        = G )
     => ( ( bNF_co2617238127527879852m_rule @ G @ ( comp_P1643750715206316048m_rule @ produc7165828336582415457m_rule @ F2 ) )
        = F2 ) ) ).

% convol_expand_snd
thf(fact_1057_convol__expand__snd_H,axiom,
    ! [F2: abstra4572470701423150929m_rule > produc164195504107695125m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > rule] :
      ( ( ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ F2 )
        = G )
     => ( ( H
          = ( comp_P1643750715206316048m_rule @ produc7165828336582415457m_rule @ F2 ) )
        = ( ( bNF_co2617238127527879852m_rule @ G @ H )
          = F2 ) ) ) ).

% convol_expand_snd'
thf(fact_1058_Collect__restrict,axiom,
    ! [X7: set_rule,P: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X5: rule] :
            ( ( member_rule @ X5 @ X7 )
            & ( P @ X5 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1059_Collect__restrict,axiom,
    ! [X7: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X5: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X5 @ X7 )
            & ( P @ X5 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1060_Collect__restrict,axiom,
    ! [X7: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X5: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X5 @ X7 )
            & ( P @ X5 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1061_Collect__restrict,axiom,
    ! [X7: set_fm,P: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X5: fm] :
            ( ( member_fm @ X5 @ X7 )
            & ( P @ X5 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1062_ssubst__Pair__rhs,axiom,
    ! [R: list_fm,S: list_fm,R3: set_Pr7058068377845519745ist_fm,S4: list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R @ S ) @ R3 )
     => ( ( S4 = S )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1063_ssubst__Pair__rhs,axiom,
    ! [R: produc1996495991257130529ist_fm,S: rule,R3: set_Pr1008144964186165195m_rule,S4: rule] :
      ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R @ S ) @ R3 )
     => ( ( S4 = S )
       => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1064_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_1065_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_1066_ssubst__Pair__rhs,axiom,
    ! [R: rule,S: produc1996495991257130529ist_fm,R3: set_Pr2241365836748572487ist_fm,S4: produc1996495991257130529ist_fm] :
      ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R @ S ) @ R3 )
     => ( ( S4 = S )
       => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1067_fst__convol,axiom,
    ! [F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > rule] :
      ( ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ ( bNF_co2617238127527879852m_rule @ F2 @ G ) )
      = F2 ) ).

% fst_convol
thf(fact_1068_prop__restrict,axiom,
    ! [X6: rule,Z3: set_rule,X7: set_rule,P: rule > $o] :
      ( ( member_rule @ X6 @ Z3 )
     => ( ( ord_less_eq_set_rule @ Z3
          @ ( collect_rule
            @ ^ [X5: rule] :
                ( ( member_rule @ X5 @ X7 )
                & ( P @ X5 ) ) ) )
       => ( P @ X6 ) ) ) ).

% prop_restrict
thf(fact_1069_prop__restrict,axiom,
    ! [X6: produc1996495991257130529ist_fm,Z3: set_Pr7058068377845519745ist_fm,X7: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X6 @ Z3 )
     => ( ( ord_le2055072402069232929ist_fm @ Z3
          @ ( collec7231575776666803724ist_fm
            @ ^ [X5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X5 @ X7 )
                & ( P @ X5 ) ) ) )
       => ( P @ X6 ) ) ) ).

% prop_restrict
thf(fact_1070_prop__restrict,axiom,
    ! [X6: produc859450856879609959at_nat,Z3: set_Pr8693737435421807431at_nat,X7: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X6 @ Z3 )
     => ( ( ord_le3000389064537975527at_nat @ Z3
          @ ( collec7088162979684241874at_nat
            @ ^ [X5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X5 @ X7 )
                & ( P @ X5 ) ) ) )
       => ( P @ X6 ) ) ) ).

% prop_restrict
thf(fact_1071_prop__restrict,axiom,
    ! [X6: fm,Z3: set_fm,X7: set_fm,P: fm > $o] :
      ( ( member_fm @ X6 @ Z3 )
     => ( ( ord_less_eq_set_fm @ Z3
          @ ( collect_fm
            @ ^ [X5: fm] :
                ( ( member_fm @ X5 @ X7 )
                & ( P @ X5 ) ) ) )
       => ( P @ X6 ) ) ) ).

% prop_restrict
thf(fact_1072_prod_Oswap__def,axiom,
    ( produc9213896635423732003ist_fm
    = ( ^ [P6: produc4630866025709511057ist_fm] : ( produc491467635432902671m_rule @ ( produc4514744107061996309ist_fm @ P6 ) @ ( produc4228417144610596823ist_fm @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1073_prod_Oswap__def,axiom,
    ( product_swap_nat_nat
    = ( ^ [P6: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P6 ) @ ( product_fst_nat_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1074_prod_Oswap__def,axiom,
    ( produc7225117575323628663at_nat
    = ( ^ [P6: produc859450856879609959at_nat] : ( produc6161850002892822231at_nat @ ( produc6408287024330202629at_nat @ P6 ) @ ( produc3213797794245857475at_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1075_prod_Oswap__def,axiom,
    ( produc2641608828089375343m_rule
    = ( ^ [P6: produc164195504107695125m_rule] : ( produc7063755442767259331ist_fm @ ( produc7165828336582415457m_rule @ P6 ) @ ( produc6879501374131015971m_rule @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1076_prod_Oswap__def,axiom,
    ( produc7936585717479832313ist_fm
    = ( ^ [P6: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc4588648349897876871ist_fm @ P6 ) @ ( produc1501393135466168645ist_fm @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1077_fst__snd__flip,axiom,
    ( product_fst_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X5: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_1078_fst__snd__flip,axiom,
    ( produc3213797794245857475at_nat
    = ( comp_P4814319827393324394at_nat @ produc6408287024330202629at_nat
      @ ( produc6744312701629110395at_nat
        @ ^ [X5: product_prod_nat_nat,Y6: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ Y6 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_1079_fst__snd__flip,axiom,
    ( produc6879501374131015971m_rule
    = ( comp_P6646430376872094402m_rule @ produc4514744107061996309ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X5: produc1996495991257130529ist_fm,Y6: rule] : ( produc7063755442767259331ist_fm @ Y6 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_1080_fst__snd__flip,axiom,
    ( produc4228417144610596823ist_fm
    = ( comp_P4888803072877744336ist_fm @ produc7165828336582415457m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X5: rule,Y6: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y6 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_1081_fst__snd__flip,axiom,
    ( produc1501393135466168645ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X5: list_fm,Y6: list_fm] : ( produc381145313068854617ist_fm @ Y6 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_1082_snd__fst__flip,axiom,
    ( product_snd_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X5: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_1083_snd__fst__flip,axiom,
    ( produc6408287024330202629at_nat
    = ( comp_P4814319827393324394at_nat @ produc3213797794245857475at_nat
      @ ( produc6744312701629110395at_nat
        @ ^ [X5: product_prod_nat_nat,Y6: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ Y6 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_1084_snd__fst__flip,axiom,
    ( produc4514744107061996309ist_fm
    = ( comp_P2707267495825058490ist_fm @ produc6879501374131015971m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X5: rule,Y6: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y6 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_1085_snd__fst__flip,axiom,
    ( produc7165828336582415457m_rule
    = ( comp_P3413117018928051920m_rule @ produc4228417144610596823ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X5: produc1996495991257130529ist_fm,Y6: rule] : ( produc7063755442767259331ist_fm @ Y6 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_1086_snd__fst__flip,axiom,
    ( produc4588648349897876871ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X5: list_fm,Y6: list_fm] : ( produc381145313068854617ist_fm @ Y6 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_1087_stake__pos__minWait,axiom,
    ! [Rs: stream_rule,S: produc1996495991257130529ist_fm,R: rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( ord_less_nat
          @ ( abstra738319790293016300ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S )
          @ ( abstract_pos_rule @ Rs @ R ) )
       => ( ( member_rule @ R @ ( sset_rule @ rules ) )
         => ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
           => ( ( abstract_pos_rule
                @ ( stl_rule
                  @ ( abstra2363873759530810705ist_fm
                    @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S2 )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ Rs
                    @ S ) )
                @ R )
              = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R )
                @ ( suc
                  @ ( abstra738319790293016300ist_fm
                    @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S2 )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ Rs
                    @ S ) ) ) ) ) ) ) ) ).

% stake_pos_minWait
thf(fact_1088_swap__simp,axiom,
    ! [X6: list_fm,Y5: list_fm] :
      ( ( produc7936585717479832313ist_fm @ ( produc381145313068854617ist_fm @ X6 @ Y5 ) )
      = ( produc381145313068854617ist_fm @ Y5 @ X6 ) ) ).

% swap_simp
thf(fact_1089_swap__simp,axiom,
    ! [X6: produc1996495991257130529ist_fm,Y5: rule] :
      ( ( produc2641608828089375343m_rule @ ( produc491467635432902671m_rule @ X6 @ Y5 ) )
      = ( produc7063755442767259331ist_fm @ Y5 @ X6 ) ) ).

% swap_simp
thf(fact_1090_swap__simp,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) )
      = ( product_Pair_nat_nat @ Y5 @ X6 ) ) ).

% swap_simp
thf(fact_1091_swap__simp,axiom,
    ! [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] :
      ( ( produc7225117575323628663at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) )
      = ( produc6161850002892822231at_nat @ Y5 @ X6 ) ) ).

% swap_simp
thf(fact_1092_swap__simp,axiom,
    ! [X6: rule,Y5: produc1996495991257130529ist_fm] :
      ( ( produc9213896635423732003ist_fm @ ( produc7063755442767259331ist_fm @ X6 @ Y5 ) )
      = ( produc491467635432902671m_rule @ Y5 @ X6 ) ) ).

% swap_simp
thf(fact_1093_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream8299795917829157543ist_fm] :
      ( ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S ) )
      = ( sdrop_4442373711808556042ist_fm @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_1094_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream3588992565182678257e_rule] :
      ( ( stl_Pr4904654673476115995e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S ) )
      = ( sdrop_2199339672054592340e_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_1095_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream1878305877988316353m_rule] :
      ( ( stl_Pr5425948690297933931m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S ) )
      = ( sdrop_1764812713384251940m_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_1096_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream2312013117288958913e_rule] :
      ( ( stl_Pr830518440239794027e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S ) )
      = ( sdrop_6392754500180887844e_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_1097_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream6494289010434245521m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S ) )
      = ( sdrop_7373388980841566196m_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_1098_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S ) )
      = ( sdrop_7224736112439592940m_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_1099_sdrop__simps_I2_J,axiom,
    ! [N: nat,S: stream_rule] :
      ( ( stl_rule @ ( sdrop_rule @ N @ S ) )
      = ( sdrop_rule @ ( suc @ N ) @ S ) ) ).

% sdrop_simps(2)
thf(fact_1100_snd__swap,axiom,
    ! [X6: produc164195504107695125m_rule] :
      ( ( produc4514744107061996309ist_fm @ ( produc2641608828089375343m_rule @ X6 ) )
      = ( produc6879501374131015971m_rule @ X6 ) ) ).

% snd_swap
thf(fact_1101_snd__swap,axiom,
    ! [X6: produc4630866025709511057ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc9213896635423732003ist_fm @ X6 ) )
      = ( produc4228417144610596823ist_fm @ X6 ) ) ).

% snd_swap
thf(fact_1102_snd__swap,axiom,
    ! [X6: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc7936585717479832313ist_fm @ X6 ) )
      = ( produc1501393135466168645ist_fm @ X6 ) ) ).

% snd_swap
thf(fact_1103_fst__swap,axiom,
    ! [X6: produc4630866025709511057ist_fm] :
      ( ( produc6879501374131015971m_rule @ ( produc9213896635423732003ist_fm @ X6 ) )
      = ( produc4514744107061996309ist_fm @ X6 ) ) ).

% fst_swap
thf(fact_1104_fst__swap,axiom,
    ! [X6: produc164195504107695125m_rule] :
      ( ( produc4228417144610596823ist_fm @ ( produc2641608828089375343m_rule @ X6 ) )
      = ( produc7165828336582415457m_rule @ X6 ) ) ).

% fst_swap
thf(fact_1105_fst__swap,axiom,
    ! [X6: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc7936585717479832313ist_fm @ X6 ) )
      = ( produc4588648349897876871ist_fm @ X6 ) ) ).

% fst_swap
thf(fact_1106_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1107_leD,axiom,
    ! [Y5: set_fm,X6: set_fm] :
      ( ( ord_less_eq_set_fm @ Y5 @ X6 )
     => ~ ( ord_less_set_fm @ X6 @ Y5 ) ) ).

% leD
thf(fact_1108_leD,axiom,
    ! [Y5: nat,X6: nat] :
      ( ( ord_less_eq_nat @ Y5 @ X6 )
     => ~ ( ord_less_nat @ X6 @ Y5 ) ) ).

% leD
thf(fact_1109_leI,axiom,
    ! [X6: nat,Y5: nat] :
      ( ~ ( ord_less_nat @ X6 @ Y5 )
     => ( ord_less_eq_nat @ Y5 @ X6 ) ) ).

% leI
thf(fact_1110_nless__le,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ~ ( ord_less_set_fm @ A @ B ) )
      = ( ~ ( ord_less_eq_set_fm @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1111_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1112_antisym__conv1,axiom,
    ! [X6: set_fm,Y5: set_fm] :
      ( ~ ( ord_less_set_fm @ X6 @ Y5 )
     => ( ( ord_less_eq_set_fm @ X6 @ Y5 )
        = ( X6 = Y5 ) ) ) ).

% antisym_conv1
thf(fact_1113_antisym__conv1,axiom,
    ! [X6: nat,Y5: nat] :
      ( ~ ( ord_less_nat @ X6 @ Y5 )
     => ( ( ord_less_eq_nat @ X6 @ Y5 )
        = ( X6 = Y5 ) ) ) ).

% antisym_conv1
thf(fact_1114_antisym__conv2,axiom,
    ! [X6: set_fm,Y5: set_fm] :
      ( ( ord_less_eq_set_fm @ X6 @ Y5 )
     => ( ( ~ ( ord_less_set_fm @ X6 @ Y5 ) )
        = ( X6 = Y5 ) ) ) ).

% antisym_conv2
thf(fact_1115_antisym__conv2,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X6 @ Y5 )
     => ( ( ~ ( ord_less_nat @ X6 @ Y5 ) )
        = ( X6 = Y5 ) ) ) ).

% antisym_conv2
thf(fact_1116_less__le__not__le,axiom,
    ( ord_less_set_fm
    = ( ^ [X5: set_fm,Y6: set_fm] :
          ( ( ord_less_eq_set_fm @ X5 @ Y6 )
          & ~ ( ord_less_eq_set_fm @ Y6 @ X5 ) ) ) ) ).

% less_le_not_le
thf(fact_1117_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X5: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y6 )
          & ~ ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ).

% less_le_not_le
thf(fact_1118_not__le__imp__less,axiom,
    ! [Y5: nat,X6: nat] :
      ( ~ ( ord_less_eq_nat @ Y5 @ X6 )
     => ( ord_less_nat @ X6 @ Y5 ) ) ).

% not_le_imp_less
thf(fact_1119_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A5: set_fm,B7: set_fm] :
          ( ( ord_less_set_fm @ A5 @ B7 )
          | ( A5 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1120_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B7: nat] :
          ( ( ord_less_nat @ A5 @ B7 )
          | ( A5 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1121_order_Ostrict__iff__order,axiom,
    ( ord_less_set_fm
    = ( ^ [A5: set_fm,B7: set_fm] :
          ( ( ord_less_eq_set_fm @ A5 @ B7 )
          & ( A5 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1122_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A5 @ B7 )
          & ( A5 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1123_order_Ostrict__trans1,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_set_fm @ B @ C2 )
       => ( ord_less_set_fm @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1124_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1125_order_Ostrict__trans2,axiom,
    ! [A: set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ord_less_set_fm @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1126_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1127_order_Ostrict__iff__not,axiom,
    ( ord_less_set_fm
    = ( ^ [A5: set_fm,B7: set_fm] :
          ( ( ord_less_eq_set_fm @ A5 @ B7 )
          & ~ ( ord_less_eq_set_fm @ B7 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1128_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A5 @ B7 )
          & ~ ( ord_less_eq_nat @ B7 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1129_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [B7: set_fm,A5: set_fm] :
          ( ( ord_less_set_fm @ B7 @ A5 )
          | ( A5 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1130_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A5: nat] :
          ( ( ord_less_nat @ B7 @ A5 )
          | ( A5 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1131_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_fm
    = ( ^ [B7: set_fm,A5: set_fm] :
          ( ( ord_less_eq_set_fm @ B7 @ A5 )
          & ( A5 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1132_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B7 @ A5 )
          & ( A5 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1133_dual__order_Ostrict__trans1,axiom,
    ! [B: set_fm,A: set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_set_fm @ C2 @ B )
       => ( ord_less_set_fm @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1134_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1135_dual__order_Ostrict__trans2,axiom,
    ! [B: set_fm,A: set_fm,C2: set_fm] :
      ( ( ord_less_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ C2 @ B )
       => ( ord_less_set_fm @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1136_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1137_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_fm
    = ( ^ [B7: set_fm,A5: set_fm] :
          ( ( ord_less_eq_set_fm @ B7 @ A5 )
          & ~ ( ord_less_eq_set_fm @ A5 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1138_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B7 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1139_order_Ostrict__implies__order,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ( ord_less_eq_set_fm @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1140_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1141_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_fm,A: set_fm] :
      ( ( ord_less_set_fm @ B @ A )
     => ( ord_less_eq_set_fm @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1142_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1143_order__le__less,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [X5: set_fm,Y6: set_fm] :
          ( ( ord_less_set_fm @ X5 @ Y6 )
          | ( X5 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_1144_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X5: nat,Y6: nat] :
          ( ( ord_less_nat @ X5 @ Y6 )
          | ( X5 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_1145_order__less__le,axiom,
    ( ord_less_set_fm
    = ( ^ [X5: set_fm,Y6: set_fm] :
          ( ( ord_less_eq_set_fm @ X5 @ Y6 )
          & ( X5 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_1146_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X5: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y6 )
          & ( X5 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_1147_linorder__not__le,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ~ ( ord_less_eq_nat @ X6 @ Y5 ) )
      = ( ord_less_nat @ Y5 @ X6 ) ) ).

% linorder_not_le
thf(fact_1148_linorder__not__less,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ~ ( ord_less_nat @ X6 @ Y5 ) )
      = ( ord_less_eq_nat @ Y5 @ X6 ) ) ).

% linorder_not_less
thf(fact_1149_order__less__imp__le,axiom,
    ! [X6: set_fm,Y5: set_fm] :
      ( ( ord_less_set_fm @ X6 @ Y5 )
     => ( ord_less_eq_set_fm @ X6 @ Y5 ) ) ).

% order_less_imp_le
thf(fact_1150_order__less__imp__le,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ord_less_nat @ X6 @ Y5 )
     => ( ord_less_eq_nat @ X6 @ Y5 ) ) ).

% order_less_imp_le
thf(fact_1151_order__le__neq__trans,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_fm @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1152_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1153_order__neq__le__trans,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ( A != B )
     => ( ( ord_less_eq_set_fm @ A @ B )
       => ( ord_less_set_fm @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1154_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1155_order__le__less__trans,axiom,
    ! [X6: set_fm,Y5: set_fm,Z2: set_fm] :
      ( ( ord_less_eq_set_fm @ X6 @ Y5 )
     => ( ( ord_less_set_fm @ Y5 @ Z2 )
       => ( ord_less_set_fm @ X6 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1156_order__le__less__trans,axiom,
    ! [X6: nat,Y5: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X6 @ Y5 )
     => ( ( ord_less_nat @ Y5 @ Z2 )
       => ( ord_less_nat @ X6 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1157_order__less__le__trans,axiom,
    ! [X6: set_fm,Y5: set_fm,Z2: set_fm] :
      ( ( ord_less_set_fm @ X6 @ Y5 )
     => ( ( ord_less_eq_set_fm @ Y5 @ Z2 )
       => ( ord_less_set_fm @ X6 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1158_order__less__le__trans,axiom,
    ! [X6: nat,Y5: nat,Z2: nat] :
      ( ( ord_less_nat @ X6 @ Y5 )
     => ( ( ord_less_eq_nat @ Y5 @ Z2 )
       => ( ord_less_nat @ X6 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1159_order__le__less__subst1,axiom,
    ! [A: set_fm,F2: nat > set_fm,B: nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_set_fm @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1160_order__le__less__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1161_order__le__less__subst2,axiom,
    ! [A: set_fm,B: set_fm,F2: set_fm > set_fm,C2: set_fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_set_fm @ ( F2 @ B ) @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_set_fm @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1162_order__le__less__subst2,axiom,
    ! [A: set_fm,B: set_fm,F2: set_fm > nat,C2: nat] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1163_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > set_fm,C2: set_fm] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_set_fm @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_set_fm @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1164_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_1165_order__less__le__subst1,axiom,
    ! [A: set_fm,F2: set_fm > set_fm,B: set_fm,C2: set_fm] :
      ( ( ord_less_set_fm @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_set_fm @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1166_order__less__le__subst1,axiom,
    ! [A: nat,F2: set_fm > nat,B: set_fm,C2: set_fm] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_set_fm @ B @ C2 )
       => ( ! [X3: set_fm,Y3: set_fm] :
              ( ( ord_less_eq_set_fm @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1167_order__less__le__subst1,axiom,
    ! [A: set_fm,F2: nat > set_fm,B: nat,C2: nat] :
      ( ( ord_less_set_fm @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_set_fm @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1168_order__less__le__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1169_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > set_fm,C2: set_fm] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_set_fm @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_set_fm @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_set_fm @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1170_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_1171_linorder__le__less__linear,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X6 @ Y5 )
      | ( ord_less_nat @ Y5 @ X6 ) ) ).

% linorder_le_less_linear
thf(fact_1172_order__le__imp__less__or__eq,axiom,
    ! [X6: set_fm,Y5: set_fm] :
      ( ( ord_less_eq_set_fm @ X6 @ Y5 )
     => ( ( ord_less_set_fm @ X6 @ Y5 )
        | ( X6 = Y5 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1173_order__le__imp__less__or__eq,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X6 @ Y5 )
     => ( ( ord_less_nat @ X6 @ Y5 )
        | ( X6 = Y5 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1174_gt__ex,axiom,
    ! [X6: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X6 @ X_12 ) ).

% gt_ex
thf(fact_1175_less__imp__neq,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ord_less_nat @ X6 @ Y5 )
     => ( X6 != Y5 ) ) ).

% less_imp_neq
thf(fact_1176_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_1177_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1178_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1179_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y7: nat] :
              ( ( ord_less_nat @ Y7 @ X3 )
             => ( P @ Y7 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_1180_antisym__conv3,axiom,
    ! [Y5: nat,X6: nat] :
      ( ~ ( ord_less_nat @ Y5 @ X6 )
     => ( ( ~ ( ord_less_nat @ X6 @ Y5 ) )
        = ( X6 = Y5 ) ) ) ).

% antisym_conv3
thf(fact_1181_linorder__cases,axiom,
    ! [X6: nat,Y5: nat] :
      ( ~ ( ord_less_nat @ X6 @ Y5 )
     => ( ( X6 != Y5 )
       => ( ord_less_nat @ Y5 @ X6 ) ) ) ).

% linorder_cases
thf(fact_1182_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1183_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1184_exists__least__iff,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X9: nat] : ( P5 @ X9 ) )
    = ( ^ [P3: nat > $o] :
        ? [N3: nat] :
          ( ( P3 @ N3 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ~ ( P3 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1185_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1186_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1187_not__less__iff__gr__or__eq,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ~ ( ord_less_nat @ X6 @ Y5 ) )
      = ( ( ord_less_nat @ Y5 @ X6 )
        | ( X6 = Y5 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1188_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1189_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1190_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1191_linorder__neqE,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( X6 != Y5 )
     => ( ~ ( ord_less_nat @ X6 @ Y5 )
       => ( ord_less_nat @ Y5 @ X6 ) ) ) ).

% linorder_neqE
thf(fact_1192_order__less__asym,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ord_less_nat @ X6 @ Y5 )
     => ~ ( ord_less_nat @ Y5 @ X6 ) ) ).

% order_less_asym
thf(fact_1193_linorder__neq__iff,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( X6 != Y5 )
      = ( ( ord_less_nat @ X6 @ Y5 )
        | ( ord_less_nat @ Y5 @ X6 ) ) ) ).

% linorder_neq_iff
thf(fact_1194_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1195_order__less__trans,axiom,
    ! [X6: nat,Y5: nat,Z2: nat] :
      ( ( ord_less_nat @ X6 @ Y5 )
     => ( ( ord_less_nat @ Y5 @ Z2 )
       => ( ord_less_nat @ X6 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_1196_ord__eq__less__subst,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1197_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1198_order__less__irrefl,axiom,
    ! [X6: nat] :
      ~ ( ord_less_nat @ X6 @ X6 ) ).

% order_less_irrefl
thf(fact_1199_order__less__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1200_order__less__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1201_order__less__not__sym,axiom,
    ! [X6: nat,Y5: nat] :
      ( ( ord_less_nat @ X6 @ Y5 )
     => ~ ( ord_less_nat @ Y5 @ X6 ) ) ).

% order_less_not_sym
thf(fact_1202_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1203_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1204_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_1205_less__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1206_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_1207_le__trans,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K2 )
       => ( ord_less_eq_nat @ I @ K2 ) ) ) ).

% le_trans
thf(fact_1208_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_1209_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_1210_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_1211_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K2: nat,B: nat] :
      ( ( P @ K2 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y7: nat] :
                ( ( P @ Y7 )
               => ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1212_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1213_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1214_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1215_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
     => ? [M5: nat] :
          ( M4
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_1216_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1217_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1218_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1219_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M6: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M6 ) @ N2 )
             => ( P @ M6 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1220_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1221_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R3: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R3 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z4: nat] :
              ( ( R3 @ X3 @ Y3 )
             => ( ( R3 @ Y3 @ Z4 )
               => ( R3 @ X3 @ Z4 ) ) )
         => ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
           => ( R3 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1222_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
          & ( M3 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_1223_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_1224_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
          | ( M3 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1225_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_1226_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_1227_less__mono__imp__le__mono,axiom,
    ! [F2: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1228_eq__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ( minus_minus_nat @ M @ K2 )
            = ( minus_minus_nat @ N @ K2 ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1229_le__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1230_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1231_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1232_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1233_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1234_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_1235_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1236_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1237_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1238_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1239_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1240_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1241_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1242_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1243_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1244_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1245_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N @ K )
         => ( P @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_1246_prod__decode__aux_Oelims,axiom,
    ! [X6: nat,Xa2: nat,Y5: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X6 @ Xa2 )
        = Y5 )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X6 )
         => ( Y5
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X6 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X6 )
         => ( Y5
            = ( nat_prod_decode_aux @ ( suc @ X6 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X6 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_1247_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K3 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_1248_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1249_per__def,axiom,
    ! [R: rule] :
      ( ( abstra4153889242452539894ist_fm
        @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R )
      = ( ! [S2: produc1996495991257130529ist_fm,R1: rule,Sl6: fset_P661503646757059847ist_fm,S8: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ S2 @ top_to728902761071950545ist_fm )
              & ( abstra3055540779341473792ist_fm
                @ ^ [R2: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R
                @ S2 )
              & ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ rules ) @ ( insert_rule @ R @ bot_bot_set_rule ) ) )
              & ( ( eff @ R1 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Sl6 ) )
              & ( fmembe3381613331217039976ist_fm @ S8 @ Sl6 ) )
           => ( abstra3055540779341473792ist_fm
              @ ^ [R2: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ T )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R
              @ S8 ) ) ) ) ).

% per_def
thf(fact_1250_NE__R,axiom,
    ( ( sset_rule @ rules )
   != bot_bot_set_rule ) ).

% NE_R
thf(fact_1251_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1252_eff_Osimps_I2_J,axiom,
    ! [P: nat,Ts: list_tm,A4: list_fm,B4: list_fm] :
      ( ( ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A4 ) )
          & ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B4 ) ) )
       => ( ( eff @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
          = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
      & ( ~ ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A4 ) )
            & ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B4 ) ) )
       => ( ( eff @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(2)
thf(fact_1253_pickEff__def,axiom,
    ! [R: rule,S: produc1996495991257130529ist_fm] :
      ( ( ( abstra3055540779341473792ist_fm
          @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S2 )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ R
          @ S )
       => ( ( abstra458490714785879965ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R
            @ S )
          = ( fChoic3892542562059901019ist_fm
            @ ^ [Sl: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Sl ) ) ) ) )
      & ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R
            @ S )
       => ( ( abstra458490714785879965ist_fm
            @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R
            @ S )
          = ( the_fs400605564276472219ist_fm @ none_f2701453242279119720ist_fm ) ) ) ) ).

% pickEff_def
thf(fact_1254_rule_Oinject_I1_J,axiom,
    ! [X21: nat,X222: list_tm,Y21: nat,Y22: list_tm] :
      ( ( ( axiom @ X21 @ X222 )
        = ( axiom @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% rule.inject(1)
thf(fact_1255_rule_Odistinct_I23_J,axiom,
    ! [X21: nat,X222: list_tm,X71: tm,X72: fm] :
      ( ( axiom @ X21 @ X222 )
     != ( uniL @ X71 @ X72 ) ) ).

% rule.distinct(23)
thf(fact_1256_eff_Osimps_I1_J,axiom,
    ! [A4: list_fm,B4: list_fm] :
      ( ( eff @ idle @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
      = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A4 @ B4 ) @ bot_bo2367426573206113139ist_fm ) ) ) ).

% eff.simps(1)
thf(fact_1257_rule_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_tm] :
      ( idle
     != ( axiom @ X21 @ X222 ) ) ).

% rule.distinct(1)
thf(fact_1258_rule_Odistinct_I11_J,axiom,
    ! [X71: tm,X72: fm] :
      ( idle
     != ( uniL @ X71 @ X72 ) ) ).

% rule.distinct(11)
thf(fact_1259_eff_Opsimps_I2_J,axiom,
    ! [P: nat,Ts: list_tm,A4: list_fm,B4: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A4 @ B4 ) ) )
     => ( ( ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A4 ) )
            & ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B4 ) ) )
         => ( ( eff @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
            = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
        & ( ~ ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A4 ) )
              & ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B4 ) ) )
         => ( ( eff @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(2)
thf(fact_1260_eff_Opsimps_I1_J,axiom,
    ! [A4: list_fm,B4: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ idle @ ( produc381145313068854617ist_fm @ A4 @ B4 ) ) )
     => ( ( eff @ idle @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
        = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A4 @ B4 ) @ bot_bo2367426573206113139ist_fm ) ) ) ) ).

% eff.psimps(1)
thf(fact_1261_eff_Opsimps_I3_J,axiom,
    ! [A4: list_fm,B4: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ flsL @ ( produc381145313068854617ist_fm @ A4 @ B4 ) ) )
     => ( ( ( member_fm @ falsity @ ( set_fm2 @ A4 ) )
         => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
            = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
        & ( ~ ( member_fm @ falsity @ ( set_fm2 @ A4 ) )
         => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(3)
thf(fact_1262_rule_Odistinct_I15_J,axiom,
    ! [X21: nat,X222: list_tm] :
      ( ( axiom @ X21 @ X222 )
     != flsL ) ).

% rule.distinct(15)
thf(fact_1263_rule_Odistinct_I33_J,axiom,
    ! [X71: tm,X72: fm] :
      ( flsL
     != ( uniL @ X71 @ X72 ) ) ).

% rule.distinct(33)
thf(fact_1264_fm_Odistinct_I5_J,axiom,
    ! [X4: fm] :
      ( falsity
     != ( uni @ X4 ) ) ).

% fm.distinct(5)
thf(fact_1265_fm_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_tm] :
      ( falsity
     != ( pre @ X21 @ X222 ) ) ).

% fm.distinct(1)
thf(fact_1266_rule_Odistinct_I3_J,axiom,
    idle != flsL ).

% rule.distinct(3)
thf(fact_1267_eff_Osimps_I3_J,axiom,
    ! [A4: list_fm,B4: list_fm] :
      ( ( ( member_fm @ falsity @ ( set_fm2 @ A4 ) )
       => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
          = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
      & ( ~ ( member_fm @ falsity @ ( set_fm2 @ A4 ) )
       => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(3)
thf(fact_1268_prod__decode__aux_Opelims,axiom,
    ! [X6: nat,Xa2: nat,Y5: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X6 @ Xa2 )
        = Y5 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X6 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X6 )
               => ( Y5
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X6 @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X6 )
               => ( Y5
                  = ( nat_prod_decode_aux @ ( suc @ X6 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X6 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X6 @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_1269_eff_Opsimps_I4_J,axiom,
    ! [A4: list_fm,B4: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ flsR @ ( produc381145313068854617ist_fm @ A4 @ B4 ) ) )
     => ( ( ( member_fm @ falsity @ ( set_fm2 @ B4 ) )
         => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
            = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A4 @ ( removeAll_fm @ falsity @ B4 ) ) @ bot_bo2367426573206113139ist_fm ) ) ) )
        & ( ~ ( member_fm @ falsity @ ( set_fm2 @ B4 ) )
         => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(4)
thf(fact_1270_eff_Osimps_I4_J,axiom,
    ! [B4: list_fm,A4: list_fm] :
      ( ( ( member_fm @ falsity @ ( set_fm2 @ B4 ) )
       => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
          = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A4 @ ( removeAll_fm @ falsity @ B4 ) ) @ bot_bo2367426573206113139ist_fm ) ) ) )
      & ( ~ ( member_fm @ falsity @ ( set_fm2 @ B4 ) )
       => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A4 @ B4 ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(4)
thf(fact_1271_rule_Odistinct_I17_J,axiom,
    ! [X21: nat,X222: list_tm] :
      ( ( axiom @ X21 @ X222 )
     != flsR ) ).

% rule.distinct(17)
thf(fact_1272_rule_Odistinct_I41_J,axiom,
    ! [X71: tm,X72: fm] :
      ( flsR
     != ( uniL @ X71 @ X72 ) ) ).

% rule.distinct(41)
thf(fact_1273_rule_Odistinct_I5_J,axiom,
    idle != flsR ).

% rule.distinct(5)
thf(fact_1274_rule_Odistinct_I27_J,axiom,
    flsL != flsR ).

% rule.distinct(27)

% Helper facts (22)
thf(help_fChoice_1_1_fChoice_001t__Syntax__Ofm_T,axiom,
    ! [P: fm > $o] :
      ( ( P @ ( fChoice_fm @ P ) )
      = ( ? [X: fm] : ( P @ X ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Syntax__Orule_T,axiom,
    ! [P: rule > $o] :
      ( ( P @ ( fChoice_rule @ P ) )
      = ( ? [X: rule] : ( P @ X ) ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X6: stream_rule,Y5: stream_rule] :
      ( ( if_stream_rule @ $false @ X6 @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X6: stream_rule,Y5: stream_rule] :
      ( ( if_stream_rule @ $true @ X6 @ Y5 )
      = X6 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X6 @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X6: product_prod_nat_nat,Y5: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X6 @ Y5 )
      = X6 ) ).

thf(help_fChoice_1_1_fChoice_001t__Abstract____Completeness__Otree_It__Syntax__Orule_J_T,axiom,
    ! [P: abstract_tree_rule > $o] :
      ( ( P @ ( fChoic8596484284512669613e_rule @ P ) )
      = ( ? [X: abstract_tree_rule] : ( P @ X ) ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
    ! [X6: produc1996495991257130529ist_fm,Y5: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $false @ X6 @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
    ! [X6: produc1996495991257130529ist_fm,Y5: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $true @ X6 @ Y5 )
      = X6 ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
    ! [P: produc1996495991257130529ist_fm > $o] :
      ( ( P @ ( fChoic1980167584695909237ist_fm @ P ) )
      = ( ? [X: produc1996495991257130529ist_fm] : ( P @ X ) ) ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [X6: fset_P661503646757059847ist_fm,Y5: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $false @ X6 @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [X6: fset_P661503646757059847ist_fm,Y5: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $true @ X6 @ Y5 )
      = X6 ) ).

thf(help_fChoice_1_1_fChoice_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o] :
      ( ( P @ ( fChoic3892542562059901019ist_fm @ P ) )
      = ( ? [X: fset_P661503646757059847ist_fm] : ( P @ X ) ) ) ).

thf(help_fChoice_1_1_fChoice_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_T,axiom,
    ! [P: produc859450856879609959at_nat > $o] :
      ( ( P @ ( fChoic6673916560228149179at_nat @ P ) )
      = ( ? [X: produc859450856879609959at_nat] : ( P @ X ) ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X6: stream727092118206550309m_rule,Y5: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $false @ X6 @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X6: stream727092118206550309m_rule,Y5: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $true @ X6 @ Y5 )
      = X6 ) ).

thf(help_If_2_1_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X6: abstra4572470701423150929m_rule,Y5: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $false @ X6 @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X6: abstra4572470701423150929m_rule,Y5: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $true @ X6 @ Y5 )
      = X6 ) ).

thf(help_fChoice_1_1_fChoice_001t__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [P: abstra4572470701423150929m_rule > $o] :
      ( ( P @ ( fChoic7543385803022166781m_rule @ P ) )
      = ( ? [X: abstra4572470701423150929m_rule] : ( P @ X ) ) ) ).

thf(help_If_3_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J_T,axiom,
    ! [X6: fset_A5396818216090053761m_rule,Y5: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $false @ X6 @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J_T,axiom,
    ! [X6: fset_A5396818216090053761m_rule,Y5: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $true @ X6 @ Y5 )
      = X6 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ? [K4: nat] :
      ( abstra3055540779341473792ist_fm
      @ ^ [R2: rule,S2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R2 @ S2 )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ ( uniL @ t @ p )
      @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ K4 @ steps ) ) ) ) ).

%------------------------------------------------------------------------------