TPTP Problem File: SLH0445^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FOL_Seq_Calc3/0011_Completeness/prob_00071_002239__12062980_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2008 ( 593 unt; 710 typ;   0 def)
%            Number of atoms       : 4291 (1113 equ;  10 cnn)
%            Maximal formula atoms :   15 (   3 avg)
%            Number of connectives : 15443 ( 293   ~;  68   |; 448   &;12651   @)
%                                         (   0 <=>;1983  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :  121 ( 120 usr)
%            Number of type conns  : 3345 (3345   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  594 ( 590 usr;  17 con; 0-4 aty)
%            Number of variables   : 4885 ( 894   ^;3723   !; 268   ?;4885   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:27:54.068
%------------------------------------------------------------------------------
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    stl_Pr3851248094077404016fm_nat: stream749138142146636976fm_nat > stream749138142146636976fm_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__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__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_Ostreams_001t__Nat__Onat,type,
    streams_nat: set_nat > set_stream_nat ).

thf(sy_c_Stream_Ostreams_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    stream8099348070969791641ist_fm: set_Pr7058068377845519745ist_fm > set_st6731251264560614279ist_fm ).

thf(sy_c_Stream_Ostreams_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,
    stream1749396743056113565m_rule: set_Pr1008144964186165195m_rule > set_st6921181922238103643m_rule ).

thf(sy_c_Stream_Ostreams_001t__Syntax__Ofm,type,
    streams_fm: set_fm > set_stream_fm ).

thf(sy_c_Stream_Ostreams_001t__Syntax__Orule,type,
    streams_rule: set_rule > set_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__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__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    szip_P2499414959592755846m_rule: stream727092118206550309m_rule > stream727092118206550309m_rule > stream6494289010434245521m_rule ).

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

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

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

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

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__Nat__Onat,type,
    fChoice_nat: ( nat > $o ) > nat ).

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__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    fChoic9196029409387999681m_rule: ( produc164195504107695125m_rule > $o ) > produc164195504107695125m_rule ).

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__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__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__Stream__Ostream_It__Nat__Onat_J,type,
    member_stream_nat: stream_nat > set_stream_nat > $o ).

thf(sy_c_member_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    member5006235884692811216ist_fm: stream8299795917829157543ist_fm > set_st6731251264560614279ist_fm > $o ).

thf(sy_c_member_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,
    member6117863459850561468m_rule: stream727092118206550309m_rule > set_st6921181922238103643m_rule > $o ).

thf(sy_c_member_001t__Stream__Ostream_It__Syntax__Ofm_J,type,
    member_stream_fm: stream_fm > set_stream_fm > $o ).

thf(sy_c_member_001t__Stream__Ostream_It__Syntax__Orule_J,type,
    member_stream_rule: stream_rule > set_stream_rule > $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_ss,type,
    ss: fset_P661503646757059847ist_fm ).

thf(sy_v_steps,type,
    steps: stream727092118206550309m_rule ).

% Relevant facts (1277)
thf(fact_0_assms_I2_J,axiom,
    ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ steps ) ) )
    = ( some_f2162114139285431020ist_fm @ ss ) ) ).

% assms(2)
thf(fact_1_stream_Oexpand,axiom,
    ! [Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( ( ( shd_Pr7235097944458474089m_rule @ Stream )
          = ( shd_Pr7235097944458474089m_rule @ Stream2 ) )
        & ( ( stl_Pr950425576149878629m_rule @ Stream )
          = ( stl_Pr950425576149878629m_rule @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_2_stream_Oexpand,axiom,
    ! [Stream: stream_rule,Stream2: stream_rule] :
      ( ( ( ( shd_rule @ Stream )
          = ( shd_rule @ Stream2 ) )
        & ( ( stl_rule @ Stream )
          = ( stl_rule @ Stream2 ) ) )
     => ( Stream = Stream2 ) ) ).

% stream.expand
thf(fact_3_stream_Ocoinduct,axiom,
    ! [R: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( R @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_4_stream_Ocoinduct,axiom,
    ! [R: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( R @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_5_stream_Ocoinduct__strong,axiom,
    ! [R: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( ( R @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) )
                | ( ( stl_Pr950425576149878629m_rule @ Stream3 )
                  = ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_6_stream_Ocoinduct__strong,axiom,
    ! [R: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( ( R @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) )
                | ( ( stl_rule @ Stream3 )
                  = ( stl_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_7_fset__eqI,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A )
          = ( fmembe3381613331217039976ist_fm @ X @ B ) )
     => ( A = B ) ) ).

% fset_eqI
thf(fact_8_fset__eqI,axiom,
    ! [A: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A )
          = ( fmembe1635983029874454474m_rule @ X @ B ) )
     => ( A = B ) ) ).

% fset_eqI
thf(fact_9_fequalityCE,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( A = B )
     => ( ( ( fmembe3381613331217039976ist_fm @ C @ A )
         => ~ ( fmembe3381613331217039976ist_fm @ C @ B ) )
       => ~ ( ~ ( fmembe3381613331217039976ist_fm @ C @ A )
           => ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ) ).

% fequalityCE
thf(fact_10_fequalityCE,axiom,
    ! [A: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( A = B )
     => ( ( ( fmembe1635983029874454474m_rule @ C @ A )
         => ~ ( fmembe1635983029874454474m_rule @ C @ B ) )
       => ~ ( ~ ( fmembe1635983029874454474m_rule @ C @ A )
           => ( fmembe1635983029874454474m_rule @ C @ B ) ) ) ) ).

% fequalityCE
thf(fact_11_eq__fmem__trans,axiom,
    ! [A2: produc1996495991257130529ist_fm,B2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( A2 = B2 )
     => ( ( fmembe3381613331217039976ist_fm @ B2 @ A )
       => ( fmembe3381613331217039976ist_fm @ A2 @ A ) ) ) ).

% eq_fmem_trans
thf(fact_12_eq__fmem__trans,axiom,
    ! [A2: abstra4572470701423150929m_rule,B2: abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule] :
      ( ( A2 = B2 )
     => ( ( fmembe1635983029874454474m_rule @ B2 @ A )
       => ( fmembe1635983029874454474m_rule @ A2 @ A ) ) ) ).

% eq_fmem_trans
thf(fact_13_eqfset__imp__iff,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,X2: produc1996495991257130529ist_fm] :
      ( ( A = B )
     => ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
        = ( fmembe3381613331217039976ist_fm @ X2 @ B ) ) ) ).

% eqfset_imp_iff
thf(fact_14_eqfset__imp__iff,axiom,
    ! [A: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,X2: abstra4572470701423150929m_rule] :
      ( ( A = B )
     => ( ( fmembe1635983029874454474m_rule @ X2 @ A )
        = ( fmembe1635983029874454474m_rule @ X2 @ B ) ) ) ).

% eqfset_imp_iff
thf(fact_15_if__split__fmem1,axiom,
    ! [Q: $o,X2: produc1996495991257130529ist_fm,Y: produc1996495991257130529ist_fm,B2: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ ( if_Pro3930376587665744871ist_fm @ Q @ X2 @ Y ) @ B2 )
      = ( ( Q
         => ( fmembe3381613331217039976ist_fm @ X2 @ B2 ) )
        & ( ~ Q
         => ( fmembe3381613331217039976ist_fm @ Y @ B2 ) ) ) ) ).

% if_split_fmem1
thf(fact_16_if__split__fmem1,axiom,
    ! [Q: $o,X2: abstra4572470701423150929m_rule,Y: abstra4572470701423150929m_rule,B2: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ ( if_Abs4566652138351786251m_rule @ Q @ X2 @ Y ) @ B2 )
      = ( ( Q
         => ( fmembe1635983029874454474m_rule @ X2 @ B2 ) )
        & ( ~ Q
         => ( fmembe1635983029874454474m_rule @ Y @ B2 ) ) ) ) ).

% if_split_fmem1
thf(fact_17_if__split__fmem2,axiom,
    ! [A2: produc1996495991257130529ist_fm,Q: $o,X2: fset_P661503646757059847ist_fm,Y: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A2 @ ( if_fse3714977293194272717ist_fm @ Q @ X2 @ Y ) )
      = ( ( Q
         => ( fmembe3381613331217039976ist_fm @ A2 @ X2 ) )
        & ( ~ Q
         => ( fmembe3381613331217039976ist_fm @ A2 @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_18_if__split__fmem2,axiom,
    ! [A2: abstra4572470701423150929m_rule,Q: $o,X2: fset_A5396818216090053761m_rule,Y: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ A2 @ ( if_fse3370020817956878651m_rule @ Q @ X2 @ Y ) )
      = ( ( Q
         => ( fmembe1635983029874454474m_rule @ A2 @ X2 ) )
        & ( ~ Q
         => ( fmembe1635983029874454474m_rule @ A2 @ Y ) ) ) ) ).

% if_split_fmem2
thf(fact_19_eqfelem__imp__iff,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( X2 = Y )
     => ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
        = ( fmembe3381613331217039976ist_fm @ Y @ A ) ) ) ).

% eqfelem_imp_iff
thf(fact_20_eqfelem__imp__iff,axiom,
    ! [X2: abstra4572470701423150929m_rule,Y: abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule] :
      ( ( X2 = Y )
     => ( ( fmembe1635983029874454474m_rule @ X2 @ A )
        = ( fmembe1635983029874454474m_rule @ Y @ A ) ) ) ).

% eqfelem_imp_iff
thf(fact_21_stream_Ocase__eq__if,axiom,
    ( case_s2494923709252486212m_rule
    = ( ^ [F: rule > stream_rule > abstra4572470701423150929m_rule,Stream5: stream_rule] : ( F @ ( shd_rule @ Stream5 ) @ ( stl_rule @ Stream5 ) ) ) ) ).

% stream.case_eq_if
thf(fact_22_option_Oinject,axiom,
    ! [X22: fset_P661503646757059847ist_fm,Y2: fset_P661503646757059847ist_fm] :
      ( ( ( some_f2162114139285431020ist_fm @ X22 )
        = ( some_f2162114139285431020ist_fm @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_23_prod__eq__iff,axiom,
    ( ( ^ [Y3: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y3 = Z ) )
    = ( ^ [S: produc164195504107695125m_rule,T: produc164195504107695125m_rule] :
          ( ( ( produc6879501374131015971m_rule @ S )
            = ( produc6879501374131015971m_rule @ T ) )
          & ( ( produc7165828336582415457m_rule @ S )
            = ( produc7165828336582415457m_rule @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_24_exE__realizer_H,axiom,
    ! [P: rule > produc1996495991257130529ist_fm > $o,P2: produc164195504107695125m_rule] :
      ( ( P @ ( produc7165828336582415457m_rule @ P2 ) @ ( produc6879501374131015971m_rule @ P2 ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y4: rule] :
            ~ ( P @ Y4 @ X ) ) ).

% exE_realizer'
thf(fact_25_prod__eqI,axiom,
    ! [P2: produc164195504107695125m_rule,Q2: produc164195504107695125m_rule] :
      ( ( ( produc6879501374131015971m_rule @ P2 )
        = ( produc6879501374131015971m_rule @ Q2 ) )
     => ( ( ( produc7165828336582415457m_rule @ P2 )
          = ( produc7165828336582415457m_rule @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_26_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_27_assms_I1_J,axiom,
    ( abstra5415202973855822231ist_fm
    @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R2 @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ steps ) ).

% assms(1)
thf(fact_28_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_29_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( stl_Pr830518440239794027e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
      = ( szip_P553291425819358518e_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_30_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr5425948690297933931m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
      = ( szip_r2254114822203693110m_rule @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_31_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( stl_Pr4904654673476115995e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_32_szip_Osimps_I2_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% szip.simps(2)
thf(fact_33_sdrop__while_Osimps,axiom,
    ( sdrop_5541336384294084785m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( if_str8948254419368749791m_rule @ ( P3 @ ( shd_Pr7235097944458474089m_rule @ S ) ) @ ( sdrop_5541336384294084785m_rule @ P3 @ ( stl_Pr950425576149878629m_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_34_sdrop__while_Osimps,axiom,
    ( sdrop_while_rule
    = ( ^ [P3: rule > $o,S: stream_rule] : ( if_stream_rule @ ( P3 @ ( shd_rule @ S ) ) @ ( sdrop_while_rule @ P3 @ ( stl_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_35_stream_Osplit__sel__asm,axiom,
    ! [P: abstra4572470701423150929m_rule > $o,F2: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( P @ ( case_s2494923709252486212m_rule @ F2 @ Stream ) )
      = ( ~ ( ( Stream
              = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) )
            & ~ ( P @ ( F2 @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ) ) ) ).

% stream.split_sel_asm
thf(fact_36_stream_Osplit__sel,axiom,
    ! [P: abstra4572470701423150929m_rule > $o,F2: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( P @ ( case_s2494923709252486212m_rule @ F2 @ Stream ) )
      = ( ( Stream
          = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) )
       => ( P @ ( F2 @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ) ) ).

% stream.split_sel
thf(fact_37_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X3: stream3409308193418444653ist_fm > $o,X2: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream3409308193418444653ist_fm] :
            ( ( X3 @ X )
           => ? [Steps: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
                ( ( X = Steps )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) @ Sl )
                & ( Eff @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr8700568296466786583ist_fm @ Steps ) )
                  | ( abstra5061764961602482947ist_fm @ Eff @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ Eff @ Rules @ X2 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_38_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X3: stream5128679079952569136_fm_fm > $o,X2: stream5128679079952569136_fm_fm,Rules: stream_fm,Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream5128679079952569136_fm_fm] :
            ( ( X3 @ X )
           => ? [Steps: stream5128679079952569136_fm_fm,Sl: fset_P661503646757059847ist_fm] :
                ( ( X = Steps )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) @ Sl )
                & ( Eff @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) )
                  | ( abstra9155489748554225612ist_fm @ Eff @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ Eff @ Rules @ X2 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_39_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X3: stream749138142146636976fm_nat > $o,X2: stream749138142146636976fm_nat,Rules: stream_nat,Eff: nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream749138142146636976fm_nat] :
            ( ( X3 @ X )
           => ? [Steps: stream749138142146636976fm_nat,Sl: fset_P661503646757059847ist_fm] :
                ( ( X = Steps )
                & ( member_nat @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps ) ) @ ( sset_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ ( stl_Pr3851248094077404016fm_nat @ Steps ) ) ) @ Sl )
                & ( Eff @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps ) ) @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr3851248094077404016fm_nat @ Steps ) )
                  | ( abstra1795404587622969676ist_fm @ Eff @ Rules @ ( stl_Pr3851248094077404016fm_nat @ Steps ) ) ) ) )
       => ( abstra1795404587622969676ist_fm @ Eff @ Rules @ X2 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_40_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X3: stream7989562319868115861ist_fm > $o,X2: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream7989562319868115861ist_fm] :
            ( ( X3 @ X )
           => ? [Steps: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
                ( ( X = Steps )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) @ Sl )
                & ( Eff @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr6727363551237990101ist_fm @ Steps ) )
                  | ( abstra8066765647771630831m_rule @ Eff @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ Eff @ Rules @ X2 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_41_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X3: stream812778442145099016ule_fm > $o,X2: stream812778442145099016ule_fm,Rules: stream_fm,Eff: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream812778442145099016ule_fm] :
            ( ( X3 @ X )
           => ? [Steps: stream812778442145099016ule_fm,Sl: fset_A5396818216090053761m_rule] :
                ( ( X = Steps )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) @ Sl )
                & ( Eff @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr2064664776838053682ule_fm @ Steps ) )
                  | ( abstra1364558218721404774m_rule @ Eff @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ Eff @ Rules @ X2 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_42_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X3: stream493580325864649352le_nat > $o,X2: stream493580325864649352le_nat,Rules: stream_nat,Eff: nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream493580325864649352le_nat] :
            ( ( X3 @ X )
           => ? [Steps: stream493580325864649352le_nat,Sl: fset_A5396818216090053761m_rule] :
                ( ( X = Steps )
                & ( member_nat @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps ) ) @ ( sset_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ ( stl_Pr2395366152698928946le_nat @ Steps ) ) ) @ Sl )
                & ( Eff @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps ) ) @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr2395366152698928946le_nat @ Steps ) )
                  | ( abstra1783251563909148134m_rule @ Eff @ Rules @ ( stl_Pr2395366152698928946le_nat @ Steps ) ) ) ) )
       => ( abstra1783251563909148134m_rule @ Eff @ Rules @ X2 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_43_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X3: stream7755889244956737277e_rule > $o,X2: stream7755889244956737277e_rule,Rules: stream_rule,Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream7755889244956737277e_rule] :
            ( ( X3 @ X )
           => ? [Steps: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
                ( ( X = Steps )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) @ Sl )
                & ( Eff @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr8695508334200742183e_rule @ Steps ) )
                  | ( abstra5823804033175701339m_rule @ Eff @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ Eff @ Rules @ X2 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_44_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X3: stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule,Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X3 @ X )
           => ? [Steps: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
                ( ( X = Steps )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl )
                & ( Eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
                  | ( abstra5415202973855822231ist_fm @ Eff @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ Eff @ Rules @ X2 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_45_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps2: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Sl2: fset_P661503646757059847ist_fm,Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) @ Sl2 )
       => ( ( Eff @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ Sl2 )
         => ( ( abstra5061764961602482947ist_fm @ Eff @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) )
           => ( abstra5061764961602482947ist_fm @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_46_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps2: stream5128679079952569136_fm_fm,Rules: stream_fm,Sl2: fset_P661503646757059847ist_fm,Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( 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 ) ) ) @ Sl2 )
       => ( ( Eff @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ Sl2 )
         => ( ( abstra9155489748554225612ist_fm @ Eff @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) )
           => ( abstra9155489748554225612ist_fm @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_47_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps2: stream749138142146636976fm_nat,Rules: stream_nat,Sl2: fset_P661503646757059847ist_fm,Eff: nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_nat @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps2 ) ) @ ( sset_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ ( stl_Pr3851248094077404016fm_nat @ Steps2 ) ) ) @ Sl2 )
       => ( ( Eff @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps2 ) ) @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps2 ) ) @ Sl2 )
         => ( ( abstra1795404587622969676ist_fm @ Eff @ Rules @ ( stl_Pr3851248094077404016fm_nat @ Steps2 ) )
           => ( abstra1795404587622969676ist_fm @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_48_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps2: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Sl2: fset_A5396818216090053761m_rule,Eff: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) @ Sl2 )
       => ( ( Eff @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ Sl2 )
         => ( ( abstra8066765647771630831m_rule @ Eff @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) )
           => ( abstra8066765647771630831m_rule @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_49_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps2: stream812778442145099016ule_fm,Rules: stream_fm,Sl2: fset_A5396818216090053761m_rule,Eff: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) @ Sl2 )
       => ( ( Eff @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ Sl2 )
         => ( ( abstra1364558218721404774m_rule @ Eff @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) )
           => ( abstra1364558218721404774m_rule @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_50_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps2: stream493580325864649352le_nat,Rules: stream_nat,Sl2: fset_A5396818216090053761m_rule,Eff: nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member_nat @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps2 ) ) @ ( sset_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ ( stl_Pr2395366152698928946le_nat @ Steps2 ) ) ) @ Sl2 )
       => ( ( Eff @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps2 ) ) @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps2 ) ) @ Sl2 )
         => ( ( abstra1783251563909148134m_rule @ Eff @ Rules @ ( stl_Pr2395366152698928946le_nat @ Steps2 ) )
           => ( abstra1783251563909148134m_rule @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_51_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps2: stream7755889244956737277e_rule,Rules: stream_rule,Sl2: fset_A5396818216090053761m_rule,Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) @ Sl2 )
       => ( ( Eff @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ Sl2 )
         => ( ( abstra5823804033175701339m_rule @ Eff @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) )
           => ( abstra5823804033175701339m_rule @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_52_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps2: stream727092118206550309m_rule,Rules: stream_rule,Sl2: fset_P661503646757059847ist_fm,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl2 )
       => ( ( Eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ Sl2 )
         => ( ( abstra5415202973855822231ist_fm @ Eff @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps2 ) )
           => ( abstra5415202973855822231ist_fm @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_53_stream_Ocollapse,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_54_stream_Ocollapse,axiom,
    ! [Stream: stream_rule] :
      ( ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_55_stream_Oset__intros_I2_J,axiom,
    ! [Y: produc1996495991257130529ist_fm,X22: stream8299795917829157543ist_fm,X1: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ Y @ ( sset_P3431374123360452149ist_fm @ X22 ) )
     => ( member8102475879199740618ist_fm @ Y @ ( sset_P3431374123360452149ist_fm @ ( sCons_307821682964077496ist_fm @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_56_stream_Oset__intros_I2_J,axiom,
    ! [Y: fm,X22: stream_fm,X1: fm] :
      ( ( member_fm @ Y @ ( sset_fm @ X22 ) )
     => ( member_fm @ Y @ ( sset_fm @ ( sCons_fm @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_57_stream_Oset__intros_I2_J,axiom,
    ! [Y: nat,X22: stream_nat,X1: nat] :
      ( ( member_nat @ Y @ ( sset_nat @ X22 ) )
     => ( member_nat @ Y @ ( sset_nat @ ( sCons_nat @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_58_stream_Oset__intros_I2_J,axiom,
    ! [Y: rule,X22: stream_rule,X1: rule] :
      ( ( member_rule @ Y @ ( sset_rule @ X22 ) )
     => ( member_rule @ Y @ ( sset_rule @ ( sCons_rule @ X1 @ X22 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_59_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc1996495991257130529ist_fm,X22: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X1 @ ( sset_P3431374123360452149ist_fm @ ( sCons_307821682964077496ist_fm @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_60_stream_Oset__intros_I1_J,axiom,
    ! [X1: fm,X22: stream_fm] : ( member_fm @ X1 @ ( sset_fm @ ( sCons_fm @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_61_stream_Oset__intros_I1_J,axiom,
    ! [X1: nat,X22: stream_nat] : ( member_nat @ X1 @ ( sset_nat @ ( sCons_nat @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_62_stream_Oset__intros_I1_J,axiom,
    ! [X1: rule,X22: stream_rule] : ( member_rule @ X1 @ ( sset_rule @ ( sCons_rule @ X1 @ X22 ) ) ) ).

% stream.set_intros(1)
thf(fact_63_stream_Oset__cases,axiom,
    ! [E: produc1996495991257130529ist_fm,A2: stream8299795917829157543ist_fm] :
      ( ( member8102475879199740618ist_fm @ E @ ( sset_P3431374123360452149ist_fm @ A2 ) )
     => ( ! [Z2: stream8299795917829157543ist_fm] :
            ( A2
           != ( sCons_307821682964077496ist_fm @ E @ Z2 ) )
       => ~ ! [Z1: produc1996495991257130529ist_fm,Z2: stream8299795917829157543ist_fm] :
              ( ( A2
                = ( sCons_307821682964077496ist_fm @ Z1 @ Z2 ) )
             => ~ ( member8102475879199740618ist_fm @ E @ ( sset_P3431374123360452149ist_fm @ Z2 ) ) ) ) ) ).

% stream.set_cases
thf(fact_64_stream_Oset__cases,axiom,
    ! [E: fm,A2: stream_fm] :
      ( ( member_fm @ E @ ( sset_fm @ A2 ) )
     => ( ! [Z2: stream_fm] :
            ( A2
           != ( sCons_fm @ E @ Z2 ) )
       => ~ ! [Z1: fm,Z2: stream_fm] :
              ( ( A2
                = ( sCons_fm @ Z1 @ Z2 ) )
             => ~ ( member_fm @ E @ ( sset_fm @ Z2 ) ) ) ) ) ).

% stream.set_cases
thf(fact_65_stream_Oset__cases,axiom,
    ! [E: nat,A2: stream_nat] :
      ( ( member_nat @ E @ ( sset_nat @ A2 ) )
     => ( ! [Z2: stream_nat] :
            ( A2
           != ( sCons_nat @ E @ Z2 ) )
       => ~ ! [Z1: nat,Z2: stream_nat] :
              ( ( A2
                = ( sCons_nat @ Z1 @ Z2 ) )
             => ~ ( member_nat @ E @ ( sset_nat @ Z2 ) ) ) ) ) ).

% stream.set_cases
thf(fact_66_stream_Oset__cases,axiom,
    ! [E: rule,A2: stream_rule] :
      ( ( member_rule @ E @ ( sset_rule @ A2 ) )
     => ( ! [Z2: stream_rule] :
            ( A2
           != ( sCons_rule @ E @ Z2 ) )
       => ~ ! [Z1: rule,Z2: stream_rule] :
              ( ( A2
                = ( sCons_rule @ Z1 @ Z2 ) )
             => ~ ( member_rule @ E @ ( sset_rule @ Z2 ) ) ) ) ) ).

% stream.set_cases
thf(fact_67_stream_Oset__induct,axiom,
    ! [X2: produc1996495991257130529ist_fm,A2: stream8299795917829157543ist_fm,P: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X2 @ ( sset_P3431374123360452149ist_fm @ A2 ) )
     => ( ! [Z1: produc1996495991257130529ist_fm,Z2: stream8299795917829157543ist_fm] : ( P @ Z1 @ ( sCons_307821682964077496ist_fm @ Z1 @ Z2 ) )
       => ( ! [Z1: produc1996495991257130529ist_fm,Z2: stream8299795917829157543ist_fm,Xa: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Xa @ ( sset_P3431374123360452149ist_fm @ Z2 ) )
             => ( ( P @ Xa @ Z2 )
               => ( P @ Xa @ ( sCons_307821682964077496ist_fm @ Z1 @ Z2 ) ) ) )
         => ( P @ X2 @ A2 ) ) ) ) ).

% stream.set_induct
thf(fact_68_stream_Oset__induct,axiom,
    ! [X2: fm,A2: stream_fm,P: fm > stream_fm > $o] :
      ( ( member_fm @ X2 @ ( sset_fm @ A2 ) )
     => ( ! [Z1: fm,Z2: stream_fm] : ( P @ Z1 @ ( sCons_fm @ Z1 @ Z2 ) )
       => ( ! [Z1: fm,Z2: stream_fm,Xa: fm] :
              ( ( member_fm @ Xa @ ( sset_fm @ Z2 ) )
             => ( ( P @ Xa @ Z2 )
               => ( P @ Xa @ ( sCons_fm @ Z1 @ Z2 ) ) ) )
         => ( P @ X2 @ A2 ) ) ) ) ).

% stream.set_induct
thf(fact_69_stream_Oset__induct,axiom,
    ! [X2: nat,A2: stream_nat,P: nat > stream_nat > $o] :
      ( ( member_nat @ X2 @ ( sset_nat @ A2 ) )
     => ( ! [Z1: nat,Z2: stream_nat] : ( P @ Z1 @ ( sCons_nat @ Z1 @ Z2 ) )
       => ( ! [Z1: nat,Z2: stream_nat,Xa: nat] :
              ( ( member_nat @ Xa @ ( sset_nat @ Z2 ) )
             => ( ( P @ Xa @ Z2 )
               => ( P @ Xa @ ( sCons_nat @ Z1 @ Z2 ) ) ) )
         => ( P @ X2 @ A2 ) ) ) ) ).

% stream.set_induct
thf(fact_70_stream_Oset__induct,axiom,
    ! [X2: rule,A2: stream_rule,P: rule > stream_rule > $o] :
      ( ( member_rule @ X2 @ ( sset_rule @ A2 ) )
     => ( ! [Z1: rule,Z2: stream_rule] : ( P @ Z1 @ ( sCons_rule @ Z1 @ Z2 ) )
       => ( ! [Z1: rule,Z2: stream_rule,Xa: rule] :
              ( ( member_rule @ Xa @ ( sset_rule @ Z2 ) )
             => ( ( P @ Xa @ Z2 )
               => ( P @ Xa @ ( sCons_rule @ Z1 @ Z2 ) ) ) )
         => ( P @ X2 @ A2 ) ) ) ) ).

% stream.set_induct
thf(fact_71_sdrop__while__SCons,axiom,
    ! [P: rule > $o,A2: rule,S3: stream_rule] :
      ( ( ( P @ A2 )
       => ( ( sdrop_while_rule @ P @ ( sCons_rule @ A2 @ S3 ) )
          = ( sdrop_while_rule @ P @ S3 ) ) )
      & ( ~ ( P @ A2 )
       => ( ( sdrop_while_rule @ P @ ( sCons_rule @ A2 @ S3 ) )
          = ( sCons_rule @ A2 @ S3 ) ) ) ) ).

% sdrop_while_SCons
thf(fact_72_RuleSystem__Defs_Oepath_Ocong,axiom,
    abstra5415202973855822231ist_fm = abstra5415202973855822231ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_73_stl__def,axiom,
    ( stl_Pr950425576149878629m_rule
    = ( case_s474603282864825704m_rule
      @ ^ [X12: produc164195504107695125m_rule,X23: stream727092118206550309m_rule] : X23 ) ) ).

% stl_def
thf(fact_74_stl__def,axiom,
    ( stl_rule
    = ( case_s5295311805034529096m_rule
      @ ^ [X12: rule,X23: stream_rule] : X23 ) ) ).

% stl_def
thf(fact_75_stream_Ocase__distrib,axiom,
    ! [H: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,F2: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( H @ ( case_s2494923709252486212m_rule @ F2 @ Stream ) )
      = ( case_s2494923709252486212m_rule
        @ ^ [X12: rule,X23: stream_rule] : ( H @ ( F2 @ X12 @ X23 ) )
        @ Stream ) ) ).

% stream.case_distrib
thf(fact_76_shd__def,axiom,
    ( shd_Pr7235097944458474089m_rule
    = ( case_s4599053733867249880m_rule
      @ ^ [X12: produc164195504107695125m_rule,X23: stream727092118206550309m_rule] : X12 ) ) ).

% shd_def
thf(fact_77_shd__def,axiom,
    ( shd_rule
    = ( case_s3980788006751379256e_rule
      @ ^ [X12: rule,X23: stream_rule] : X12 ) ) ).

% shd_def
thf(fact_78_stl__sset,axiom,
    ! [X2: produc1996495991257130529ist_fm,A2: stream8299795917829157543ist_fm] :
      ( ( member8102475879199740618ist_fm @ X2 @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A2 ) ) )
     => ( member8102475879199740618ist_fm @ X2 @ ( sset_P3431374123360452149ist_fm @ A2 ) ) ) ).

% stl_sset
thf(fact_79_stl__sset,axiom,
    ! [X2: fm,A2: stream_fm] :
      ( ( member_fm @ X2 @ ( sset_fm @ ( stl_fm @ A2 ) ) )
     => ( member_fm @ X2 @ ( sset_fm @ A2 ) ) ) ).

% stl_sset
thf(fact_80_stl__sset,axiom,
    ! [X2: nat,A2: stream_nat] :
      ( ( member_nat @ X2 @ ( sset_nat @ ( stl_nat @ A2 ) ) )
     => ( member_nat @ X2 @ ( sset_nat @ A2 ) ) ) ).

% stl_sset
thf(fact_81_stl__sset,axiom,
    ! [X2: produc164195504107695125m_rule,A2: stream727092118206550309m_rule] :
      ( ( member4220325220686508332m_rule @ X2 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ A2 ) ) )
     => ( member4220325220686508332m_rule @ X2 @ ( sset_P4979403222752902913m_rule @ A2 ) ) ) ).

% stl_sset
thf(fact_82_stl__sset,axiom,
    ! [X2: rule,A2: stream_rule] :
      ( ( member_rule @ X2 @ ( sset_rule @ ( stl_rule @ A2 ) ) )
     => ( member_rule @ X2 @ ( sset_rule @ A2 ) ) ) ).

% stl_sset
thf(fact_83_shd__sset,axiom,
    ! [A2: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ A2 ) @ ( sset_P3431374123360452149ist_fm @ A2 ) ) ).

% shd_sset
thf(fact_84_shd__sset,axiom,
    ! [A2: stream_fm] : ( member_fm @ ( shd_fm @ A2 ) @ ( sset_fm @ A2 ) ) ).

% shd_sset
thf(fact_85_shd__sset,axiom,
    ! [A2: stream_nat] : ( member_nat @ ( shd_nat @ A2 ) @ ( sset_nat @ A2 ) ) ).

% shd_sset
thf(fact_86_shd__sset,axiom,
    ! [A2: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) @ ( sset_P4979403222752902913m_rule @ A2 ) ) ).

% shd_sset
thf(fact_87_shd__sset,axiom,
    ! [A2: stream_rule] : ( member_rule @ ( shd_rule @ A2 ) @ ( sset_rule @ A2 ) ) ).

% shd_sset
thf(fact_88_stream_Osel_I2_J,axiom,
    ! [X1: produc164195504107695125m_rule,X22: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X22 ) )
      = X22 ) ).

% stream.sel(2)
thf(fact_89_stream_Osel_I2_J,axiom,
    ! [X1: rule,X22: stream_rule] :
      ( ( stl_rule @ ( sCons_rule @ X1 @ X22 ) )
      = X22 ) ).

% stream.sel(2)
thf(fact_90_stream_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X22: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X22 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_91_stream_Osel_I1_J,axiom,
    ! [X1: rule,X22: stream_rule] :
      ( ( shd_rule @ ( sCons_rule @ X1 @ X22 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_92_stream_Ocase,axiom,
    ! [F2: rule > stream_rule > abstra4572470701423150929m_rule,X1: rule,X22: stream_rule] :
      ( ( case_s2494923709252486212m_rule @ F2 @ ( sCons_rule @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% stream.case
thf(fact_93_sset__induct,axiom,
    ! [Y: produc1996495991257130529ist_fm,S3: stream8299795917829157543ist_fm,P: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Y @ ( sset_P3431374123360452149ist_fm @ S3 ) )
     => ( ! [S4: stream8299795917829157543ist_fm] : ( P @ ( shd_Pr772355297128350925ist_fm @ S4 ) @ S4 )
       => ( ! [S4: stream8299795917829157543ist_fm,Y4: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Y4 @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ S4 ) ) )
             => ( ( P @ Y4 @ ( stl_Pr5027778045650968273ist_fm @ S4 ) )
               => ( P @ Y4 @ S4 ) ) )
         => ( P @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_94_sset__induct,axiom,
    ! [Y: fm,S3: stream_fm,P: fm > stream_fm > $o] :
      ( ( member_fm @ Y @ ( sset_fm @ S3 ) )
     => ( ! [S4: stream_fm] : ( P @ ( shd_fm @ S4 ) @ S4 )
       => ( ! [S4: stream_fm,Y4: fm] :
              ( ( member_fm @ Y4 @ ( sset_fm @ ( stl_fm @ S4 ) ) )
             => ( ( P @ Y4 @ ( stl_fm @ S4 ) )
               => ( P @ Y4 @ S4 ) ) )
         => ( P @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_95_sset__induct,axiom,
    ! [Y: nat,S3: stream_nat,P: nat > stream_nat > $o] :
      ( ( member_nat @ Y @ ( sset_nat @ S3 ) )
     => ( ! [S4: stream_nat] : ( P @ ( shd_nat @ S4 ) @ S4 )
       => ( ! [S4: stream_nat,Y4: nat] :
              ( ( member_nat @ Y4 @ ( sset_nat @ ( stl_nat @ S4 ) ) )
             => ( ( P @ Y4 @ ( stl_nat @ S4 ) )
               => ( P @ Y4 @ S4 ) ) )
         => ( P @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_96_sset__induct,axiom,
    ! [Y: produc164195504107695125m_rule,S3: stream727092118206550309m_rule,P: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ Y @ ( sset_P4979403222752902913m_rule @ S3 ) )
     => ( ! [S4: stream727092118206550309m_rule] : ( P @ ( shd_Pr7235097944458474089m_rule @ S4 ) @ S4 )
       => ( ! [S4: stream727092118206550309m_rule,Y4: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Y4 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ S4 ) ) )
             => ( ( P @ Y4 @ ( stl_Pr950425576149878629m_rule @ S4 ) )
               => ( P @ Y4 @ S4 ) ) )
         => ( P @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_97_sset__induct,axiom,
    ! [Y: rule,S3: stream_rule,P: rule > stream_rule > $o] :
      ( ( member_rule @ Y @ ( sset_rule @ S3 ) )
     => ( ! [S4: stream_rule] : ( P @ ( shd_rule @ S4 ) @ S4 )
       => ( ! [S4: stream_rule,Y4: rule] :
              ( ( member_rule @ Y4 @ ( sset_rule @ ( stl_rule @ S4 ) ) )
             => ( ( P @ Y4 @ ( stl_rule @ S4 ) )
               => ( P @ Y4 @ S4 ) ) )
         => ( P @ Y @ S3 ) ) ) ) ).

% sset_induct
thf(fact_98_stream_Oexhaust__sel,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( Stream
      = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_99_stream_Oexhaust__sel,axiom,
    ! [Stream: stream_rule] :
      ( Stream
      = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_100_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8299795917829157543ist_fm,A2: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ Eff @ Rules @ A2 )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl3: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ A2 ) ) ) @ Sl3 )
             => ( ( Eff @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A2 ) ) @ Sl3 )
               => ~ ( abstra5061764961602482947ist_fm @ Eff @ Rules @ ( stl_Pr8700568296466786583ist_fm @ A2 ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_101_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,A2: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ Eff @ Rules @ A2 )
     => ~ ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A2 ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl3: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ A2 ) ) ) @ Sl3 )
             => ( ( Eff @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A2 ) ) @ Sl3 )
               => ~ ( abstra9155489748554225612ist_fm @ Eff @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ A2 ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_102_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_nat,A2: stream749138142146636976fm_nat] :
      ( ( abstra1795404587622969676ist_fm @ Eff @ Rules @ A2 )
     => ~ ( ( member_nat @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ A2 ) ) @ ( sset_nat @ Rules ) )
         => ! [Sl3: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ ( stl_Pr3851248094077404016fm_nat @ A2 ) ) ) @ Sl3 )
             => ( ( Eff @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ A2 ) ) @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ A2 ) ) @ Sl3 )
               => ~ ( abstra1795404587622969676ist_fm @ Eff @ Rules @ ( stl_Pr3851248094077404016fm_nat @ A2 ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_103_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream8299795917829157543ist_fm,A2: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ Eff @ Rules @ A2 )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl3: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ A2 ) ) ) @ Sl3 )
             => ( ( Eff @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ A2 ) ) @ Sl3 )
               => ~ ( abstra8066765647771630831m_rule @ Eff @ Rules @ ( stl_Pr6727363551237990101ist_fm @ A2 ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_104_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_fm,A2: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ Eff @ Rules @ A2 )
     => ~ ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A2 ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl3: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ A2 ) ) ) @ Sl3 )
             => ( ( Eff @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ A2 ) ) @ Sl3 )
               => ~ ( abstra1364558218721404774m_rule @ Eff @ Rules @ ( stl_Pr2064664776838053682ule_fm @ A2 ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_105_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_nat,A2: stream493580325864649352le_nat] :
      ( ( abstra1783251563909148134m_rule @ Eff @ Rules @ A2 )
     => ~ ( ( member_nat @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ A2 ) ) @ ( sset_nat @ Rules ) )
         => ! [Sl3: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ ( stl_Pr2395366152698928946le_nat @ A2 ) ) ) @ Sl3 )
             => ( ( Eff @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ A2 ) ) @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ A2 ) ) @ Sl3 )
               => ~ ( abstra1783251563909148134m_rule @ Eff @ Rules @ ( stl_Pr2395366152698928946le_nat @ A2 ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_106_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_rule,A2: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ Eff @ Rules @ A2 )
     => ~ ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl3: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ A2 ) ) ) @ Sl3 )
             => ( ( Eff @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ A2 ) ) @ Sl3 )
               => ~ ( abstra5823804033175701339m_rule @ Eff @ Rules @ ( stl_Pr8695508334200742183e_rule @ A2 ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_107_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A2: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ Eff @ Rules @ A2 )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl3: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A2 ) ) ) @ Sl3 )
             => ( ( Eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) ) @ Sl3 )
               => ~ ( abstra5415202973855822231ist_fm @ Eff @ Rules @ ( stl_Pr950425576149878629m_rule @ A2 ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_108_mem__Collect__eq,axiom,
    ! [A2: rule,P: rule > $o] :
      ( ( member_rule @ A2 @ ( collect_rule @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
    ! [A2: produc1996495991257130529ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ A2 @ ( collec7231575776666803724ist_fm @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
    ! [A2: fm,P: fm > $o] :
      ( ( member_fm @ A2 @ ( collect_fm @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_112_Collect__mem__eq,axiom,
    ! [A: set_rule] :
      ( ( collect_rule
        @ ^ [X4: rule] : ( member_rule @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( collec7231575776666803724ist_fm
        @ ^ [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
    ! [A: set_fm] :
      ( ( collect_fm
        @ ^ [X4: fm] : ( member_fm @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_116_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X: nat] :
          ( ( P @ X )
          = ( Q @ X ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_117_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5061764961602482947ist_fm
    = ( ^ [Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8299795917829157543ist_fm,A3: stream3409308193418444653ist_fm] :
        ? [Steps3: stream3409308193418444653ist_fm,Sl4: fset_P661503646757059847ist_fm] :
          ( ( A3 = Steps3 )
          & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl4 )
          & ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl4 )
          & ( abstra5061764961602482947ist_fm @ Eff2 @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_118_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra9155489748554225612ist_fm
    = ( ^ [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,A3: stream5128679079952569136_fm_fm] :
        ? [Steps3: stream5128679079952569136_fm_fm,Sl4: fset_P661503646757059847ist_fm] :
          ( ( A3 = 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 ) ) ) @ Sl4 )
          & ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl4 )
          & ( abstra9155489748554225612ist_fm @ Eff2 @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_119_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1795404587622969676ist_fm
    = ( ^ [Eff2: nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_nat,A3: stream749138142146636976fm_nat] :
        ? [Steps3: stream749138142146636976fm_nat,Sl4: fset_P661503646757059847ist_fm] :
          ( ( A3 = Steps3 )
          & ( member_nat @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps3 ) ) @ ( sset_nat @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ ( stl_Pr3851248094077404016fm_nat @ Steps3 ) ) ) @ Sl4 )
          & ( Eff2 @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps3 ) ) @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps3 ) ) @ Sl4 )
          & ( abstra1795404587622969676ist_fm @ Eff2 @ Rules2 @ ( stl_Pr3851248094077404016fm_nat @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_120_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra8066765647771630831m_rule
    = ( ^ [Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8299795917829157543ist_fm,A3: stream7989562319868115861ist_fm] :
        ? [Steps3: stream7989562319868115861ist_fm,Sl4: fset_A5396818216090053761m_rule] :
          ( ( A3 = Steps3 )
          & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl4 )
          & ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl4 )
          & ( abstra8066765647771630831m_rule @ Eff2 @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_121_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1364558218721404774m_rule
    = ( ^ [Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_fm,A3: stream812778442145099016ule_fm] :
        ? [Steps3: stream812778442145099016ule_fm,Sl4: fset_A5396818216090053761m_rule] :
          ( ( A3 = Steps3 )
          & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl4 )
          & ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl4 )
          & ( abstra1364558218721404774m_rule @ Eff2 @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_122_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1783251563909148134m_rule
    = ( ^ [Eff2: nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_nat,A3: stream493580325864649352le_nat] :
        ? [Steps3: stream493580325864649352le_nat,Sl4: fset_A5396818216090053761m_rule] :
          ( ( A3 = Steps3 )
          & ( member_nat @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps3 ) ) @ ( sset_nat @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ ( stl_Pr2395366152698928946le_nat @ Steps3 ) ) ) @ Sl4 )
          & ( Eff2 @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps3 ) ) @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps3 ) ) @ Sl4 )
          & ( abstra1783251563909148134m_rule @ Eff2 @ Rules2 @ ( stl_Pr2395366152698928946le_nat @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_123_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5823804033175701339m_rule
    = ( ^ [Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_rule,A3: stream7755889244956737277e_rule] :
        ? [Steps3: stream7755889244956737277e_rule,Sl4: fset_A5396818216090053761m_rule] :
          ( ( A3 = Steps3 )
          & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl4 )
          & ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl4 )
          & ( abstra5823804033175701339m_rule @ Eff2 @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_124_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5415202973855822231ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A3: stream727092118206550309m_rule] :
        ? [Steps3: stream727092118206550309m_rule,Sl4: fset_P661503646757059847ist_fm] :
          ( ( A3 = Steps3 )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl4 )
          & ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl4 )
          & ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_125_epath_Ocases,axiom,
    ! [A2: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A2 )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) ) @ ( sset_rule @ rules ) )
         => ! [Sl3: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A2 ) ) ) @ Sl3 )
             => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) ) )
                  = ( some_f2162114139285431020ist_fm @ Sl3 ) )
               => ~ ( abstra5415202973855822231ist_fm
                    @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ ( stl_Pr950425576149878629m_rule @ A2 ) ) ) ) ) ) ).

% epath.cases
thf(fact_126_epath_Osimps,axiom,
    ! [A2: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A2 )
      = ( ? [Steps3: stream727092118206550309m_rule,Sl4: fset_P661503646757059847ist_fm] :
            ( ( A2 = Steps3 )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl4 )
            & ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) )
              = ( some_f2162114139285431020ist_fm @ Sl4 ) )
            & ( abstra5415202973855822231ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) ) ).

% epath.simps
thf(fact_127_epath_Ointros,axiom,
    ! [Steps2: stream727092118206550309m_rule,Sl2: fset_P661503646757059847ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl2 )
       => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) )
            = ( some_f2162114139285431020ist_fm @ Sl2 ) )
         => ( ( abstra5415202973855822231ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ ( stl_Pr950425576149878629m_rule @ Steps2 ) )
           => ( abstra5415202973855822231ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps2 ) ) ) ) ) ).

% epath.intros
thf(fact_128_epath_Ocoinduct,axiom,
    ! [X3: stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule] :
      ( ( X3 @ X2 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X3 @ X )
           => ? [Steps: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
                ( ( X = Steps )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
                  = ( some_f2162114139285431020ist_fm @ Sl ) )
                & ( ( X3 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
                  | ( abstra5415202973855822231ist_fm
                    @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) ) )
       => ( abstra5415202973855822231ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ X2 ) ) ) ).

% epath.coinduct
thf(fact_129_i_Oepath_Ocases,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8299795917829157543ist_fm,A2: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ A2 )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl3: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ A2 ) ) ) @ Sl3 )
             => ( ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A2 ) ) @ Sl3 )
               => ~ ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ A2 ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_130_i_Oepath_Ocases,axiom,
    ! [Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_fm,A2: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ A2 )
     => ~ ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A2 ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl3: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ A2 ) ) ) @ Sl3 )
             => ( ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A2 ) ) @ Sl3 )
               => ~ ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ A2 ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_131_i_Oepath_Ocases,axiom,
    ! [Eff3: nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_nat,A2: stream749138142146636976fm_nat] :
      ( ( abstra1795404587622969676ist_fm @ ( abstra7291082400731692943ist_fm @ Eff3 ) @ Rules @ A2 )
     => ~ ( ( member_nat @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ A2 ) ) @ ( sset_nat @ Rules ) )
         => ! [Sl3: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ ( stl_Pr3851248094077404016fm_nat @ A2 ) ) ) @ Sl3 )
             => ( ( abstra7291082400731692943ist_fm @ Eff3 @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ A2 ) ) @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ A2 ) ) @ Sl3 )
               => ~ ( abstra1795404587622969676ist_fm @ ( abstra7291082400731692943ist_fm @ Eff3 ) @ Rules @ ( stl_Pr3851248094077404016fm_nat @ A2 ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_132_i_Oepath_Ocases,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8299795917829157543ist_fm,A2: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ A2 )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl3: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ A2 ) ) ) @ Sl3 )
             => ( ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ A2 ) ) @ Sl3 )
               => ~ ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ A2 ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_133_i_Oepath_Ocases,axiom,
    ! [Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_fm,A2: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ A2 )
     => ~ ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A2 ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl3: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ A2 ) ) ) @ Sl3 )
             => ( ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ A2 ) ) @ Sl3 )
               => ~ ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ A2 ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_134_i_Oepath_Ocases,axiom,
    ! [Eff3: nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_nat,A2: stream493580325864649352le_nat] :
      ( ( abstra1783251563909148134m_rule @ ( abstra8187868039879872355m_rule @ Eff3 ) @ Rules @ A2 )
     => ~ ( ( member_nat @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ A2 ) ) @ ( sset_nat @ Rules ) )
         => ! [Sl3: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ ( stl_Pr2395366152698928946le_nat @ A2 ) ) ) @ Sl3 )
             => ( ( abstra8187868039879872355m_rule @ Eff3 @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ A2 ) ) @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ A2 ) ) @ Sl3 )
               => ~ ( abstra1783251563909148134m_rule @ ( abstra8187868039879872355m_rule @ Eff3 ) @ Rules @ ( stl_Pr2395366152698928946le_nat @ A2 ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_135_i_Oepath_Ocases,axiom,
    ! [Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_rule,A2: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ A2 )
     => ~ ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl3: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ A2 ) ) ) @ Sl3 )
             => ( ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ A2 ) ) @ Sl3 )
               => ~ ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ A2 ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_136_i_Oepath_Ocases,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A2: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A2 )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl3: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A2 ) ) ) @ Sl3 )
             => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A2 ) ) @ Sl3 )
               => ~ ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ A2 ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_137_i_Oepath_Osimps,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8299795917829157543ist_fm,A2: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ A2 )
      = ( ? [Steps3: stream3409308193418444653ist_fm,Sl4: fset_P661503646757059847ist_fm] :
            ( ( A2 = Steps3 )
            & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl4 )
            & ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl4 )
            & ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_138_i_Oepath_Osimps,axiom,
    ! [Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_fm,A2: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ A2 )
      = ( ? [Steps3: stream5128679079952569136_fm_fm,Sl4: fset_P661503646757059847ist_fm] :
            ( ( A2 = Steps3 )
            & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) @ Sl4 )
            & ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl4 )
            & ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_139_i_Oepath_Osimps,axiom,
    ! [Eff3: nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_nat,A2: stream749138142146636976fm_nat] :
      ( ( abstra1795404587622969676ist_fm @ ( abstra7291082400731692943ist_fm @ Eff3 ) @ Rules @ A2 )
      = ( ? [Steps3: stream749138142146636976fm_nat,Sl4: fset_P661503646757059847ist_fm] :
            ( ( A2 = Steps3 )
            & ( member_nat @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps3 ) ) @ ( sset_nat @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ ( stl_Pr3851248094077404016fm_nat @ Steps3 ) ) ) @ Sl4 )
            & ( abstra7291082400731692943ist_fm @ Eff3 @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps3 ) ) @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps3 ) ) @ Sl4 )
            & ( abstra1795404587622969676ist_fm @ ( abstra7291082400731692943ist_fm @ Eff3 ) @ Rules @ ( stl_Pr3851248094077404016fm_nat @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_140_i_Oepath_Osimps,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8299795917829157543ist_fm,A2: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ A2 )
      = ( ? [Steps3: stream7989562319868115861ist_fm,Sl4: fset_A5396818216090053761m_rule] :
            ( ( A2 = Steps3 )
            & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl4 )
            & ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl4 )
            & ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_141_i_Oepath_Osimps,axiom,
    ! [Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_fm,A2: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ A2 )
      = ( ? [Steps3: stream812778442145099016ule_fm,Sl4: fset_A5396818216090053761m_rule] :
            ( ( A2 = Steps3 )
            & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl4 )
            & ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl4 )
            & ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_142_i_Oepath_Osimps,axiom,
    ! [Eff3: nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_nat,A2: stream493580325864649352le_nat] :
      ( ( abstra1783251563909148134m_rule @ ( abstra8187868039879872355m_rule @ Eff3 ) @ Rules @ A2 )
      = ( ? [Steps3: stream493580325864649352le_nat,Sl4: fset_A5396818216090053761m_rule] :
            ( ( A2 = Steps3 )
            & ( member_nat @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps3 ) ) @ ( sset_nat @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ ( stl_Pr2395366152698928946le_nat @ Steps3 ) ) ) @ Sl4 )
            & ( abstra8187868039879872355m_rule @ Eff3 @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps3 ) ) @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps3 ) ) @ Sl4 )
            & ( abstra1783251563909148134m_rule @ ( abstra8187868039879872355m_rule @ Eff3 ) @ Rules @ ( stl_Pr2395366152698928946le_nat @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_143_i_Oepath_Osimps,axiom,
    ! [Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_rule,A2: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ A2 )
      = ( ? [Steps3: stream7755889244956737277e_rule,Sl4: fset_A5396818216090053761m_rule] :
            ( ( A2 = Steps3 )
            & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl4 )
            & ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl4 )
            & ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_144_i_Oepath_Osimps,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A2: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A2 )
      = ( ? [Steps3: stream727092118206550309m_rule,Sl4: fset_P661503646757059847ist_fm] :
            ( ( A2 = Steps3 )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl4 )
            & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl4 )
            & ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) ) ).

% i.epath.simps
thf(fact_145_i_Oepath_Ointros,axiom,
    ! [Steps2: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Sl2: fset_P661503646757059847ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) @ Sl2 )
       => ( ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ Sl2 )
         => ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) )
           => ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ Steps2 ) ) ) ) ) ).

% i.epath.intros
thf(fact_146_i_Oepath_Ointros,axiom,
    ! [Steps2: stream5128679079952569136_fm_fm,Rules: stream_fm,Sl2: fset_P661503646757059847ist_fm,Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( 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 ) ) ) @ Sl2 )
       => ( ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ Sl2 )
         => ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) )
           => ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ Steps2 ) ) ) ) ) ).

% i.epath.intros
thf(fact_147_i_Oepath_Ointros,axiom,
    ! [Steps2: stream749138142146636976fm_nat,Rules: stream_nat,Sl2: fset_P661503646757059847ist_fm,Eff3: nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_nat @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps2 ) ) @ ( sset_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ ( stl_Pr3851248094077404016fm_nat @ Steps2 ) ) ) @ Sl2 )
       => ( ( abstra7291082400731692943ist_fm @ Eff3 @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps2 ) ) @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps2 ) ) @ Sl2 )
         => ( ( abstra1795404587622969676ist_fm @ ( abstra7291082400731692943ist_fm @ Eff3 ) @ Rules @ ( stl_Pr3851248094077404016fm_nat @ Steps2 ) )
           => ( abstra1795404587622969676ist_fm @ ( abstra7291082400731692943ist_fm @ Eff3 ) @ Rules @ Steps2 ) ) ) ) ) ).

% i.epath.intros
thf(fact_148_i_Oepath_Ointros,axiom,
    ! [Steps2: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Sl2: fset_A5396818216090053761m_rule,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) @ Sl2 )
       => ( ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ Sl2 )
         => ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) )
           => ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ Steps2 ) ) ) ) ) ).

% i.epath.intros
thf(fact_149_i_Oepath_Ointros,axiom,
    ! [Steps2: stream812778442145099016ule_fm,Rules: stream_fm,Sl2: fset_A5396818216090053761m_rule,Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) @ Sl2 )
       => ( ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ Sl2 )
         => ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) )
           => ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ Steps2 ) ) ) ) ) ).

% i.epath.intros
thf(fact_150_i_Oepath_Ointros,axiom,
    ! [Steps2: stream493580325864649352le_nat,Rules: stream_nat,Sl2: fset_A5396818216090053761m_rule,Eff3: nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member_nat @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps2 ) ) @ ( sset_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ ( stl_Pr2395366152698928946le_nat @ Steps2 ) ) ) @ Sl2 )
       => ( ( abstra8187868039879872355m_rule @ Eff3 @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps2 ) ) @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps2 ) ) @ Sl2 )
         => ( ( abstra1783251563909148134m_rule @ ( abstra8187868039879872355m_rule @ Eff3 ) @ Rules @ ( stl_Pr2395366152698928946le_nat @ Steps2 ) )
           => ( abstra1783251563909148134m_rule @ ( abstra8187868039879872355m_rule @ Eff3 ) @ Rules @ Steps2 ) ) ) ) ) ).

% i.epath.intros
thf(fact_151_i_Oepath_Ointros,axiom,
    ! [Steps2: stream7755889244956737277e_rule,Rules: stream_rule,Sl2: fset_A5396818216090053761m_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) @ Sl2 )
       => ( ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ Sl2 )
         => ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) )
           => ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ Steps2 ) ) ) ) ) ).

% i.epath.intros
thf(fact_152_i_Oepath_Ointros,axiom,
    ! [Steps2: stream727092118206550309m_rule,Rules: stream_rule,Sl2: fset_P661503646757059847ist_fm,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl2 )
       => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ Sl2 )
         => ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps2 ) )
           => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps2 ) ) ) ) ) ).

% i.epath.intros
thf(fact_153_i_Oepath_Ocoinduct,axiom,
    ! [X3: stream3409308193418444653ist_fm > $o,X2: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X3 @ X2 )
     => ( ! [X: stream3409308193418444653ist_fm] :
            ( ( X3 @ X )
           => ? [Steps: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
                ( ( X = Steps )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) @ Sl )
                & ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr8700568296466786583ist_fm @ Steps ) )
                  | ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ X2 ) ) ) ).

% i.epath.coinduct
thf(fact_154_i_Oepath_Ocoinduct,axiom,
    ! [X3: stream5128679079952569136_fm_fm > $o,X2: stream5128679079952569136_fm_fm,Rules: stream_fm,Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X3 @ X2 )
     => ( ! [X: stream5128679079952569136_fm_fm] :
            ( ( X3 @ X )
           => ? [Steps: stream5128679079952569136_fm_fm,Sl: fset_P661503646757059847ist_fm] :
                ( ( X = Steps )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) @ Sl )
                & ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) )
                  | ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ X2 ) ) ) ).

% i.epath.coinduct
thf(fact_155_i_Oepath_Ocoinduct,axiom,
    ! [X3: stream749138142146636976fm_nat > $o,X2: stream749138142146636976fm_nat,Rules: stream_nat,Eff3: nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X3 @ X2 )
     => ( ! [X: stream749138142146636976fm_nat] :
            ( ( X3 @ X )
           => ? [Steps: stream749138142146636976fm_nat,Sl: fset_P661503646757059847ist_fm] :
                ( ( X = Steps )
                & ( member_nat @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps ) ) @ ( sset_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ ( stl_Pr3851248094077404016fm_nat @ Steps ) ) ) @ Sl )
                & ( abstra7291082400731692943ist_fm @ Eff3 @ ( produc7775955371974503660fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps ) ) @ ( produc8908898514464566190fm_nat @ ( shd_Pr1651134658286383732fm_nat @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr3851248094077404016fm_nat @ Steps ) )
                  | ( abstra1795404587622969676ist_fm @ ( abstra7291082400731692943ist_fm @ Eff3 ) @ Rules @ ( stl_Pr3851248094077404016fm_nat @ Steps ) ) ) ) )
       => ( abstra1795404587622969676ist_fm @ ( abstra7291082400731692943ist_fm @ Eff3 ) @ Rules @ X2 ) ) ) ).

% i.epath.coinduct
thf(fact_156_i_Oepath_Ocoinduct,axiom,
    ! [X3: stream7989562319868115861ist_fm > $o,X2: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X3 @ X2 )
     => ( ! [X: stream7989562319868115861ist_fm] :
            ( ( X3 @ X )
           => ? [Steps: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
                ( ( X = Steps )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) @ Sl )
                & ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr6727363551237990101ist_fm @ Steps ) )
                  | ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ X2 ) ) ) ).

% i.epath.coinduct
thf(fact_157_i_Oepath_Ocoinduct,axiom,
    ! [X3: stream812778442145099016ule_fm > $o,X2: stream812778442145099016ule_fm,Rules: stream_fm,Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X3 @ X2 )
     => ( ! [X: stream812778442145099016ule_fm] :
            ( ( X3 @ X )
           => ? [Steps: stream812778442145099016ule_fm,Sl: fset_A5396818216090053761m_rule] :
                ( ( X = Steps )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) @ Sl )
                & ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr2064664776838053682ule_fm @ Steps ) )
                  | ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ X2 ) ) ) ).

% i.epath.coinduct
thf(fact_158_i_Oepath_Ocoinduct,axiom,
    ! [X3: stream493580325864649352le_nat > $o,X2: stream493580325864649352le_nat,Rules: stream_nat,Eff3: nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X3 @ X2 )
     => ( ! [X: stream493580325864649352le_nat] :
            ( ( X3 @ X )
           => ? [Steps: stream493580325864649352le_nat,Sl: fset_A5396818216090053761m_rule] :
                ( ( X = Steps )
                & ( member_nat @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps ) ) @ ( sset_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ ( stl_Pr2395366152698928946le_nat @ Steps ) ) ) @ Sl )
                & ( abstra8187868039879872355m_rule @ Eff3 @ ( produc5752276462712533480le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps ) ) @ ( produc6777997145283087782le_nat @ ( shd_Pr1787092732227814702le_nat @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr2395366152698928946le_nat @ Steps ) )
                  | ( abstra1783251563909148134m_rule @ ( abstra8187868039879872355m_rule @ Eff3 ) @ Rules @ ( stl_Pr2395366152698928946le_nat @ Steps ) ) ) ) )
       => ( abstra1783251563909148134m_rule @ ( abstra8187868039879872355m_rule @ Eff3 ) @ Rules @ X2 ) ) ) ).

% i.epath.coinduct
thf(fact_159_i_Oepath_Ocoinduct,axiom,
    ! [X3: stream7755889244956737277e_rule > $o,X2: stream7755889244956737277e_rule,Rules: stream_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X3 @ X2 )
     => ( ! [X: stream7755889244956737277e_rule] :
            ( ( X3 @ X )
           => ? [Steps: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
                ( ( X = Steps )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) @ Sl )
                & ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr8695508334200742183e_rule @ Steps ) )
                  | ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ X2 ) ) ) ).

% i.epath.coinduct
thf(fact_160_i_Oepath_Ocoinduct,axiom,
    ! [X3: stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X3 @ X2 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X3 @ X )
           => ? [Steps: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
                ( ( X = Steps )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ Sl )
                & ( ( X3 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
                  | ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X2 ) ) ) ).

% i.epath.coinduct
thf(fact_161_sinterleave_Ocode,axiom,
    ( sinter8084399408526207096m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( sinter8084399408526207096m_rule @ S22 @ ( stl_Pr950425576149878629m_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_162_sinterleave_Ocode,axiom,
    ( sinterleave_rule
    = ( ^ [S12: stream_rule,S22: stream_rule] : ( sCons_rule @ ( shd_rule @ S12 ) @ ( sinterleave_rule @ S22 @ ( stl_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_163_epath__sdrop,axiom,
    ! [Steps2: stream727092118206550309m_rule,N: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps2 )
     => ( abstra5415202973855822231ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ ( sdrop_7224736112439592940m_rule @ N @ Steps2 ) ) ) ).

% epath_sdrop
thf(fact_164_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( szip_P2499414959592755846m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_165_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( sdrop_6392754500180887844e_rule @ N @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
      = ( szip_P553291425819358518e_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_166_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
      = ( szip_r2254114822203693110m_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_167_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( szip_rule_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_168_sdrop__szip,axiom,
    ! [N: nat,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( szip_P1977448745965526924m_rule @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_169_sdrop__stl,axiom,
    ! [N: nat,S3: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( stl_Pr950425576149878629m_rule @ S3 ) )
      = ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_170_sdrop__stl,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S3 ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_171_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S2 ) )
      = ( sinter8084399408526207096m_rule @ S2 @ ( stl_Pr950425576149878629m_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_172_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( stl_rule @ ( sinterleave_rule @ S1 @ S2 ) )
      = ( sinterleave_rule @ S2 @ ( stl_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_173_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S2 ) )
      = ( shd_Pr7235097944458474089m_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_174_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( shd_rule @ ( sinterleave_rule @ S1 @ S2 ) )
      = ( shd_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_175_Stream_Osmember__def,axiom,
    ( smembe2449974647026456671ist_fm
    = ( ^ [X4: produc1996495991257130529ist_fm,S: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X4 @ ( sset_P3431374123360452149ist_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_176_Stream_Osmember__def,axiom,
    ( smember_fm
    = ( ^ [X4: fm,S: stream_fm] : ( member_fm @ X4 @ ( sset_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_177_Stream_Osmember__def,axiom,
    ( smember_nat
    = ( ^ [X4: nat,S: stream_nat] : ( member_nat @ X4 @ ( sset_nat @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_178_Stream_Osmember__def,axiom,
    ( smember_rule
    = ( ^ [X4: rule,S: stream_rule] : ( member_rule @ X4 @ ( sset_rule @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_179_eff__S,axiom,
    ! [S3: produc1996495991257130529ist_fm,R3: rule,Sl2: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( member_rule @ R3 @ ( sset_rule @ rules ) )
       => ( ( ( eff @ R3 @ S3 )
            = ( some_f2162114139285431020ist_fm @ Sl2 ) )
         => ( ( fmembe3381613331217039976ist_fm @ S5 @ Sl2 )
           => ( member8102475879199740618ist_fm @ S5 @ top_to728902761071950545ist_fm ) ) ) ) ) ).

% eff_S
thf(fact_180_enabled__R,axiom,
    ! [S3: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ? [X: rule] :
          ( ( member_rule @ X @ ( sset_rule @ rules ) )
          & ? [Sl3: fset_P661503646757059847ist_fm] :
              ( ( eff @ X @ S3 )
              = ( some_f2162114139285431020ist_fm @ Sl3 ) ) ) ) ).

% enabled_R
thf(fact_181_snd__swap,axiom,
    ! [X2: produc164195504107695125m_rule] :
      ( ( produc4514744107061996309ist_fm @ ( produc2641608828089375343m_rule @ X2 ) )
      = ( produc6879501374131015971m_rule @ X2 ) ) ).

% snd_swap
thf(fact_182_snd__swap,axiom,
    ! [X2: produc4630866025709511057ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc9213896635423732003ist_fm @ X2 ) )
      = ( produc4228417144610596823ist_fm @ X2 ) ) ).

% snd_swap
thf(fact_183_fst__swap,axiom,
    ! [X2: produc4630866025709511057ist_fm] :
      ( ( produc6879501374131015971m_rule @ ( produc9213896635423732003ist_fm @ X2 ) )
      = ( produc4514744107061996309ist_fm @ X2 ) ) ).

% fst_swap
thf(fact_184_fst__swap,axiom,
    ! [X2: produc164195504107695125m_rule] :
      ( ( produc4228417144610596823ist_fm @ ( produc2641608828089375343m_rule @ X2 ) )
      = ( produc7165828336582415457m_rule @ X2 ) ) ).

% fst_swap
thf(fact_185_sset__fenum,axiom,
    ( ( sset_rule @ ( abstra7284221463285775110m_rule @ rules ) )
    = ( sset_rule @ rules ) ) ).

% sset_fenum
thf(fact_186_Saturated__def,axiom,
    ! [Steps2: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps2 )
      = ( ! [X4: rule] :
            ( ( member_rule @ X4 @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ X4
              @ Steps2 ) ) ) ) ).

% Saturated_def
thf(fact_187_szip_Osimps_I1_J,axiom,
    ! [S1: stream_list_fm,S2: stream_list_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
      = ( produc381145313068854617ist_fm @ ( shd_list_fm @ S1 ) @ ( shd_list_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_188_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) )
      = ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_189_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( shd_Pr4631506658928855911e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) )
      = ( produc4383365014245906419e_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_190_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr3564872132220007m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) )
      = ( produc6084188410630241011m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_191_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr4573454438745646103e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
      = ( produc1769597910952761123e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

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

% szip.simps(1)
thf(fact_193_szip_Ocode,axiom,
    ( szip_list_fm_list_fm
    = ( ^ [S12: stream_list_fm,S22: stream_list_fm] : ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ ( shd_list_fm @ S12 ) @ ( shd_list_fm @ S22 ) ) @ ( szip_list_fm_list_fm @ ( stl_list_fm @ S12 ) @ ( stl_list_fm @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_194_szip_Ocode,axiom,
    ( szip_P1977448745965526924m_rule
    = ( ^ [S12: stream8299795917829157543ist_fm,S22: stream_rule] : ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_195_szip_Ocode,axiom,
    ( szip_P2499414959592755846m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] : ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) @ ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_196_szip_Ocode,axiom,
    ( szip_P553291425819358518e_rule
    = ( ^ [S12: stream727092118206550309m_rule,S22: stream_rule] : ( sCons_6835861040388153298e_rule @ ( produc4383365014245906419e_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_P553291425819358518e_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_197_szip_Ocode,axiom,
    ( szip_r2254114822203693110m_rule
    = ( ^ [S12: stream_rule,S22: stream727092118206550309m_rule] : ( sCons_2207919253591517394m_rule @ ( produc6084188410630241011m_rule @ ( shd_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) @ ( szip_r2254114822203693110m_rule @ ( stl_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_198_szip_Ocode,axiom,
    ( szip_rule_rule
    = ( ^ [S12: stream_rule,S22: stream_rule] : ( sCons_7940479310257918722e_rule @ ( produc1769597910952761123e_rule @ ( shd_rule @ S12 ) @ ( shd_rule @ S22 ) ) @ ( szip_rule_rule @ ( stl_rule @ S12 ) @ ( stl_rule @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_199_sfilter__P,axiom,
    ! [P: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ( P @ ( shd_Pr7235097944458474089m_rule @ S3 ) )
     => ( ( sfilte7579662470208054517m_rule @ P @ S3 )
        = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S3 ) @ ( sfilte7579662470208054517m_rule @ P @ ( stl_Pr950425576149878629m_rule @ S3 ) ) ) ) ) ).

% sfilter_P
thf(fact_200_sfilter__P,axiom,
    ! [P: rule > $o,S3: stream_rule] :
      ( ( P @ ( shd_rule @ S3 ) )
     => ( ( sfilter_rule @ P @ S3 )
        = ( sCons_rule @ ( shd_rule @ S3 ) @ ( sfilter_rule @ P @ ( stl_rule @ S3 ) ) ) ) ) ).

% sfilter_P
thf(fact_201_prod_Oinject,axiom,
    ! [X1: list_fm,X22: list_fm,Y1: list_fm,Y2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ X1 @ X22 )
        = ( produc381145313068854617ist_fm @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_202_prod_Oinject,axiom,
    ! [X1: produc1996495991257130529ist_fm,X22: rule,Y1: produc1996495991257130529ist_fm,Y2: rule] :
      ( ( ( produc491467635432902671m_rule @ X1 @ X22 )
        = ( produc491467635432902671m_rule @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_203_old_Oprod_Oinject,axiom,
    ! [A2: list_fm,B2: list_fm,A4: list_fm,B3: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A2 @ B2 )
        = ( produc381145313068854617ist_fm @ A4 @ B3 ) )
      = ( ( A2 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_204_old_Oprod_Oinject,axiom,
    ! [A2: produc1996495991257130529ist_fm,B2: rule,A4: produc1996495991257130529ist_fm,B3: rule] :
      ( ( ( produc491467635432902671m_rule @ A2 @ B2 )
        = ( produc491467635432902671m_rule @ A4 @ B3 ) )
      = ( ( A2 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_205_swap__simp,axiom,
    ! [X2: rule,Y: produc1996495991257130529ist_fm] :
      ( ( produc9213896635423732003ist_fm @ ( produc7063755442767259331ist_fm @ X2 @ Y ) )
      = ( produc491467635432902671m_rule @ Y @ X2 ) ) ).

% swap_simp
thf(fact_206_swap__simp,axiom,
    ! [X2: list_fm,Y: list_fm] :
      ( ( produc7936585717479832313ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y ) )
      = ( produc381145313068854617ist_fm @ Y @ X2 ) ) ).

% swap_simp
thf(fact_207_swap__simp,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y: rule] :
      ( ( produc2641608828089375343m_rule @ ( produc491467635432902671m_rule @ X2 @ Y ) )
      = ( produc7063755442767259331ist_fm @ Y @ X2 ) ) ).

% swap_simp
thf(fact_208_sdrop__smap2,axiom,
    ! [N: nat,F2: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( smap2_3865418324013074943m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_3865418324013074943m_rule @ F2 @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_209_sdrop__smap2,axiom,
    ! [N: nat,F2: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( smap2_3092065398501458735m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_3092065398501458735m_rule @ F2 @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_210_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( smap2_7722561687528711727m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_7722561687528711727m_rule @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_211_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( smap2_4438185555641665375m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_4438185555641665375m_rule @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_212_sdrop__smap2,axiom,
    ! [N: nat,F2: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_rule @ N @ ( smap2_1391242002117124143e_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_1391242002117124143e_rule @ F2 @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_213_sdrop__smap2,axiom,
    ! [N: nat,F2: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_190827247425732959e_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_190827247425732959e_rule @ F2 @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_214_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( sdrop_rule @ N @ ( smap2_2737362159257330783e_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_2737362159257330783e_rule @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_215_sdrop__smap2,axiom,
    ! [N: nat,F2: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_rule_rule_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_rule_rule_rule @ F2 @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_216_prod_Ocollapse,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_217_prod_Ocollapse,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_218_sfilter__not__P,axiom,
    ! [P: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ~ ( P @ ( shd_Pr7235097944458474089m_rule @ S3 ) )
     => ( ( sfilte7579662470208054517m_rule @ P @ S3 )
        = ( sfilte7579662470208054517m_rule @ P @ ( stl_Pr950425576149878629m_rule @ S3 ) ) ) ) ).

% sfilter_not_P
thf(fact_219_sfilter__not__P,axiom,
    ! [P: rule > $o,S3: stream_rule] :
      ( ~ ( P @ ( shd_rule @ S3 ) )
     => ( ( sfilter_rule @ P @ S3 )
        = ( sfilter_rule @ P @ ( stl_rule @ S3 ) ) ) ) ).

% sfilter_not_P
thf(fact_220_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1996495991257130529ist_fm] :
      ~ ! [A5: list_fm,B4: list_fm] :
          ( Y
         != ( produc381145313068854617ist_fm @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_221_old_Oprod_Oexhaust,axiom,
    ! [Y: produc164195504107695125m_rule] :
      ~ ! [A5: produc1996495991257130529ist_fm,B4: rule] :
          ( Y
         != ( produc491467635432902671m_rule @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_222_surj__pair,axiom,
    ! [P2: produc1996495991257130529ist_fm] :
    ? [X: list_fm,Y4: list_fm] :
      ( P2
      = ( produc381145313068854617ist_fm @ X @ Y4 ) ) ).

% surj_pair
thf(fact_223_surj__pair,axiom,
    ! [P2: produc164195504107695125m_rule] :
    ? [X: produc1996495991257130529ist_fm,Y4: rule] :
      ( P2
      = ( produc491467635432902671m_rule @ X @ Y4 ) ) ).

% surj_pair
thf(fact_224_prod__cases,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm] :
      ( ! [A5: list_fm,B4: list_fm] : ( P @ ( produc381145313068854617ist_fm @ A5 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_225_prod__cases,axiom,
    ! [P: produc164195504107695125m_rule > $o,P2: produc164195504107695125m_rule] :
      ( ! [A5: produc1996495991257130529ist_fm,B4: rule] : ( P @ ( produc491467635432902671m_rule @ A5 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_226_Pair__inject,axiom,
    ! [A2: list_fm,B2: list_fm,A4: list_fm,B3: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A2 @ B2 )
        = ( produc381145313068854617ist_fm @ A4 @ B3 ) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_227_Pair__inject,axiom,
    ! [A2: produc1996495991257130529ist_fm,B2: rule,A4: produc1996495991257130529ist_fm,B3: rule] :
      ( ( ( produc491467635432902671m_rule @ A2 @ B2 )
        = ( produc491467635432902671m_rule @ A4 @ B3 ) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_228_RuleSystem__Defs_Osaturated_Ocong,axiom,
    abstra7252176601000738572ist_fm = abstra7252176601000738572ist_fm ).

% RuleSystem_Defs.saturated.cong
thf(fact_229_RuleSystem__Defs_Ofenum_Ocong,axiom,
    abstra7284221463285775110m_rule = abstra7284221463285775110m_rule ).

% RuleSystem_Defs.fenum.cong
thf(fact_230_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra8647779336736900908ist_fm = abstra8647779336736900908ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_231_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra8647779336736900908ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,Steps3: stream727092118206550309m_rule] :
        ! [X4: rule] :
          ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
         => ( abstra7252176601000738572ist_fm @ Eff2 @ X4 @ Steps3 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_232_fst__eqD,axiom,
    ! [X2: list_fm,Y: list_fm,A2: list_fm] :
      ( ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_233_fst__eqD,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y: rule,A2: produc1996495991257130529ist_fm] :
      ( ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_234_fst__conv,axiom,
    ! [X1: list_fm,X22: list_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_235_fst__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X22: rule] :
      ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_236_snd__conv,axiom,
    ! [X1: list_fm,X22: list_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_237_snd__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X22: rule] :
      ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_238_snd__eqD,axiom,
    ! [X2: list_fm,Y: list_fm,A2: list_fm] :
      ( ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_239_snd__eqD,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y: rule,A2: rule] :
      ( ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_240_prod_Oswap__def,axiom,
    ( produc7936585717479832313ist_fm
    = ( ^ [P4: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc4588648349897876871ist_fm @ P4 ) @ ( produc1501393135466168645ist_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_241_prod_Oswap__def,axiom,
    ( produc9213896635423732003ist_fm
    = ( ^ [P4: produc4630866025709511057ist_fm] : ( produc491467635432902671m_rule @ ( produc4514744107061996309ist_fm @ P4 ) @ ( produc4228417144610596823ist_fm @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_242_prod_Oswap__def,axiom,
    ( produc2641608828089375343m_rule
    = ( ^ [P4: produc164195504107695125m_rule] : ( produc7063755442767259331ist_fm @ ( produc7165828336582415457m_rule @ P4 ) @ ( produc6879501374131015971m_rule @ P4 ) ) ) ) ).

% prod.swap_def
thf(fact_243_smap2_Osimps_I2_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3865418324013074943m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_3865418324013074943m_rule @ F2 @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_244_smap2_Osimps_I2_J,axiom,
    ! [F2: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3092065398501458735m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_3092065398501458735m_rule @ F2 @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_245_smap2_Osimps_I2_J,axiom,
    ! [F2: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_7722561687528711727m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_7722561687528711727m_rule @ F2 @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_246_smap2_Osimps_I2_J,axiom,
    ! [F2: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_4438185555641665375m_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_4438185555641665375m_rule @ F2 @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_247_smap2_Osimps_I2_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_1391242002117124143e_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_1391242002117124143e_rule @ F2 @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_248_smap2_Osimps_I2_J,axiom,
    ! [F2: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( stl_rule @ ( smap2_190827247425732959e_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_190827247425732959e_rule @ F2 @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_249_smap2_Osimps_I2_J,axiom,
    ! [F2: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_2737362159257330783e_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_2737362159257330783e_rule @ F2 @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_250_smap2_Osimps_I2_J,axiom,
    ! [F2: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( stl_rule @ ( smap2_rule_rule_rule @ F2 @ S1 @ S2 ) )
      = ( smap2_rule_rule_rule @ F2 @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_251_smap2_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3865418324013074943m_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_252_smap2_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3092065398501458735m_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_253_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_7722561687528711727m_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_254_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_4438185555641665375m_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_255_smap2_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_1391242002117124143e_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_256_smap2_Osimps_I1_J,axiom,
    ! [F2: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_190827247425732959e_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_257_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_2737362159257330783e_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_258_smap2_Osimps_I1_J,axiom,
    ! [F2: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_rule_rule_rule @ F2 @ S1 @ S2 ) )
      = ( F2 @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_259_i_OSaturated__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,Steps2: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps2 )
      = ( ! [X4: rule] :
            ( ( member_rule @ X4 @ ( sset_rule @ Rules ) )
           => ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ X4 @ Steps2 ) ) ) ) ).

% i.Saturated_def
thf(fact_260_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( Prod
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_261_prod_Oexhaust__sel,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( Prod
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_262_exI__realizer,axiom,
    ! [P: list_fm > list_fm > $o,Y: list_fm,X2: list_fm] :
      ( ( P @ Y @ X2 )
     => ( P @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y ) ) @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_263_exI__realizer,axiom,
    ! [P: rule > produc1996495991257130529ist_fm > $o,Y: rule,X2: produc1996495991257130529ist_fm] :
      ( ( P @ Y @ X2 )
     => ( P @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X2 @ Y ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_264_conjI__realizer,axiom,
    ! [P: list_fm > $o,P2: list_fm,Q: list_fm > $o,Q2: list_fm] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ P2 @ Q2 ) ) )
          & ( Q @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_265_conjI__realizer,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm,Q: rule > $o,Q2: rule] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_266_surjective__pairing,axiom,
    ! [T2: produc1996495991257130529ist_fm] :
      ( T2
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ T2 ) @ ( produc4588648349897876871ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_267_surjective__pairing,axiom,
    ! [T2: produc164195504107695125m_rule] :
      ( T2
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ T2 ) @ ( produc7165828336582415457m_rule @ T2 ) ) ) ).

% surjective_pairing
thf(fact_268_RuleSystem__Defs_Osset__fenum,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
      = ( sset_rule @ Rules ) ) ).

% RuleSystem_Defs.sset_fenum
thf(fact_269_i_Osset__fenum,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
      = ( sset_rule @ Rules ) ) ).

% i.sset_fenum
thf(fact_270_szip__unfold,axiom,
    ! [A2: list_fm,S1: stream_list_fm,B2: list_fm,S2: stream_list_fm] :
      ( ( szip_list_fm_list_fm @ ( sCons_list_fm @ A2 @ S1 ) @ ( sCons_list_fm @ B2 @ S2 ) )
      = ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ A2 @ B2 ) @ ( szip_list_fm_list_fm @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_271_szip__unfold,axiom,
    ! [A2: produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,B2: rule,S2: stream_rule] :
      ( ( szip_P1977448745965526924m_rule @ ( sCons_307821682964077496ist_fm @ A2 @ S1 ) @ ( sCons_rule @ B2 @ S2 ) )
      = ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ A2 @ B2 ) @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_272_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_273_saturated__effG__uu__def,axiom,
    ( abstra7873564507807625473ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% saturated_effG_uu_def
thf(fact_274_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: list_fm > list_fm > $o,X2: list_fm,Y: list_fm,A2: produc1996495991257130529ist_fm] :
      ( ( P @ X2 @ Y )
     => ( ( A2
          = ( produc381145313068854617ist_fm @ X2 @ Y ) )
       => ( P @ ( produc1501393135466168645ist_fm @ A2 ) @ ( produc4588648349897876871ist_fm @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_275_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: produc1996495991257130529ist_fm > rule > $o,X2: produc1996495991257130529ist_fm,Y: rule,A2: produc164195504107695125m_rule] :
      ( ( P @ X2 @ Y )
     => ( ( A2
          = ( produc491467635432902671m_rule @ X2 @ Y ) )
       => ( P @ ( produc6879501374131015971m_rule @ A2 ) @ ( produc7165828336582415457m_rule @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_276_fenum__uu__def,axiom,
    abstra8121460317444261103u_rule = abstra7284221463285775110m_rule ).

% fenum_uu_def
thf(fact_277_per,axiom,
    ! [R3: rule] :
      ( ( member_rule @ R3 @ ( sset_rule @ rules ) )
     => ( abstra4153889242452539894ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R3 ) ) ).

% per
thf(fact_278_PersistentRuleSystem__axioms,axiom,
    ( abstra7887161495092839066ist_fm
    @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R2 @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ top_to728902761071950545ist_fm ) ).

% PersistentRuleSystem_axioms
thf(fact_279_iso__tuple__UNIV__I,axiom,
    ! [X2: fm] : ( member_fm @ X2 @ top_top_set_fm ) ).

% iso_tuple_UNIV_I
thf(fact_280_iso__tuple__UNIV__I,axiom,
    ! [X2: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X2 @ top_to728902761071950545ist_fm ) ).

% iso_tuple_UNIV_I
thf(fact_281_iso__tuple__UNIV__I,axiom,
    ! [X2: rule] : ( member_rule @ X2 @ top_top_set_rule ) ).

% iso_tuple_UNIV_I
thf(fact_282_iso__tuple__UNIV__I,axiom,
    ! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_283_UNIV__I,axiom,
    ! [X2: fm] : ( member_fm @ X2 @ top_top_set_fm ) ).

% UNIV_I
thf(fact_284_UNIV__I,axiom,
    ! [X2: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X2 @ top_to728902761071950545ist_fm ) ).

% UNIV_I
thf(fact_285_UNIV__I,axiom,
    ! [X2: rule] : ( member_rule @ X2 @ top_top_set_rule ) ).

% UNIV_I
thf(fact_286_UNIV__I,axiom,
    ! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_287_top__set__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm @ top_to6798613745154499404t_fm_o ) ) ).

% top_set_def
thf(fact_288_top__set__def,axiom,
    ( top_top_set_rule
    = ( collect_rule @ top_top_rule_o ) ) ).

% top_set_def
thf(fact_289_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_290_RuleSystem_Oper_Ocong,axiom,
    abstra4153889242452539894ist_fm = abstra4153889242452539894ist_fm ).

% RuleSystem.per.cong
thf(fact_291_PersistentRuleSystem_Oper,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,R3: rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S6 )
     => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
       => ( abstra4153889242452539894ist_fm @ Eff @ Rules @ S6 @ R3 ) ) ) ).

% PersistentRuleSystem.per
thf(fact_292_UNIV__rules,axiom,
    ( ( sset_rule @ rules )
    = top_top_set_rule ) ).

% UNIV_rules
thf(fact_293_UNIV__eq__I,axiom,
    ! [A: set_fm] :
      ( ! [X: fm] : ( member_fm @ X @ A )
     => ( top_top_set_fm = A ) ) ).

% UNIV_eq_I
thf(fact_294_UNIV__eq__I,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A )
     => ( top_to728902761071950545ist_fm = A ) ) ).

% UNIV_eq_I
thf(fact_295_UNIV__eq__I,axiom,
    ! [A: set_rule] :
      ( ! [X: rule] : ( member_rule @ X @ A )
     => ( top_top_set_rule = A ) ) ).

% UNIV_eq_I
thf(fact_296_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X: nat] : ( member_nat @ X @ A )
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_297_UNIV__witness,axiom,
    ? [X: fm] : ( member_fm @ X @ top_top_set_fm ) ).

% UNIV_witness
thf(fact_298_UNIV__witness,axiom,
    ? [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ top_to728902761071950545ist_fm ) ).

% UNIV_witness
thf(fact_299_UNIV__witness,axiom,
    ? [X: rule] : ( member_rule @ X @ top_top_set_rule ) ).

% UNIV_witness
thf(fact_300_UNIV__witness,axiom,
    ? [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_301_UNIV__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm
      @ ^ [X4: produc1996495991257130529ist_fm] : $true ) ) ).

% UNIV_def
thf(fact_302_UNIV__def,axiom,
    ( top_top_set_rule
    = ( collect_rule
      @ ^ [X4: rule] : $true ) ) ).

% UNIV_def
thf(fact_303_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X4: nat] : $true ) ) ).

% UNIV_def
thf(fact_304_per__rules,axiom,
    ! [R3: rule] :
      ( abstra4153889242452539894ist_fm
      @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R2 @ S )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ rules
      @ top_to728902761071950545ist_fm
      @ R3 ) ).

% per_rules
thf(fact_305_top__empty__eq2,axiom,
    ( top_to1955928418919630078rule_o
    = ( ^ [X4: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X4 @ Y5 ) @ top_to1603951219661559419m_rule ) ) ) ).

% top_empty_eq2
thf(fact_306_top__empty__eq2,axiom,
    ( top_to7326020031456794390t_fm_o
    = ( ^ [X4: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y5 ) @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_307_sndI,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y: list_fm,Z3: list_fm] :
      ( ( X2
        = ( produc381145313068854617ist_fm @ Y @ Z3 ) )
     => ( ( produc4588648349897876871ist_fm @ X2 )
        = Z3 ) ) ).

% sndI
thf(fact_308_sndI,axiom,
    ! [X2: produc164195504107695125m_rule,Y: produc1996495991257130529ist_fm,Z3: rule] :
      ( ( X2
        = ( produc491467635432902671m_rule @ Y @ Z3 ) )
     => ( ( produc7165828336582415457m_rule @ X2 )
        = Z3 ) ) ).

% sndI
thf(fact_309_eq__snd__iff,axiom,
    ! [B2: list_fm,P2: produc1996495991257130529ist_fm] :
      ( ( B2
        = ( produc4588648349897876871ist_fm @ P2 ) )
      = ( ? [A3: list_fm] :
            ( P2
            = ( produc381145313068854617ist_fm @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_310_eq__snd__iff,axiom,
    ! [B2: rule,P2: produc164195504107695125m_rule] :
      ( ( B2
        = ( produc7165828336582415457m_rule @ P2 ) )
      = ( ? [A3: produc1996495991257130529ist_fm] :
            ( P2
            = ( produc491467635432902671m_rule @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_311_eq__fst__iff,axiom,
    ! [A2: list_fm,P2: produc1996495991257130529ist_fm] :
      ( ( A2
        = ( produc1501393135466168645ist_fm @ P2 ) )
      = ( ? [B5: list_fm] :
            ( P2
            = ( produc381145313068854617ist_fm @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_312_eq__fst__iff,axiom,
    ! [A2: produc1996495991257130529ist_fm,P2: produc164195504107695125m_rule] :
      ( ( A2
        = ( produc6879501374131015971m_rule @ P2 ) )
      = ( ? [B5: rule] :
            ( P2
            = ( produc491467635432902671m_rule @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_313_fstI,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y: list_fm,Z3: list_fm] :
      ( ( X2
        = ( produc381145313068854617ist_fm @ Y @ Z3 ) )
     => ( ( produc1501393135466168645ist_fm @ X2 )
        = Y ) ) ).

% fstI
thf(fact_314_fstI,axiom,
    ! [X2: produc164195504107695125m_rule,Y: produc1996495991257130529ist_fm,Z3: rule] :
      ( ( X2
        = ( produc491467635432902671m_rule @ Y @ Z3 ) )
     => ( ( produc6879501374131015971m_rule @ X2 )
        = Y ) ) ).

% fstI
thf(fact_315_RuleSystem__axioms,axiom,
    ( abstra6455275554694158129ist_fm
    @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R2 @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ top_to728902761071950545ist_fm ) ).

% RuleSystem_axioms
thf(fact_316_PersistentRuleSystem__axioms_Ointro,axiom,
    ! [Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,S6: set_Pr7058068377845519745ist_fm] :
      ( ! [R4: rule] :
          ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
         => ( abstra4153889242452539894ist_fm @ Eff @ Rules @ S6 @ R4 ) )
     => ( abstra2778581409224112183ist_fm @ Eff @ Rules @ S6 ) ) ).

% PersistentRuleSystem_axioms.intro
thf(fact_317_PersistentRuleSystem__axioms__def,axiom,
    ( abstra2778581409224112183ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S7: set_Pr7058068377845519745ist_fm] :
        ! [R2: rule] :
          ( ( member_rule @ R2 @ ( sset_rule @ Rules2 ) )
         => ( abstra4153889242452539894ist_fm @ Eff2 @ Rules2 @ S7 @ R2 ) ) ) ) ).

% PersistentRuleSystem_axioms_def
thf(fact_318_sfilter_Ocode,axiom,
    ( sfilte7579662470208054517m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P3 ) @ S ) ) @ ( sfilte7579662470208054517m_rule @ P3 @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P3 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_319_sfilter_Ocode,axiom,
    ( sfilter_rule
    = ( ^ [P3: rule > $o,S: stream_rule] : ( sCons_rule @ ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P3 ) @ S ) ) @ ( sfilter_rule @ P3 @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P3 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_320_K__record__comp,axiom,
    ! [C: produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P4378417847848123130m_rule
        @ ^ [X4: produc164195504107695125m_rule] : C
        @ F2 )
      = ( ^ [X4: abstra4572470701423150929m_rule] : C ) ) ).

% K_record_comp
thf(fact_321_PersistentRuleSystem_Ointro,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( abstra2778581409224112183ist_fm @ Eff @ Rules @ S6 )
       => ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S6 ) ) ) ).

% PersistentRuleSystem.intro
thf(fact_322_PersistentRuleSystem__def,axiom,
    ( abstra7887161495092839066ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S7: set_Pr7058068377845519745ist_fm] :
          ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S7 )
          & ( abstra2778581409224112183ist_fm @ Eff2 @ Rules2 @ S7 ) ) ) ) ).

% PersistentRuleSystem_def
thf(fact_323_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_rule] : ( Eff @ X @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_324_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_fm] : ( Eff @ X @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_325_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_nat] : ( Eff @ X @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_326_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P661503646757059847ist_fm] : ( Eff @ X @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_327_PersistentRuleSystem_Oaxioms_I1_J,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S6 )
     => ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 ) ) ).

% PersistentRuleSystem.axioms(1)
thf(fact_328_RuleSystem_Oeff__S,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S6: set_rule,S3: rule,R3: fm,Sl2: fset_rule,S5: rule] :
      ( ( abstra5167362717900046400m_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_rule @ S5 @ Sl2 )
             => ( member_rule @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_329_RuleSystem_Oeff__S,axiom,
    ! [Eff: nat > rule > fset_rule > $o,Rules: stream_nat,S6: set_rule,S3: rule,R3: nat,Sl2: fset_rule,S5: rule] :
      ( ( abstra5092216761188309440t_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_rule @ S5 @ Sl2 )
             => ( member_rule @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_330_RuleSystem_Oeff__S,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S6: set_fm,S3: fm,R3: fm,Sl2: fset_fm,S5: fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_fm @ S5 @ Sl2 )
             => ( member_fm @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_331_RuleSystem_Oeff__S,axiom,
    ! [Eff: nat > fm > fset_fm > $o,Rules: stream_nat,S6: set_fm,S3: fm,R3: nat,Sl2: fset_fm,S5: fm] :
      ( ( abstra1720523912436379467nat_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_fm @ S5 @ Sl2 )
             => ( member_fm @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_332_RuleSystem_Oeff__S,axiom,
    ! [Eff: fm > nat > fset_nat > $o,Rules: stream_fm,S6: set_nat,S3: nat,R3: fm,Sl2: fset_nat,S5: nat] :
      ( ( abstra9084600350906980299fm_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_nat @ S5 @ Sl2 )
             => ( member_nat @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_333_RuleSystem_Oeff__S,axiom,
    ! [Eff: nat > nat > fset_nat > $o,Rules: stream_nat,S6: set_nat,S3: nat,R3: nat,Sl2: fset_nat,S5: nat] :
      ( ( abstra5088622135990055243at_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_nat @ S5 @ Sl2 )
             => ( member_nat @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_334_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,R3: rule,Sl2: fset_rule,S5: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_rule @ S5 @ Sl2 )
             => ( member_rule @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_335_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,R3: rule,Sl2: fset_fm,S5: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_fm @ S5 @ Sl2 )
             => ( member_fm @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_336_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,R3: rule,Sl2: fset_nat,S5: nat] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_nat @ S5 @ Sl2 )
             => ( member_nat @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_337_RuleSystem_Oeff__S,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S6: set_rule,S3: rule,R3: produc1996495991257130529ist_fm,Sl2: fset_rule,S5: rule] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ( ( Eff @ R3 @ S3 @ Sl2 )
           => ( ( fmember_rule @ S5 @ Sl2 )
             => ( member_rule @ S5 @ S6 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_338_RuleSystem_Ointro,axiom,
    ! [S6: set_rule,Rules: stream_fm,Eff: fm > rule > fset_rule > $o] :
      ( ! [S4: rule] :
          ( ( member_rule @ S4 @ S6 )
         => ! [R4: fm] :
              ( ( member_fm @ R4 @ ( sset_fm @ Rules ) )
             => ! [Sl3: fset_rule] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: rule] :
                      ( ( fmember_rule @ S8 @ Sl3 )
                     => ( member_rule @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: rule] :
            ( ( member_rule @ S4 @ S6 )
           => ? [X5: fm] :
                ( ( member_fm @ X5 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra5167362717900046400m_rule @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_339_RuleSystem_Ointro,axiom,
    ! [S6: set_rule,Rules: stream_nat,Eff: nat > rule > fset_rule > $o] :
      ( ! [S4: rule] :
          ( ( member_rule @ S4 @ S6 )
         => ! [R4: nat] :
              ( ( member_nat @ R4 @ ( sset_nat @ Rules ) )
             => ! [Sl3: fset_rule] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: rule] :
                      ( ( fmember_rule @ S8 @ Sl3 )
                     => ( member_rule @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: rule] :
            ( ( member_rule @ S4 @ S6 )
           => ? [X5: nat] :
                ( ( member_nat @ X5 @ ( sset_nat @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra5092216761188309440t_rule @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_340_RuleSystem_Ointro,axiom,
    ! [S6: set_fm,Rules: stream_fm,Eff: fm > fm > fset_fm > $o] :
      ( ! [S4: fm] :
          ( ( member_fm @ S4 @ S6 )
         => ! [R4: fm] :
              ( ( member_fm @ R4 @ ( sset_fm @ Rules ) )
             => ! [Sl3: fset_fm] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: fm] :
                      ( ( fmember_fm @ S8 @ Sl3 )
                     => ( member_fm @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: fm] :
            ( ( member_fm @ S4 @ S6 )
           => ? [X5: fm] :
                ( ( member_fm @ X5 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra886934874521163211_fm_fm @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_341_RuleSystem_Ointro,axiom,
    ! [S6: set_fm,Rules: stream_nat,Eff: nat > fm > fset_fm > $o] :
      ( ! [S4: fm] :
          ( ( member_fm @ S4 @ S6 )
         => ! [R4: nat] :
              ( ( member_nat @ R4 @ ( sset_nat @ Rules ) )
             => ! [Sl3: fset_fm] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: fm] :
                      ( ( fmember_fm @ S8 @ Sl3 )
                     => ( member_fm @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: fm] :
            ( ( member_fm @ S4 @ S6 )
           => ? [X5: nat] :
                ( ( member_nat @ X5 @ ( sset_nat @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra1720523912436379467nat_fm @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_342_RuleSystem_Ointro,axiom,
    ! [S6: set_nat,Rules: stream_fm,Eff: fm > nat > fset_nat > $o] :
      ( ! [S4: nat] :
          ( ( member_nat @ S4 @ S6 )
         => ! [R4: fm] :
              ( ( member_fm @ R4 @ ( sset_fm @ Rules ) )
             => ! [Sl3: fset_nat] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: nat] :
                      ( ( fmember_nat @ S8 @ Sl3 )
                     => ( member_nat @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: nat] :
            ( ( member_nat @ S4 @ S6 )
           => ? [X5: fm] :
                ( ( member_fm @ X5 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_nat] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra9084600350906980299fm_nat @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_343_RuleSystem_Ointro,axiom,
    ! [S6: set_nat,Rules: stream_nat,Eff: nat > nat > fset_nat > $o] :
      ( ! [S4: nat] :
          ( ( member_nat @ S4 @ S6 )
         => ! [R4: nat] :
              ( ( member_nat @ R4 @ ( sset_nat @ Rules ) )
             => ! [Sl3: fset_nat] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: nat] :
                      ( ( fmember_nat @ S8 @ Sl3 )
                     => ( member_nat @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: nat] :
            ( ( member_nat @ S4 @ S6 )
           => ? [X5: nat] :
                ( ( member_nat @ X5 @ ( sset_nat @ Rules ) )
                & ? [X_12: fset_nat] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra5088622135990055243at_nat @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_344_RuleSystem_Ointro,axiom,
    ! [S6: set_rule,Rules: stream_rule,Eff: rule > rule > fset_rule > $o] :
      ( ! [S4: rule] :
          ( ( member_rule @ S4 @ S6 )
         => ! [R4: rule] :
              ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
             => ! [Sl3: fset_rule] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: rule] :
                      ( ( fmember_rule @ S8 @ Sl3 )
                     => ( member_rule @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: rule] :
            ( ( member_rule @ S4 @ S6 )
           => ? [X5: rule] :
                ( ( member_rule @ X5 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_345_RuleSystem_Ointro,axiom,
    ! [S6: set_fm,Rules: stream_rule,Eff: rule > fm > fset_fm > $o] :
      ( ! [S4: fm] :
          ( ( member_fm @ S4 @ S6 )
         => ! [R4: rule] :
              ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
             => ! [Sl3: fset_fm] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: fm] :
                      ( ( fmember_fm @ S8 @ Sl3 )
                     => ( member_fm @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: fm] :
            ( ( member_fm @ S4 @ S6 )
           => ? [X5: rule] :
                ( ( member_rule @ X5 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_346_RuleSystem_Ointro,axiom,
    ! [S6: set_nat,Rules: stream_rule,Eff: rule > nat > fset_nat > $o] :
      ( ! [S4: nat] :
          ( ( member_nat @ S4 @ S6 )
         => ! [R4: rule] :
              ( ( member_rule @ R4 @ ( sset_rule @ Rules ) )
             => ! [Sl3: fset_nat] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: nat] :
                      ( ( fmember_nat @ S8 @ Sl3 )
                     => ( member_nat @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: nat] :
            ( ( member_nat @ S4 @ S6 )
           => ? [X5: rule] :
                ( ( member_rule @ X5 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_nat] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_347_RuleSystem_Ointro,axiom,
    ! [S6: set_rule,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o] :
      ( ! [S4: rule] :
          ( ( member_rule @ S4 @ S6 )
         => ! [R4: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R4 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ! [Sl3: fset_rule] :
                  ( ( Eff @ R4 @ S4 @ Sl3 )
                 => ! [S8: rule] :
                      ( ( fmember_rule @ S8 @ Sl3 )
                     => ( member_rule @ S8 @ S6 ) ) ) ) )
     => ( ! [S4: rule] :
            ( ( member_rule @ S4 @ S6 )
           => ? [X5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X5 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X5 @ S4 @ X_12 ) ) )
       => ( abstra9106359784214577277m_rule @ Eff @ Rules @ S6 ) ) ) ).

% RuleSystem.intro
thf(fact_348_RuleSystem__def,axiom,
    ( abstra5167362717900046400m_rule
    = ( ^ [Eff2: fm > rule > fset_rule > $o,Rules2: stream_fm,S7: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S7 )
             => ! [R2: fm] :
                  ( ( member_fm @ R2 @ ( sset_fm @ Rules2 ) )
                 => ! [Sl4: fset_rule] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl4 )
                         => ( member_rule @ S9 @ S7 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S7 )
             => ? [X4: fm] :
                  ( ( member_fm @ X4 @ ( sset_fm @ Rules2 ) )
                  & ? [X6: fset_rule] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_349_RuleSystem__def,axiom,
    ( abstra5092216761188309440t_rule
    = ( ^ [Eff2: nat > rule > fset_rule > $o,Rules2: stream_nat,S7: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S7 )
             => ! [R2: nat] :
                  ( ( member_nat @ R2 @ ( sset_nat @ Rules2 ) )
                 => ! [Sl4: fset_rule] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl4 )
                         => ( member_rule @ S9 @ S7 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S7 )
             => ? [X4: nat] :
                  ( ( member_nat @ X4 @ ( sset_nat @ Rules2 ) )
                  & ? [X6: fset_rule] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_350_RuleSystem__def,axiom,
    ( abstra886934874521163211_fm_fm
    = ( ^ [Eff2: fm > fm > fset_fm > $o,Rules2: stream_fm,S7: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S7 )
             => ! [R2: fm] :
                  ( ( member_fm @ R2 @ ( sset_fm @ Rules2 ) )
                 => ! [Sl4: fset_fm] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl4 )
                         => ( member_fm @ S9 @ S7 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S7 )
             => ? [X4: fm] :
                  ( ( member_fm @ X4 @ ( sset_fm @ Rules2 ) )
                  & ? [X6: fset_fm] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_351_RuleSystem__def,axiom,
    ( abstra1720523912436379467nat_fm
    = ( ^ [Eff2: nat > fm > fset_fm > $o,Rules2: stream_nat,S7: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S7 )
             => ! [R2: nat] :
                  ( ( member_nat @ R2 @ ( sset_nat @ Rules2 ) )
                 => ! [Sl4: fset_fm] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl4 )
                         => ( member_fm @ S9 @ S7 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S7 )
             => ? [X4: nat] :
                  ( ( member_nat @ X4 @ ( sset_nat @ Rules2 ) )
                  & ? [X6: fset_fm] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_352_RuleSystem__def,axiom,
    ( abstra9084600350906980299fm_nat
    = ( ^ [Eff2: fm > nat > fset_nat > $o,Rules2: stream_fm,S7: set_nat] :
          ( ! [S: nat] :
              ( ( member_nat @ S @ S7 )
             => ! [R2: fm] :
                  ( ( member_fm @ R2 @ ( sset_fm @ Rules2 ) )
                 => ! [Sl4: fset_nat] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: nat] :
                          ( ( fmember_nat @ S9 @ Sl4 )
                         => ( member_nat @ S9 @ S7 ) ) ) ) )
          & ! [S: nat] :
              ( ( member_nat @ S @ S7 )
             => ? [X4: fm] :
                  ( ( member_fm @ X4 @ ( sset_fm @ Rules2 ) )
                  & ? [X6: fset_nat] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_353_RuleSystem__def,axiom,
    ( abstra5088622135990055243at_nat
    = ( ^ [Eff2: nat > nat > fset_nat > $o,Rules2: stream_nat,S7: set_nat] :
          ( ! [S: nat] :
              ( ( member_nat @ S @ S7 )
             => ! [R2: nat] :
                  ( ( member_nat @ R2 @ ( sset_nat @ Rules2 ) )
                 => ! [Sl4: fset_nat] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: nat] :
                          ( ( fmember_nat @ S9 @ Sl4 )
                         => ( member_nat @ S9 @ S7 ) ) ) ) )
          & ! [S: nat] :
              ( ( member_nat @ S @ S7 )
             => ? [X4: nat] :
                  ( ( member_nat @ X4 @ ( sset_nat @ Rules2 ) )
                  & ? [X6: fset_nat] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_354_RuleSystem__def,axiom,
    ( abstra2725590884229999157e_rule
    = ( ^ [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S7: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S7 )
             => ! [R2: rule] :
                  ( ( member_rule @ R2 @ ( sset_rule @ Rules2 ) )
                 => ! [Sl4: fset_rule] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl4 )
                         => ( member_rule @ S9 @ S7 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S7 )
             => ? [X4: rule] :
                  ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
                  & ? [X6: fset_rule] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_355_RuleSystem__def,axiom,
    ( abstra916514217863227968ule_fm
    = ( ^ [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S7: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S7 )
             => ! [R2: rule] :
                  ( ( member_rule @ R2 @ ( sset_rule @ Rules2 ) )
                 => ! [Sl4: fset_fm] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: fm] :
                          ( ( fmember_fm @ S9 @ Sl4 )
                         => ( member_fm @ S9 @ S7 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S7 )
             => ? [X4: rule] :
                  ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
                  & ? [X6: fset_fm] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_356_RuleSystem__def,axiom,
    ( abstra8329481176008169152le_nat
    = ( ^ [Eff2: rule > nat > fset_nat > $o,Rules2: stream_rule,S7: set_nat] :
          ( ! [S: nat] :
              ( ( member_nat @ S @ S7 )
             => ! [R2: rule] :
                  ( ( member_rule @ R2 @ ( sset_rule @ Rules2 ) )
                 => ! [Sl4: fset_nat] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: nat] :
                          ( ( fmember_nat @ S9 @ Sl4 )
                         => ( member_nat @ S9 @ S7 ) ) ) ) )
          & ! [S: nat] :
              ( ( member_nat @ S @ S7 )
             => ? [X4: rule] :
                  ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
                  & ? [X6: fset_nat] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_357_RuleSystem__def,axiom,
    ( abstra9106359784214577277m_rule
    = ( ^ [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S7: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S7 )
             => ! [R2: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                 => ! [Sl4: fset_rule] :
                      ( ( Eff2 @ R2 @ S @ Sl4 )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl4 )
                         => ( member_rule @ S9 @ S7 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S7 )
             => ? [X4: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X4 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                  & ? [X6: fset_rule] : ( Eff2 @ X4 @ S @ X6 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_358_top__empty__eq,axiom,
    ( top_top_fm_o
    = ( ^ [X4: fm] : ( member_fm @ X4 @ top_top_set_fm ) ) ) ).

% top_empty_eq
thf(fact_359_top__empty__eq,axiom,
    ( top_to6798613745154499404t_fm_o
    = ( ^ [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq
thf(fact_360_top__empty__eq,axiom,
    ( top_top_rule_o
    = ( ^ [X4: rule] : ( member_rule @ X4 @ top_top_set_rule ) ) ) ).

% top_empty_eq
thf(fact_361_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_362_sfilter_Osimps_I2_J,axiom,
    ! [P: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sfilte7579662470208054517m_rule @ P @ S3 ) )
      = ( sfilte7579662470208054517m_rule @ P @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P ) @ S3 ) ) ) ) ).

% sfilter.simps(2)
thf(fact_363_sfilter_Osimps_I2_J,axiom,
    ! [P: rule > $o,S3: stream_rule] :
      ( ( stl_rule @ ( sfilter_rule @ P @ S3 ) )
      = ( sfilter_rule @ P @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P ) @ S3 ) ) ) ) ).

% sfilter.simps(2)
thf(fact_364_sfilter_Osimps_I1_J,axiom,
    ! [P: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sfilte7579662470208054517m_rule @ P @ S3 ) )
      = ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P ) @ S3 ) ) ) ).

% sfilter.simps(1)
thf(fact_365_sfilter_Osimps_I1_J,axiom,
    ! [P: rule > $o,S3: stream_rule] :
      ( ( shd_rule @ ( sfilter_rule @ P @ S3 ) )
      = ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P ) @ S3 ) ) ) ).

% sfilter.simps(1)
thf(fact_366_PersistentRuleSystem_Oaxioms_I2_J,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S6 )
     => ( abstra2778581409224112183ist_fm @ Eff @ Rules @ S6 ) ) ).

% PersistentRuleSystem.axioms(2)
thf(fact_367_pred__equals__eq2,axiom,
    ! [R: set_Pr7058068377845519745ist_fm,S6: set_Pr7058068377845519745ist_fm] :
      ( ( ( ^ [X4: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y5 ) @ R ) )
        = ( ^ [X4: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_368_pred__equals__eq2,axiom,
    ! [R: set_Pr1008144964186165195m_rule,S6: set_Pr1008144964186165195m_rule] :
      ( ( ( ^ [X4: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X4 @ Y5 ) @ R ) )
        = ( ^ [X4: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X4 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_369_per__rules_H,axiom,
    ! [R3: rule,A: list_fm,B: list_fm,A6: list_fm,B6: list_fm,R5: rule,Ss2: fset_P661503646757059847ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R3
        @ ( produc381145313068854617ist_fm @ A @ B ) )
     => ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R3
            @ ( produc381145313068854617ist_fm @ A6 @ B6 ) )
       => ( ( ( eff @ R5 @ ( produc381145313068854617ist_fm @ A @ B ) )
            = ( some_f2162114139285431020ist_fm @ Ss2 ) )
         => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B6 ) @ Ss2 )
           => ( R5 = R3 ) ) ) ) ) ).

% per_rules'
thf(fact_370_epath__completeness__Saturated,axiom,
    ! [S3: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ? [T3: abstra4572470701423150929m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T3 ) )
              = S3 )
            & ( abstra4476735159789068068ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ T3 )
            & ( abstra3507476289341504315m_rule @ T3 ) )
        | ? [Steps4: stream727092118206550309m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps4 ) )
              = S3 )
            & ( abstra5415202973855822231ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps4 )
            & ( abstra8647779336736900908ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps4 ) ) ) ) ).

% epath_completeness_Saturated
thf(fact_371_sdrop__while_Oraw__induct,axiom,
    ! [Pa: produc6030424929543929934m_rule > stream727092118206550309m_rule > $o,P: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule,Y: stream727092118206550309m_rule] :
      ( ! [Sdrop_while: ( produc164195504107695125m_rule > $o ) > stream727092118206550309m_rule > stream727092118206550309m_rule] :
          ( ! [S10: produc164195504107695125m_rule > $o,B7: stream727092118206550309m_rule] :
              ( ( ( Sdrop_while @ S10 @ B7 )
               != undefi856395438905169085m_rule )
             => ( Pa @ ( produc911768729731935304m_rule @ S10 @ B7 ) @ ( Sdrop_while @ S10 @ B7 ) ) )
         => ! [P5: produc164195504107695125m_rule > $o,S4: stream727092118206550309m_rule,Pa2: stream727092118206550309m_rule] :
              ( ( ( ( P5 @ ( shd_Pr7235097944458474089m_rule @ S4 ) )
                 => ( ( Sdrop_while @ P5 @ ( stl_Pr950425576149878629m_rule @ S4 ) )
                    = Pa2 ) )
                & ( ~ ( P5 @ ( shd_Pr7235097944458474089m_rule @ S4 ) )
                 => ( S4 = Pa2 ) ) )
             => ( ( Pa2 != undefi856395438905169085m_rule )
               => ( Pa @ ( produc911768729731935304m_rule @ P5 @ S4 ) @ Pa2 ) ) ) )
     => ( ( ( sdrop_5541336384294084785m_rule @ P @ S3 )
          = Y )
       => ( ( Y != undefi856395438905169085m_rule )
         => ( Pa @ ( produc911768729731935304m_rule @ P @ S3 ) @ Y ) ) ) ) ).

% sdrop_while.raw_induct
thf(fact_372_sdrop__while_Oraw__induct,axiom,
    ! [Pa: produc8882594798121145038m_rule > stream_rule > $o,P: rule > $o,S3: stream_rule,Y: stream_rule] :
      ( ! [Sdrop_while: ( rule > $o ) > stream_rule > stream_rule] :
          ( ! [S10: rule > $o,B7: stream_rule] :
              ( ( ( Sdrop_while @ S10 @ B7 )
               != undefi4918021367320992109m_rule )
             => ( Pa @ ( produc1357633833106916296m_rule @ S10 @ B7 ) @ ( Sdrop_while @ S10 @ B7 ) ) )
         => ! [P5: rule > $o,S4: stream_rule,Pa2: stream_rule] :
              ( ( ( ( P5 @ ( shd_rule @ S4 ) )
                 => ( ( Sdrop_while @ P5 @ ( stl_rule @ S4 ) )
                    = Pa2 ) )
                & ( ~ ( P5 @ ( shd_rule @ S4 ) )
                 => ( S4 = Pa2 ) ) )
             => ( ( Pa2 != undefi4918021367320992109m_rule )
               => ( Pa @ ( produc1357633833106916296m_rule @ P5 @ S4 ) @ Pa2 ) ) ) )
     => ( ( ( sdrop_while_rule @ P @ S3 )
          = Y )
       => ( ( Y != undefi4918021367320992109m_rule )
         => ( Pa @ ( produc1357633833106916296m_rule @ P @ S3 ) @ Y ) ) ) ) ).

% sdrop_while.raw_induct
thf(fact_373_nxt_Oelims,axiom,
    ! [X2: stream727092118206550309m_rule > $o,Xa2: stream727092118206550309m_rule,Y: $o] :
      ( ( ( linear7357459635800698592rule_o @ X2 @ Xa2 )
        = Y )
     => ( Y
        = ( X2 @ ( stl_Pr950425576149878629m_rule @ Xa2 ) ) ) ) ).

% nxt.elims
thf(fact_374_RuleSystem__Defs_Oenabled_Ocong,axiom,
    abstra3055540779341473792ist_fm = abstra3055540779341473792ist_fm ).

% RuleSystem_Defs.enabled.cong
thf(fact_375_RuleSystem__Defs_Oenabled__def,axiom,
    ( abstra3055540779341473792ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R2: rule,S: produc1996495991257130529ist_fm] :
        ? [X6: fset_P661503646757059847ist_fm] : ( Eff2 @ R2 @ S @ X6 ) ) ) ).

% RuleSystem_Defs.enabled_def
thf(fact_376_RuleSystem__Defs_Owf_Ocong,axiom,
    abstra4476735159789068068ist_fm = abstra4476735159789068068ist_fm ).

% RuleSystem_Defs.wf.cong
thf(fact_377_fst__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc5894184038225292483ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X4: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X4 @ X4 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% fst_diag_fst
thf(fact_378_snd__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc6725306161561501521e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X4: rule] : ( produc1769597910952761123e_rule @ X4 @ X4 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% snd_diag_snd
thf(fact_379_snd__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc1639249265332130309ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X4: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X4 @ X4 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% snd_diag_fst
thf(fact_380_fst__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc7264528503596239631e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X4: rule] : ( produc1769597910952761123e_rule @ X4 @ X4 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% fst_diag_snd
thf(fact_381_i_Oenabled__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R3: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R3 @ S3 )
      = ( ? [X6: fset_P661503646757059847ist_fm] : ( abstra5459341389059251418ist_fm @ Eff3 @ R3 @ S3 @ X6 ) ) ) ).

% i.enabled_def
thf(fact_382_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ? [T3: abstra4572470701423150929m_rule] :
              ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T3 ) )
                = S3 )
              & ( abstra4476735159789068068ist_fm @ Eff @ Rules @ T3 )
              & ( abstra3507476289341504315m_rule @ T3 ) )
          | ? [Steps4: stream727092118206550309m_rule] :
              ( ( ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps4 ) )
                = S3 )
              & ( abstra5415202973855822231ist_fm @ Eff @ Rules @ Steps4 )
              & ( abstra8647779336736900908ist_fm @ Eff @ Rules @ Steps4 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_383_enabled__def,axiom,
    ! [R3: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R3
        @ S3 )
      = ( ? [Sl4: fset_P661503646757059847ist_fm] :
            ( ( eff @ R3 @ S3 )
            = ( some_f2162114139285431020ist_fm @ Sl4 ) ) ) ) ).

% enabled_def
thf(fact_384_nxt_Osimps,axiom,
    ( linear7357459635800698592rule_o
    = ( ^ [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule] : ( Phi @ ( stl_Pr950425576149878629m_rule @ Xs ) ) ) ) ).

% nxt.simps
thf(fact_385_wf__ipath__epath,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps2: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps2 )
       => ( abstra5415202973855822231ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ Steps2 ) ) ) ).

% wf_ipath_epath
thf(fact_386_minWait__ex,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ? [N2: nat] :
            ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) )
            @ S3 ) ) ) ).

% minWait_ex
thf(fact_387_per__alw,axiom,
    ! [R3: rule,Steps2: stream727092118206550309m_rule] :
      ( ( abstra4153889242452539894ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R3 )
     => ( ( ( abstra5415202973855822231ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ Steps2 )
          & ( member8102475879199740618ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ top_to728902761071950545ist_fm ) )
       => ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( linear7082939723987557601m_rule
                @ ^ [Step: produc164195504107695125m_rule] :
                    ( abstra3055540779341473792ist_fm
                    @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ R3
                    @ ( produc6879501374131015971m_rule @ Step ) )
                @ Xs )
             => ( ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] :
                      ( ( produc7165828336582415457m_rule @ Step )
                      = R3 )
                  @ Xs )
                | ( linear7357459635800698592rule_o
                  @ ( linear7082939723987557601m_rule
                    @ ^ [Step: produc164195504107695125m_rule] :
                        ( abstra3055540779341473792ist_fm
                        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                            ( ( eff @ R2 @ S )
                            = ( some_f2162114139285431020ist_fm @ Ss ) )
                        @ R3
                        @ ( produc6879501374131015971m_rule @ Step ) ) )
                  @ Xs ) ) )
          @ Steps2 ) ) ) ).

% per_alw
thf(fact_388_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S6: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra2524698237922071414e_rule @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_389_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7200325324992684464ist_fm @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_390_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S6: set_fm,S3: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7152516257876272129ule_fm @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_391_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc164195504107695125m_rule > nat > fset_nat > $o,Rules: stream727092118206550309m_rule,S6: set_nat,S3: nat,Rs: stream727092118206550309m_rule] :
      ( ( abstra8087887464339558544le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra174609082569646337le_nat @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_392_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3154316310710804774e_rule @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_393_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7038331712872434097ule_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_394_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra4466204623219695281le_nat @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_395_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3055540779341473792ist_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_396_i_Owf__ipath__epath,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,T2: abstra4572470701423150929m_rule,Steps2: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps2 )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps2 ) ) ) ).

% i.wf_ipath_epath
thf(fact_397_pickEff,axiom,
    ! [R3: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R3
        @ S3 )
     => ( ( eff @ R3 @ S3 )
        = ( some_f2162114139285431020ist_fm
          @ ( abstra458490714785879965ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R3
            @ S3 ) ) ) ) ).

% pickEff
thf(fact_398_konig_Osimps_I1_J,axiom,
    ! [T2: abstract_tree_rule] :
      ( ( shd_rule @ ( abstract_konig_rule @ T2 ) )
      = ( abstract_root_rule @ T2 ) ) ).

% konig.simps(1)
thf(fact_399_konig_Osimps_I1_J,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( abstra3079604473823497664m_rule @ T2 ) )
      = ( abstra3503897793822247108m_rule @ T2 ) ) ).

% konig.simps(1)
thf(fact_400_i_Otrim__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( sdrop_while_rule
        @ ^ [R2: rule] :
            ~ ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S3 )
        @ Rs ) ) ).

% i.trim_def
thf(fact_401_alw__alw,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ ( linear5684299538428340377m_rule @ Phi2 ) )
      = ( linear5684299538428340377m_rule @ Phi2 ) ) ).

% alw_alw
thf(fact_402_alw__alw,axiom,
    ! [Phi2: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ ( linear7625967296368300489w_rule @ Phi2 ) )
      = ( linear7625967296368300489w_rule @ Phi2 ) ) ).

% alw_alw
thf(fact_403_nxt__mono,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear7357459635800698592rule_o @ Phi2 @ Xs2 )
     => ( ! [Xs3: stream727092118206550309m_rule] :
            ( ( Phi2 @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear7357459635800698592rule_o @ Psi @ Xs2 ) ) ) ).

% nxt_mono
thf(fact_404_alw__nxt,axiom,
    ( linear7625967296368300489w_rule
    = ( ^ [Phi: stream_rule > $o,Xs: stream_rule] :
          ( ( Phi @ Xs )
          & ( linear860818938954496176rule_o @ ( linear7625967296368300489w_rule @ Phi ) @ Xs ) ) ) ) ).

% alw_nxt
thf(fact_405_alw__nxt,axiom,
    ( linear5684299538428340377m_rule
    = ( ^ [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule] :
          ( ( Phi @ Xs )
          & ( linear7357459635800698592rule_o @ ( linear5684299538428340377m_rule @ Phi ) @ Xs ) ) ) ) ).

% alw_nxt
thf(fact_406_alw__invar,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule] :
      ( ( Phi2 @ Xs2 )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi2 @ Xs )
             => ( linear860818938954496176rule_o @ Phi2 @ Xs ) )
          @ Xs2 )
       => ( linear7625967296368300489w_rule @ Phi2 @ Xs2 ) ) ) ).

% alw_invar
thf(fact_407_alw__invar,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( Phi2 @ Xs2 )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( Phi2 @ Xs )
             => ( linear7357459635800698592rule_o @ Phi2 @ Xs ) )
          @ Xs2 )
       => ( linear5684299538428340377m_rule @ Phi2 @ Xs2 ) ) ) ).

% alw_invar
thf(fact_408_holds__aand,axiom,
    ! [P: produc164195504107695125m_rule > $o,Steps2: stream727092118206550309m_rule,Q: produc164195504107695125m_rule > $o] :
      ( ( ( linear7082939723987557601m_rule @ P @ Steps2 )
        & ( linear7082939723987557601m_rule @ Q @ Steps2 ) )
      = ( linear7082939723987557601m_rule
        @ ^ [Step: produc164195504107695125m_rule] :
            ( ( P @ Step )
            & ( Q @ Step ) )
        @ Steps2 ) ) ).

% holds_aand
thf(fact_409_holds__aand,axiom,
    ! [P: rule > $o,Steps2: stream_rule,Q: rule > $o] :
      ( ( ( linear4670012741531116049s_rule @ P @ Steps2 )
        & ( linear4670012741531116049s_rule @ Q @ Steps2 ) )
      = ( linear4670012741531116049s_rule
        @ ^ [Step: rule] :
            ( ( P @ Step )
            & ( Q @ Step ) )
        @ Steps2 ) ) ).

% holds_aand
thf(fact_410_alw__holds,axiom,
    ! [P: produc164195504107695125m_rule > $o,H: produc164195504107695125m_rule,T2: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear7082939723987557601m_rule @ P ) @ ( sCons_5731567480799343806m_rule @ H @ T2 ) )
      = ( ( P @ H )
        & ( linear5684299538428340377m_rule @ ( linear7082939723987557601m_rule @ P ) @ T2 ) ) ) ).

% alw_holds
thf(fact_411_alw__holds,axiom,
    ! [P: rule > $o,H: rule,T2: stream_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear4670012741531116049s_rule @ P ) @ ( sCons_rule @ H @ T2 ) )
      = ( ( P @ H )
        & ( linear7625967296368300489w_rule @ ( linear4670012741531116049s_rule @ P ) @ T2 ) ) ) ).

% alw_holds
thf(fact_412_RuleSystem__Defs_Ofair_Ocong,axiom,
    abstra1114877437032970289r_rule = abstra1114877437032970289r_rule ).

% RuleSystem_Defs.fair.cong
thf(fact_413_RuleSystem__Defs_Otrim_Ocong,axiom,
    abstra2363873759530810705ist_fm = abstra2363873759530810705ist_fm ).

% RuleSystem_Defs.trim.cong
thf(fact_414_RuleSystem__Defs_OpickEff_Ocong,axiom,
    abstra458490714785879965ist_fm = abstra458490714785879965ist_fm ).

% RuleSystem_Defs.pickEff.cong
thf(fact_415_alw__False,axiom,
    ! [Omega: stream727092118206550309m_rule] :
      ~ ( linear5684299538428340377m_rule
        @ ^ [X4: stream727092118206550309m_rule] : $false
        @ Omega ) ).

% alw_False
thf(fact_416_alw__False,axiom,
    ! [Omega: stream_rule] :
      ~ ( linear7625967296368300489w_rule
        @ ^ [X4: stream_rule] : $false
        @ Omega ) ).

% alw_False
thf(fact_417_alw__aand,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( Phi2 @ Xs )
            & ( Psi @ Xs ) ) )
      = ( ^ [Xs: stream727092118206550309m_rule] :
            ( ( linear5684299538428340377m_rule @ Phi2 @ Xs )
            & ( linear5684299538428340377m_rule @ Psi @ Xs ) ) ) ) ).

% alw_aand
thf(fact_418_alw__aand,axiom,
    ! [Phi2: stream_rule > $o,Psi: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule
        @ ^ [Xs: stream_rule] :
            ( ( Phi2 @ Xs )
            & ( Psi @ Xs ) ) )
      = ( ^ [Xs: stream_rule] :
            ( ( linear7625967296368300489w_rule @ Phi2 @ Xs )
            & ( linear7625967296368300489w_rule @ Psi @ Xs ) ) ) ) ).

% alw_aand
thf(fact_419_alw__mp,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ Phi2 @ Xs2 )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( Phi2 @ Xs )
             => ( Psi @ Xs ) )
          @ Xs2 )
       => ( linear5684299538428340377m_rule @ Psi @ Xs2 ) ) ) ).

% alw_mp
thf(fact_420_alw__mp,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ Phi2 @ Xs2 )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi2 @ Xs )
             => ( Psi @ Xs ) )
          @ Xs2 )
       => ( linear7625967296368300489w_rule @ Psi @ Xs2 ) ) ) ).

% alw_mp
thf(fact_421_alwD,axiom,
    ! [P: stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ P @ X2 )
     => ( P @ X2 ) ) ).

% alwD
thf(fact_422_alwD,axiom,
    ! [P: stream_rule > $o,X2: stream_rule] :
      ( ( linear7625967296368300489w_rule @ P @ X2 )
     => ( P @ X2 ) ) ).

% alwD
thf(fact_423_alw__alwD,axiom,
    ! [P: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ P @ Omega )
     => ( linear5684299538428340377m_rule @ ( linear5684299538428340377m_rule @ P ) @ Omega ) ) ).

% alw_alwD
thf(fact_424_alw__alwD,axiom,
    ! [P: stream_rule > $o,Omega: stream_rule] :
      ( ( linear7625967296368300489w_rule @ P @ Omega )
     => ( linear7625967296368300489w_rule @ ( linear7625967296368300489w_rule @ P ) @ Omega ) ) ).

% alw_alwD
thf(fact_425_alw__cong,axiom,
    ! [P: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule,Q1: stream727092118206550309m_rule > $o,Q22: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ P @ Omega )
     => ( ! [Omega2: stream727092118206550309m_rule] :
            ( ( P @ Omega2 )
           => ( ( Q1 @ Omega2 )
              = ( Q22 @ Omega2 ) ) )
       => ( ( linear5684299538428340377m_rule @ Q1 @ Omega )
          = ( linear5684299538428340377m_rule @ Q22 @ Omega ) ) ) ) ).

% alw_cong
thf(fact_426_alw__cong,axiom,
    ! [P: stream_rule > $o,Omega: stream_rule,Q1: stream_rule > $o,Q22: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ P @ Omega )
     => ( ! [Omega2: stream_rule] :
            ( ( P @ Omega2 )
           => ( ( Q1 @ Omega2 )
              = ( Q22 @ Omega2 ) ) )
       => ( ( linear7625967296368300489w_rule @ Q1 @ Omega )
          = ( linear7625967296368300489w_rule @ Q22 @ Omega ) ) ) ) ).

% alw_cong
thf(fact_427_alw__mono,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ Phi2 @ Xs2 )
     => ( ! [Xs3: stream727092118206550309m_rule] :
            ( ( Phi2 @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear5684299538428340377m_rule @ Psi @ Xs2 ) ) ) ).

% alw_mono
thf(fact_428_alw__mono,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ Phi2 @ Xs2 )
     => ( ! [Xs3: stream_rule] :
            ( ( Phi2 @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear7625967296368300489w_rule @ Psi @ Xs2 ) ) ) ).

% alw_mono
thf(fact_429_holds__mono,axiom,
    ! [P: produc164195504107695125m_rule > $o,Xs2: stream727092118206550309m_rule,Q: produc164195504107695125m_rule > $o] :
      ( ( linear7082939723987557601m_rule @ P @ Xs2 )
     => ( ! [X: produc164195504107695125m_rule] :
            ( ( P @ X )
           => ( Q @ X ) )
       => ( linear7082939723987557601m_rule @ Q @ Xs2 ) ) ) ).

% holds_mono
thf(fact_430_holds__mono,axiom,
    ! [P: rule > $o,Xs2: stream_rule,Q: rule > $o] :
      ( ( linear4670012741531116049s_rule @ P @ Xs2 )
     => ( ! [X: rule] :
            ( ( P @ X )
           => ( Q @ X ) )
       => ( linear4670012741531116049s_rule @ Q @ Xs2 ) ) ) ).

% holds_mono
thf(fact_431_all__imp__alw,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ! [X_1: stream727092118206550309m_rule] : ( Phi2 @ X_1 )
     => ( linear5684299538428340377m_rule @ Phi2 @ Xs2 ) ) ).

% all_imp_alw
thf(fact_432_all__imp__alw,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule] :
      ( ! [X_1: stream_rule] : ( Phi2 @ X_1 )
     => ( linear7625967296368300489w_rule @ Phi2 @ Xs2 ) ) ).

% all_imp_alw
thf(fact_433_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_434_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_435_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra4161464353856247968le_nat @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_436_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_437_alw__holds2,axiom,
    ! [P: produc164195504107695125m_rule > $o,Ss3: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear7082939723987557601m_rule @ P ) @ Ss3 )
      = ( ( P @ ( shd_Pr7235097944458474089m_rule @ Ss3 ) )
        & ( linear5684299538428340377m_rule @ ( linear7082939723987557601m_rule @ P ) @ ( stl_Pr950425576149878629m_rule @ Ss3 ) ) ) ) ).

% alw_holds2
thf(fact_438_alw__holds2,axiom,
    ! [P: rule > $o,Ss3: stream_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear4670012741531116049s_rule @ P ) @ Ss3 )
      = ( ( P @ ( shd_rule @ Ss3 ) )
        & ( linear7625967296368300489w_rule @ ( linear4670012741531116049s_rule @ P ) @ ( stl_rule @ Ss3 ) ) ) ) ).

% alw_holds2
thf(fact_439_alw_Ocases,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,A2: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ Phi2 @ A2 )
     => ~ ( ( Phi2 @ A2 )
         => ~ ( linear5684299538428340377m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ A2 ) ) ) ) ).

% alw.cases
thf(fact_440_alw_Ocases,axiom,
    ! [Phi2: stream_rule > $o,A2: stream_rule] :
      ( ( linear7625967296368300489w_rule @ Phi2 @ A2 )
     => ~ ( ( Phi2 @ A2 )
         => ~ ( linear7625967296368300489w_rule @ Phi2 @ ( stl_rule @ A2 ) ) ) ) ).

% alw.cases
thf(fact_441_alw_Osimps,axiom,
    ( linear5684299538428340377m_rule
    = ( ^ [Phi: stream727092118206550309m_rule > $o,A3: stream727092118206550309m_rule] :
        ? [Xs: stream727092118206550309m_rule] :
          ( ( A3 = Xs )
          & ( Phi @ Xs )
          & ( linear5684299538428340377m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ Xs ) ) ) ) ) ).

% alw.simps
thf(fact_442_alw_Osimps,axiom,
    ( linear7625967296368300489w_rule
    = ( ^ [Phi: stream_rule > $o,A3: stream_rule] :
        ? [Xs: stream_rule] :
          ( ( A3 = Xs )
          & ( Phi @ Xs )
          & ( linear7625967296368300489w_rule @ Phi @ ( stl_rule @ Xs ) ) ) ) ) ).

% alw.simps
thf(fact_443_alw_Ointros,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( Phi2 @ Xs2 )
     => ( ( linear5684299538428340377m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ Xs2 ) )
       => ( linear5684299538428340377m_rule @ Phi2 @ Xs2 ) ) ) ).

% alw.intros
thf(fact_444_alw_Ointros,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule] :
      ( ( Phi2 @ Xs2 )
     => ( ( linear7625967296368300489w_rule @ Phi2 @ ( stl_rule @ Xs2 ) )
       => ( linear7625967296368300489w_rule @ Phi2 @ Xs2 ) ) ) ).

% alw.intros
thf(fact_445_alw_Ocoinduct,axiom,
    ! [X3: stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule,Phi2: stream727092118206550309m_rule > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X3 @ X )
           => ? [Xs4: stream727092118206550309m_rule] :
                ( ( X = Xs4 )
                & ( Phi2 @ Xs4 )
                & ( ( X3 @ ( stl_Pr950425576149878629m_rule @ Xs4 ) )
                  | ( linear5684299538428340377m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ Xs4 ) ) ) ) )
       => ( linear5684299538428340377m_rule @ Phi2 @ X2 ) ) ) ).

% alw.coinduct
thf(fact_446_alw_Ocoinduct,axiom,
    ! [X3: stream_rule > $o,X2: stream_rule,Phi2: stream_rule > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream_rule] :
            ( ( X3 @ X )
           => ? [Xs4: stream_rule] :
                ( ( X = Xs4 )
                & ( Phi2 @ Xs4 )
                & ( ( X3 @ ( stl_rule @ Xs4 ) )
                  | ( linear7625967296368300489w_rule @ Phi2 @ ( stl_rule @ Xs4 ) ) ) ) )
       => ( linear7625967296368300489w_rule @ Phi2 @ X2 ) ) ) ).

% alw.coinduct
thf(fact_447_alw__coinduct,axiom,
    ! [X3: stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule,Phi2: stream727092118206550309m_rule > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X3 @ X )
           => ( Phi2 @ X ) )
       => ( ! [X: stream727092118206550309m_rule] :
              ( ( X3 @ X )
             => ( ~ ( linear5684299538428340377m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ X ) )
               => ( X3 @ ( stl_Pr950425576149878629m_rule @ X ) ) ) )
         => ( linear5684299538428340377m_rule @ Phi2 @ X2 ) ) ) ) ).

% alw_coinduct
thf(fact_448_alw__coinduct,axiom,
    ! [X3: stream_rule > $o,X2: stream_rule,Phi2: stream_rule > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: stream_rule] :
            ( ( X3 @ X )
           => ( Phi2 @ X ) )
       => ( ! [X: stream_rule] :
              ( ( X3 @ X )
             => ( ~ ( linear7625967296368300489w_rule @ Phi2 @ ( stl_rule @ X ) )
               => ( X3 @ ( stl_rule @ X ) ) ) )
         => ( linear7625967296368300489w_rule @ Phi2 @ X2 ) ) ) ) ).

% alw_coinduct
thf(fact_449_alw__iff__sdrop,axiom,
    ( linear5684299538428340377m_rule
    = ( ^ [P3: stream727092118206550309m_rule > $o,Omega3: stream727092118206550309m_rule] :
        ! [M: nat] : ( P3 @ ( sdrop_7224736112439592940m_rule @ M @ Omega3 ) ) ) ) ).

% alw_iff_sdrop
thf(fact_450_alw__iff__sdrop,axiom,
    ( linear7625967296368300489w_rule
    = ( ^ [P3: stream_rule > $o,Omega3: stream_rule] :
        ! [M: nat] : ( P3 @ ( sdrop_rule @ M @ Omega3 ) ) ) ) ).

% alw_iff_sdrop
thf(fact_451_alw__sdrop,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,N: nat] :
      ( ( linear5684299538428340377m_rule @ Phi2 @ Xs2 )
     => ( linear5684299538428340377m_rule @ Phi2 @ ( sdrop_7224736112439592940m_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_452_alw__sdrop,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,N: nat] :
      ( ( linear7625967296368300489w_rule @ Phi2 @ Xs2 )
     => ( linear7625967296368300489w_rule @ Phi2 @ ( sdrop_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_453_sdrop__fair,axiom,
    ! [Rs: stream_rule,M2: nat] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% sdrop_fair
thf(fact_454_fair__stl,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).

% fair_stl
thf(fact_455_alw__inv,axiom,
    ! [F2: stream727092118206550309m_rule > stream727092118206550309m_rule,P: stream727092118206550309m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ! [S4: stream727092118206550309m_rule] :
          ( ( F2 @ ( stl_Pr950425576149878629m_rule @ S4 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F2 @ S4 ) ) )
     => ( ( linear5684299538428340377m_rule @ P @ ( F2 @ S3 ) )
        = ( linear5684299538428340377m_rule
          @ ^ [X4: stream727092118206550309m_rule] : ( P @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% alw_inv
thf(fact_456_alw__inv,axiom,
    ! [F2: stream_rule > stream727092118206550309m_rule,P: stream727092118206550309m_rule > $o,S3: stream_rule] :
      ( ! [S4: stream_rule] :
          ( ( F2 @ ( stl_rule @ S4 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F2 @ S4 ) ) )
     => ( ( linear5684299538428340377m_rule @ P @ ( F2 @ S3 ) )
        = ( linear7625967296368300489w_rule
          @ ^ [X4: stream_rule] : ( P @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% alw_inv
thf(fact_457_alw__inv,axiom,
    ! [F2: stream727092118206550309m_rule > stream_rule,P: stream_rule > $o,S3: stream727092118206550309m_rule] :
      ( ! [S4: stream727092118206550309m_rule] :
          ( ( F2 @ ( stl_Pr950425576149878629m_rule @ S4 ) )
          = ( stl_rule @ ( F2 @ S4 ) ) )
     => ( ( linear7625967296368300489w_rule @ P @ ( F2 @ S3 ) )
        = ( linear5684299538428340377m_rule
          @ ^ [X4: stream727092118206550309m_rule] : ( P @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% alw_inv
thf(fact_458_alw__inv,axiom,
    ! [F2: stream_rule > stream_rule,P: stream_rule > $o,S3: stream_rule] :
      ( ! [S4: stream_rule] :
          ( ( F2 @ ( stl_rule @ S4 ) )
          = ( stl_rule @ ( F2 @ S4 ) ) )
     => ( ( linear7625967296368300489w_rule @ P @ ( F2 @ S3 ) )
        = ( linear7625967296368300489w_rule
          @ ^ [X4: stream_rule] : ( P @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% alw_inv
thf(fact_459_holds_Oelims_I3_J,axiom,
    ! [X2: produc164195504107695125m_rule > $o,Xa2: stream727092118206550309m_rule] :
      ( ~ ( linear7082939723987557601m_rule @ X2 @ Xa2 )
     => ~ ( X2 @ ( shd_Pr7235097944458474089m_rule @ Xa2 ) ) ) ).

% holds.elims(3)
thf(fact_460_holds_Oelims_I3_J,axiom,
    ! [X2: rule > $o,Xa2: stream_rule] :
      ( ~ ( linear4670012741531116049s_rule @ X2 @ Xa2 )
     => ~ ( X2 @ ( shd_rule @ Xa2 ) ) ) ).

% holds.elims(3)
thf(fact_461_holds_Oelims_I2_J,axiom,
    ! [X2: produc164195504107695125m_rule > $o,Xa2: stream727092118206550309m_rule] :
      ( ( linear7082939723987557601m_rule @ X2 @ Xa2 )
     => ( X2 @ ( shd_Pr7235097944458474089m_rule @ Xa2 ) ) ) ).

% holds.elims(2)
thf(fact_462_holds_Oelims_I2_J,axiom,
    ! [X2: rule > $o,Xa2: stream_rule] :
      ( ( linear4670012741531116049s_rule @ X2 @ Xa2 )
     => ( X2 @ ( shd_rule @ Xa2 ) ) ) ).

% holds.elims(2)
thf(fact_463_holds_Oelims_I1_J,axiom,
    ! [X2: produc164195504107695125m_rule > $o,Xa2: stream727092118206550309m_rule,Y: $o] :
      ( ( ( linear7082939723987557601m_rule @ X2 @ Xa2 )
        = Y )
     => ( Y
        = ( X2 @ ( shd_Pr7235097944458474089m_rule @ Xa2 ) ) ) ) ).

% holds.elims(1)
thf(fact_464_holds_Oelims_I1_J,axiom,
    ! [X2: rule > $o,Xa2: stream_rule,Y: $o] :
      ( ( ( linear4670012741531116049s_rule @ X2 @ Xa2 )
        = Y )
     => ( Y
        = ( X2 @ ( shd_rule @ Xa2 ) ) ) ) ).

% holds.elims(1)
thf(fact_465_holds_Osimps,axiom,
    ( linear7082939723987557601m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o,Xs: stream727092118206550309m_rule] : ( P3 @ ( shd_Pr7235097944458474089m_rule @ Xs ) ) ) ) ).

% holds.simps
thf(fact_466_holds_Osimps,axiom,
    ( linear4670012741531116049s_rule
    = ( ^ [P3: rule > $o,Xs: stream_rule] : ( P3 @ ( shd_rule @ Xs ) ) ) ) ).

% holds.simps
thf(fact_467_Konig,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ~ ( abstra3507476289341504315m_rule @ T2 )
     => ( abstra2285467793620430502m_rule @ T2 @ ( abstra3079604473823497664m_rule @ T2 ) ) ) ).

% Konig
thf(fact_468_holds__Stream,axiom,
    ! [P: produc164195504107695125m_rule > $o,X2: produc164195504107695125m_rule,S3: stream727092118206550309m_rule] :
      ( ( linear7082939723987557601m_rule @ P @ ( sCons_5731567480799343806m_rule @ X2 @ S3 ) )
      = ( P @ X2 ) ) ).

% holds_Stream
thf(fact_469_holds__Stream,axiom,
    ! [P: rule > $o,X2: rule,S3: stream_rule] :
      ( ( linear4670012741531116049s_rule @ P @ ( sCons_rule @ X2 @ S3 ) )
      = ( P @ X2 ) ) ).

% holds_Stream
thf(fact_470_i_Ofair__stl,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( stl_Pr950425576149878629m_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_471_i_Ofair__stl,axiom,
    ! [Rules: stream_rule,Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( stl_rule @ Rs ) ) ) ).

% i.fair_stl
thf(fact_472_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( stl_Pr950425576149878629m_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_473_RuleSystem__Defs_Ofair__stl,axiom,
    ! [Rules: stream_rule,Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( stl_rule @ Rs ) ) ) ).

% RuleSystem_Defs.fair_stl
thf(fact_474_i_Osdrop__fair,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M2: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( sdrop_7224736112439592940m_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_475_i_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M2: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_476_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M2: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( sdrop_7224736112439592940m_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_477_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M2: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_478_i_Ofair__fenum,axiom,
    ! [Rules: stream_rule] : ( abstra1114877437032970289r_rule @ Rules @ ( abstra7284221463285775110m_rule @ Rules ) ) ).

% i.fair_fenum
thf(fact_479_RuleSystem__Defs_Ofair__fenum,axiom,
    ! [Rules: stream_rule] : ( abstra1114877437032970289r_rule @ Rules @ ( abstra7284221463285775110m_rule @ Rules ) ) ).

% RuleSystem_Defs.fair_fenum
thf(fact_480_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S6: set_rule,S3: rule,Rs: stream_fm] :
      ( ( abstra5167362717900046400m_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra8496492133263135264m_rule @ Eff @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_481_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: nat > rule > fset_rule > $o,Rules: stream_nat,S6: set_rule,S3: rule,Rs: stream_nat] :
      ( ( abstra5092216761188309440t_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
         => ( member_nat @ ( shd_nat @ ( abstra924199939036388256t_rule @ Eff @ Rs @ S3 ) ) @ ( sset_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_482_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S6: set_fm,S3: fm,Rs: stream_fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra5942290089224017835_fm_fm @ Eff @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_483_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: nat > fm > fset_fm > $o,Rules: stream_nat,S6: set_fm,S3: fm,Rs: stream_nat] :
      ( ( abstra1720523912436379467nat_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
         => ( member_nat @ ( shd_nat @ ( abstra189054280229527851nat_fm @ Eff @ Rs @ S3 ) ) @ ( sset_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_484_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > nat > fset_nat > $o,Rules: stream_fm,S6: set_nat,S3: nat,Rs: stream_fm] :
      ( ( abstra9084600350906980299fm_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra7553130718700128683fm_nat @ Eff @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_485_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: nat > nat > fset_nat > $o,Rules: stream_nat,S6: set_nat,S3: nat,Rs: stream_nat] :
      ( ( abstra5088622135990055243at_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
         => ( member_nat @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff @ Rs @ S3 ) ) @ ( sset_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_486_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_487_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_488_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra4161464353856247968le_nat @ Eff @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_489_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S6: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs @ S3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_490_RuleSystem__Defs_OpickEff,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R3: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ Eff @ R3 @ S3 )
     => ( Eff @ R3 @ S3 @ ( abstra458490714785879965ist_fm @ Eff @ R3 @ S3 ) ) ) ).

% RuleSystem_Defs.pickEff
thf(fact_491_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S6: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra2524698237922071414e_rule @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_492_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra7200325324992684464ist_fm @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_493_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S6: set_fm,S3: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra7152516257876272129ule_fm @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_494_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > nat > fset_nat > $o,Rules: stream727092118206550309m_rule,S6: set_nat,S3: nat,Rs: stream727092118206550309m_rule] :
      ( ( abstra8087887464339558544le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra174609082569646337le_nat @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra7715883167740232304le_nat @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_495_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra3154316310710804774e_rule @ Eff @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_496_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra7038331712872434097ule_fm @ Eff @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_497_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra4466204623219695281le_nat @ Eff @ ( shd_rule @ ( abstra4161464353856247968le_nat @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_498_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra3055540779341473792ist_fm @ Eff @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_499_trim__fair,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra1114877437032970289r_rule @ rules
          @ ( abstra2363873759530810705ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 ) ) ) ) ).

% trim_fair
thf(fact_500_ftree__no__ipath,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps2: stream727092118206550309m_rule] :
      ( ( abstra3507476289341504315m_rule @ T2 )
     => ~ ( abstra2285467793620430502m_rule @ T2 @ Steps2 ) ) ).

% ftree_no_ipath
thf(fact_501_trim__in__R,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( member_rule
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) )
          @ ( sset_rule @ rules ) ) ) ) ).

% trim_in_R
thf(fact_502_trim__enabled,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra3055540779341473792ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) )
          @ S3 ) ) ) ).

% trim_enabled
thf(fact_503_fair__fenum,axiom,
    abstra1114877437032970289r_rule @ rules @ ( abstra7284221463285775110m_rule @ rules ) ).

% fair_fenum
thf(fact_504_i_OpickEff,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R3: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R3 @ S3 )
     => ( abstra5459341389059251418ist_fm @ Eff3 @ R3 @ S3 @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R3 @ S3 ) ) ) ).

% i.pickEff
thf(fact_505_RuleSystem__Defs_Otrim__def,axiom,
    ( abstra2363873759530810705ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] :
          ( sdrop_while_rule
          @ ^ [R2: rule] :
              ~ ( abstra3055540779341473792ist_fm @ Eff2 @ R2 @ S )
          @ Rs2 ) ) ) ).

% RuleSystem_Defs.trim_def
thf(fact_506_RuleSystem__Defs_Owf__ipath__epath,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,T2: abstra4572470701423150929m_rule,Steps2: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff @ Rules @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps2 )
       => ( abstra5415202973855822231ist_fm @ Eff @ Rules @ Steps2 ) ) ) ).

% RuleSystem_Defs.wf_ipath_epath
thf(fact_507_RuleSystem_Oper__alw,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,R3: rule,Steps2: stream727092118206550309m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( abstra4153889242452539894ist_fm @ Eff @ Rules @ S6 @ R3 )
       => ( ( ( abstra5415202973855822231ist_fm @ Eff @ Rules @ Steps2 )
            & ( member8102475879199740618ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ S6 ) )
         => ( linear5684299538428340377m_rule
            @ ^ [Xs: stream727092118206550309m_rule] :
                ( ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] : ( abstra3055540779341473792ist_fm @ Eff @ R3 @ ( produc6879501374131015971m_rule @ Step ) )
                  @ Xs )
               => ( ( linear7082939723987557601m_rule
                    @ ^ [Step: produc164195504107695125m_rule] :
                        ( ( produc7165828336582415457m_rule @ Step )
                        = R3 )
                    @ Xs )
                  | ( linear7357459635800698592rule_o
                    @ ( linear7082939723987557601m_rule
                      @ ^ [Step: produc164195504107695125m_rule] : ( abstra3055540779341473792ist_fm @ Eff @ R3 @ ( produc6879501374131015971m_rule @ Step ) ) )
                    @ Xs ) ) )
            @ Steps2 ) ) ) ) ).

% RuleSystem.per_alw
thf(fact_508_trim__def,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra2363873759530810705ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( sdrop_while_rule
        @ ^ [R2: rule] :
            ~ ( abstra3055540779341473792ist_fm
              @ ^ [S: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ S @ T )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R2
              @ S3 )
        @ Rs ) ) ).

% trim_def
thf(fact_509_ipath__mkTree__sdrop,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps2: stream727092118206550309m_rule,M2: nat] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps2 )
         => ? [N2: nat,S8: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S8 @ top_to728902761071950545ist_fm )
              & ( abstra2285467793620430502m_rule
                @ ( abstra178918799166095863ist_fm
                  @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R2 @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ ( sdrop_rule @ N2 @ Rs )
                  @ S8 )
                @ ( sdrop_7224736112439592940m_rule @ M2 @ Steps2 ) ) ) ) ) ) ).

% ipath_mkTree_sdrop
thf(fact_510_ipath__mkTree__saturated,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps2: stream727092118206550309m_rule,R3: rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps2 )
         => ( ( member_rule @ R3 @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R3
              @ Steps2 ) ) ) ) ) ).

% ipath_mkTree_saturated
thf(fact_511_trim__alt,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2363873759530810705ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 )
          = ( sdrop_rule
            @ ( abstra738319790293016300ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Rs ) ) ) ) ).

% trim_alt
thf(fact_512_RuleSystem_Opos,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R3: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
         => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R3 ) @ Rs ) )
            = R3 ) ) ) ) ).

% RuleSystem.pos
thf(fact_513_pos,axiom,
    ! [Rs: stream_rule,R3: rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( member_rule @ R3 @ ( sset_rule @ rules ) )
       => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R3 ) @ Rs ) )
          = R3 ) ) ) ).

% pos
thf(fact_514_ipath__mkTree__Saturated,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps2: stream727092118206550309m_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps2 )
         => ( abstra8647779336736900908ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ Steps2 ) ) ) ) ).

% ipath_mkTree_Saturated
thf(fact_515_saturated__def,axiom,
    ! [R3: rule] :
      ( ( abstra7252176601000738572ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R3 )
      = ( linear5684299538428340377m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( linear7082939723987557601m_rule
              @ ^ [Step: produc164195504107695125m_rule] :
                  ( abstra3055540779341473792ist_fm
                  @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R2 @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ R3
                  @ ( produc6879501374131015971m_rule @ Step ) )
              @ Xs )
           => ( linear2722748333883190224m_rule
              @ ( linear7082939723987557601m_rule
                @ ^ [Step: produc164195504107695125m_rule] :
                    ( ( produc7165828336582415457m_rule @ Step )
                    = R3 ) )
              @ Xs ) ) ) ) ).

% saturated_def
thf(fact_516_fair__uu__def,axiom,
    abstra1913438439042302728u_rule = abstra1114877437032970289r_rule ).

% fair_uu_def
thf(fact_517_ev__ev,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ ( linear2722748333883190224m_rule @ Phi2 ) )
      = ( linear2722748333883190224m_rule @ Phi2 ) ) ).

% ev_ev
thf(fact_518_ev__ev,axiom,
    ! [Phi2: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ ( linear6831045468625226112v_rule @ Phi2 ) )
      = ( linear6831045468625226112v_rule @ Phi2 ) ) ).

% ev_ev
thf(fact_519_not__ev__not,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o] :
      ( ( ^ [Xs: stream727092118206550309m_rule] :
            ~ ( linear2722748333883190224m_rule
              @ ^ [Xt: stream727092118206550309m_rule] :
                  ~ ( Phi2 @ Xt )
              @ Xs ) )
      = ( linear5684299538428340377m_rule @ Phi2 ) ) ).

% not_ev_not
thf(fact_520_not__ev__not,axiom,
    ! [Phi2: stream_rule > $o] :
      ( ( ^ [Xs: stream_rule] :
            ~ ( linear6831045468625226112v_rule
              @ ^ [Xt: stream_rule] :
                  ~ ( Phi2 @ Xt )
              @ Xs ) )
      = ( linear7625967296368300489w_rule @ Phi2 ) ) ).

% not_ev_not
thf(fact_521_not__alw__not,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o] :
      ( ( ^ [Xs: stream727092118206550309m_rule] :
            ~ ( linear5684299538428340377m_rule
              @ ^ [Xt: stream727092118206550309m_rule] :
                  ~ ( Phi2 @ Xt )
              @ Xs ) )
      = ( linear2722748333883190224m_rule @ Phi2 ) ) ).

% not_alw_not
thf(fact_522_not__alw__not,axiom,
    ! [Phi2: stream_rule > $o] :
      ( ( ^ [Xs: stream_rule] :
            ~ ( linear7625967296368300489w_rule
              @ ^ [Xt: stream_rule] :
                  ~ ( Phi2 @ Xt )
              @ Xs ) )
      = ( linear6831045468625226112v_rule @ Phi2 ) ) ).

% not_alw_not
thf(fact_523_ev__alw__stl,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi2 ) @ ( stl_Pr950425576149878629m_rule @ X2 ) )
      = ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi2 ) @ X2 ) ) ).

% ev_alw_stl
thf(fact_524_ev__alw__stl,axiom,
    ! [Phi2: stream_rule > $o,X2: stream_rule] :
      ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi2 ) @ ( stl_rule @ X2 ) )
      = ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi2 ) @ X2 ) ) ).

% ev_alw_stl
thf(fact_525_ev__mono,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ Xs2 )
     => ( ! [Xs3: stream727092118206550309m_rule] :
            ( ( Phi2 @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear2722748333883190224m_rule @ Psi @ Xs2 ) ) ) ).

% ev_mono
thf(fact_526_ev__mono,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ Xs2 )
     => ( ! [Xs3: stream_rule] :
            ( ( Phi2 @ Xs3 )
           => ( Psi @ Xs3 ) )
       => ( linear6831045468625226112v_rule @ Psi @ Xs2 ) ) ) ).

% ev_mono
thf(fact_527_ev_Obase,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( Phi2 @ Xs2 )
     => ( linear2722748333883190224m_rule @ Phi2 @ Xs2 ) ) ).

% ev.base
thf(fact_528_ev_Obase,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule] :
      ( ( Phi2 @ Xs2 )
     => ( linear6831045468625226112v_rule @ Phi2 @ Xs2 ) ) ).

% ev.base
thf(fact_529_ev__or,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( Phi2 @ Xs )
            | ( Psi @ Xs ) ) )
      = ( ^ [Xs: stream727092118206550309m_rule] :
            ( ( linear2722748333883190224m_rule @ Phi2 @ Xs )
            | ( linear2722748333883190224m_rule @ Psi @ Xs ) ) ) ) ).

% ev_or
thf(fact_530_ev__or,axiom,
    ! [Phi2: stream_rule > $o,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule
        @ ^ [Xs: stream_rule] :
            ( ( Phi2 @ Xs )
            | ( Psi @ Xs ) ) )
      = ( ^ [Xs: stream_rule] :
            ( ( linear6831045468625226112v_rule @ Phi2 @ Xs )
            | ( linear6831045468625226112v_rule @ Psi @ Xs ) ) ) ) ).

% ev_or
thf(fact_531_ev__False,axiom,
    ! [Omega: stream727092118206550309m_rule] :
      ~ ( linear2722748333883190224m_rule
        @ ^ [X4: stream727092118206550309m_rule] : $false
        @ Omega ) ).

% ev_False
thf(fact_532_ev__False,axiom,
    ! [Omega: stream_rule] :
      ~ ( linear6831045468625226112v_rule
        @ ^ [X4: stream_rule] : $false
        @ Omega ) ).

% ev_False
thf(fact_533_RuleSystem__Defs_OmkTree_Ocong,axiom,
    abstra178918799166095863ist_fm = abstra178918799166095863ist_fm ).

% RuleSystem_Defs.mkTree.cong
thf(fact_534_RuleSystem_OminWait_Ocong,axiom,
    abstra738319790293016300ist_fm = abstra738319790293016300ist_fm ).

% RuleSystem.minWait.cong
thf(fact_535_ev__induct__strong,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule,P: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ X2 )
     => ( ! [Xs3: stream727092118206550309m_rule] :
            ( ( Phi2 @ Xs3 )
           => ( P @ Xs3 ) )
       => ( ! [Xs3: stream727092118206550309m_rule] :
              ( ( linear2722748333883190224m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ Xs3 ) )
             => ( ~ ( Phi2 @ Xs3 )
               => ( ( P @ ( stl_Pr950425576149878629m_rule @ Xs3 ) )
                 => ( P @ Xs3 ) ) ) )
         => ( P @ X2 ) ) ) ) ).

% ev_induct_strong
thf(fact_536_ev__induct__strong,axiom,
    ! [Phi2: stream_rule > $o,X2: stream_rule,P: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ X2 )
     => ( ! [Xs3: stream_rule] :
            ( ( Phi2 @ Xs3 )
           => ( P @ Xs3 ) )
       => ( ! [Xs3: stream_rule] :
              ( ( linear6831045468625226112v_rule @ Phi2 @ ( stl_rule @ Xs3 ) )
             => ( ~ ( Phi2 @ Xs3 )
               => ( ( P @ ( stl_rule @ Xs3 ) )
                 => ( P @ Xs3 ) ) ) )
         => ( P @ X2 ) ) ) ) ).

% ev_induct_strong
thf(fact_537_ev_Osimps,axiom,
    ( linear2722748333883190224m_rule
    = ( ^ [Phi: stream727092118206550309m_rule > $o,A3: stream727092118206550309m_rule] :
          ( ? [Xs: stream727092118206550309m_rule] :
              ( ( A3 = Xs )
              & ( Phi @ Xs ) )
          | ? [Xs: stream727092118206550309m_rule] :
              ( ( A3 = Xs )
              & ( linear2722748333883190224m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ Xs ) ) ) ) ) ) ).

% ev.simps
thf(fact_538_ev_Osimps,axiom,
    ( linear6831045468625226112v_rule
    = ( ^ [Phi: stream_rule > $o,A3: stream_rule] :
          ( ? [Xs: stream_rule] :
              ( ( A3 = Xs )
              & ( Phi @ Xs ) )
          | ? [Xs: stream_rule] :
              ( ( A3 = Xs )
              & ( linear6831045468625226112v_rule @ Phi @ ( stl_rule @ Xs ) ) ) ) ) ) ).

% ev.simps
thf(fact_539_ev_Ocases,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,A2: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ A2 )
     => ( ~ ( Phi2 @ A2 )
       => ( linear2722748333883190224m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ A2 ) ) ) ) ).

% ev.cases
thf(fact_540_ev_Ocases,axiom,
    ! [Phi2: stream_rule > $o,A2: stream_rule] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ A2 )
     => ( ~ ( Phi2 @ A2 )
       => ( linear6831045468625226112v_rule @ Phi2 @ ( stl_rule @ A2 ) ) ) ) ).

% ev.cases
thf(fact_541_ev_Ostep,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ Xs2 ) )
     => ( linear2722748333883190224m_rule @ Phi2 @ Xs2 ) ) ).

% ev.step
thf(fact_542_ev_Ostep,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ ( stl_rule @ Xs2 ) )
     => ( linear6831045468625226112v_rule @ Phi2 @ Xs2 ) ) ).

% ev.step
thf(fact_543_ev__alw__imp__alw__ev,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi2 ) @ Xs2 )
     => ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ Phi2 ) @ Xs2 ) ) ).

% ev_alw_imp_alw_ev
thf(fact_544_ev__alw__imp__alw__ev,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi2 ) @ Xs2 )
     => ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ Phi2 ) @ Xs2 ) ) ).

% ev_alw_imp_alw_ev
thf(fact_545_ev__cong,axiom,
    ! [P: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule,Q1: stream727092118206550309m_rule > $o,Q22: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ P @ Omega )
     => ( ! [Omega2: stream727092118206550309m_rule] :
            ( ( P @ Omega2 )
           => ( ( Q1 @ Omega2 )
              = ( Q22 @ Omega2 ) ) )
       => ( ( linear2722748333883190224m_rule @ Q1 @ Omega )
          = ( linear2722748333883190224m_rule @ Q22 @ Omega ) ) ) ) ).

% ev_cong
thf(fact_546_ev__cong,axiom,
    ! [P: stream_rule > $o,Omega: stream_rule,Q1: stream_rule > $o,Q22: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ P @ Omega )
     => ( ! [Omega2: stream_rule] :
            ( ( P @ Omega2 )
           => ( ( Q1 @ Omega2 )
              = ( Q22 @ Omega2 ) ) )
       => ( ( linear6831045468625226112v_rule @ Q1 @ Omega )
          = ( linear6831045468625226112v_rule @ Q22 @ Omega ) ) ) ) ).

% ev_cong
thf(fact_547_ev__Stream,axiom,
    ! [P: stream727092118206550309m_rule > $o,X2: produc164195504107695125m_rule,S3: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ P @ ( sCons_5731567480799343806m_rule @ X2 @ S3 ) )
      = ( ( P @ ( sCons_5731567480799343806m_rule @ X2 @ S3 ) )
        | ( linear2722748333883190224m_rule @ P @ S3 ) ) ) ).

% ev_Stream
thf(fact_548_ev__Stream,axiom,
    ! [P: stream_rule > $o,X2: rule,S3: stream_rule] :
      ( ( linear6831045468625226112v_rule @ P @ ( sCons_rule @ X2 @ S3 ) )
      = ( ( P @ ( sCons_rule @ X2 @ S3 ) )
        | ( linear6831045468625226112v_rule @ P @ S3 ) ) ) ).

% ev_Stream
thf(fact_549_ev__iff__sdrop,axiom,
    ( linear2722748333883190224m_rule
    = ( ^ [P3: stream727092118206550309m_rule > $o,Omega3: stream727092118206550309m_rule] :
        ? [M: nat] : ( P3 @ ( sdrop_7224736112439592940m_rule @ M @ Omega3 ) ) ) ) ).

% ev_iff_sdrop
thf(fact_550_ev__iff__sdrop,axiom,
    ( linear6831045468625226112v_rule
    = ( ^ [P3: stream_rule > $o,Omega3: stream_rule] :
        ? [M: nat] : ( P3 @ ( sdrop_rule @ M @ Omega3 ) ) ) ) ).

% ev_iff_sdrop
thf(fact_551_ev__nxt,axiom,
    ( linear6831045468625226112v_rule
    = ( ^ [Phi: stream_rule > $o,Xs: stream_rule] :
          ( ( Phi @ Xs )
          | ( linear860818938954496176rule_o @ ( linear6831045468625226112v_rule @ Phi ) @ Xs ) ) ) ) ).

% ev_nxt
thf(fact_552_ev__nxt,axiom,
    ( linear2722748333883190224m_rule
    = ( ^ [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule] :
          ( ( Phi @ Xs )
          | ( linear7357459635800698592rule_o @ ( linear2722748333883190224m_rule @ Phi ) @ Xs ) ) ) ) ).

% ev_nxt
thf(fact_553_ev__inv,axiom,
    ! [F2: stream727092118206550309m_rule > stream727092118206550309m_rule,P: stream727092118206550309m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ! [S4: stream727092118206550309m_rule] :
          ( ( F2 @ ( stl_Pr950425576149878629m_rule @ S4 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F2 @ S4 ) ) )
     => ( ( linear2722748333883190224m_rule @ P @ ( F2 @ S3 ) )
        = ( linear2722748333883190224m_rule
          @ ^ [X4: stream727092118206550309m_rule] : ( P @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% ev_inv
thf(fact_554_ev__inv,axiom,
    ! [F2: stream_rule > stream727092118206550309m_rule,P: stream727092118206550309m_rule > $o,S3: stream_rule] :
      ( ! [S4: stream_rule] :
          ( ( F2 @ ( stl_rule @ S4 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F2 @ S4 ) ) )
     => ( ( linear2722748333883190224m_rule @ P @ ( F2 @ S3 ) )
        = ( linear6831045468625226112v_rule
          @ ^ [X4: stream_rule] : ( P @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% ev_inv
thf(fact_555_ev__inv,axiom,
    ! [F2: stream727092118206550309m_rule > stream_rule,P: stream_rule > $o,S3: stream727092118206550309m_rule] :
      ( ! [S4: stream727092118206550309m_rule] :
          ( ( F2 @ ( stl_Pr950425576149878629m_rule @ S4 ) )
          = ( stl_rule @ ( F2 @ S4 ) ) )
     => ( ( linear6831045468625226112v_rule @ P @ ( F2 @ S3 ) )
        = ( linear2722748333883190224m_rule
          @ ^ [X4: stream727092118206550309m_rule] : ( P @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% ev_inv
thf(fact_556_ev__inv,axiom,
    ! [F2: stream_rule > stream_rule,P: stream_rule > $o,S3: stream_rule] :
      ( ! [S4: stream_rule] :
          ( ( F2 @ ( stl_rule @ S4 ) )
          = ( stl_rule @ ( F2 @ S4 ) ) )
     => ( ( linear6831045468625226112v_rule @ P @ ( F2 @ S3 ) )
        = ( linear6831045468625226112v_rule
          @ ^ [X4: stream_rule] : ( P @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% ev_inv
thf(fact_557_not__ev,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o] :
      ( ( ^ [Xs: stream727092118206550309m_rule] :
            ~ ( linear2722748333883190224m_rule @ Phi2 @ Xs ) )
      = ( linear5684299538428340377m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ~ ( Phi2 @ Xs ) ) ) ).

% not_ev
thf(fact_558_not__ev,axiom,
    ! [Phi2: stream_rule > $o] :
      ( ( ^ [Xs: stream_rule] :
            ~ ( linear6831045468625226112v_rule @ Phi2 @ Xs ) )
      = ( linear7625967296368300489w_rule
        @ ^ [Xs: stream_rule] :
            ~ ( Phi2 @ Xs ) ) ) ).

% not_ev
thf(fact_559_not__alw,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o] :
      ( ( ^ [Xs: stream727092118206550309m_rule] :
            ~ ( linear5684299538428340377m_rule @ Phi2 @ Xs ) )
      = ( linear2722748333883190224m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ~ ( Phi2 @ Xs ) ) ) ).

% not_alw
thf(fact_560_not__alw,axiom,
    ! [Phi2: stream_rule > $o] :
      ( ( ^ [Xs: stream_rule] :
            ~ ( linear7625967296368300489w_rule @ Phi2 @ Xs ) )
      = ( linear6831045468625226112v_rule
        @ ^ [Xs: stream_rule] :
            ~ ( Phi2 @ Xs ) ) ) ).

% not_alw
thf(fact_561_not__ev__iff,axiom,
    ! [P: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ( ~ ( linear2722748333883190224m_rule @ P @ Omega ) )
      = ( linear5684299538428340377m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ~ ( P @ Xs )
        @ Omega ) ) ).

% not_ev_iff
thf(fact_562_not__ev__iff,axiom,
    ! [P: stream_rule > $o,Omega: stream_rule] :
      ( ( ~ ( linear6831045468625226112v_rule @ P @ Omega ) )
      = ( linear7625967296368300489w_rule
        @ ^ [Xs: stream_rule] :
            ~ ( P @ Xs )
        @ Omega ) ) ).

% not_ev_iff
thf(fact_563_ev__alw__aand,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi2 ) @ Xs2 )
     => ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Psi ) @ Xs2 )
       => ( linear2722748333883190224m_rule
          @ ( linear5684299538428340377m_rule
            @ ^ [Xs: stream727092118206550309m_rule] :
                ( ( Phi2 @ Xs )
                & ( Psi @ Xs ) ) )
          @ Xs2 ) ) ) ).

% ev_alw_aand
thf(fact_564_ev__alw__aand,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi2 ) @ Xs2 )
     => ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Psi ) @ Xs2 )
       => ( linear6831045468625226112v_rule
          @ ( linear7625967296368300489w_rule
            @ ^ [Xs: stream_rule] :
                ( ( Phi2 @ Xs )
                & ( Psi @ Xs ) ) )
          @ Xs2 ) ) ) ).

% ev_alw_aand
thf(fact_565_ev__alw__impl,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ Xs2 )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( Phi2 @ Xs )
             => ( Psi @ Xs ) )
          @ Xs2 )
       => ( linear2722748333883190224m_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_impl
thf(fact_566_ev__alw__impl,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ Xs2 )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi2 @ Xs )
             => ( Psi @ Xs ) )
          @ Xs2 )
       => ( linear6831045468625226112v_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_impl
thf(fact_567_not__alw__iff,axiom,
    ! [P: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ( ~ ( linear5684299538428340377m_rule @ P @ Omega ) )
      = ( linear2722748333883190224m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ~ ( P @ Xs )
        @ Omega ) ) ).

% not_alw_iff
thf(fact_568_not__alw__iff,axiom,
    ! [P: stream_rule > $o,Omega: stream_rule] :
      ( ( ~ ( linear7625967296368300489w_rule @ P @ Omega ) )
      = ( linear6831045468625226112v_rule
        @ ^ [Xs: stream_rule] :
            ~ ( P @ Xs )
        @ Omega ) ) ).

% not_alw_iff
thf(fact_569_ev__alw__impl__ev,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ Xs2 )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( Phi2 @ Xs )
             => ( linear2722748333883190224m_rule @ Psi @ Xs ) )
          @ Xs2 )
       => ( linear2722748333883190224m_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_impl_ev
thf(fact_570_ev__alw__impl__ev,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ Xs2 )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi2 @ Xs )
             => ( linear6831045468625226112v_rule @ Psi @ Xs ) )
          @ Xs2 )
       => ( linear6831045468625226112v_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_impl_ev
thf(fact_571_alw__alw__impl__ev,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( linear5684299538428340377m_rule @ Phi2 @ Xs )
           => ( linear2722748333883190224m_rule @ Psi @ Xs ) ) )
      = ( ^ [Xs: stream727092118206550309m_rule] :
            ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi2 ) @ Xs )
           => ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ Psi ) @ Xs ) ) ) ) ).

% alw_alw_impl_ev
thf(fact_572_alw__alw__impl__ev,axiom,
    ! [Phi2: stream_rule > $o,Psi: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule
        @ ^ [Xs: stream_rule] :
            ( ( linear7625967296368300489w_rule @ Phi2 @ Xs )
           => ( linear6831045468625226112v_rule @ Psi @ Xs ) ) )
      = ( ^ [Xs: stream_rule] :
            ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi2 ) @ Xs )
           => ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ Psi ) @ Xs ) ) ) ) ).

% alw_alw_impl_ev
thf(fact_573_alw__impl__ev__alw,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( Phi2 @ Xs )
           => ( linear2722748333883190224m_rule @ Psi @ Xs ) )
        @ Xs2 )
     => ( linear5684299538428340377m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( linear2722748333883190224m_rule @ Phi2 @ Xs )
           => ( linear2722748333883190224m_rule @ Psi @ Xs ) )
        @ Xs2 ) ) ).

% alw_impl_ev_alw
thf(fact_574_alw__impl__ev__alw,axiom,
    ! [Phi2: stream_rule > $o,Psi: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear7625967296368300489w_rule
        @ ^ [Xs: stream_rule] :
            ( ( Phi2 @ Xs )
           => ( linear6831045468625226112v_rule @ Psi @ Xs ) )
        @ Xs2 )
     => ( linear7625967296368300489w_rule
        @ ^ [Xs: stream_rule] :
            ( ( linear6831045468625226112v_rule @ Phi2 @ Xs )
           => ( linear6831045468625226112v_rule @ Psi @ Xs ) )
        @ Xs2 ) ) ).

% alw_impl_ev_alw
thf(fact_575_ev__alw__alw__impl,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi2 ) @ Xs2 )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( linear5684299538428340377m_rule @ Phi2 @ Xs )
             => ( linear2722748333883190224m_rule @ Psi @ Xs ) )
          @ Xs2 )
       => ( linear2722748333883190224m_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_alw_impl
thf(fact_576_ev__alw__alw__impl,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi2 ) @ Xs2 )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs: stream_rule] :
              ( ( linear7625967296368300489w_rule @ Phi2 @ Xs )
             => ( linear6831045468625226112v_rule @ Psi @ Xs ) )
          @ Xs2 )
       => ( linear6831045468625226112v_rule @ Psi @ Xs2 ) ) ) ).

% ev_alw_alw_impl
thf(fact_577_alw__ev__imp__ev__alw,axiom,
    ! [P: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ P ) @ Omega )
     => ( linear2722748333883190224m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( P @ Xs )
            & ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ P ) @ Xs ) )
        @ Omega ) ) ).

% alw_ev_imp_ev_alw
thf(fact_578_alw__ev__imp__ev__alw,axiom,
    ! [P: stream_rule > $o,Omega: stream_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ P ) @ Omega )
     => ( linear6831045468625226112v_rule
        @ ^ [Xs: stream_rule] :
            ( ( P @ Xs )
            & ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ P ) @ Xs ) )
        @ Omega ) ) ).

% alw_ev_imp_ev_alw
thf(fact_579_alw__ev__stl,axiom,
    ! [P: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ P ) @ ( stl_Pr950425576149878629m_rule @ Omega ) )
      = ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ P ) @ Omega ) ) ).

% alw_ev_stl
thf(fact_580_alw__ev__stl,axiom,
    ! [P: stream_rule > $o,Omega: stream_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ P ) @ ( stl_rule @ Omega ) )
      = ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ P ) @ Omega ) ) ).

% alw_ev_stl
thf(fact_581_alw__ev__sdrop,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,M2: nat,Xs2: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ Phi2 ) @ ( sdrop_7224736112439592940m_rule @ M2 @ Xs2 ) )
     => ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ Phi2 ) @ Xs2 ) ) ).

% alw_ev_sdrop
thf(fact_582_alw__ev__sdrop,axiom,
    ! [Phi2: stream_rule > $o,M2: nat,Xs2: stream_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ Phi2 ) @ ( sdrop_rule @ M2 @ Xs2 ) )
     => ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ Phi2 ) @ Xs2 ) ) ).

% alw_ev_sdrop
thf(fact_583_ev__holds__sset,axiom,
    ! [P: produc164195504107695125m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ ( linear7082939723987557601m_rule @ P ) @ Xs2 )
      = ( ? [X4: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ X4 @ ( sset_P4979403222752902913m_rule @ Xs2 ) )
            & ( P @ X4 ) ) ) ) ).

% ev_holds_sset
thf(fact_584_ev__holds__sset,axiom,
    ! [P: rule > $o,Xs2: stream_rule] :
      ( ( linear6831045468625226112v_rule @ ( linear4670012741531116049s_rule @ P ) @ Xs2 )
      = ( ? [X4: rule] :
            ( ( member_rule @ X4 @ ( sset_rule @ Xs2 ) )
            & ( P @ X4 ) ) ) ) ).

% ev_holds_sset
thf(fact_585_ev__alw__imp__nxt,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ Xs2 )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi2 @ Xs )
             => ( linear860818938954496176rule_o @ Phi2 @ Xs ) )
          @ Xs2 )
       => ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi2 ) @ Xs2 ) ) ) ).

% ev_alw_imp_nxt
thf(fact_586_ev__alw__imp__nxt,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ Xs2 )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( Phi2 @ Xs )
             => ( linear7357459635800698592rule_o @ Phi2 @ Xs ) )
          @ Xs2 )
       => ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi2 ) @ Xs2 ) ) ) ).

% ev_alw_imp_nxt
thf(fact_587_variance,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,Psi: stream_rule > $o] :
      ( ( Phi2 @ Xs2 )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs: stream_rule] :
              ( ( Phi2 @ Xs )
             => ( ( Psi @ Xs )
                | ( linear860818938954496176rule_o @ Phi2 @ Xs ) ) )
          @ Xs2 )
       => ( ( linear7625967296368300489w_rule @ Phi2 @ Xs2 )
          | ( linear6831045468625226112v_rule @ Psi @ Xs2 ) ) ) ) ).

% variance
thf(fact_588_variance,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( Phi2 @ Xs2 )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( Phi2 @ Xs )
             => ( ( Psi @ Xs )
                | ( linear7357459635800698592rule_o @ Phi2 @ Xs ) ) )
          @ Xs2 )
       => ( ( linear5684299538428340377m_rule @ Phi2 @ Xs2 )
          | ( linear2722748333883190224m_rule @ Psi @ Xs2 ) ) ) ) ).

% variance
thf(fact_589_ev__sfilter,axiom,
    ! [P: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule,Q: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ ( linear7082939723987557601m_rule @ P ) ) @ S3 )
     => ( ( linear2722748333883190224m_rule @ Q @ ( sfilte7579662470208054517m_rule @ P @ S3 ) )
        = ( linear2722748333883190224m_rule
          @ ^ [X4: stream727092118206550309m_rule] : ( Q @ ( sfilte7579662470208054517m_rule @ P @ X4 ) )
          @ S3 ) ) ) ).

% ev_sfilter
thf(fact_590_ev__sfilter,axiom,
    ! [P: rule > $o,S3: stream_rule,Q: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ ( linear4670012741531116049s_rule @ P ) ) @ S3 )
     => ( ( linear6831045468625226112v_rule @ Q @ ( sfilter_rule @ P @ S3 ) )
        = ( linear6831045468625226112v_rule
          @ ^ [X4: stream_rule] : ( Q @ ( sfilter_rule @ P @ X4 ) )
          @ S3 ) ) ) ).

% ev_sfilter
thf(fact_591_alw__sfilter,axiom,
    ! [P: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule,Q: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ ( linear7082939723987557601m_rule @ P ) ) @ S3 )
     => ( ( linear5684299538428340377m_rule @ Q @ ( sfilte7579662470208054517m_rule @ P @ S3 ) )
        = ( linear5684299538428340377m_rule
          @ ^ [X4: stream727092118206550309m_rule] : ( Q @ ( sfilte7579662470208054517m_rule @ P @ X4 ) )
          @ S3 ) ) ) ).

% alw_sfilter
thf(fact_592_alw__sfilter,axiom,
    ! [P: rule > $o,S3: stream_rule,Q: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ ( linear4670012741531116049s_rule @ P ) ) @ S3 )
     => ( ( linear7625967296368300489w_rule @ Q @ ( sfilter_rule @ P @ S3 ) )
        = ( linear7625967296368300489w_rule
          @ ^ [X4: stream_rule] : ( Q @ ( sfilter_rule @ P @ X4 ) )
          @ S3 ) ) ) ).

% alw_sfilter
thf(fact_593_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra5602818997816140546e_rule @ Eff @ Rules @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_594_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra2794488811626936717ule_fm @ Eff @ Rules @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_595_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra520652830744493197le_nat @ Eff @ Rules @ ( abstra3504266011406229370le_nat @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_596_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_597_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: list_fm > list_fm > fset_list_fm > $o,Rs: stream_list_fm,S3: list_fm] :
      ( ( abstra778364809706815794ist_fm @ ( abstra3846921640779263781ist_fm @ Eff @ Rs @ S3 ) )
      = ( produc381145313068854617ist_fm @ S3 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_598_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) )
      = ( produc491467635432902671m_rule @ S3 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_599_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S6: set_rule,S3: rule,Rs: stream_fm,Steps2: stream1800043145262542076ule_fm,R3: fm] :
      ( ( abstra5167362717900046400m_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra8780395760585530021ule_fm @ ( abstra8242076542743448570m_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
             => ( ( linear2894481046950705138ule_fm
                  @ ( linear543948517440606634ule_fm
                    @ ^ [Step: product_prod_rule_fm] : ( abstra2065808176054476721m_rule @ Eff @ R3 @ ( product_fst_rule_fm @ Step ) ) )
                  @ Steps2 )
               => ( linear6327378476458846971ule_fm
                  @ ( linear543948517440606634ule_fm
                    @ ^ [Step: product_prod_rule_fm] :
                        ( ( product_snd_rule_fm @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_600_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: nat > rule > fset_rule > $o,Rules: stream_nat,S6: set_rule,S3: rule,Rs: stream_nat,Steps2: stream2660830046501416060le_nat,R3: nat] :
      ( ( abstra5092216761188309440t_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
         => ( ( abstra4272207280279176229le_nat @ ( abstra267001596586369658t_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
             => ( ( linear9115425191959525362le_nat
                  @ ( linear7086005427038662058le_nat
                    @ ^ [Step: produc6595031365954185078le_nat] : ( abstra1228940208399835569t_rule @ Eff @ R3 @ ( product_fst_rule_nat @ Step ) ) )
                  @ Steps2 )
               => ( linear2305749637662859131le_nat
                  @ ( linear7086005427038662058le_nat
                    @ ^ [Step: produc6595031365954185078le_nat] :
                        ( ( product_snd_rule_nat @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_601_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S6: set_fm,S3: fm,Rs: stream_fm,Steps2: stream1375025385734440967_fm_fm,R3: fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra724614664783131952_fm_fm @ ( abstra5285091746773999237_fm_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
             => ( ( linear5567832576463481085_fm_fm
                  @ ( linear3538412811542617781_fm_fm
                    @ ^ [Step: product_prod_fm_fm] : ( abstra6247030358587465148_fm_fm @ Eff @ R3 @ ( product_fst_fm_fm @ Step ) ) )
                  @ Steps2 )
               => ( linear7981529059021590662_fm_fm
                  @ ( linear3538412811542617781_fm_fm
                    @ ^ [Step: product_prod_fm_fm] :
                        ( ( product_snd_fm_fm @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_602_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: nat > fm > fset_fm > $o,Rules: stream_nat,S6: set_fm,S3: fm,Rs: stream_nat,Steps2: stream2847032186174250247fm_nat,R3: nat] :
      ( ( abstra1720523912436379467nat_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
         => ( ( abstra7849342000906253360fm_nat @ ( abstra945010636717552901nat_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
             => ( ( linear2221084222810013309fm_nat
                  @ ( linear3248762273433014325fm_nat
                    @ ^ [Step: product_prod_fm_nat] : ( abstra5667281435656691644nat_fm @ Eff @ R3 @ ( product_fst_fm_nat @ Step ) ) )
                  @ Steps2 )
               => ( linear7848696306570402438fm_nat
                  @ ( linear3248762273433014325fm_nat
                    @ ^ [Step: product_prod_fm_nat] :
                        ( ( product_snd_fm_nat @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_603_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: fm > nat > fset_nat > $o,Rules: stream_fm,S6: set_nat,S3: nat,Rs: stream_fm,Steps2: stream1515660912037602951nat_fm,R3: fm] :
      ( ( abstra9084600350906980299fm_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra4380219248426614192nat_fm @ ( abstra8309087075188153733fm_nat @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
             => ( ( linear7975333507185149949nat_fm
                  @ ( linear9003011557808150965nat_fm
                    @ ^ [Step: product_prod_nat_fm] : ( abstra3807985837272516668fm_nat @ Eff @ R3 @ ( product_fst_nat_fm @ Step ) ) )
                  @ Steps2 )
               => ( linear4379573554090763270nat_fm
                  @ ( linear9003011557808150965nat_fm
                    @ ^ [Step: product_prod_nat_fm] :
                        ( ( product_snd_nat_fm @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_604_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: nat > nat > fset_nat > $o,Rules: stream_nat,S6: set_nat,S3: nat,Rs: stream_nat,Steps2: stream6724221391990029191at_nat,R3: nat] :
      ( ( abstra5088622135990055243at_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
         => ( ( abstra5838625251101670576at_nat @ ( abstra1192804020863382533at_nat @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
             => ( ( linear1521848693801000317at_nat
                  @ ( linear2787706585215634229at_nat
                    @ ^ [Step: product_prod_nat_nat] : ( abstra2407503598237092412at_nat @ Eff @ R3 @ ( product_fst_nat_nat @ Step ) ) )
                  @ Steps2 )
               => ( linear869931615226441222at_nat
                  @ ( linear2787706585215634229at_nat
                    @ ^ [Step: product_prod_nat_nat] :
                        ( ( product_snd_nat_nat @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_605_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule,Steps2: stream3588992565182678257e_rule,R3: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ( ( linear6469244230239975911e_rule
                  @ ( linear7974216874357323679e_rule
                    @ ^ [Step: produc780299800278082923e_rule] : ( abstra3154316310710804774e_rule @ Eff @ R3 @ ( produc7264528503596239631e_rule @ Step ) ) )
                  @ Steps2 )
               => ( linear7219246396390979696e_rule
                  @ ( linear7974216874357323679e_rule
                    @ ^ [Step: produc780299800278082923e_rule] :
                        ( ( produc6725306161561501521e_rule @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_606_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule,Steps2: stream8579757915198507644m_rule,R3: rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ( ( linear1831006749712209266m_rule
                  @ ( linear8703846257056886570m_rule
                    @ ^ [Step: product_prod_fm_rule] : ( abstra7038331712872434097ule_fm @ Eff @ R3 @ ( product_fst_fm_rule @ Step ) ) )
                  @ Steps2 )
               => ( linear5263904179220351099m_rule
                  @ ( linear8703846257056886570m_rule
                    @ ^ [Step: product_prod_fm_rule] :
                        ( ( product_snd_fm_rule @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_607_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule,Steps2: stream3532830303409350908t_rule,R3: rule] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra8838945791029739685t_rule @ ( abstra3504266011406229370le_nat @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ( ( linear4458791665855313010t_rule
                  @ ( linear2429371900934449706t_rule
                    @ ^ [Step: produc1938397839849972726t_rule] : ( abstra4466204623219695281le_nat @ Eff @ R3 @ ( product_fst_nat_rule @ Step ) ) )
                  @ Steps2 )
               => ( linear6872488148413422587t_rule
                  @ ( linear2429371900934449706t_rule
                    @ ^ [Step: produc1938397839849972726t_rule] :
                        ( ( product_snd_nat_rule @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_608_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S6: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm,Steps2: stream1960312990768957601ist_fm,R3: produc1996495991257130529ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( ( linear927598023175380501ist_fm
                  @ ( linear2326238208734597725ist_fm
                    @ ^ [Step: produc4630866025709511057ist_fm] : ( abstra5706625008861892940m_rule @ Eff @ R3 @ ( produc4228417144610596823ist_fm @ Step ) ) )
                  @ Steps2 )
               => ( linear7189418855485006156ist_fm
                  @ ( linear2326238208734597725ist_fm
                    @ ^ [Step: produc4630866025709511057ist_fm] :
                        ( ( produc4514744107061996309ist_fm @ Step )
                        = R3 ) )
                  @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_609_mkTree_Osimps_I1_J,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 ) )
      = ( produc491467635432902671m_rule @ S3
        @ ( shd_rule
          @ ( abstra2363873759530810705ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 ) ) ) ) ).

% mkTree.simps(1)
thf(fact_610_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule,Steps2: stream3588992565182678257e_rule] :
      ( ( abstra8115150628584006924e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( abstra4228102177642093818e_rule @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_611_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule,Steps2: stream8579757915198507644m_rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( abstra7168897066996117381ule_fm @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_612_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule,Steps2: stream3532830303409350908t_rule] :
      ( ( abstra209963545474467991le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra8838945791029739685t_rule @ ( abstra3504266011406229370le_nat @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( abstra8935584723450605189le_nat @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_613_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps2: stream727092118206550309m_rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( abstra8647779336736900908ist_fm @ Eff @ Rules @ Steps2 ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_614_RuleSystem__Defs_Osaturated__def,axiom,
    ( abstra7252176601000738572ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R2: rule] :
          ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( linear7082939723987557601m_rule
                @ ^ [Step: produc164195504107695125m_rule] : ( abstra3055540779341473792ist_fm @ Eff2 @ R2 @ ( produc6879501374131015971m_rule @ Step ) )
                @ Xs )
             => ( linear2722748333883190224m_rule
                @ ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] :
                      ( ( produc7165828336582415457m_rule @ Step )
                      = R2 ) )
                @ Xs ) ) ) ) ) ).

% RuleSystem_Defs.saturated_def
thf(fact_615_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S3: list_fm] :
      ( ( abstra778364809706815794ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) )
      = ( produc381145313068854617ist_fm @ S3 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_616_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) )
      = ( produc491467635432902671m_rule @ S3 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_617_ipath__mkTree__ev,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps2: stream727092118206550309m_rule,R3: rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps2 )
         => ( ( member_rule @ R3 @ ( sset_rule @ rules ) )
           => ( ( linear5684299538428340377m_rule
                @ ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] :
                      ( abstra3055540779341473792ist_fm
                      @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R2 @ S )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ R3
                      @ ( produc6879501374131015971m_rule @ Step ) ) )
                @ Steps2 )
             => ( linear2722748333883190224m_rule
                @ ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] :
                      ( ( produc7165828336582415457m_rule @ Step )
                      = R3 ) )
                @ Steps2 ) ) ) ) ) ) ).

% ipath_mkTree_ev
thf(fact_618_i_Osaturated__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R3: rule] :
      ( ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R3 )
      = ( linear5684299538428340377m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( linear7082939723987557601m_rule
              @ ^ [Step: produc164195504107695125m_rule] : ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R3 @ ( produc6879501374131015971m_rule @ Step ) )
              @ Xs )
           => ( linear2722748333883190224m_rule
              @ ( linear7082939723987557601m_rule
                @ ^ [Step: produc164195504107695125m_rule] :
                    ( ( produc7165828336582415457m_rule @ Step )
                    = R3 ) )
              @ Xs ) ) ) ) ).

% i.saturated_def
thf(fact_619_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S6: set_rule,S3: rule,Rs: stream_fm,Steps2: stream1800043145262542076ule_fm,R3: fm] :
      ( ( abstra4039608574777915031m_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra8780395760585530021ule_fm @ ( abstra8242076542743448570m_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
             => ( abstra5359274141611169189m_rule @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_620_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: nat > rule > fset_rule > $o,Rules: stream_nat,S6: set_rule,S3: rule,Rs: stream_nat,Steps2: stream2660830046501416060le_nat,R3: nat] :
      ( ( abstra6196071167509384087t_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
         => ( ( abstra4272207280279176229le_nat @ ( abstra267001596586369658t_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
             => ( abstra4286361020537442725t_rule @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_621_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S6: set_fm,S3: fm,Rs: stream_fm,Steps2: stream1375025385734440967_fm_fm,R3: fm] :
      ( ( abstra1990789280842237858_fm_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra724614664783131952_fm_fm @ ( abstra5285091746773999237_fm_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
             => ( abstra81079133870296496_fm_fm @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_622_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: nat > fm > fset_fm > $o,Rules: stream_nat,S6: set_fm,S3: fm,Rs: stream_nat,Steps2: stream2847032186174250247fm_nat,R3: nat] :
      ( ( abstra1839785910314048674nat_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
         => ( ( abstra7849342000906253360fm_nat @ ( abstra945010636717552901nat_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
             => ( abstra6007160153220221360nat_fm @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_623_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > nat > fset_nat > $o,Rules: stream_fm,S6: set_nat,S3: nat,Rs: stream_fm,Steps2: stream1515660912037602951nat_fm,R3: fm] :
      ( ( abstra9203862348784649506fm_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra4380219248426614192nat_fm @ ( abstra8309087075188153733fm_nat @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
             => ( abstra4147864554836046384fm_nat @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_624_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: nat > nat > fset_nat > $o,Rules: stream_nat,S6: set_nat,S3: nat,Rs: stream_nat,Steps2: stream6724221391990029191at_nat,R3: nat] :
      ( ( abstra2374624847373867554at_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
         => ( ( abstra5838625251101670576at_nat @ ( abstra1192804020863382533at_nat @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
             => ( abstra5577456715138805296at_nat @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_625_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule,Steps2: stream3588992565182678257e_rule,R3: rule] :
      ( ( abstra8115150628584006924e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ( abstra6372423936129899802e_rule @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_626_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule,Steps2: stream8579757915198507644m_rule,R3: rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ( abstra1108425641574350757ule_fm @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_627_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule,Steps2: stream3532830303409350908t_rule,R3: rule] :
      ( ( abstra209963545474467991le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra8838945791029739685t_rule @ ( abstra3504266011406229370le_nat @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ( abstra7523625435357302437le_nat @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_628_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S6: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm,Steps2: stream1960312990768957601ist_fm,R3: produc1996495991257130529ist_fm] :
      ( ( abstra1314873687758482406m_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( abstra679888793666381912m_rule @ Eff @ R3 @ Steps2 ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_629_wf__mkTree,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra4476735159789068068ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ ( abstra178918799166095863ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 ) ) ) ) ).

% wf_mkTree
thf(fact_630_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S6: set_rule,S3: rule,Rs: stream727092118206550309m_rule,Steps2: stream1878305877988316353m_rule,M2: nat] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ? [N2: nat,S8: rule] :
                ( ( member_rule @ S8 @ S6 )
                & ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S8 ) @ ( sdrop_1764812713384251940m_rule @ M2 @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_631_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule,Steps2: stream642806101564963573m_rule,M2: nat] :
      ( ( abstra3408323020314982753ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra7135721583251150454m_rule @ ( abstra2161496378874750759ist_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ? [N2: nat,S8: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S8 @ S6 )
                & ( abstra7135721583251150454m_rule @ ( abstra2161496378874750759ist_fm @ Eff @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S8 ) @ ( sdrop_9094843686636319420m_rule @ M2 @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_632_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S6: set_fm,S3: fm,Rs: stream727092118206550309m_rule,Steps2: stream5001952251264143308m_rule,M2: nat] :
      ( ( abstra3764363818050485264ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ? [N2: nat,S8: fm] :
                ( ( member_fm @ S8 @ S6 )
                & ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S8 ) @ ( sdrop_6203755225909610927m_rule @ M2 @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_633_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc164195504107695125m_rule > nat > fset_nat > $o,Rules: stream727092118206550309m_rule,S6: set_nat,S3: nat,Rs: stream727092118206550309m_rule,Steps2: stream3365801610433932876m_rule,M2: nat] :
      ( ( abstra8087887464339558544le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra5284986315758917621m_rule @ ( abstra8179680689149056586le_nat @ Eff @ Rs @ S3 ) @ Steps2 )
           => ? [N2: nat,S8: nat] :
                ( ( member_nat @ S8 @ S6 )
                & ( abstra5284986315758917621m_rule @ ( abstra8179680689149056586le_nat @ Eff @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S8 ) @ ( sdrop_6386509040039022383m_rule @ M2 @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_634_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule,Steps2: stream3588992565182678257e_rule,M2: nat] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) @ Steps2 )
           => ? [N2: nat,S8: rule] :
                ( ( member_rule @ S8 @ S6 )
                & ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S8 ) @ ( sdrop_2199339672054592340e_rule @ M2 @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_635_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule,Steps2: stream8579757915198507644m_rule,M2: nat] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ? [N2: nat,S8: fm] :
                ( ( member_fm @ S8 @ S6 )
                & ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S8 ) @ ( sdrop_432990894411838815m_rule @ M2 @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_636_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule,Steps2: stream3532830303409350908t_rule,M2: nat] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra8838945791029739685t_rule @ ( abstra3504266011406229370le_nat @ Eff @ Rs @ S3 ) @ Steps2 )
           => ? [N2: nat,S8: nat] :
                ( ( member_nat @ S8 @ S6 )
                & ( abstra8838945791029739685t_rule @ ( abstra3504266011406229370le_nat @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S8 ) @ ( sdrop_6826213527187012319t_rule @ M2 @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_637_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps2: stream727092118206550309m_rule,M2: nat] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) @ Steps2 )
           => ? [N2: nat,S8: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S8 @ S6 )
                & ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S8 ) @ ( sdrop_7224736112439592940m_rule @ M2 @ Steps2 ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_638_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S6: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra6412047235177443722e_rule @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_639_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra6201285654312316801ist_fm @ Eff @ Rs @ S3 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra5265644873707572892ist_fm @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_640_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S6: set_fm,S3: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra2315704231547878896ule_fm @ Eff @ Rs @ S3 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra6093700014593756181ule_fm @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_641_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc164195504107695125m_rule > nat > fset_nat > $o,Rules: stream727092118206550309m_rule,S6: set_nat,S3: nat,Rs: stream727092118206550309m_rule] :
      ( ( abstra8087887464339558544le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra7715883167740232304le_nat @ Eff @ Rs @ S3 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra3327483289227891477le_nat @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_642_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra5682536783049438522e_rule @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_643_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra2294468009765627845ule_fm @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_644_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra4161464353856247968le_nat @ Eff @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra2107806536733918917le_nat @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_645_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra738319790293016300ist_fm @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_646_prover__def,axiom,
    ( prover
    = ( abstra178918799166095863ist_fm
      @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R2 @ S )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ rules ) ) ).

% prover_def
thf(fact_647_mkTree__effG__uu__def,axiom,
    ( abstra3917079402916513406ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% mkTree_effG_uu_def
thf(fact_648_in__cont__mkTree,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,T4: abstra4572470701423150929m_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( fmembe1635983029874454474m_rule @ T4
            @ ( abstra4962253831105676628m_rule
              @ ( abstra178918799166095863ist_fm
                @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S3 ) ) )
         => ? [Sl5: fset_P661503646757059847ist_fm,S8: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S8 @ top_to728902761071950545ist_fm )
              & ( ( eff
                  @ ( shd_rule
                    @ ( abstra2363873759530810705ist_fm
                      @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R2 @ S )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ Rs
                      @ S3 ) )
                  @ S3 )
                = ( some_f2162114139285431020ist_fm @ Sl5 ) )
              & ( fmembe3381613331217039976ist_fm @ S8 @ Sl5 )
              & ( T4
                = ( abstra178918799166095863ist_fm
                  @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R2 @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ ( stl_rule
                    @ ( abstra2363873759530810705ist_fm
                      @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R2 @ S )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ Rs
                      @ S3 ) )
                  @ S8 ) ) ) ) ) ) ).

% in_cont_mkTree
thf(fact_649_minWait__le__pos,axiom,
    ! [Rs: stream_rule,R3: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( member_rule @ R3 @ ( sset_rule @ rules ) )
       => ( ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R3
            @ S3 )
         => ( ord_less_eq_nat
            @ ( abstra738319790293016300ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ ( abstract_pos_rule @ Rs @ R3 ) ) ) ) ) ).

% minWait_le_pos
thf(fact_650_minWait__def,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra738319790293016300ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( ord_Least_nat
        @ ^ [N3: nat] :
            ( abstra3055540779341473792ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
            @ S3 ) ) ) ).

% minWait_def
thf(fact_651_minWait__least,axiom,
    ! [N: nat,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
        @ S3 )
     => ( ord_less_eq_nat
        @ ( abstra738319790293016300ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 )
        @ N ) ) ).

% minWait_least
thf(fact_652_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S6: set_rule,S3: rule,Rs: stream727092118206550309m_rule,T4: abstra4791331342562632597m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe1981849019698968796m_rule @ T4 @ ( abstra5403976286072314556m_rule @ ( abstra1231468025953361855e_rule @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_rule,S8: rule] :
                ( ( member_rule @ S8 @ S6 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_rule @ S8 @ Sl5 )
                & ( T4
                  = ( abstra1231468025953361855e_rule @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_653_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S6: set_fm,S3: fm,Rs: stream727092118206550309m_rule,T4: abstra4575282202256230560m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe8691679578407899495m_rule @ T4 @ ( abstra2433431901973683783m_rule @ ( abstra1207708450097661130ule_fm @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_fm,S8: fm] :
                ( ( member_fm @ S8 @ S6 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_fm @ S8 @ Sl5 )
                & ( T4
                  = ( abstra1207708450097661130ule_fm @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra2315704231547878896ule_fm @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_654_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > nat > fset_nat > $o,Rules: stream727092118206550309m_rule,S6: set_nat,S3: nat,Rs: stream727092118206550309m_rule,T4: abstra8588637103743308064m_rule] :
      ( ( abstra8087887464339558544le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe6442932449457638759m_rule @ T4 @ ( abstra6185963983136976839m_rule @ ( abstra8179680689149056586le_nat @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_nat,S8: nat] :
                ( ( member_nat @ S8 @ S6 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra7715883167740232304le_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_nat @ S8 @ Sl5 )
                & ( T4
                  = ( abstra8179680689149056586le_nat @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra7715883167740232304le_nat @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_655_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule,T4: abstra4658698708444432161m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe2925761000581315610m_rule @ T4 @ ( abstra5924671108449608740m_rule @ ( abstra2161496378874750759ist_fm @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_P661503646757059847ist_fm,S8: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S8 @ S6 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe3381613331217039976ist_fm @ S8 @ Sl5 )
                & ( T4
                  = ( abstra2161496378874750759ist_fm @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra6201285654312316801ist_fm @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_656_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream727092118206550309m_rule,S6: set_Ab3744519486624674951m_rule,S3: abstra4572470701423150929m_rule,Rs: stream727092118206550309m_rule,T4: abstra5426939140630736801m_rule] :
      ( ( abstra6898608545798587409m_rule @ Eff @ Rules @ S6 )
     => ( ( member1590433669991072232m_rule @ S3 @ S6 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe8791921695509808616m_rule @ T4 @ ( abstra3071294317365906120m_rule @ ( abstra1631510761171731915m_rule @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_A5396818216090053761m_rule,S8: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S8 @ S6 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra7609338345072424433m_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe1635983029874454474m_rule @ S8 @ Sl5 )
                & ( T4
                  = ( abstra1631510761171731915m_rule @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra7609338345072424433m_rule @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_657_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,S3: rule,Rs: stream_rule,T4: abstra1071693965251717573e_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( member_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe5852619498312149900e_rule @ T4 @ ( abstra2651384284180525036e_rule @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_rule,S8: rule] :
                ( ( member_rule @ S8 @ S6 )
                & ( Eff @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_rule @ S8 @ Sl5 )
                & ( T4
                  = ( abstra7443870075974203631e_rule @ Eff @ ( stl_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_658_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,S3: fm,Rs: stream_rule,T4: abstra4524330545568143696m_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( member_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe2914889872492170903m_rule @ T4 @ ( abstra5134701107988495863m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_fm,S8: fm] :
                ( ( member_fm @ S8 @ S6 )
                & ( Eff @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_fm @ S8 @ Sl5 )
                & ( T4
                  = ( abstra3991228042706630138ule_fm @ Eff @ ( stl_rule @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_659_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,S3: nat,Rs: stream_rule,T4: abstra1428203607558414800t_rule] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( member_nat @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe9011880150065154711t_rule @ T4 @ ( abstra1446102021933600631t_rule @ ( abstra3504266011406229370le_nat @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_nat,S8: nat] :
                ( ( member_nat @ S8 @ S6 )
                & ( Eff @ ( shd_rule @ ( abstra4161464353856247968le_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmember_nat @ S8 @ Sl5 )
                & ( T4
                  = ( abstra3504266011406229370le_nat @ Eff @ ( stl_rule @ ( abstra4161464353856247968le_nat @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_660_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_rule,S6: set_Ab3744519486624674951m_rule,S3: abstra4572470701423150929m_rule,Rs: stream_rule,T4: abstra4125161011840266193e_rule] :
      ( ( abstra6244190358723783233m_rule @ Eff @ Rules @ S6 )
     => ( ( member1590433669991072232m_rule @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe8010993199532995224e_rule @ T4 @ ( abstra6162512231279892984e_rule @ ( abstra2861652164185304827m_rule @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_A5396818216090053761m_rule,S8: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S8 @ S6 )
                & ( Eff @ ( shd_rule @ ( abstra1865002795007470625m_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe1635983029874454474m_rule @ S8 @ Sl5 )
                & ( T4
                  = ( abstra2861652164185304827m_rule @ Eff @ ( stl_rule @ ( abstra1865002795007470625m_rule @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_661_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,T4: abstra4572470701423150929m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S6 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) ) )
           => ? [Sl5: fset_P661503646757059847ist_fm,S8: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S8 @ S6 )
                & ( Eff @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl5 )
                & ( fmembe3381613331217039976ist_fm @ S8 @ Sl5 )
                & ( T4
                  = ( abstra178918799166095863ist_fm @ Eff @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ S8 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_662_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_663_order__refl,axiom,
    ! [X2: set_rule] : ( ord_less_eq_set_rule @ X2 @ X2 ) ).

% order_refl
thf(fact_664_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_665_dual__order_Orefl,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_666_LeastI,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI
thf(fact_667_Least__le,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ K ) ) ).

% Least_le
thf(fact_668_Least1I,axiom,
    ! [P: nat > $o] :
      ( ? [X5: nat] :
          ( ( P @ X5 )
          & ! [Y4: nat] :
              ( ( P @ Y4 )
             => ( ord_less_eq_nat @ X5 @ Y4 ) )
          & ! [Y4: nat] :
              ( ( ( P @ Y4 )
                & ! [Ya: nat] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_nat @ Y4 @ Ya ) ) )
             => ( Y4 = X5 ) ) )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% Least1I
thf(fact_669_Least1I,axiom,
    ! [P: set_rule > $o] :
      ( ? [X5: set_rule] :
          ( ( P @ X5 )
          & ! [Y4: set_rule] :
              ( ( P @ Y4 )
             => ( ord_less_eq_set_rule @ X5 @ Y4 ) )
          & ! [Y4: set_rule] :
              ( ( ( P @ Y4 )
                & ! [Ya: set_rule] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_set_rule @ Y4 @ Ya ) ) )
             => ( Y4 = X5 ) ) )
     => ( P @ ( ord_Least_set_rule @ P ) ) ) ).

% Least1I
thf(fact_670_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_671_Least1__le,axiom,
    ! [P: nat > $o,Z3: nat] :
      ( ? [X5: nat] :
          ( ( P @ X5 )
          & ! [Y4: nat] :
              ( ( P @ Y4 )
             => ( ord_less_eq_nat @ X5 @ Y4 ) )
          & ! [Y4: nat] :
              ( ( ( P @ Y4 )
                & ! [Ya: nat] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_nat @ Y4 @ Ya ) ) )
             => ( Y4 = X5 ) ) )
     => ( ( P @ Z3 )
       => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ Z3 ) ) ) ).

% Least1_le
thf(fact_672_Least1__le,axiom,
    ! [P: set_rule > $o,Z3: set_rule] :
      ( ? [X5: set_rule] :
          ( ( P @ X5 )
          & ! [Y4: set_rule] :
              ( ( P @ Y4 )
             => ( ord_less_eq_set_rule @ X5 @ Y4 ) )
          & ! [Y4: set_rule] :
              ( ( ( P @ Y4 )
                & ! [Ya: set_rule] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_set_rule @ Y4 @ Ya ) ) )
             => ( Y4 = X5 ) ) )
     => ( ( P @ Z3 )
       => ( ord_less_eq_set_rule @ ( ord_Least_set_rule @ P ) @ Z3 ) ) ) ).

% Least1_le
thf(fact_673_LeastI2,axiom,
    ! [P: nat > $o,A2: nat,Q: nat > $o] :
      ( ( P @ A2 )
     => ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2
thf(fact_674_le__cases3,axiom,
    ! [X2: nat,Y: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_675_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_676_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_rule,Z: set_rule] : ( Y3 = Z ) )
    = ( ^ [X4: set_rule,Y5: set_rule] :
          ( ( ord_less_eq_set_rule @ X4 @ Y5 )
          & ( ord_less_eq_set_rule @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_677_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_678_ord__eq__le__trans,axiom,
    ! [A2: set_rule,B2: set_rule,C: set_rule] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_rule @ B2 @ C )
       => ( ord_less_eq_set_rule @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_679_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_680_ord__le__eq__trans,axiom,
    ! [A2: set_rule,B2: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_rule @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_681_LeastI2__order,axiom,
    ! [P: nat > $o,X2: nat,Q: nat > $o] :
      ( ( P @ X2 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ X2 @ Y4 ) )
       => ( ! [X: nat] :
              ( ( P @ X )
             => ( ! [Y6: nat] :
                    ( ( P @ Y6 )
                   => ( ord_less_eq_nat @ X @ Y6 ) )
               => ( Q @ X ) ) )
         => ( Q @ ( ord_Least_nat @ P ) ) ) ) ) ).

% LeastI2_order
thf(fact_682_LeastI2__order,axiom,
    ! [P: set_rule > $o,X2: set_rule,Q: set_rule > $o] :
      ( ( P @ X2 )
     => ( ! [Y4: set_rule] :
            ( ( P @ Y4 )
           => ( ord_less_eq_set_rule @ X2 @ Y4 ) )
       => ( ! [X: set_rule] :
              ( ( P @ X )
             => ( ! [Y6: set_rule] :
                    ( ( P @ Y6 )
                   => ( ord_less_eq_set_rule @ X @ Y6 ) )
               => ( Q @ X ) ) )
         => ( Q @ ( ord_Least_set_rule @ P ) ) ) ) ) ).

% LeastI2_order
thf(fact_683_order__antisym,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_684_order__antisym,axiom,
    ! [X2: set_rule,Y: set_rule] :
      ( ( ord_less_eq_set_rule @ X2 @ Y )
     => ( ( ord_less_eq_set_rule @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_685_LeastI__ex,axiom,
    ! [P: nat > $o] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI_ex
thf(fact_686_Least__equality,axiom,
    ! [P: nat > $o,X2: nat] :
      ( ( P @ X2 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ X2 @ Y4 ) )
       => ( ( ord_Least_nat @ P )
          = X2 ) ) ) ).

% Least_equality
thf(fact_687_Least__equality,axiom,
    ! [P: set_rule > $o,X2: set_rule] :
      ( ( P @ X2 )
     => ( ! [Y4: set_rule] :
            ( ( P @ Y4 )
           => ( ord_less_eq_set_rule @ X2 @ Y4 ) )
       => ( ( ord_Least_set_rule @ P )
          = X2 ) ) ) ).

% Least_equality
thf(fact_688_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_689_order_Otrans,axiom,
    ! [A2: set_rule,B2: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( ord_less_eq_set_rule @ B2 @ C )
       => ( ord_less_eq_set_rule @ A2 @ C ) ) ) ).

% order.trans
thf(fact_690_order__trans,axiom,
    ! [X2: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).

% order_trans
thf(fact_691_order__trans,axiom,
    ! [X2: set_rule,Y: set_rule,Z3: set_rule] :
      ( ( ord_less_eq_set_rule @ X2 @ Y )
     => ( ( ord_less_eq_set_rule @ Y @ Z3 )
       => ( ord_less_eq_set_rule @ X2 @ Z3 ) ) ) ).

% order_trans
thf(fact_692_LeastI2__ex,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_ex
thf(fact_693_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: nat,B4: nat] :
            ( ( P @ B4 @ A5 )
           => ( P @ A5 @ B4 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_694_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [A3: nat,B5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_695_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_rule,Z: set_rule] : ( Y3 = Z ) )
    = ( ^ [A3: set_rule,B5: set_rule] :
          ( ( ord_less_eq_set_rule @ B5 @ A3 )
          & ( ord_less_eq_set_rule @ A3 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_696_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_697_dual__order_Oantisym,axiom,
    ! [B2: set_rule,A2: set_rule] :
      ( ( ord_less_eq_set_rule @ B2 @ A2 )
     => ( ( ord_less_eq_set_rule @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_698_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_699_dual__order_Otrans,axiom,
    ! [B2: set_rule,A2: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ B2 @ A2 )
     => ( ( ord_less_eq_set_rule @ C @ B2 )
       => ( ord_less_eq_set_rule @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_700_LeastI2__wellorder,axiom,
    ! [P: nat > $o,A2: nat,Q: nat > $o] :
      ( ( P @ A2 )
     => ( ! [A5: nat] :
            ( ( P @ A5 )
           => ( ! [B7: nat] :
                  ( ( P @ B7 )
                 => ( ord_less_eq_nat @ A5 @ B7 ) )
             => ( Q @ A5 ) ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_wellorder
thf(fact_701_LeastI2__wellorder__ex,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [A5: nat] :
            ( ( P @ A5 )
           => ( ! [B7: nat] :
                  ( ( P @ B7 )
                 => ( ord_less_eq_nat @ A5 @ B7 ) )
             => ( Q @ A5 ) ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_wellorder_ex
thf(fact_702_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_703_antisym,axiom,
    ! [A2: set_rule,B2: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( ord_less_eq_set_rule @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_704_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [A3: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A3 @ B5 )
          & ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_705_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_rule,Z: set_rule] : ( Y3 = Z ) )
    = ( ^ [A3: set_rule,B5: set_rule] :
          ( ( ord_less_eq_set_rule @ A3 @ B5 )
          & ( ord_less_eq_set_rule @ B5 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_706_order__subst1,axiom,
    ! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_707_order__subst1,axiom,
    ! [A2: nat,F2: set_rule > nat,B2: set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_rule @ B2 @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_708_order__subst1,axiom,
    ! [A2: set_rule,F2: nat > set_rule,B2: nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_709_order__subst1,axiom,
    ! [A2: set_rule,F2: set_rule > set_rule,B2: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_rule @ B2 @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_710_order__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_711_order__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_rule @ ( F2 @ B2 ) @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_712_order__subst2,axiom,
    ! [A2: set_rule,B2: set_rule,F2: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_713_order__subst2,axiom,
    ! [A2: set_rule,B2: set_rule,F2: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( ord_less_eq_set_rule @ ( F2 @ B2 ) @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_714_order__eq__refl,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 = Y )
     => ( ord_less_eq_nat @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_715_order__eq__refl,axiom,
    ! [X2: set_rule,Y: set_rule] :
      ( ( X2 = Y )
     => ( ord_less_eq_set_rule @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_716_linorder__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
      | ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_717_ord__eq__le__subst,axiom,
    ! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_718_ord__eq__le__subst,axiom,
    ! [A2: set_rule,F2: nat > set_rule,B2: nat,C: nat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A2 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_719_ord__eq__le__subst,axiom,
    ! [A2: nat,F2: set_rule > nat,B2: set_rule,C: set_rule] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_rule @ B2 @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_720_ord__eq__le__subst,axiom,
    ! [A2: set_rule,F2: set_rule > set_rule,B2: set_rule,C: set_rule] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_rule @ B2 @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ A2 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_721_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_722_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F2: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F2 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_723_ord__le__eq__subst,axiom,
    ! [A2: set_rule,B2: set_rule,F2: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_724_ord__le__eq__subst,axiom,
    ! [A2: set_rule,B2: set_rule,F2: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_eq_set_rule @ ( F2 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_725_linorder__le__cases,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_726_order__antisym__conv,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_727_order__antisym__conv,axiom,
    ! [Y: set_rule,X2: set_rule] :
      ( ( ord_less_eq_set_rule @ Y @ X2 )
     => ( ( ord_less_eq_set_rule @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_728_top_Oextremum__uniqueI,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A2 )
     => ( A2 = top_to728902761071950545ist_fm ) ) ).

% top.extremum_uniqueI
thf(fact_729_top_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
     => ( A2 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_730_top_Oextremum__uniqueI,axiom,
    ! [A2: set_rule] :
      ( ( ord_less_eq_set_rule @ top_top_set_rule @ A2 )
     => ( A2 = top_top_set_rule ) ) ).

% top.extremum_uniqueI
thf(fact_731_top_Oextremum__unique,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A2 )
      = ( A2 = top_to728902761071950545ist_fm ) ) ).

% top.extremum_unique
thf(fact_732_top_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
      = ( A2 = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_733_top_Oextremum__unique,axiom,
    ! [A2: set_rule] :
      ( ( ord_less_eq_set_rule @ top_top_set_rule @ A2 )
      = ( A2 = top_top_set_rule ) ) ).

% top.extremum_unique
thf(fact_734_top__greatest,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A2 @ top_to728902761071950545ist_fm ) ).

% top_greatest
thf(fact_735_top__greatest,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% top_greatest
thf(fact_736_top__greatest,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ top_top_set_rule ) ).

% top_greatest
thf(fact_737_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_738_tfinite,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ! [T5: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T2 ) )
         => ( abstra3507476289341504315m_rule @ T5 ) )
     => ( abstra3507476289341504315m_rule @ T2 ) ) ).

% tfinite
thf(fact_739_tfinite_Osimps,axiom,
    ( abstra3507476289341504315m_rule
    = ( ^ [A3: abstra4572470701423150929m_rule] :
        ? [T: abstra4572470701423150929m_rule] :
          ( ( A3 = T )
          & ! [X4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X4 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra3507476289341504315m_rule @ X4 ) ) ) ) ) ).

% tfinite.simps
thf(fact_740_tfinite_Ocases,axiom,
    ! [A2: abstra4572470701423150929m_rule] :
      ( ( abstra3507476289341504315m_rule @ A2 )
     => ! [T6: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ A2 ) )
         => ( abstra3507476289341504315m_rule @ T6 ) ) ) ).

% tfinite.cases
thf(fact_741_pos__least,axiom,
    ! [N: nat,Rs: stream727092118206550309m_rule,R3: produc164195504107695125m_rule] :
      ( ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Rs ) )
        = R3 )
     => ( ord_less_eq_nat @ ( abstra1882453022147397137m_rule @ Rs @ R3 ) @ N ) ) ).

% pos_least
thf(fact_742_pos__least,axiom,
    ! [N: nat,Rs: stream_rule,R3: rule] :
      ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
        = R3 )
     => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R3 ) @ N ) ) ).

% pos_least
thf(fact_743_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_744_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_745_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream727092118206550309m_rule,R3: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Rs ) )
          = R3 )
       => ( ord_less_eq_nat @ ( abstra1882453022147397137m_rule @ Rs @ R3 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_746_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,R3: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
          = R3 )
       => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R3 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_747_ipath_Ocases,axiom,
    ! [A1: abstract_tree_rule,A22: stream_rule] :
      ( ( abstract_ipath_rule @ A1 @ A22 )
     => ~ ( ( ( abstract_root_rule @ A1 )
            = ( shd_rule @ A22 ) )
         => ! [T5: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T5 @ ( abstract_cont_rule @ A1 ) )
             => ~ ( abstract_ipath_rule @ T5 @ ( stl_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_748_ipath_Ocases,axiom,
    ! [A1: abstra4572470701423150929m_rule,A22: stream727092118206550309m_rule] :
      ( ( abstra2285467793620430502m_rule @ A1 @ A22 )
     => ~ ( ( ( abstra3503897793822247108m_rule @ A1 )
            = ( shd_Pr7235097944458474089m_rule @ A22 ) )
         => ! [T5: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ A1 ) )
             => ~ ( abstra2285467793620430502m_rule @ T5 @ ( stl_Pr950425576149878629m_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_749_ipath_Osimps,axiom,
    ( abstract_ipath_rule
    = ( ^ [A12: abstract_tree_rule,A23: stream_rule] :
        ? [T: abstract_tree_rule,Steps3: stream_rule,T7: abstract_tree_rule] :
          ( ( A12 = T )
          & ( A23 = Steps3 )
          & ( ( abstract_root_rule @ T )
            = ( shd_rule @ Steps3 ) )
          & ( fmembe7964111328986570234e_rule @ T7 @ ( abstract_cont_rule @ T ) )
          & ( abstract_ipath_rule @ T7 @ ( stl_rule @ Steps3 ) ) ) ) ) ).

% ipath.simps
thf(fact_750_ipath_Osimps,axiom,
    ( abstra2285467793620430502m_rule
    = ( ^ [A12: abstra4572470701423150929m_rule,A23: stream727092118206550309m_rule] :
        ? [T: abstra4572470701423150929m_rule,Steps3: stream727092118206550309m_rule,T7: abstra4572470701423150929m_rule] :
          ( ( A12 = T )
          & ( A23 = Steps3 )
          & ( ( abstra3503897793822247108m_rule @ T )
            = ( shd_Pr7235097944458474089m_rule @ Steps3 ) )
          & ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ T ) )
          & ( abstra2285467793620430502m_rule @ T7 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) ) ).

% ipath.simps
thf(fact_751_ipath_Ointros,axiom,
    ! [T2: abstract_tree_rule,Steps2: stream_rule,T4: abstract_tree_rule] :
      ( ( ( abstract_root_rule @ T2 )
        = ( shd_rule @ Steps2 ) )
     => ( ( fmembe7964111328986570234e_rule @ T4 @ ( abstract_cont_rule @ T2 ) )
       => ( ( abstract_ipath_rule @ T4 @ ( stl_rule @ Steps2 ) )
         => ( abstract_ipath_rule @ T2 @ Steps2 ) ) ) ) ).

% ipath.intros
thf(fact_752_ipath_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps2: stream727092118206550309m_rule,T4: abstra4572470701423150929m_rule] :
      ( ( ( abstra3503897793822247108m_rule @ T2 )
        = ( shd_Pr7235097944458474089m_rule @ Steps2 ) )
     => ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T2 ) )
       => ( ( abstra2285467793620430502m_rule @ T4 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) )
         => ( abstra2285467793620430502m_rule @ T2 @ Steps2 ) ) ) ) ).

% ipath.intros
thf(fact_753_ipath_Ocoinduct,axiom,
    ! [X3: abstract_tree_rule > stream_rule > $o,X1: abstract_tree_rule,X22: stream_rule] :
      ( ( X3 @ X1 @ X22 )
     => ( ! [X13: abstract_tree_rule,X24: stream_rule] :
            ( ( X3 @ X13 @ X24 )
           => ? [T8: abstract_tree_rule,Steps: stream_rule,T6: abstract_tree_rule] :
                ( ( X13 = T8 )
                & ( X24 = Steps )
                & ( ( abstract_root_rule @ T8 )
                  = ( shd_rule @ Steps ) )
                & ( fmembe7964111328986570234e_rule @ T6 @ ( abstract_cont_rule @ T8 ) )
                & ( ( X3 @ T6 @ ( stl_rule @ Steps ) )
                  | ( abstract_ipath_rule @ T6 @ ( stl_rule @ Steps ) ) ) ) )
       => ( abstract_ipath_rule @ X1 @ X22 ) ) ) ).

% ipath.coinduct
thf(fact_754_ipath_Ocoinduct,axiom,
    ! [X3: abstra4572470701423150929m_rule > stream727092118206550309m_rule > $o,X1: abstra4572470701423150929m_rule,X22: stream727092118206550309m_rule] :
      ( ( X3 @ X1 @ X22 )
     => ( ! [X13: abstra4572470701423150929m_rule,X24: stream727092118206550309m_rule] :
            ( ( X3 @ X13 @ X24 )
           => ? [T8: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule,T6: abstra4572470701423150929m_rule] :
                ( ( X13 = T8 )
                & ( X24 = Steps )
                & ( ( abstra3503897793822247108m_rule @ T8 )
                  = ( shd_Pr7235097944458474089m_rule @ Steps ) )
                & ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ T8 ) )
                & ( ( X3 @ T6 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
                  | ( abstra2285467793620430502m_rule @ T6 @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) ) )
       => ( abstra2285467793620430502m_rule @ X1 @ X22 ) ) ) ).

% ipath.coinduct
thf(fact_755_RuleSystem_Opos__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream727092118206550309m_rule,R3: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( abstra1882453022147397137m_rule @ Rs @ R3 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N3 @ Rs ) )
              = R3 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_756_RuleSystem_Opos__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R3: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( abstract_pos_rule @ Rs @ R3 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
              = R3 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_757_RuleSystem_OminWait__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( abstra3055540779341473792ist_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S3 )
       => ( ord_less_eq_nat @ ( abstra738319790293016300ist_fm @ Eff @ Rs @ S3 ) @ N ) ) ) ).

% RuleSystem.minWait_least
thf(fact_758_RuleSystem_OminWait__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( abstra738319790293016300ist_fm @ Eff @ Rs @ S3 )
        = ( ord_Least_nat
          @ ^ [N3: nat] : ( abstra3055540779341473792ist_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_def
thf(fact_759_RuleSystem_OminWait__le__pos,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S6: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R3: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S6 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
         => ( ( abstra3055540779341473792ist_fm @ Eff @ R3 @ S3 )
           => ( ord_less_eq_nat @ ( abstra738319790293016300ist_fm @ Eff @ Rs @ S3 ) @ ( abstract_pos_rule @ Rs @ R3 ) ) ) ) ) ) ).

% RuleSystem.minWait_le_pos
thf(fact_760_wf_Ocases,axiom,
    ! [A2: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A2 )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A2 ) ) @ ( sset_rule @ rules ) )
         => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A2 ) ) )
              = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A2 ) ) ) )
           => ~ ! [T6: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ A2 ) )
                 => ( abstra4476735159789068068ist_fm
                    @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ T6 ) ) ) ) ) ).

% wf.cases
thf(fact_761_wf_Osimps,axiom,
    ! [A2: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A2 )
      = ( ? [T: abstra4572470701423150929m_rule] :
            ( ( A2 = 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 ) ) ) )
            & ! [X4: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X4 @ ( abstra4962253831105676628m_rule @ T ) )
               => ( abstra4476735159789068068ist_fm
                  @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R2 @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ rules
                  @ X4 ) ) ) ) ) ).

% wf.simps
thf(fact_762_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,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ rules
                @ T5 ) )
         => ( abstra4476735159789068068ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ T2 ) ) ) ) ).

% wf.intros
thf(fact_763_wf_Ocoinduct,axiom,
    ! [X3: abstra4572470701423150929m_rule > $o,X2: abstra4572470701423150929m_rule] :
      ( ( X3 @ X2 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X3 @ X )
           => ? [T8: abstra4572470701423150929m_rule] :
                ( ( X = T8 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( sset_rule @ rules ) )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) )
                  = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T8 ) ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T8 ) )
                   => ( ( X3 @ Xa )
                      | ( abstra4476735159789068068ist_fm
                        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                            ( ( eff @ R2 @ S )
                            = ( some_f2162114139285431020ist_fm @ Ss ) )
                        @ rules
                        @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ X2 ) ) ) ).

% wf.coinduct
thf(fact_764_mkTree_Osimps_I2_J,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 ) )
      = ( fimage748043113761278389m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( stl_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) ) )
        @ ( abstra458490714785879965ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) )
          @ S3 ) ) ) ).

% mkTree.simps(2)
thf(fact_765_i_Owf_Ocoinduct,axiom,
    ! [X3: abstra4572470701423150929m_rule > $o,X2: abstra4572470701423150929m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X3 @ X2 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X3 @ X )
           => ? [T8: abstra4572470701423150929m_rule] :
                ( ( X = T8 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( sset_rule @ Rules ) )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T8 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T8 ) )
                   => ( ( X3 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X2 ) ) ) ).

% i.wf.coinduct
thf(fact_766_fsubsetI,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A )
         => ( fmembe3381613331217039976ist_fm @ X @ B ) )
     => ( ord_le2064643713053750439ist_fm @ A @ B ) ) ).

% fsubsetI
thf(fact_767_fsubsetI,axiom,
    ! [A: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A )
         => ( fmembe1635983029874454474m_rule @ X @ B ) )
     => ( ord_le6951374404268211425m_rule @ A @ B ) ) ).

% fsubsetI
thf(fact_768_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_769_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_770_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_771_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_772_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_773_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_774_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_775_fimage__eqI,axiom,
    ! [B2: produc1996495991257130529ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( B2
        = ( F2 @ X2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
       => ( fmembe3381613331217039976ist_fm @ B2 @ ( fimage8674584286907012797ist_fm @ F2 @ A ) ) ) ) ).

% fimage_eqI
thf(fact_776_fimage__eqI,axiom,
    ! [B2: abstra4572470701423150929m_rule,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,X2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( B2
        = ( F2 @ X2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
       => ( fmembe1635983029874454474m_rule @ B2 @ ( fimage748043113761278389m_rule @ F2 @ A ) ) ) ) ).

% fimage_eqI
thf(fact_777_fimage__eqI,axiom,
    ! [B2: produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,X2: abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule] :
      ( ( B2
        = ( F2 @ X2 ) )
     => ( ( fmembe1635983029874454474m_rule @ X2 @ A )
       => ( fmembe3381613331217039976ist_fm @ B2 @ ( fimage9164954772359936849ist_fm @ F2 @ A ) ) ) ) ).

% fimage_eqI
thf(fact_778_fimage__eqI,axiom,
    ! [B2: abstra4572470701423150929m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,X2: abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule] :
      ( ( B2
        = ( F2 @ X2 ) )
     => ( ( fmembe1635983029874454474m_rule @ X2 @ A )
       => ( fmembe1635983029874454474m_rule @ B2 @ ( fimage3349360726776475169m_rule @ F2 @ A ) ) ) ) ).

% fimage_eqI
thf(fact_779_fimage__mono,axiom,
    ! [A: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ A @ B )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F2 @ A ) @ ( fimage9164954772359936849ist_fm @ F2 @ B ) ) ) ).

% fimage_mono
thf(fact_780_fimage__mono,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ A @ B )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F2 @ A ) @ ( fimage748043113761278389m_rule @ F2 @ B ) ) ) ).

% fimage_mono
thf(fact_781_subset__fimage__iff,axiom,
    ! [B: fset_P661503646757059847ist_fm,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A: fset_A5396818216090053761m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F2 @ A ) )
      = ( ? [AA: fset_A5396818216090053761m_rule] :
            ( ( ord_le6951374404268211425m_rule @ AA @ A )
            & ( B
              = ( fimage9164954772359936849ist_fm @ F2 @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_782_subset__fimage__iff,axiom,
    ! [B: fset_A5396818216090053761m_rule,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A: fset_P661503646757059847ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ B @ ( fimage748043113761278389m_rule @ F2 @ A ) )
      = ( ? [AA: fset_P661503646757059847ist_fm] :
            ( ( ord_le2064643713053750439ist_fm @ AA @ A )
            & ( B
              = ( fimage748043113761278389m_rule @ F2 @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_783_fimage__fsubsetI,axiom,
    ! [A: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A )
         => ( fmembe3381613331217039976ist_fm @ ( F2 @ X ) @ B ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage8674584286907012797ist_fm @ F2 @ A ) @ B ) ) ).

% fimage_fsubsetI
thf(fact_784_fimage__fsubsetI,axiom,
    ! [A: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,B: fset_A5396818216090053761m_rule] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A )
         => ( fmembe1635983029874454474m_rule @ ( F2 @ X ) @ B ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F2 @ A ) @ B ) ) ).

% fimage_fsubsetI
thf(fact_785_fimage__fsubsetI,axiom,
    ! [A: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A )
         => ( fmembe3381613331217039976ist_fm @ ( F2 @ X ) @ B ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F2 @ A ) @ B ) ) ).

% fimage_fsubsetI
thf(fact_786_fimage__fsubsetI,axiom,
    ! [A: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,B: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A )
         => ( fmembe1635983029874454474m_rule @ ( F2 @ X ) @ B ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage3349360726776475169m_rule @ F2 @ A ) @ B ) ) ).

% fimage_fsubsetI
thf(fact_787_pred__subset__eq2,axiom,
    ! [R: set_Pr7058068377845519745ist_fm,S6: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le1774903256965159270t_fm_o
        @ ^ [X4: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y5 ) @ R )
        @ ^ [X4: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y5 ) @ S6 ) )
      = ( ord_le2055072402069232929ist_fm @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_788_pred__subset__eq2,axiom,
    ! [R: set_Pr1008144964186165195m_rule,S6: set_Pr1008144964186165195m_rule] :
      ( ( ord_le4891813983447887534rule_o
        @ ^ [X4: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X4 @ Y5 ) @ R )
        @ ^ [X4: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X4 @ Y5 ) @ S6 ) )
      = ( ord_le5052251716133888043m_rule @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_789_fimage__fimage,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A: fset_A5396818216090053761m_rule] :
      ( ( fimage8674584286907012797ist_fm @ F2 @ ( fimage9164954772359936849ist_fm @ G @ A ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X4: abstra4572470701423150929m_rule] : ( F2 @ ( G @ X4 ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_790_fimage__fimage,axiom,
    ! [F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A: fset_P661503646757059847ist_fm] :
      ( ( fimage3349360726776475169m_rule @ F2 @ ( fimage748043113761278389m_rule @ G @ A ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X4: produc1996495991257130529ist_fm] : ( F2 @ ( G @ X4 ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_791_fimage__fimage,axiom,
    ! [F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule] :
      ( ( fimage9164954772359936849ist_fm @ F2 @ ( fimage3349360726776475169m_rule @ G @ A ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X4: abstra4572470701423150929m_rule] : ( F2 @ ( G @ X4 ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_792_fimage__fimage,axiom,
    ! [F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A: fset_P661503646757059847ist_fm] :
      ( ( fimage9164954772359936849ist_fm @ F2 @ ( fimage748043113761278389m_rule @ G @ A ) )
      = ( fimage8674584286907012797ist_fm
        @ ^ [X4: produc1996495991257130529ist_fm] : ( F2 @ ( G @ X4 ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_793_fimage__fimage,axiom,
    ! [F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fimage748043113761278389m_rule @ F2 @ ( fimage8674584286907012797ist_fm @ G @ A ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X4: produc1996495991257130529ist_fm] : ( F2 @ ( G @ X4 ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_794_fimage__fimage,axiom,
    ! [F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A: fset_A5396818216090053761m_rule] :
      ( ( fimage748043113761278389m_rule @ F2 @ ( fimage9164954772359936849ist_fm @ G @ A ) )
      = ( fimage3349360726776475169m_rule
        @ ^ [X4: abstra4572470701423150929m_rule] : ( F2 @ ( G @ X4 ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_795_rev__fimage__eqI,axiom,
    ! [X2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B2: produc1996495991257130529ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
     => ( ( B2
          = ( F2 @ X2 ) )
       => ( fmembe3381613331217039976ist_fm @ B2 @ ( fimage8674584286907012797ist_fm @ F2 @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_796_rev__fimage__eqI,axiom,
    ! [X2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,B2: abstra4572470701423150929m_rule,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
     => ( ( B2
          = ( F2 @ X2 ) )
       => ( fmembe1635983029874454474m_rule @ B2 @ ( fimage748043113761278389m_rule @ F2 @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_797_rev__fimage__eqI,axiom,
    ! [X2: abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule,B2: produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X2 @ A )
     => ( ( B2
          = ( F2 @ X2 ) )
       => ( fmembe3381613331217039976ist_fm @ B2 @ ( fimage9164954772359936849ist_fm @ F2 @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_798_rev__fimage__eqI,axiom,
    ! [X2: abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule,B2: abstra4572470701423150929m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X2 @ A )
     => ( ( B2
          = ( F2 @ X2 ) )
       => ( fmembe1635983029874454474m_rule @ B2 @ ( fimage3349360726776475169m_rule @ F2 @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_799_fimage__cong,axiom,
    ! [M3: fset_P661503646757059847ist_fm,N4: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( M3 = N4 )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( fmembe3381613331217039976ist_fm @ X @ N4 )
           => ( ( F2 @ X )
              = ( G @ X ) ) )
       => ( ( fimage748043113761278389m_rule @ F2 @ M3 )
          = ( fimage748043113761278389m_rule @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_800_fimage__cong,axiom,
    ! [M3: fset_A5396818216090053761m_rule,N4: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( M3 = N4 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( fmembe1635983029874454474m_rule @ X @ N4 )
           => ( ( F2 @ X )
              = ( G @ X ) ) )
       => ( ( fimage9164954772359936849ist_fm @ F2 @ M3 )
          = ( fimage9164954772359936849ist_fm @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_801_fimageI,axiom,
    ! [X2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
     => ( fmembe3381613331217039976ist_fm @ ( F2 @ X2 ) @ ( fimage8674584286907012797ist_fm @ F2 @ A ) ) ) ).

% fimageI
thf(fact_802_fimageI,axiom,
    ! [X2: produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
     => ( fmembe1635983029874454474m_rule @ ( F2 @ X2 ) @ ( fimage748043113761278389m_rule @ F2 @ A ) ) ) ).

% fimageI
thf(fact_803_fimageI,axiom,
    ! [X2: abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X2 @ A )
     => ( fmembe3381613331217039976ist_fm @ ( F2 @ X2 ) @ ( fimage9164954772359936849ist_fm @ F2 @ A ) ) ) ).

% fimageI
thf(fact_804_fimageI,axiom,
    ! [X2: abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X2 @ A )
     => ( fmembe1635983029874454474m_rule @ ( F2 @ X2 ) @ ( fimage3349360726776475169m_rule @ F2 @ A ) ) ) ).

% fimageI
thf(fact_805_fimageE,axiom,
    ! [B2: produc1996495991257130529ist_fm,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ B2 @ ( fimage8674584286907012797ist_fm @ F2 @ A ) )
     => ~ ! [X: produc1996495991257130529ist_fm] :
            ( ( B2
              = ( F2 @ X ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X @ A ) ) ) ).

% fimageE
thf(fact_806_fimageE,axiom,
    ! [B2: produc1996495991257130529ist_fm,F2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A: fset_A5396818216090053761m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ B2 @ ( fimage9164954772359936849ist_fm @ F2 @ A ) )
     => ~ ! [X: abstra4572470701423150929m_rule] :
            ( ( B2
              = ( F2 @ X ) )
           => ~ ( fmembe1635983029874454474m_rule @ X @ A ) ) ) ).

% fimageE
thf(fact_807_fimageE,axiom,
    ! [B2: abstra4572470701423150929m_rule,F2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A: fset_P661503646757059847ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ B2 @ ( fimage748043113761278389m_rule @ F2 @ A ) )
     => ~ ! [X: produc1996495991257130529ist_fm] :
            ( ( B2
              = ( F2 @ X ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X @ A ) ) ) ).

% fimageE
thf(fact_808_fimageE,axiom,
    ! [B2: abstra4572470701423150929m_rule,F2: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ B2 @ ( fimage3349360726776475169m_rule @ F2 @ A ) )
     => ~ ! [X: abstra4572470701423150929m_rule] :
            ( ( B2
              = ( F2 @ X ) )
           => ~ ( fmembe1635983029874454474m_rule @ X @ A ) ) ) ).

% fimageE
thf(fact_809_subrelI,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S3: set_Pr7058068377845519745ist_fm] :
      ( ! [X: list_fm,Y4: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ R3 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y4 ) @ S3 ) )
     => ( ord_le2055072402069232929ist_fm @ R3 @ S3 ) ) ).

% subrelI
thf(fact_810_subrelI,axiom,
    ! [R3: set_Pr1008144964186165195m_rule,S3: set_Pr1008144964186165195m_rule] :
      ( ! [X: produc1996495991257130529ist_fm,Y4: rule] :
          ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X @ Y4 ) @ R3 )
         => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X @ Y4 ) @ S3 ) )
     => ( ord_le5052251716133888043m_rule @ R3 @ S3 ) ) ).

% subrelI
thf(fact_811_subset__UNIV,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A @ top_to728902761071950545ist_fm ) ).

% subset_UNIV
thf(fact_812_subset__UNIV,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_813_subset__UNIV,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ top_top_set_rule ) ).

% subset_UNIV
thf(fact_814_fsubsetD,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A @ B )
     => ( ( fmembe3381613331217039976ist_fm @ C @ A )
       => ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ).

% fsubsetD
thf(fact_815_fsubsetD,axiom,
    ! [A: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A @ B )
     => ( ( fmembe1635983029874454474m_rule @ C @ A )
       => ( fmembe1635983029874454474m_rule @ C @ B ) ) ) ).

% fsubsetD
thf(fact_816_fin__mono,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,X2: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A @ B )
     => ( ( fmembe3381613331217039976ist_fm @ X2 @ A )
       => ( fmembe3381613331217039976ist_fm @ X2 @ B ) ) ) ).

% fin_mono
thf(fact_817_fin__mono,axiom,
    ! [A: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,X2: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A @ B )
     => ( ( fmembe1635983029874454474m_rule @ X2 @ A )
       => ( fmembe1635983029874454474m_rule @ X2 @ B ) ) ) ).

% fin_mono
thf(fact_818_sset__sdrop,axiom,
    ! [N: nat,S3: stream727092118206550309m_rule] : ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S3 ) ) @ ( sset_P4979403222752902913m_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_819_sset__sdrop,axiom,
    ! [N: nat,S3: stream_rule] : ( ord_less_eq_set_rule @ ( sset_rule @ ( sdrop_rule @ N @ S3 ) ) @ ( sset_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_820_Prover_Ofair__def,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
      = ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs ) @ ( sset_rule @ rules ) )
        & ! [X4: rule] :
            ( ( member_rule @ X4 @ ( sset_rule @ rules ) )
           => ( linear7625967296368300489w_rule
              @ ( linear6831045468625226112v_rule
                @ ( linear4670012741531116049s_rule
                  @ ( ^ [Y3: rule,Z: rule] : ( Y3 = Z )
                    @ X4 ) ) )
              @ Rs ) ) ) ) ).

% Prover.fair_def
thf(fact_821_i_Ofair__def,axiom,
    ( abstra4997497885011821313m_rule
    = ( ^ [Rules2: stream727092118206550309m_rule,Rs2: stream727092118206550309m_rule] :
          ( ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ Rs2 ) @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
          & ! [X4: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ X4 @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
             => ( linear5684299538428340377m_rule
                @ ( linear2722748333883190224m_rule
                  @ ( linear7082939723987557601m_rule
                    @ ( ^ [Y3: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y3 = Z )
                      @ X4 ) ) )
                @ Rs2 ) ) ) ) ) ).

% i.fair_def
thf(fact_822_i_Ofair__def,axiom,
    ( abstra1114877437032970289r_rule
    = ( ^ [Rules2: stream_rule,Rs2: stream_rule] :
          ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs2 ) @ ( sset_rule @ Rules2 ) )
          & ! [X4: rule] :
              ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
             => ( linear7625967296368300489w_rule
                @ ( linear6831045468625226112v_rule
                  @ ( linear4670012741531116049s_rule
                    @ ( ^ [Y3: rule,Z: rule] : ( Y3 = Z )
                      @ X4 ) ) )
                @ Rs2 ) ) ) ) ) ).

% i.fair_def
thf(fact_823_RuleSystem__Defs_Ofair__def,axiom,
    ( abstra4997497885011821313m_rule
    = ( ^ [Rules2: stream727092118206550309m_rule,Rs2: stream727092118206550309m_rule] :
          ( ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ Rs2 ) @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
          & ! [X4: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ X4 @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
             => ( linear5684299538428340377m_rule
                @ ( linear2722748333883190224m_rule
                  @ ( linear7082939723987557601m_rule
                    @ ( ^ [Y3: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y3 = Z )
                      @ X4 ) ) )
                @ Rs2 ) ) ) ) ) ).

% RuleSystem_Defs.fair_def
thf(fact_824_RuleSystem__Defs_Ofair__def,axiom,
    ( abstra1114877437032970289r_rule
    = ( ^ [Rules2: stream_rule,Rs2: stream_rule] :
          ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs2 ) @ ( sset_rule @ Rules2 ) )
          & ! [X4: rule] :
              ( ( member_rule @ X4 @ ( sset_rule @ Rules2 ) )
             => ( linear7625967296368300489w_rule
                @ ( linear6831045468625226112v_rule
                  @ ( linear4670012741531116049s_rule
                    @ ( ^ [Y3: rule,Z: rule] : ( Y3 = Z )
                      @ X4 ) ) )
                @ Rs2 ) ) ) ) ) ).

% RuleSystem_Defs.fair_def
thf(fact_825_RuleSystem__Defs_OmkTree_Osimps_I2_J,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) )
      = ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) ) @ ( abstra458490714785879965ist_fm @ Eff @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ).

% RuleSystem_Defs.mkTree.simps(2)
thf(fact_826_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X3: abstra4572470701423150929m_rule > $o,X2: abstra4572470701423150929m_rule,Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X3 @ X2 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X3 @ X )
           => ? [T8: abstra4572470701423150929m_rule] :
                ( ( X = T8 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( sset_rule @ Rules ) )
                & ( Eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T8 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T8 ) )
                   => ( ( X3 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ Eff @ Rules @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ X2 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_827_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( Eff @ ( 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 @ Eff @ Rules @ T5 ) )
         => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ T2 ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_828_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra4476735159789068068ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A3: abstra4572470701423150929m_rule] :
        ? [T: abstra4572470701423150929m_rule] :
          ( ( A3 = T )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ Rules2 ) )
          & ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) )
          & ! [X4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X4 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ X4 ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_829_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A2: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff @ Rules @ A2 )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
         => ( ( Eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A2 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A2 ) ) )
           => ~ ! [T6: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ A2 ) )
                 => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ T6 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_830_i_OmkTree_Osimps_I2_J,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) )
      = ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ).

% i.mkTree.simps(2)
thf(fact_831_i_Owf_Ocases,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A2: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A2 )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A2 ) ) @ ( sset_rule @ Rules ) )
         => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A2 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A2 ) ) )
           => ~ ! [T6: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ A2 ) )
                 => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T6 ) ) ) ) ) ).

% i.wf.cases
thf(fact_832_i_Owf_Osimps,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A2: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A2 )
      = ( ? [T: abstra4572470701423150929m_rule] :
            ( ( A2 = T )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ Rules ) )
            & ( 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 ) ) )
            & ! [X4: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X4 @ ( abstra4962253831105676628m_rule @ T ) )
               => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X4 ) ) ) ) ) ).

% i.wf.simps
thf(fact_833_i_Owf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( 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 ) @ Rules @ T5 ) )
         => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T2 ) ) ) ) ).

% i.wf.intros
thf(fact_834_mkTree_Ocode,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( abstra7467780926338743076m_rule
        @ ( produc491467635432902671m_rule @ S3
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) ) )
        @ ( fimage748043113761278389m_rule
          @ ( abstra178918799166095863ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( stl_rule
              @ ( abstra2363873759530810705ist_fm
                @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S3 ) ) )
          @ ( abstra458490714785879965ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule
              @ ( abstra2363873759530810705ist_fm
                @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S3 ) )
            @ S3 ) ) ) ) ).

% mkTree.code
thf(fact_835_mkTree__unfold,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( case_s2494923709252486212m_rule
        @ ^ [R2: rule,S9: stream_rule] :
            ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S3 @ R2 )
            @ ( fimage748043113761278389m_rule
              @ ( abstra178918799166095863ist_fm
                @ ^ [S: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ S9 )
              @ ( abstra458490714785879965ist_fm
                @ ^ [S: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R2
                @ S3 ) ) )
        @ ( abstra2363873759530810705ist_fm
          @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R2 @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 ) ) ) ).

% mkTree_unfold
thf(fact_836_i_OmkTree_Ocode,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S3: list_fm] :
      ( ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S3 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ ( stl_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_837_i_OmkTree_Ocode,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S3 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_838_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra3846921640779263781ist_fm
    = ( ^ [Eff2: list_fm > list_fm > fset_list_fm > $o,Rs2: stream_list_fm,S: list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff2 @ Rs2 @ S ) ) ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ ( stl_list_fm @ ( abstra2983386638146376267ist_fm @ Eff2 @ Rs2 @ S ) ) ) @ ( abstra1097029126060538623ist_fm @ Eff2 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff2 @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_839_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra178918799166095863ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs2 @ S ) ) ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs2 @ S ) ) ) @ ( abstra458490714785879965ist_fm @ Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_840_i_OmkTree__unfold,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S3: list_fm] :
      ( ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( case_s5690291548269216621ist_fm
        @ ^ [R2: list_fm,S9: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S3 @ R2 ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ S9 ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ R2 @ S3 ) ) )
        @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_841_i_OmkTree__unfold,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 )
      = ( case_s2494923709252486212m_rule
        @ ^ [R2: rule,S9: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S3 @ R2 ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ S9 ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S3 ) ) )
        @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_842_subsetI,axiom,
    ! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X @ A )
         => ( member8102475879199740618ist_fm @ X @ B ) )
     => ( ord_le2055072402069232929ist_fm @ A @ B ) ) ).

% subsetI
thf(fact_843_subsetI,axiom,
    ! [A: set_fm,B: set_fm] :
      ( ! [X: fm] :
          ( ( member_fm @ X @ A )
         => ( member_fm @ X @ B ) )
     => ( ord_less_eq_set_fm @ A @ B ) ) ).

% subsetI
thf(fact_844_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X: nat] :
          ( ( member_nat @ X @ A )
         => ( member_nat @ X @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% subsetI
thf(fact_845_subsetI,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ! [X: rule] :
          ( ( member_rule @ X @ A )
         => ( member_rule @ X @ B ) )
     => ( ord_less_eq_set_rule @ A @ B ) ) ).

% subsetI
thf(fact_846_subset__antisym,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_847_tree_Oinject,axiom,
    ! [X1: produc164195504107695125m_rule,X22: fset_A5396818216090053761m_rule,Y1: produc164195504107695125m_rule,Y2: fset_A5396818216090053761m_rule] :
      ( ( ( abstra7467780926338743076m_rule @ X1 @ X22 )
        = ( abstra7467780926338743076m_rule @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% tree.inject
thf(fact_848_tree_Ocollapse,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) )
      = Tree ) ).

% tree.collapse
thf(fact_849_conj__subset__def,axiom,
    ! [A: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_850_conj__subset__def,axiom,
    ! [A: set_rule,P: rule > $o,Q: rule > $o] :
      ( ( ord_less_eq_set_rule @ A
        @ ( collect_rule
          @ ^ [X4: rule] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) )
      = ( ( ord_less_eq_set_rule @ A @ ( collect_rule @ P ) )
        & ( ord_less_eq_set_rule @ A @ ( collect_rule @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_851_less__eq__set__def,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B8: set_Pr7058068377845519745ist_fm] :
          ( ord_le8518180823940269308t_fm_o
          @ ^ [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ A7 )
          @ ^ [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_852_less__eq__set__def,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B8: set_fm] :
          ( ord_less_eq_fm_o
          @ ^ [X4: fm] : ( member_fm @ X4 @ A7 )
          @ ^ [X4: fm] : ( member_fm @ X4 @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_853_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A7 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_854_less__eq__set__def,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
          ( ord_less_eq_rule_o
          @ ^ [X4: rule] : ( member_rule @ X4 @ A7 )
          @ ^ [X4: rule] : ( member_rule @ X4 @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_855_in__mono,axiom,
    ! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,X2: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A @ B )
     => ( ( member8102475879199740618ist_fm @ X2 @ A )
       => ( member8102475879199740618ist_fm @ X2 @ B ) ) ) ).

% in_mono
thf(fact_856_in__mono,axiom,
    ! [A: set_fm,B: set_fm,X2: fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( member_fm @ X2 @ A )
       => ( member_fm @ X2 @ B ) ) ) ).

% in_mono
thf(fact_857_in__mono,axiom,
    ! [A: set_nat,B: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ X2 @ A )
       => ( member_nat @ X2 @ B ) ) ) ).

% in_mono
thf(fact_858_in__mono,axiom,
    ! [A: set_rule,B: set_rule,X2: rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( member_rule @ X2 @ A )
       => ( member_rule @ X2 @ B ) ) ) ).

% in_mono
thf(fact_859_subsetD,axiom,
    ! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A @ B )
     => ( ( member8102475879199740618ist_fm @ C @ A )
       => ( member8102475879199740618ist_fm @ C @ B ) ) ) ).

% subsetD
thf(fact_860_subsetD,axiom,
    ! [A: set_fm,B: set_fm,C: fm] :
      ( ( ord_less_eq_set_fm @ A @ B )
     => ( ( member_fm @ C @ A )
       => ( member_fm @ C @ B ) ) ) ).

% subsetD
thf(fact_861_subsetD,axiom,
    ! [A: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_862_subsetD,axiom,
    ! [A: set_rule,B: set_rule,C: rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( member_rule @ C @ A )
       => ( member_rule @ C @ B ) ) ) ).

% subsetD
thf(fact_863_equalityE,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_rule @ A @ B )
         => ~ ( ord_less_eq_set_rule @ B @ A ) ) ) ).

% equalityE
thf(fact_864_subset__eq,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B8: set_Pr7058068377845519745ist_fm] :
        ! [X4: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X4 @ A7 )
         => ( member8102475879199740618ist_fm @ X4 @ B8 ) ) ) ) ).

% subset_eq
thf(fact_865_subset__eq,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B8: set_fm] :
        ! [X4: fm] :
          ( ( member_fm @ X4 @ A7 )
         => ( member_fm @ X4 @ B8 ) ) ) ) ).

% subset_eq
thf(fact_866_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A7 )
         => ( member_nat @ X4 @ B8 ) ) ) ) ).

% subset_eq
thf(fact_867_subset__eq,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
        ! [X4: rule] :
          ( ( member_rule @ X4 @ A7 )
         => ( member_rule @ X4 @ B8 ) ) ) ) ).

% subset_eq
thf(fact_868_equalityD1,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( A = B )
     => ( ord_less_eq_set_rule @ A @ B ) ) ).

% equalityD1
thf(fact_869_equalityD2,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( A = B )
     => ( ord_less_eq_set_rule @ B @ A ) ) ).

% equalityD2
thf(fact_870_subset__iff,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B8: set_Pr7058068377845519745ist_fm] :
        ! [T: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ T @ A7 )
         => ( member8102475879199740618ist_fm @ T @ B8 ) ) ) ) ).

% subset_iff
thf(fact_871_subset__iff,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B8: set_fm] :
        ! [T: fm] :
          ( ( member_fm @ T @ A7 )
         => ( member_fm @ T @ B8 ) ) ) ) ).

% subset_iff
thf(fact_872_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
        ! [T: nat] :
          ( ( member_nat @ T @ A7 )
         => ( member_nat @ T @ B8 ) ) ) ) ).

% subset_iff
thf(fact_873_subset__iff,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
        ! [T: rule] :
          ( ( member_rule @ T @ A7 )
         => ( member_rule @ T @ B8 ) ) ) ) ).

% subset_iff
thf(fact_874_subset__refl,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ A ) ).

% subset_refl
thf(fact_875_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X: nat] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_876_Collect__mono,axiom,
    ! [P: rule > $o,Q: rule > $o] :
      ( ! [X: rule] :
          ( ( P @ X )
         => ( Q @ X ) )
     => ( ord_less_eq_set_rule @ ( collect_rule @ P ) @ ( collect_rule @ Q ) ) ) ).

% Collect_mono
thf(fact_877_subset__trans,axiom,
    ! [A: set_rule,B: set_rule,C2: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ B @ C2 )
       => ( ord_less_eq_set_rule @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_878_set__eq__subset,axiom,
    ( ( ^ [Y3: set_rule,Z: set_rule] : ( Y3 = Z ) )
    = ( ^ [A7: set_rule,B8: set_rule] :
          ( ( ord_less_eq_set_rule @ A7 @ B8 )
          & ( ord_less_eq_set_rule @ B8 @ A7 ) ) ) ) ).

% set_eq_subset
thf(fact_879_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_880_Collect__mono__iff,axiom,
    ! [P: rule > $o,Q: rule > $o] :
      ( ( ord_less_eq_set_rule @ ( collect_rule @ P ) @ ( collect_rule @ Q ) )
      = ( ! [X4: rule] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_881_Collect__subset,axiom,
    ! [A: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X4: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_882_Collect__subset,axiom,
    ! [A: set_fm,P: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X4: fm] :
            ( ( member_fm @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_883_Collect__subset,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_884_Collect__subset,axiom,
    ! [A: set_rule,P: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X4: rule] :
            ( ( member_rule @ X4 @ A )
            & ( P @ X4 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_885_subset__CollectI,axiom,
    ! [B: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm,Q: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B @ A )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le2055072402069232929ist_fm
          @ ( collec7231575776666803724ist_fm
            @ ^ [X4: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collec7231575776666803724ist_fm
            @ ^ [X4: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_886_subset__CollectI,axiom,
    ! [B: set_fm,A: set_fm,Q: fm > $o,P: fm > $o] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ! [X: fm] :
            ( ( member_fm @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_less_eq_set_fm
          @ ( collect_fm
            @ ^ [X4: fm] :
                ( ( member_fm @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collect_fm
            @ ^ [X4: fm] :
                ( ( member_fm @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_887_subset__CollectI,axiom,
    ! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ! [X: nat] :
            ( ( member_nat @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_888_subset__CollectI,axiom,
    ! [B: set_rule,A: set_rule,Q: rule > $o,P: rule > $o] :
      ( ( ord_less_eq_set_rule @ B @ A )
     => ( ! [X: rule] :
            ( ( member_rule @ X @ B )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_less_eq_set_rule
          @ ( collect_rule
            @ ^ [X4: rule] :
                ( ( member_rule @ X4 @ B )
                & ( Q @ X4 ) ) )
          @ ( collect_rule
            @ ^ [X4: rule] :
                ( ( member_rule @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_889_subset__Collect__iff,axiom,
    ! [B: set_Pr7058068377845519745ist_fm,A: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B @ A )
     => ( ( ord_le2055072402069232929ist_fm @ B
          @ ( collec7231575776666803724ist_fm
            @ ^ [X4: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_890_subset__Collect__iff,axiom,
    ! [B: set_fm,A: set_fm,P: fm > $o] :
      ( ( ord_less_eq_set_fm @ B @ A )
     => ( ( ord_less_eq_set_fm @ B
          @ ( collect_fm
            @ ^ [X4: fm] :
                ( ( member_fm @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: fm] :
              ( ( member_fm @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_891_subset__Collect__iff,axiom,
    ! [B: set_nat,A: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ B
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_892_subset__Collect__iff,axiom,
    ! [B: set_rule,A: set_rule,P: rule > $o] :
      ( ( ord_less_eq_set_rule @ B @ A )
     => ( ( ord_less_eq_set_rule @ B
          @ ( collect_rule
            @ ^ [X4: rule] :
                ( ( member_rule @ X4 @ A )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: rule] :
              ( ( member_rule @ X4 @ B )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_893_pred__subset__eq,axiom,
    ! [R: set_Pr7058068377845519745ist_fm,S6: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le8518180823940269308t_fm_o
        @ ^ [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ R )
        @ ^ [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ S6 ) )
      = ( ord_le2055072402069232929ist_fm @ R @ S6 ) ) ).

% pred_subset_eq
thf(fact_894_pred__subset__eq,axiom,
    ! [R: set_fm,S6: set_fm] :
      ( ( ord_less_eq_fm_o
        @ ^ [X4: fm] : ( member_fm @ X4 @ R )
        @ ^ [X4: fm] : ( member_fm @ X4 @ S6 ) )
      = ( ord_less_eq_set_fm @ R @ S6 ) ) ).

% pred_subset_eq
thf(fact_895_pred__subset__eq,axiom,
    ! [R: set_nat,S6: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S6 ) )
      = ( ord_less_eq_set_nat @ R @ S6 ) ) ).

% pred_subset_eq
thf(fact_896_pred__subset__eq,axiom,
    ! [R: set_rule,S6: set_rule] :
      ( ( ord_less_eq_rule_o
        @ ^ [X4: rule] : ( member_rule @ X4 @ R )
        @ ^ [X4: rule] : ( member_rule @ X4 @ S6 ) )
      = ( ord_less_eq_set_rule @ R @ S6 ) ) ).

% pred_subset_eq
thf(fact_897_tree_Oexhaust,axiom,
    ! [Y: abstra4572470701423150929m_rule] :
      ~ ! [X13: produc164195504107695125m_rule,X24: fset_A5396818216090053761m_rule] :
          ( Y
         != ( abstra7467780926338743076m_rule @ X13 @ X24 ) ) ).

% tree.exhaust
thf(fact_898_tree_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X22: fset_A5396818216090053761m_rule] :
      ( ( abstra3503897793822247108m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X22 ) )
      = X1 ) ).

% tree.sel(1)
thf(fact_899_tree_Osel_I2_J,axiom,
    ! [X1: produc164195504107695125m_rule,X22: fset_A5396818216090053761m_rule] :
      ( ( abstra4962253831105676628m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X22 ) )
      = X22 ) ).

% tree.sel(2)
thf(fact_900_tree_Oexhaust__sel,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( Tree
      = ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) ) ) ).

% tree.exhaust_sel
thf(fact_901_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra3846921640779263781ist_fm
    = ( ^ [Eff2: list_fm > list_fm > fset_list_fm > $o,Rs2: stream_list_fm,S: list_fm] :
          ( case_s5690291548269216621ist_fm
          @ ^ [R2: list_fm,S9: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S @ R2 ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ S9 ) @ ( abstra1097029126060538623ist_fm @ Eff2 @ R2 @ S ) ) )
          @ ( abstra2983386638146376267ist_fm @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_902_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra178918799166095863ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] :
          ( case_s2494923709252486212m_rule
          @ ^ [R2: rule,S9: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ R2 ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ S9 ) @ ( abstra458490714785879965ist_fm @ Eff2 @ R2 @ S ) ) )
          @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_903_konig_Ocode,axiom,
    ( abstra3079604473823497664m_rule
    = ( ^ [T: abstra4572470701423150929m_rule] :
          ( sCons_5731567480799343806m_rule @ ( abstra3503897793822247108m_rule @ T )
          @ ( abstra3079604473823497664m_rule
            @ ( fChoic7543385803022166781m_rule
              @ ^ [T7: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ T ) )
                  & ~ ( abstra3507476289341504315m_rule @ T7 ) ) ) ) ) ) ) ).

% konig.code
thf(fact_904_sdrop__wait__least,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,N: nat] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ Xs2 )
     => ( ( Phi2 @ ( sdrop_7224736112439592940m_rule @ N @ Xs2 ) )
       => ( ord_less_eq_nat @ ( linear2497734134072303348m_rule @ Phi2 @ Xs2 ) @ N ) ) ) ).

% sdrop_wait_least
thf(fact_905_sdrop__wait__least,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,N: nat] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ Xs2 )
     => ( ( Phi2 @ ( sdrop_rule @ N @ Xs2 ) )
       => ( ord_less_eq_nat @ ( linear25361906725448356t_rule @ Phi2 @ Xs2 ) @ N ) ) ) ).

% sdrop_wait_least
thf(fact_906_konig_Osimps_I2_J,axiom,
    ! [T2: abstract_tree_rule] :
      ( ( stl_rule @ ( abstract_konig_rule @ T2 ) )
      = ( abstract_konig_rule
        @ ( fChoic8596484284512669613e_rule
          @ ^ [T7: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T7 @ ( abstract_cont_rule @ T2 ) )
              & ~ ( abstra6797871900832723819e_rule @ T7 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_907_konig_Osimps_I2_J,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( abstra3079604473823497664m_rule @ T2 ) )
      = ( abstra3079604473823497664m_rule
        @ ( fChoic7543385803022166781m_rule
          @ ^ [T7: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ T2 ) )
              & ~ ( abstra3507476289341504315m_rule @ T7 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_908_snd__sndOp,axiom,
    ! [P: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm > rule > $o] :
      ( produc7165828336582415457m_rule
      = ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( bNF_sn6617447237779055481m_rule @ P @ Q ) ) ) ).

% snd_sndOp
thf(fact_909_fst__fstOp,axiom,
    ! [P: produc1996495991257130529ist_fm > rule > $o,Q: rule > rule > $o] :
      ( produc6879501374131015971m_rule
      = ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( bNF_fs2143400664727635963e_rule @ P @ Q ) ) ) ).

% fst_fstOp
thf(fact_910_sdrop__wait,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ Xs2 )
     => ( Phi2 @ ( sdrop_7224736112439592940m_rule @ ( linear2497734134072303348m_rule @ Phi2 @ Xs2 ) @ Xs2 ) ) ) ).

% sdrop_wait
thf(fact_911_sdrop__wait,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ Xs2 )
     => ( Phi2 @ ( sdrop_rule @ ( linear25361906725448356t_rule @ Phi2 @ Xs2 ) @ Xs2 ) ) ) ).

% sdrop_wait
thf(fact_912_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_913_someI2__bex,axiom,
    ! [A: set_rule,P: rule > $o,Q: rule > $o] :
      ( ? [X5: rule] :
          ( ( member_rule @ X5 @ A )
          & ( P @ X5 ) )
     => ( ! [X: rule] :
            ( ( ( member_rule @ X @ A )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoice_rule
            @ ^ [X4: rule] :
                ( ( member_rule @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_914_someI2__bex,axiom,
    ! [A: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm > $o] :
      ( ? [X5: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X5 @ A )
          & ( P @ X5 ) )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ X @ A )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoic1980167584695909237ist_fm
            @ ^ [X4: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_915_someI2__bex,axiom,
    ! [A: set_fm,P: fm > $o,Q: fm > $o] :
      ( ? [X5: fm] :
          ( ( member_fm @ X5 @ A )
          & ( P @ X5 ) )
     => ( ! [X: fm] :
            ( ( ( member_fm @ X @ A )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoice_fm
            @ ^ [X4: fm] :
                ( ( member_fm @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_916_someI2__bex,axiom,
    ! [A: set_nat,P: nat > $o,Q: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ A )
          & ( P @ X5 ) )
     => ( ! [X: nat] :
            ( ( ( member_nat @ X @ A )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoice_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A )
                & ( P @ X4 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_917_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_918_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_919_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P1712579846118164081m_rule @ S3 @ N5 ) )
     => ( ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P ) @ S3 )
        = ( sdrop_7224736112439592940m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P1712579846118164081m_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_920_sdrop__while__sdrop__LEAST,axiom,
    ! [P: rule > $o,S3: stream_rule] :
      ( ? [N5: nat] : ( P @ ( snth_rule @ S3 @ N5 ) )
     => ( ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P ) @ S3 )
        = ( sdrop_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_921_apsnd__conv,axiom,
    ! [F2: list_fm > list_fm,X2: list_fm,Y: list_fm] :
      ( ( produc9047611296996456864ist_fm @ F2 @ ( produc381145313068854617ist_fm @ X2 @ Y ) )
      = ( produc381145313068854617ist_fm @ X2 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_922_apsnd__conv,axiom,
    ! [F2: rule > rule,X2: produc1996495991257130529ist_fm,Y: rule] :
      ( ( produc3063135074001183255ist_fm @ F2 @ ( produc491467635432902671m_rule @ X2 @ Y ) )
      = ( produc491467635432902671m_rule @ X2 @ ( F2 @ Y ) ) ) ).

% apsnd_conv
thf(fact_923_fst__apsnd,axiom,
    ! [F2: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F2 @ X2 ) )
      = ( produc6879501374131015971m_rule @ X2 ) ) ).

% fst_apsnd
thf(fact_924_snd__apsnd,axiom,
    ! [F2: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F2 @ X2 ) )
      = ( F2 @ ( produc7165828336582415457m_rule @ X2 ) ) ) ).

% snd_apsnd
thf(fact_925_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S3 ) )
      = ( snth_P1712579846118164081m_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_926_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( shd_rule @ ( sdrop_rule @ N @ S3 ) )
      = ( snth_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_927_snth__szip,axiom,
    ! [S1: stream_list_fm,S2: stream_list_fm,N: nat] :
      ( ( snth_P2624749845746815685ist_fm @ ( szip_list_fm_list_fm @ S1 @ S2 ) @ N )
      = ( produc381145313068854617ist_fm @ ( snth_list_fm @ S1 @ N ) @ ( snth_list_fm @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_928_snth__szip,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) @ N )
      = ( produc491467635432902671m_rule @ ( snth_P2624749845746815685ist_fm @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_929_fst__comp__apsnd,axiom,
    ! [F2: rule > rule] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F2 ) )
      = produc6879501374131015971m_rule ) ).

% fst_comp_apsnd
thf(fact_930_snth__sset,axiom,
    ! [S3: stream8299795917829157543ist_fm,N: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S3 @ N ) @ ( sset_P3431374123360452149ist_fm @ S3 ) ) ).

% snth_sset
thf(fact_931_snth__sset,axiom,
    ! [S3: stream_fm,N: nat] : ( member_fm @ ( snth_fm @ S3 @ N ) @ ( sset_fm @ S3 ) ) ).

% snth_sset
thf(fact_932_snth__sset,axiom,
    ! [S3: stream_nat,N: nat] : ( member_nat @ ( snth_nat @ S3 @ N ) @ ( sset_nat @ S3 ) ) ).

% snth_sset
thf(fact_933_snth__sset,axiom,
    ! [S3: stream_rule,N: nat] : ( member_rule @ ( snth_rule @ S3 @ N ) @ ( sset_rule @ S3 ) ) ).

% snth_sset
thf(fact_934_cont__def,axiom,
    ( abstra4962253831105676628m_rule
    = ( abstra1216198984563783442m_rule
      @ ^ [X12: produc164195504107695125m_rule,X23: fset_A5396818216090053761m_rule] : X23 ) ) ).

% cont_def
thf(fact_935_tree_Oroot__def,axiom,
    ( abstra3503897793822247108m_rule
    = ( abstra6581154509029876902m_rule
      @ ^ [X12: produc164195504107695125m_rule,X23: fset_A5396818216090053761m_rule] : X12 ) ) ).

% tree.root_def
thf(fact_936_snth__sset__smerge,axiom,
    ! [Ss3: stream8183306005803988909ist_fm,N: nat,M2: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ ( snth_s4276058539312847179ist_fm @ Ss3 @ N ) @ M2 ) @ ( sset_P3431374123360452149ist_fm @ ( smerge8067294617398470191ist_fm @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_937_snth__sset__smerge,axiom,
    ! [Ss3: stream_stream_fm,N: nat,M2: nat] : ( member_fm @ ( snth_fm @ ( snth_stream_fm @ Ss3 @ N ) @ M2 ) @ ( sset_fm @ ( smerge_fm @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_938_snth__sset__smerge,axiom,
    ! [Ss3: stream_stream_nat,N: nat,M2: nat] : ( member_nat @ ( snth_nat @ ( snth_stream_nat @ Ss3 @ N ) @ M2 ) @ ( sset_nat @ ( smerge_nat @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_939_snth__sset__smerge,axiom,
    ! [Ss3: stream_stream_rule,N: nat,M2: nat] : ( member_rule @ ( snth_rule @ ( snth_stream_rule @ Ss3 @ N ) @ M2 ) @ ( sset_rule @ ( smerge_rule @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_940_apfst__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,X2: produc164195504107695125m_rule] :
      ( ( produc6153989603652719015ist_fm @ F2 @ ( produc2413407552729308604ist_fm @ G @ X2 ) )
      = ( produc381145313068854617ist_fm @ ( F2 @ ( produc6879501374131015971m_rule @ X2 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X2 ) ) ) ) ).

% apfst_apsnd
thf(fact_941_apfst__apsnd,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( produc3731403639160201281m_rule @ F2 @ ( produc3063135074001183255ist_fm @ G @ X2 ) )
      = ( produc491467635432902671m_rule @ ( F2 @ ( produc6879501374131015971m_rule @ X2 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X2 ) ) ) ) ).

% apfst_apsnd
thf(fact_942_apsnd__apfst,axiom,
    ! [F2: rule > list_fm,G: produc1996495991257130529ist_fm > list_fm,X2: produc164195504107695125m_rule] :
      ( ( produc7464737497110386437ist_fm @ F2 @ ( produc7276395001879883020m_rule @ G @ X2 ) )
      = ( produc381145313068854617ist_fm @ ( G @ ( produc6879501374131015971m_rule @ X2 ) ) @ ( F2 @ ( produc7165828336582415457m_rule @ X2 ) ) ) ) ).

% apsnd_apfst
thf(fact_943_apsnd__apfst,axiom,
    ! [F2: rule > rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc164195504107695125m_rule] :
      ( ( produc3063135074001183255ist_fm @ F2 @ ( produc3731403639160201281m_rule @ G @ X2 ) )
      = ( produc491467635432902671m_rule @ ( G @ ( produc6879501374131015971m_rule @ X2 ) ) @ ( F2 @ ( produc7165828336582415457m_rule @ X2 ) ) ) ) ).

% apsnd_apfst
thf(fact_944_sfilter__streams,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,Omega: stream8299795917829157543ist_fm,A: set_Pr7058068377845519745ist_fm] :
      ( ( linear1269638131612670621ist_fm @ ( linear1231927432584843558ist_fm @ ( linear7000341289576568917ist_fm @ P ) ) @ Omega )
     => ( ( member5006235884692811216ist_fm @ Omega @ ( stream8099348070969791641ist_fm @ A ) )
       => ( member5006235884692811216ist_fm @ ( sfilte646686417537525057ist_fm @ P @ Omega )
          @ ( stream8099348070969791641ist_fm
            @ ( collec7231575776666803724ist_fm
              @ ^ [X4: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X4 @ A )
                  & ( P @ X4 ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_945_sfilter__streams,axiom,
    ! [P: fm > $o,Omega: stream_fm,A: set_fm] :
      ( ( linear3923485788533267412alw_fm @ ( linear5758214424494685963_ev_fm @ ( linear2965097450329772060lds_fm @ P ) ) @ Omega )
     => ( ( member_stream_fm @ Omega @ ( streams_fm @ A ) )
       => ( member_stream_fm @ ( sfilter_fm @ P @ Omega )
          @ ( streams_fm
            @ ( collect_fm
              @ ^ [X4: fm] :
                  ( ( member_fm @ X4 @ A )
                  & ( P @ X4 ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_946_sfilter__streams,axiom,
    ! [P: nat > $o,Omega: stream_nat,A: set_nat] :
      ( ( linear1448867875752120916lw_nat @ ( linear5502553777413005579ev_nat @ ( linear3087052677352696476ds_nat @ P ) ) @ Omega )
     => ( ( member_stream_nat @ Omega @ ( streams_nat @ A ) )
       => ( member_stream_nat @ ( sfilter_nat @ P @ Omega )
          @ ( streams_nat
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( member_nat @ X4 @ A )
                  & ( P @ X4 ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_947_sfilter__streams,axiom,
    ! [P: produc164195504107695125m_rule > $o,Omega: stream727092118206550309m_rule,A: set_Pr1008144964186165195m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ ( linear7082939723987557601m_rule @ P ) ) @ Omega )
     => ( ( member6117863459850561468m_rule @ Omega @ ( stream1749396743056113565m_rule @ A ) )
       => ( member6117863459850561468m_rule @ ( sfilte7579662470208054517m_rule @ P @ Omega )
          @ ( stream1749396743056113565m_rule
            @ ( collec2069506698588392810m_rule
              @ ^ [X4: produc164195504107695125m_rule] :
                  ( ( member4220325220686508332m_rule @ X4 @ A )
                  & ( P @ X4 ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_948_sfilter__streams,axiom,
    ! [P: rule > $o,Omega: stream_rule,A: set_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ ( linear4670012741531116049s_rule @ P ) ) @ Omega )
     => ( ( member_stream_rule @ Omega @ ( streams_rule @ A ) )
       => ( member_stream_rule @ ( sfilter_rule @ P @ Omega )
          @ ( streams_rule
            @ ( collect_rule
              @ ^ [X4: rule] :
                  ( ( member_rule @ X4 @ A )
                  & ( P @ X4 ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_949_streams__UNIV,axiom,
    ( ( stream8099348070969791641ist_fm @ top_to728902761071950545ist_fm )
    = top_to3943610876133046487ist_fm ) ).

% streams_UNIV
thf(fact_950_streams__UNIV,axiom,
    ( ( streams_rule @ top_top_set_rule )
    = top_to3705917391389534779m_rule ) ).

% streams_UNIV
thf(fact_951_streams__UNIV,axiom,
    ( ( streams_nat @ top_top_set_nat )
    = top_to7548458143485696966am_nat ) ).

% streams_UNIV
thf(fact_952_apfst__conv,axiom,
    ! [F2: list_fm > list_fm,X2: list_fm,Y: list_fm] :
      ( ( produc1130293176790411102ist_fm @ F2 @ ( produc381145313068854617ist_fm @ X2 @ Y ) )
      = ( produc381145313068854617ist_fm @ ( F2 @ X2 ) @ Y ) ) ).

% apfst_conv
thf(fact_953_apfst__conv,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc1996495991257130529ist_fm,Y: rule] :
      ( ( produc3731403639160201281m_rule @ F2 @ ( produc491467635432902671m_rule @ X2 @ Y ) )
      = ( produc491467635432902671m_rule @ ( F2 @ X2 ) @ Y ) ) ).

% apfst_conv
thf(fact_954_fst__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F2 @ X2 ) )
      = ( F2 @ ( produc6879501374131015971m_rule @ X2 ) ) ) ).

% fst_apfst
thf(fact_955_snd__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X2: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F2 @ X2 ) )
      = ( produc7165828336582415457m_rule @ X2 ) ) ).

% snd_apfst
thf(fact_956_snd__comp__apfst,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F2 ) )
      = produc7165828336582415457m_rule ) ).

% snd_comp_apfst
thf(fact_957_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_958_streams__mono2,axiom,
    ! [S6: set_rule,T9: set_rule] :
      ( ( ord_less_eq_set_rule @ S6 @ T9 )
     => ( ord_le5832554575816053739m_rule @ ( streams_rule @ S6 ) @ ( streams_rule @ T9 ) ) ) ).

% streams_mono2
thf(fact_959_streams__mono,axiom,
    ! [S3: stream_rule,A: set_rule,B: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A ) )
     => ( ( ord_less_eq_set_rule @ A @ B )
       => ( member_stream_rule @ S3 @ ( streams_rule @ B ) ) ) ) ).

% streams_mono
thf(fact_960_streams__stl,axiom,
    ! [S3: stream727092118206550309m_rule,A: set_Pr1008144964186165195m_rule] :
      ( ( member6117863459850561468m_rule @ S3 @ ( stream1749396743056113565m_rule @ A ) )
     => ( member6117863459850561468m_rule @ ( stl_Pr950425576149878629m_rule @ S3 ) @ ( stream1749396743056113565m_rule @ A ) ) ) ).

% streams_stl
thf(fact_961_streams__stl,axiom,
    ! [S3: stream_rule,A: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A ) )
     => ( member_stream_rule @ ( stl_rule @ S3 ) @ ( streams_rule @ A ) ) ) ).

% streams_stl
thf(fact_962_streams__shd,axiom,
    ! [S3: stream8299795917829157543ist_fm,A: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ S3 @ ( stream8099348070969791641ist_fm @ A ) )
     => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ S3 ) @ A ) ) ).

% streams_shd
thf(fact_963_streams__shd,axiom,
    ! [S3: stream_fm,A: set_fm] :
      ( ( member_stream_fm @ S3 @ ( streams_fm @ A ) )
     => ( member_fm @ ( shd_fm @ S3 ) @ A ) ) ).

% streams_shd
thf(fact_964_streams__shd,axiom,
    ! [S3: stream_nat,A: set_nat] :
      ( ( member_stream_nat @ S3 @ ( streams_nat @ A ) )
     => ( member_nat @ ( shd_nat @ S3 ) @ A ) ) ).

% streams_shd
thf(fact_965_streams__shd,axiom,
    ! [S3: stream727092118206550309m_rule,A: set_Pr1008144964186165195m_rule] :
      ( ( member6117863459850561468m_rule @ S3 @ ( stream1749396743056113565m_rule @ A ) )
     => ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ S3 ) @ A ) ) ).

% streams_shd
thf(fact_966_streams__shd,axiom,
    ! [S3: stream_rule,A: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A ) )
     => ( member_rule @ ( shd_rule @ S3 ) @ A ) ) ).

% streams_shd
thf(fact_967_streams_Ocases,axiom,
    ! [A2: stream_rule,A: set_rule] :
      ( ( member_stream_rule @ A2 @ ( streams_rule @ A ) )
     => ~ ! [A5: rule,S4: stream_rule] :
            ( ( A2
              = ( sCons_rule @ A5 @ S4 ) )
           => ( ( member_rule @ A5 @ A )
             => ~ ( member_stream_rule @ S4 @ ( streams_rule @ A ) ) ) ) ) ).

% streams.cases
thf(fact_968_streams_Ocases,axiom,
    ! [A2: stream8299795917829157543ist_fm,A: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ A2 @ ( stream8099348070969791641ist_fm @ A ) )
     => ~ ! [A5: produc1996495991257130529ist_fm,S4: stream8299795917829157543ist_fm] :
            ( ( A2
              = ( sCons_307821682964077496ist_fm @ A5 @ S4 ) )
           => ( ( member8102475879199740618ist_fm @ A5 @ A )
             => ~ ( member5006235884692811216ist_fm @ S4 @ ( stream8099348070969791641ist_fm @ A ) ) ) ) ) ).

% streams.cases
thf(fact_969_streams_Ocases,axiom,
    ! [A2: stream_fm,A: set_fm] :
      ( ( member_stream_fm @ A2 @ ( streams_fm @ A ) )
     => ~ ! [A5: fm,S4: stream_fm] :
            ( ( A2
              = ( sCons_fm @ A5 @ S4 ) )
           => ( ( member_fm @ A5 @ A )
             => ~ ( member_stream_fm @ S4 @ ( streams_fm @ A ) ) ) ) ) ).

% streams.cases
thf(fact_970_streams_Ocases,axiom,
    ! [A2: stream_nat,A: set_nat] :
      ( ( member_stream_nat @ A2 @ ( streams_nat @ A ) )
     => ~ ! [A5: nat,S4: stream_nat] :
            ( ( A2
              = ( sCons_nat @ A5 @ S4 ) )
           => ( ( member_nat @ A5 @ A )
             => ~ ( member_stream_nat @ S4 @ ( streams_nat @ A ) ) ) ) ) ).

% streams.cases
thf(fact_971_streams_Osimps,axiom,
    ! [A2: stream_rule,A: set_rule] :
      ( ( member_stream_rule @ A2 @ ( streams_rule @ A ) )
      = ( ? [A3: rule,S: stream_rule] :
            ( ( A2
              = ( sCons_rule @ A3 @ S ) )
            & ( member_rule @ A3 @ A )
            & ( member_stream_rule @ S @ ( streams_rule @ A ) ) ) ) ) ).

% streams.simps
thf(fact_972_streams_Osimps,axiom,
    ! [A2: stream8299795917829157543ist_fm,A: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ A2 @ ( stream8099348070969791641ist_fm @ A ) )
      = ( ? [A3: produc1996495991257130529ist_fm,S: stream8299795917829157543ist_fm] :
            ( ( A2
              = ( sCons_307821682964077496ist_fm @ A3 @ S ) )
            & ( member8102475879199740618ist_fm @ A3 @ A )
            & ( member5006235884692811216ist_fm @ S @ ( stream8099348070969791641ist_fm @ A ) ) ) ) ) ).

% streams.simps
thf(fact_973_streams_Osimps,axiom,
    ! [A2: stream_fm,A: set_fm] :
      ( ( member_stream_fm @ A2 @ ( streams_fm @ A ) )
      = ( ? [A3: fm,S: stream_fm] :
            ( ( A2
              = ( sCons_fm @ A3 @ S ) )
            & ( member_fm @ A3 @ A )
            & ( member_stream_fm @ S @ ( streams_fm @ A ) ) ) ) ) ).

% streams.simps
thf(fact_974_streams_Osimps,axiom,
    ! [A2: stream_nat,A: set_nat] :
      ( ( member_stream_nat @ A2 @ ( streams_nat @ A ) )
      = ( ? [A3: nat,S: stream_nat] :
            ( ( A2
              = ( sCons_nat @ A3 @ S ) )
            & ( member_nat @ A3 @ A )
            & ( member_stream_nat @ S @ ( streams_nat @ A ) ) ) ) ) ).

% streams.simps
thf(fact_975_streams_Ocoinduct,axiom,
    ! [X3: stream_rule > $o,X2: stream_rule,A: set_rule] :
      ( ( X3 @ X2 )
     => ( ! [X: stream_rule] :
            ( ( X3 @ X )
           => ? [A8: rule,S10: stream_rule] :
                ( ( X
                  = ( sCons_rule @ A8 @ S10 ) )
                & ( member_rule @ A8 @ A )
                & ( ( X3 @ S10 )
                  | ( member_stream_rule @ S10 @ ( streams_rule @ A ) ) ) ) )
       => ( member_stream_rule @ X2 @ ( streams_rule @ A ) ) ) ) ).

% streams.coinduct
thf(fact_976_streams_Ocoinduct,axiom,
    ! [X3: stream8299795917829157543ist_fm > $o,X2: stream8299795917829157543ist_fm,A: set_Pr7058068377845519745ist_fm] :
      ( ( X3 @ X2 )
     => ( ! [X: stream8299795917829157543ist_fm] :
            ( ( X3 @ X )
           => ? [A8: produc1996495991257130529ist_fm,S10: stream8299795917829157543ist_fm] :
                ( ( X
                  = ( sCons_307821682964077496ist_fm @ A8 @ S10 ) )
                & ( member8102475879199740618ist_fm @ A8 @ A )
                & ( ( X3 @ S10 )
                  | ( member5006235884692811216ist_fm @ S10 @ ( stream8099348070969791641ist_fm @ A ) ) ) ) )
       => ( member5006235884692811216ist_fm @ X2 @ ( stream8099348070969791641ist_fm @ A ) ) ) ) ).

% streams.coinduct
thf(fact_977_streams_Ocoinduct,axiom,
    ! [X3: stream_fm > $o,X2: stream_fm,A: set_fm] :
      ( ( X3 @ X2 )
     => ( ! [X: stream_fm] :
            ( ( X3 @ X )
           => ? [A8: fm,S10: stream_fm] :
                ( ( X
                  = ( sCons_fm @ A8 @ S10 ) )
                & ( member_fm @ A8 @ A )
                & ( ( X3 @ S10 )
                  | ( member_stream_fm @ S10 @ ( streams_fm @ A ) ) ) ) )
       => ( member_stream_fm @ X2 @ ( streams_fm @ A ) ) ) ) ).

% streams.coinduct
thf(fact_978_streams_Ocoinduct,axiom,
    ! [X3: stream_nat > $o,X2: stream_nat,A: set_nat] :
      ( ( X3 @ X2 )
     => ( ! [X: stream_nat] :
            ( ( X3 @ X )
           => ? [A8: nat,S10: stream_nat] :
                ( ( X
                  = ( sCons_nat @ A8 @ S10 ) )
                & ( member_nat @ A8 @ A )
                & ( ( X3 @ S10 )
                  | ( member_stream_nat @ S10 @ ( streams_nat @ A ) ) ) ) )
       => ( member_stream_nat @ X2 @ ( streams_nat @ A ) ) ) ) ).

% streams.coinduct
thf(fact_979_streams__Stream,axiom,
    ! [X2: rule,S3: stream_rule,A: set_rule] :
      ( ( member_stream_rule @ ( sCons_rule @ X2 @ S3 ) @ ( streams_rule @ A ) )
      = ( ( member_rule @ X2 @ A )
        & ( member_stream_rule @ S3 @ ( streams_rule @ A ) ) ) ) ).

% streams_Stream
thf(fact_980_streams__Stream,axiom,
    ! [X2: produc1996495991257130529ist_fm,S3: stream8299795917829157543ist_fm,A: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ ( sCons_307821682964077496ist_fm @ X2 @ S3 ) @ ( stream8099348070969791641ist_fm @ A ) )
      = ( ( member8102475879199740618ist_fm @ X2 @ A )
        & ( member5006235884692811216ist_fm @ S3 @ ( stream8099348070969791641ist_fm @ A ) ) ) ) ).

% streams_Stream
thf(fact_981_streams__Stream,axiom,
    ! [X2: fm,S3: stream_fm,A: set_fm] :
      ( ( member_stream_fm @ ( sCons_fm @ X2 @ S3 ) @ ( streams_fm @ A ) )
      = ( ( member_fm @ X2 @ A )
        & ( member_stream_fm @ S3 @ ( streams_fm @ A ) ) ) ) ).

% streams_Stream
thf(fact_982_streams__Stream,axiom,
    ! [X2: nat,S3: stream_nat,A: set_nat] :
      ( ( member_stream_nat @ ( sCons_nat @ X2 @ S3 ) @ ( streams_nat @ A ) )
      = ( ( member_nat @ X2 @ A )
        & ( member_stream_nat @ S3 @ ( streams_nat @ A ) ) ) ) ).

% streams_Stream
thf(fact_983_snth__in,axiom,
    ! [S3: stream_rule,X3: set_rule,N: nat] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ X3 ) )
     => ( member_rule @ ( snth_rule @ S3 @ N ) @ X3 ) ) ).

% snth_in
thf(fact_984_snth__in,axiom,
    ! [S3: stream8299795917829157543ist_fm,X3: set_Pr7058068377845519745ist_fm,N: nat] :
      ( ( member5006235884692811216ist_fm @ S3 @ ( stream8099348070969791641ist_fm @ X3 ) )
     => ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S3 @ N ) @ X3 ) ) ).

% snth_in
thf(fact_985_snth__in,axiom,
    ! [S3: stream_fm,X3: set_fm,N: nat] :
      ( ( member_stream_fm @ S3 @ ( streams_fm @ X3 ) )
     => ( member_fm @ ( snth_fm @ S3 @ N ) @ X3 ) ) ).

% snth_in
thf(fact_986_snth__in,axiom,
    ! [S3: stream_nat,X3: set_nat,N: nat] :
      ( ( member_stream_nat @ S3 @ ( streams_nat @ X3 ) )
     => ( member_nat @ ( snth_nat @ S3 @ N ) @ X3 ) ) ).

% snth_in
thf(fact_987_streams__iff__snth,axiom,
    ! [S3: stream_rule,X3: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ X3 ) )
      = ( ! [N3: nat] : ( member_rule @ ( snth_rule @ S3 @ N3 ) @ X3 ) ) ) ).

% streams_iff_snth
thf(fact_988_streams__iff__snth,axiom,
    ! [S3: stream8299795917829157543ist_fm,X3: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ S3 @ ( stream8099348070969791641ist_fm @ X3 ) )
      = ( ! [N3: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S3 @ N3 ) @ X3 ) ) ) ).

% streams_iff_snth
thf(fact_989_streams__iff__snth,axiom,
    ! [S3: stream_fm,X3: set_fm] :
      ( ( member_stream_fm @ S3 @ ( streams_fm @ X3 ) )
      = ( ! [N3: nat] : ( member_fm @ ( snth_fm @ S3 @ N3 ) @ X3 ) ) ) ).

% streams_iff_snth
thf(fact_990_streams__iff__snth,axiom,
    ! [S3: stream_nat,X3: set_nat] :
      ( ( member_stream_nat @ S3 @ ( streams_nat @ X3 ) )
      = ( ! [N3: nat] : ( member_nat @ ( snth_nat @ S3 @ N3 ) @ X3 ) ) ) ).

% streams_iff_snth
thf(fact_991_sset__streams,axiom,
    ! [S3: stream_rule,A: set_rule] :
      ( ( ord_less_eq_set_rule @ ( sset_rule @ S3 ) @ A )
     => ( member_stream_rule @ S3 @ ( streams_rule @ A ) ) ) ).

% sset_streams
thf(fact_992_streams__sset,axiom,
    ! [S3: stream_rule,A: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A ) )
     => ( ord_less_eq_set_rule @ ( sset_rule @ S3 ) @ A ) ) ).

% streams_sset
thf(fact_993_streams__iff__sset,axiom,
    ! [S3: stream_rule,A: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A ) )
      = ( ord_less_eq_set_rule @ ( sset_rule @ S3 ) @ A ) ) ).

% streams_iff_sset
thf(fact_994_streamsE,axiom,
    ! [S3: stream8299795917829157543ist_fm,A: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ S3 @ ( stream8099348070969791641ist_fm @ A ) )
     => ~ ( ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ S3 ) @ A )
         => ~ ( member5006235884692811216ist_fm @ ( stl_Pr5027778045650968273ist_fm @ S3 ) @ ( stream8099348070969791641ist_fm @ A ) ) ) ) ).

% streamsE
thf(fact_995_streamsE,axiom,
    ! [S3: stream_fm,A: set_fm] :
      ( ( member_stream_fm @ S3 @ ( streams_fm @ A ) )
     => ~ ( ( member_fm @ ( shd_fm @ S3 ) @ A )
         => ~ ( member_stream_fm @ ( stl_fm @ S3 ) @ ( streams_fm @ A ) ) ) ) ).

% streamsE
thf(fact_996_streamsE,axiom,
    ! [S3: stream_nat,A: set_nat] :
      ( ( member_stream_nat @ S3 @ ( streams_nat @ A ) )
     => ~ ( ( member_nat @ ( shd_nat @ S3 ) @ A )
         => ~ ( member_stream_nat @ ( stl_nat @ S3 ) @ ( streams_nat @ A ) ) ) ) ).

% streamsE
thf(fact_997_streamsE,axiom,
    ! [S3: stream727092118206550309m_rule,A: set_Pr1008144964186165195m_rule] :
      ( ( member6117863459850561468m_rule @ S3 @ ( stream1749396743056113565m_rule @ A ) )
     => ~ ( ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ S3 ) @ A )
         => ~ ( member6117863459850561468m_rule @ ( stl_Pr950425576149878629m_rule @ S3 ) @ ( stream1749396743056113565m_rule @ A ) ) ) ) ).

% streamsE
thf(fact_998_streamsE,axiom,
    ! [S3: stream_rule,A: set_rule] :
      ( ( member_stream_rule @ S3 @ ( streams_rule @ A ) )
     => ~ ( ( member_rule @ ( shd_rule @ S3 ) @ A )
         => ~ ( member_stream_rule @ ( stl_rule @ S3 ) @ ( streams_rule @ A ) ) ) ) ).

% streamsE
thf(fact_999_in__streams,axiom,
    ! [S3: stream8299795917829157543ist_fm,S6: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ ( stl_Pr5027778045650968273ist_fm @ S3 ) @ ( stream8099348070969791641ist_fm @ S6 ) )
     => ( ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ S3 ) @ S6 )
       => ( member5006235884692811216ist_fm @ S3 @ ( stream8099348070969791641ist_fm @ S6 ) ) ) ) ).

% in_streams
thf(fact_1000_in__streams,axiom,
    ! [S3: stream_fm,S6: set_fm] :
      ( ( member_stream_fm @ ( stl_fm @ S3 ) @ ( streams_fm @ S6 ) )
     => ( ( member_fm @ ( shd_fm @ S3 ) @ S6 )
       => ( member_stream_fm @ S3 @ ( streams_fm @ S6 ) ) ) ) ).

% in_streams
thf(fact_1001_in__streams,axiom,
    ! [S3: stream_nat,S6: set_nat] :
      ( ( member_stream_nat @ ( stl_nat @ S3 ) @ ( streams_nat @ S6 ) )
     => ( ( member_nat @ ( shd_nat @ S3 ) @ S6 )
       => ( member_stream_nat @ S3 @ ( streams_nat @ S6 ) ) ) ) ).

% in_streams
thf(fact_1002_in__streams,axiom,
    ! [S3: stream727092118206550309m_rule,S6: set_Pr1008144964186165195m_rule] :
      ( ( member6117863459850561468m_rule @ ( stl_Pr950425576149878629m_rule @ S3 ) @ ( stream1749396743056113565m_rule @ S6 ) )
     => ( ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ S3 ) @ S6 )
       => ( member6117863459850561468m_rule @ S3 @ ( stream1749396743056113565m_rule @ S6 ) ) ) ) ).

% in_streams
thf(fact_1003_in__streams,axiom,
    ! [S3: stream_rule,S6: set_rule] :
      ( ( member_stream_rule @ ( stl_rule @ S3 ) @ ( streams_rule @ S6 ) )
     => ( ( member_rule @ ( shd_rule @ S3 ) @ S6 )
       => ( member_stream_rule @ S3 @ ( streams_rule @ S6 ) ) ) ) ).

% in_streams
thf(fact_1004_apfst__convE,axiom,
    ! [Q2: produc1996495991257130529ist_fm,F2: list_fm > list_fm,P2: produc1996495991257130529ist_fm] :
      ( ( Q2
        = ( produc1130293176790411102ist_fm @ F2 @ P2 ) )
     => ~ ! [X: list_fm,Y4: list_fm] :
            ( ( P2
              = ( produc381145313068854617ist_fm @ X @ Y4 ) )
           => ( Q2
             != ( produc381145313068854617ist_fm @ ( F2 @ X ) @ Y4 ) ) ) ) ).

% apfst_convE
thf(fact_1005_apfst__convE,axiom,
    ! [Q2: produc164195504107695125m_rule,F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,P2: produc164195504107695125m_rule] :
      ( ( Q2
        = ( produc3731403639160201281m_rule @ F2 @ P2 ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y4: rule] :
            ( ( P2
              = ( produc491467635432902671m_rule @ X @ Y4 ) )
           => ( Q2
             != ( produc491467635432902671m_rule @ ( F2 @ X ) @ Y4 ) ) ) ) ).

% apfst_convE
thf(fact_1006_Fair__Stream_Ofair__def,axiom,
    ( fair_fair_rule
    = ( ^ [S: stream_rule] :
        ! [X4: rule] :
          ( ( member_rule @ X4 @ ( sset_rule @ S ) )
         => ! [M: nat] :
            ? [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
              & ( ( snth_rule @ S @ N3 )
                = X4 ) ) ) ) ) ).

% Fair_Stream.fair_def
thf(fact_1007_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_1008_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_1009_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_1010_Collect__restrict,axiom,
    ! [X3: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X4: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X4 @ X3 )
            & ( P @ X4 ) ) )
      @ X3 ) ).

% Collect_restrict
thf(fact_1011_Collect__restrict,axiom,
    ! [X3: set_fm,P: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X4: fm] :
            ( ( member_fm @ X4 @ X3 )
            & ( P @ X4 ) ) )
      @ X3 ) ).

% Collect_restrict
thf(fact_1012_Collect__restrict,axiom,
    ! [X3: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ X3 )
            & ( P @ X4 ) ) )
      @ X3 ) ).

% Collect_restrict
thf(fact_1013_Collect__restrict,axiom,
    ! [X3: set_rule,P: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X4: rule] :
            ( ( member_rule @ X4 @ X3 )
            & ( P @ X4 ) ) )
      @ X3 ) ).

% Collect_restrict
thf(fact_1014_prop__restrict,axiom,
    ! [X2: produc1996495991257130529ist_fm,Z4: set_Pr7058068377845519745ist_fm,X3: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X2 @ Z4 )
     => ( ( ord_le2055072402069232929ist_fm @ Z4
          @ ( collec7231575776666803724ist_fm
            @ ^ [X4: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X4 @ X3 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_1015_prop__restrict,axiom,
    ! [X2: fm,Z4: set_fm,X3: set_fm,P: fm > $o] :
      ( ( member_fm @ X2 @ Z4 )
     => ( ( ord_less_eq_set_fm @ Z4
          @ ( collect_fm
            @ ^ [X4: fm] :
                ( ( member_fm @ X4 @ X3 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_1016_prop__restrict,axiom,
    ! [X2: nat,Z4: set_nat,X3: set_nat,P: nat > $o] :
      ( ( member_nat @ X2 @ Z4 )
     => ( ( ord_less_eq_set_nat @ Z4
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ X3 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_1017_prop__restrict,axiom,
    ! [X2: rule,Z4: set_rule,X3: set_rule,P: rule > $o] :
      ( ( member_rule @ X2 @ Z4 )
     => ( ( ord_less_eq_set_rule @ Z4
          @ ( collect_rule
            @ ^ [X4: rule] :
                ( ( member_rule @ X4 @ X3 )
                & ( P @ X4 ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_1018_eff__preserves__Pre_I2_J,axiom,
    ! [A: list_fm,B: list_fm,R3: rule,Ss3: fset_P661503646757059847ist_fm,A6: list_fm,B6: list_fm,P: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A @ B ) @ R3 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A @ B ) @ R3 ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss3 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B6 ) @ Ss3 )
       => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B ) )
         => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B6 ) ) ) ) ) ).

% eff_preserves_Pre(2)
thf(fact_1019_eff__preserves__Pre_I1_J,axiom,
    ! [A: list_fm,B: list_fm,R3: rule,Ss3: fset_P661503646757059847ist_fm,A6: list_fm,B6: list_fm,P: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A @ B ) @ R3 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A @ B ) @ R3 ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss3 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B6 ) @ Ss3 )
       => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A ) )
         => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A6 ) ) ) ) ) ).

% eff_preserves_Pre(1)
thf(fact_1020_ssubst__Pair__rhs,axiom,
    ! [R3: list_fm,S3: list_fm,R: set_Pr7058068377845519745ist_fm,S5: list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R3 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R3 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1021_ssubst__Pair__rhs,axiom,
    ! [R3: produc1996495991257130529ist_fm,S3: rule,R: set_Pr1008144964186165195m_rule,S5: rule] :
      ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R3 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R3 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1022_fset__of__list__subset,axiom,
    ! [Xs2: list_fm,Ys: list_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Ys ) )
     => ( ord_less_eq_fset_fm @ ( fset_of_list_fm @ Xs2 ) @ ( fset_of_list_fm @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_1023_fset__of__list__subset,axiom,
    ! [Xs2: list_rule,Ys: list_rule] :
      ( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs2 ) @ ( set_rule2 @ Ys ) )
     => ( ord_le7171078382243635157t_rule @ ( fset_of_list_rule @ Xs2 ) @ ( fset_of_list_rule @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_1024_snd__fst__flip,axiom,
    ( produc4588648349897876871ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X4: list_fm,Y5: list_fm] : ( produc381145313068854617ist_fm @ Y5 @ X4 ) ) ) ) ).

% snd_fst_flip
thf(fact_1025_snd__fst__flip,axiom,
    ( produc4514744107061996309ist_fm
    = ( comp_P2707267495825058490ist_fm @ produc6879501374131015971m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X4: rule,Y5: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y5 @ X4 ) ) ) ) ).

% snd_fst_flip
thf(fact_1026_snd__fst__flip,axiom,
    ( produc7165828336582415457m_rule
    = ( comp_P3413117018928051920m_rule @ produc4228417144610596823ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X4: produc1996495991257130529ist_fm,Y5: rule] : ( produc7063755442767259331ist_fm @ Y5 @ X4 ) ) ) ) ).

% snd_fst_flip
thf(fact_1027_fst__snd__flip,axiom,
    ( produc1501393135466168645ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X4: list_fm,Y5: list_fm] : ( produc381145313068854617ist_fm @ Y5 @ X4 ) ) ) ) ).

% fst_snd_flip
thf(fact_1028_fst__snd__flip,axiom,
    ( produc6879501374131015971m_rule
    = ( comp_P6646430376872094402m_rule @ produc4514744107061996309ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X4: produc1996495991257130529ist_fm,Y5: rule] : ( produc7063755442767259331ist_fm @ Y5 @ X4 ) ) ) ) ).

% fst_snd_flip
thf(fact_1029_fst__snd__flip,axiom,
    ( produc4228417144610596823ist_fm
    = ( comp_P4888803072877744336ist_fm @ produc7165828336582415457m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X4: rule,Y5: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y5 @ X4 ) ) ) ) ).

% fst_snd_flip
thf(fact_1030_stake__pos__minWait,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm,R3: rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( ord_less_nat
          @ ( abstra738319790293016300ist_fm
            @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 )
          @ ( abstract_pos_rule @ Rs @ R3 ) )
       => ( ( member_rule @ R3 @ ( sset_rule @ rules ) )
         => ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
           => ( ( abstract_pos_rule
                @ ( stl_rule
                  @ ( abstra2363873759530810705ist_fm
                    @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ Rs
                    @ S3 ) )
                @ R3 )
              = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R3 )
                @ ( suc
                  @ ( abstra738319790293016300ist_fm
                    @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R2 @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ Rs
                    @ S3 ) ) ) ) ) ) ) ) ).

% stake_pos_minWait
thf(fact_1031_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S3 ) )
      = ( sdrop_7224736112439592940m_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1032_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( stl_rule @ ( sdrop_rule @ N @ S3 ) )
      = ( sdrop_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1033_top_Onot__eq__extremum,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] :
      ( ( A2 != top_to728902761071950545ist_fm )
      = ( ord_le5784518313675440685ist_fm @ A2 @ top_to728902761071950545ist_fm ) ) ).

% top.not_eq_extremum
thf(fact_1034_top_Onot__eq__extremum,axiom,
    ! [A2: set_rule] :
      ( ( A2 != top_top_set_rule )
      = ( ord_less_set_rule @ A2 @ top_top_set_rule ) ) ).

% top.not_eq_extremum
thf(fact_1035_top_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != top_top_set_nat )
      = ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).

% top.not_eq_extremum
thf(fact_1036_top_Oextremum__strict,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] :
      ~ ( ord_le5784518313675440685ist_fm @ top_to728902761071950545ist_fm @ A2 ) ).

% top.extremum_strict
thf(fact_1037_top_Oextremum__strict,axiom,
    ! [A2: set_rule] :
      ~ ( ord_less_set_rule @ top_top_set_rule @ A2 ) ).

% top.extremum_strict
thf(fact_1038_top_Oextremum__strict,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).

% top.extremum_strict
thf(fact_1039_order__le__imp__less__or__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_nat @ X2 @ Y )
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1040_order__le__imp__less__or__eq,axiom,
    ! [X2: set_rule,Y: set_rule] :
      ( ( ord_less_eq_set_rule @ X2 @ Y )
     => ( ( ord_less_set_rule @ X2 @ Y )
        | ( X2 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1041_linorder__le__less__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
      | ( ord_less_nat @ Y @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_1042_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1043_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > set_rule,C: set_rule] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_rule @ ( F2 @ B2 ) @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_set_rule @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1044_order__less__le__subst1,axiom,
    ! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1045_order__less__le__subst1,axiom,
    ! [A2: set_rule,F2: nat > set_rule,B2: nat,C: nat] :
      ( ( ord_less_set_rule @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_set_rule @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1046_order__less__le__subst1,axiom,
    ! [A2: nat,F2: set_rule > nat,B2: set_rule,C: set_rule] :
      ( ( ord_less_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_rule @ B2 @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1047_order__less__le__subst1,axiom,
    ! [A2: set_rule,F2: set_rule > set_rule,B2: set_rule,C: set_rule] :
      ( ( ord_less_set_rule @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_rule @ B2 @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_set_rule @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1048_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1049_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_set_rule @ ( F2 @ B2 ) @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_set_rule @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1050_order__le__less__subst2,axiom,
    ! [A2: set_rule,B2: set_rule,F2: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1051_order__le__less__subst2,axiom,
    ! [A2: set_rule,B2: set_rule,F2: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( ord_less_set_rule @ ( F2 @ B2 ) @ C )
       => ( ! [X: set_rule,Y4: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y4 )
             => ( ord_less_eq_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_set_rule @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1052_order__le__less__subst1,axiom,
    ! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1053_order__le__less__subst1,axiom,
    ! [A2: set_rule,F2: nat > set_rule,B2: nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_set_rule @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_set_rule @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1054_order__less__le__trans,axiom,
    ! [X2: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_nat @ X2 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_1055_order__less__le__trans,axiom,
    ! [X2: set_rule,Y: set_rule,Z3: set_rule] :
      ( ( ord_less_set_rule @ X2 @ Y )
     => ( ( ord_less_eq_set_rule @ Y @ Z3 )
       => ( ord_less_set_rule @ X2 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_1056_order__le__less__trans,axiom,
    ! [X2: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X2 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_1057_order__le__less__trans,axiom,
    ! [X2: set_rule,Y: set_rule,Z3: set_rule] :
      ( ( ord_less_eq_set_rule @ X2 @ Y )
     => ( ( ord_less_set_rule @ Y @ Z3 )
       => ( ord_less_set_rule @ X2 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_1058_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1059_order__neq__le__trans,axiom,
    ! [A2: set_rule,B2: set_rule] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_set_rule @ A2 @ B2 )
       => ( ord_less_set_rule @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_1060_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1061_order__le__neq__trans,axiom,
    ! [A2: set_rule,B2: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_rule @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_1062_order__less__imp__le,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_1063_order__less__imp__le,axiom,
    ! [X2: set_rule,Y: set_rule] :
      ( ( ord_less_set_rule @ X2 @ Y )
     => ( ord_less_eq_set_rule @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_1064_linorder__not__less,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_1065_linorder__not__le,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
      = ( ord_less_nat @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_1066_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1067_order__less__le,axiom,
    ( ord_less_set_rule
    = ( ^ [X4: set_rule,Y5: set_rule] :
          ( ( ord_less_eq_set_rule @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1068_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1069_order__le__less,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [X4: set_rule,Y5: set_rule] :
          ( ( ord_less_set_rule @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1070_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1071_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_rule,A2: set_rule] :
      ( ( ord_less_set_rule @ B2 @ A2 )
     => ( ord_less_eq_set_rule @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_1072_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1073_order_Ostrict__implies__order,axiom,
    ! [A2: set_rule,B2: set_rule] :
      ( ( ord_less_set_rule @ A2 @ B2 )
     => ( ord_less_eq_set_rule @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_1074_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B5 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1075_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_rule
    = ( ^ [B5: set_rule,A3: set_rule] :
          ( ( ord_less_eq_set_rule @ B5 @ A3 )
          & ~ ( ord_less_eq_set_rule @ A3 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1076_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1077_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_rule,A2: set_rule,C: set_rule] :
      ( ( ord_less_set_rule @ B2 @ A2 )
     => ( ( ord_less_eq_set_rule @ C @ B2 )
       => ( ord_less_set_rule @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1078_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1079_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_rule,A2: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ B2 @ A2 )
     => ( ( ord_less_set_rule @ C @ B2 )
       => ( ord_less_set_rule @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1080_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B5 @ A3 )
          & ( A3 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1081_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_rule
    = ( ^ [B5: set_rule,A3: set_rule] :
          ( ( ord_less_eq_set_rule @ B5 @ A3 )
          & ( A3 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1082_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A3: nat] :
          ( ( ord_less_nat @ B5 @ A3 )
          | ( A3 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1083_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [B5: set_rule,A3: set_rule] :
          ( ( ord_less_set_rule @ B5 @ A3 )
          | ( A3 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1084_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A3 @ B5 )
          & ~ ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1085_order_Ostrict__iff__not,axiom,
    ( ord_less_set_rule
    = ( ^ [A3: set_rule,B5: set_rule] :
          ( ( ord_less_eq_set_rule @ A3 @ B5 )
          & ~ ( ord_less_eq_set_rule @ B5 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1086_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_1087_order_Ostrict__trans2,axiom,
    ! [A2: set_rule,B2: set_rule,C: set_rule] :
      ( ( ord_less_set_rule @ A2 @ B2 )
     => ( ( ord_less_eq_set_rule @ B2 @ C )
       => ( ord_less_set_rule @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_1088_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_1089_order_Ostrict__trans1,axiom,
    ! [A2: set_rule,B2: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B2 )
     => ( ( ord_less_set_rule @ B2 @ C )
       => ( ord_less_set_rule @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_1090_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A3 @ B5 )
          & ( A3 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1091_order_Ostrict__iff__order,axiom,
    ( ord_less_set_rule
    = ( ^ [A3: set_rule,B5: set_rule] :
          ( ( ord_less_eq_set_rule @ A3 @ B5 )
          & ( A3 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1092_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B5: nat] :
          ( ( ord_less_nat @ A3 @ B5 )
          | ( A3 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1093_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A3: set_rule,B5: set_rule] :
          ( ( ord_less_set_rule @ A3 @ B5 )
          | ( A3 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1094_not__le__imp__less,axiom,
    ! [Y: nat,X2: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X2 )
     => ( ord_less_nat @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_1095_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1096_less__le__not__le,axiom,
    ( ord_less_set_rule
    = ( ^ [X4: set_rule,Y5: set_rule] :
          ( ( ord_less_eq_set_rule @ X4 @ Y5 )
          & ~ ( ord_less_eq_set_rule @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1097_antisym__conv2,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_1098_antisym__conv2,axiom,
    ! [X2: set_rule,Y: set_rule] :
      ( ( ord_less_eq_set_rule @ X2 @ Y )
     => ( ( ~ ( ord_less_set_rule @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_1099_antisym__conv1,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_1100_antisym__conv1,axiom,
    ! [X2: set_rule,Y: set_rule] :
      ( ~ ( ord_less_set_rule @ X2 @ Y )
     => ( ( ord_less_eq_set_rule @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_1101_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_1102_nless__le,axiom,
    ! [A2: set_rule,B2: set_rule] :
      ( ( ~ ( ord_less_set_rule @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_set_rule @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_1103_leI,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ Y @ X2 ) ) ).

% leI
thf(fact_1104_leD,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ~ ( ord_less_nat @ X2 @ Y ) ) ).

% leD
thf(fact_1105_leD,axiom,
    ! [Y: set_rule,X2: set_rule] :
      ( ( ord_less_eq_set_rule @ Y @ X2 )
     => ~ ( ord_less_set_rule @ X2 @ Y ) ) ).

% leD
thf(fact_1106_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_1107_less__imp__neq,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( X2 != Y ) ) ).

% less_imp_neq
thf(fact_1108_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1109_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1110_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1111_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X )
             => ( P @ Y6 ) )
         => ( P @ X ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_1112_antisym__conv3,axiom,
    ! [Y: nat,X2: nat] :
      ( ~ ( ord_less_nat @ Y @ X2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv3
thf(fact_1113_linorder__cases,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y )
     => ( ( X2 != Y )
       => ( ord_less_nat @ Y @ X2 ) ) ) ).

% linorder_cases
thf(fact_1114_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1115_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1116_exists__least__iff,axiom,
    ( ( ^ [P6: nat > $o] :
        ? [X7: nat] : ( P6 @ X7 ) )
    = ( ^ [P3: nat > $o] :
        ? [N3: nat] :
          ( ( P3 @ N3 )
          & ! [M: nat] :
              ( ( ord_less_nat @ M @ N3 )
             => ~ ( P3 @ M ) ) ) ) ) ).

% exists_least_iff
thf(fact_1117_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
         => ( P @ A5 @ B4 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B4: nat] :
              ( ( P @ B4 @ A5 )
             => ( P @ A5 @ B4 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1118_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_1119_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y ) )
      = ( ( ord_less_nat @ Y @ X2 )
        | ( X2 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1120_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1121_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1122_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1123_linorder__neqE,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 != Y )
     => ( ~ ( ord_less_nat @ X2 @ Y )
       => ( ord_less_nat @ Y @ X2 ) ) ) ).

% linorder_neqE
thf(fact_1124_order__less__asym,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ~ ( ord_less_nat @ Y @ X2 ) ) ).

% order_less_asym
thf(fact_1125_linorder__neq__iff,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 != Y )
      = ( ( ord_less_nat @ X2 @ Y )
        | ( ord_less_nat @ Y @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_1126_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1127_order__less__trans,axiom,
    ! [X2: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X2 @ Z3 ) ) ) ).

% order_less_trans
thf(fact_1128_ord__eq__less__subst,axiom,
    ! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1129_ord__less__eq__subst,axiom,
    ! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1130_order__less__irrefl,axiom,
    ! [X2: nat] :
      ~ ( ord_less_nat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_1131_order__less__subst1,axiom,
    ! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1132_order__less__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y4 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1133_order__less__not__sym,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ~ ( ord_less_nat @ Y @ X2 ) ) ).

% order_less_not_sym
thf(fact_1134_order__less__imp__triv,axiom,
    ! [X2: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_nat @ Y @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1135_linorder__less__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
      | ( X2 = Y )
      | ( ord_less_nat @ Y @ X2 ) ) ).

% linorder_less_linear
thf(fact_1136_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( X2 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1137_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( Y != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_1138_order__less__imp__not__less,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ~ ( ord_less_nat @ Y @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_1139_case__prod__Pair__iden,axiom,
    ! [P2: produc1996495991257130529ist_fm] :
      ( ( produc4544959197559615795ist_fm @ produc381145313068854617ist_fm @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1140_case__prod__Pair__iden,axiom,
    ! [P2: produc164195504107695125m_rule] :
      ( ( produc6532370182875420753m_rule @ produc491467635432902671m_rule @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1141_fst__def,axiom,
    ( produc6879501374131015971m_rule
    = ( produc8807799049144449253ist_fm
      @ ^ [X12: produc1996495991257130529ist_fm,X23: rule] : X12 ) ) ).

% fst_def
thf(fact_1142_snd__def,axiom,
    ( produc7165828336582415457m_rule
    = ( produc1213517383330063745e_rule
      @ ^ [X12: produc1996495991257130529ist_fm,X23: rule] : X23 ) ) ).

% snd_def
thf(fact_1143_not__less__Least,axiom,
    ! [K: nat,P: nat > $o] :
      ( ( ord_less_nat @ K @ ( ord_Least_nat @ P ) )
     => ~ ( P @ K ) ) ).

% not_less_Least
thf(fact_1144_sdrop_Osimps_I2_J,axiom,
    ! [N: nat,S3: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ ( suc @ N ) @ S3 )
      = ( sdrop_7224736112439592940m_rule @ N @ ( stl_Pr950425576149878629m_rule @ S3 ) ) ) ).

% sdrop.simps(2)
thf(fact_1145_sdrop_Osimps_I2_J,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( sdrop_rule @ ( suc @ N ) @ S3 )
      = ( sdrop_rule @ N @ ( stl_rule @ S3 ) ) ) ).

% sdrop.simps(2)
thf(fact_1146_snth_Osimps_I2_J,axiom,
    ! [S3: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ S3 @ ( suc @ N ) )
      = ( snth_P1712579846118164081m_rule @ ( stl_Pr950425576149878629m_rule @ S3 ) @ N ) ) ).

% snth.simps(2)
thf(fact_1147_snth_Osimps_I2_J,axiom,
    ! [S3: stream_rule,N: nat] :
      ( ( snth_rule @ S3 @ ( suc @ N ) )
      = ( snth_rule @ ( stl_rule @ S3 ) @ N ) ) ).

% snth.simps(2)
thf(fact_1148_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S6: set_rule,Rs: stream_fm,S3: rule,R3: fm] :
      ( ( abstra5167362717900046400m_rule @ Eff @ Rules @ S6 )
     => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra6545316509802446277m_rule @ Eff @ Rs @ S3 ) @ ( abstract_pos_fm @ Rs @ R3 ) )
         => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
           => ( ( member_rule @ S3 @ S6 )
             => ( ( abstract_pos_fm @ ( stl_fm @ ( abstra8496492133263135264m_rule @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstract_pos_fm @ Rs @ R3 ) @ ( suc @ ( abstra6545316509802446277m_rule @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1149_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S6: set_fm,Rs: stream_fm,S3: fm,R3: fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff @ Rules @ S6 )
     => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra3888632272101688784_fm_fm @ Eff @ Rs @ S3 ) @ ( abstract_pos_fm @ Rs @ R3 ) )
         => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
           => ( ( member_fm @ S3 @ S6 )
             => ( ( abstract_pos_fm @ ( stl_fm @ ( abstra5942290089224017835_fm_fm @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstract_pos_fm @ Rs @ R3 ) @ ( suc @ ( abstra3888632272101688784_fm_fm @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1150_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: fm > nat > fset_nat > $o,Rules: stream_fm,S6: set_nat,Rs: stream_fm,S3: nat,R3: fm] :
      ( ( abstra9084600350906980299fm_nat @ Eff @ Rules @ S6 )
     => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra1419524965967231568fm_nat @ Eff @ Rs @ S3 ) @ ( abstract_pos_fm @ Rs @ R3 ) )
         => ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
           => ( ( member_nat @ S3 @ S6 )
             => ( ( abstract_pos_fm @ ( stl_fm @ ( abstra7553130718700128683fm_nat @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstract_pos_fm @ Rs @ R3 ) @ ( suc @ ( abstra1419524965967231568fm_nat @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1151_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: nat > rule > fset_rule > $o,Rules: stream_nat,S6: set_rule,Rs: stream_nat,S3: rule,R3: nat] :
      ( ( abstra5092216761188309440t_rule @ Eff @ Rules @ S6 )
     => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra8093914158768835013t_rule @ Eff @ Rs @ S3 ) @ ( abstract_pos_nat @ Rs @ R3 ) )
         => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
           => ( ( member_rule @ S3 @ S6 )
             => ( ( abstract_pos_nat @ ( stl_nat @ ( abstra924199939036388256t_rule @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstract_pos_nat @ Rs @ R3 ) @ ( suc @ ( abstra8093914158768835013t_rule @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1152_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: nat > fm > fset_fm > $o,Rules: stream_nat,S6: set_fm,Rs: stream_nat,S3: fm,R3: nat] :
      ( ( abstra1720523912436379467nat_fm @ Eff @ Rules @ S6 )
     => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra3278820564351406544nat_fm @ Eff @ Rs @ S3 ) @ ( abstract_pos_nat @ Rs @ R3 ) )
         => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
           => ( ( member_fm @ S3 @ S6 )
             => ( ( abstract_pos_nat @ ( stl_nat @ ( abstra189054280229527851nat_fm @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstract_pos_nat @ Rs @ R3 ) @ ( suc @ ( abstra3278820564351406544nat_fm @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1153_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: nat > nat > fset_nat > $o,Rules: stream_nat,S6: set_nat,Rs: stream_nat,S3: nat,R3: nat] :
      ( ( abstra5088622135990055243at_nat @ Eff @ Rules @ S6 )
     => ( ( abstra7018062486307922108ir_nat @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra5877043044917184080at_nat @ Eff @ Rs @ S3 ) @ ( abstract_pos_nat @ Rs @ R3 ) )
         => ( ( member_nat @ R3 @ ( sset_nat @ Rules ) )
           => ( ( member_nat @ S3 @ S6 )
             => ( ( abstract_pos_nat @ ( stl_nat @ ( abstra7573781713853518635at_nat @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstract_pos_nat @ Rs @ R3 ) @ ( suc @ ( abstra5877043044917184080at_nat @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1154_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S6: set_rule,Rs: stream_rule,S3: rule,R3: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S6 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra5682536783049438522e_rule @ Eff @ Rs @ S3 ) @ ( abstract_pos_rule @ Rs @ R3 ) )
         => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
           => ( ( member_rule @ S3 @ S6 )
             => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R3 ) @ ( suc @ ( abstra5682536783049438522e_rule @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1155_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S6: set_fm,Rs: stream_rule,S3: fm,R3: rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S6 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra2294468009765627845ule_fm @ Eff @ Rs @ S3 ) @ ( abstract_pos_rule @ Rs @ R3 ) )
         => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
           => ( ( member_fm @ S3 @ S6 )
             => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R3 ) @ ( suc @ ( abstra2294468009765627845ule_fm @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1156_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: rule > nat > fset_nat > $o,Rules: stream_rule,S6: set_nat,Rs: stream_rule,S3: nat,R3: rule] :
      ( ( abstra8329481176008169152le_nat @ Eff @ Rules @ S6 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra2107806536733918917le_nat @ Eff @ Rs @ S3 ) @ ( abstract_pos_rule @ Rs @ R3 ) )
         => ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
           => ( ( member_nat @ S3 @ S6 )
             => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra4161464353856247968le_nat @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R3 ) @ ( suc @ ( abstra2107806536733918917le_nat @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1157_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S6: set_rule,Rs: stream8299795917829157543ist_fm,S3: rule,R3: produc1996495991257130529ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S6 )
     => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra3389404019813435448m_rule @ Eff @ Rs @ S3 ) @ ( abstra3513618393749879589ist_fm @ Rs @ R3 ) )
         => ( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
           => ( ( member_rule @ S3 @ S6 )
             => ( ( abstra3513618393749879589ist_fm @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs @ S3 ) ) @ R3 )
                = ( minus_minus_nat @ ( abstra3513618393749879589ist_fm @ Rs @ R3 ) @ ( suc @ ( abstra3389404019813435448m_rule @ Eff @ Rs @ S3 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1158_fset__of__list__elem,axiom,
    ! [X2: rule,Xs2: list_rule] :
      ( ( fmember_rule @ X2 @ ( fset_of_list_rule @ Xs2 ) )
      = ( member_rule @ X2 @ ( set_rule2 @ Xs2 ) ) ) ).

% fset_of_list_elem
thf(fact_1159_fset__of__list__elem,axiom,
    ! [X2: nat,Xs2: list_nat] :
      ( ( fmember_nat @ X2 @ ( fset_of_list_nat @ Xs2 ) )
      = ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) ) ) ).

% fset_of_list_elem
thf(fact_1160_fset__of__list__elem,axiom,
    ! [X2: produc1996495991257130529ist_fm,Xs2: list_P5616295576739893671ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X2 @ ( fset_o3706400737857578983ist_fm @ Xs2 ) )
      = ( member8102475879199740618ist_fm @ X2 @ ( set_Pr8767716839810916150ist_fm @ Xs2 ) ) ) ).

% fset_of_list_elem
thf(fact_1161_fset__of__list__elem,axiom,
    ! [X2: abstra4572470701423150929m_rule,Xs2: list_A1849579841884561377m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X2 @ ( fset_o8294491781805274379m_rule @ Xs2 ) )
      = ( member1590433669991072232m_rule @ X2 @ ( set_Ab5312821370972875900m_rule @ Xs2 ) ) ) ).

% fset_of_list_elem
thf(fact_1162_fset__of__list__elem,axiom,
    ! [X2: fm,Xs2: list_fm] :
      ( ( fmember_fm @ X2 @ ( fset_of_list_fm @ Xs2 ) )
      = ( member_fm @ X2 @ ( set_fm2 @ Xs2 ) ) ) ).

% fset_of_list_elem
thf(fact_1163_top_Oordering__top__axioms,axiom,
    orderi8258964890003709106ist_fm @ ord_le2055072402069232929ist_fm @ ord_le5784518313675440685ist_fm @ top_to728902761071950545ist_fm ).

% top.ordering_top_axioms
thf(fact_1164_top_Oordering__top__axioms,axiom,
    ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).

% top.ordering_top_axioms
thf(fact_1165_top_Oordering__top__axioms,axiom,
    orderi2038897200410189450t_rule @ ord_less_eq_set_rule @ ord_less_set_rule @ top_top_set_rule ).

% top.ordering_top_axioms
thf(fact_1166_holds__sfilter,axiom,
    ! [Q: produc164195504107695125m_rule > $o,S3: stream727092118206550309m_rule,P: produc164195504107695125m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ ( linear7082939723987557601m_rule @ Q ) @ S3 )
     => ( ( linear7082939723987557601m_rule @ P @ ( sfilte7579662470208054517m_rule @ Q @ S3 ) )
        = ( linear6209517582250783242m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ~ ( linear7082939723987557601m_rule @ Q @ Xs )
          @ ( linear7082939723987557601m_rule
            @ ^ [Xs: produc164195504107695125m_rule] :
                ( ( Q @ Xs )
                & ( P @ Xs ) ) )
          @ S3 ) ) ) ).

% holds_sfilter
thf(fact_1167_holds__sfilter,axiom,
    ! [Q: rule > $o,S3: stream_rule,P: rule > $o] :
      ( ( linear6831045468625226112v_rule @ ( linear4670012741531116049s_rule @ Q ) @ S3 )
     => ( ( linear4670012741531116049s_rule @ P @ ( sfilter_rule @ Q @ S3 ) )
        = ( linear8374973388053238458l_rule
          @ ^ [Xs: stream_rule] :
              ~ ( linear4670012741531116049s_rule @ Q @ Xs )
          @ ( linear4670012741531116049s_rule
            @ ^ [Xs: rule] :
                ( ( Q @ Xs )
                & ( P @ Xs ) ) )
          @ S3 ) ) ) ).

% holds_sfilter
thf(fact_1168_psubsetI,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_rule @ A @ B ) ) ) ).

% psubsetI
thf(fact_1169_case__prodI2,axiom,
    ! [P2: produc1996495991257130529ist_fm,C: list_fm > list_fm > $o] :
      ( ! [A5: list_fm,B4: list_fm] :
          ( ( P2
            = ( produc381145313068854617ist_fm @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc2011568048573023402t_fm_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_1170_case__prodI2,axiom,
    ! [P2: produc164195504107695125m_rule,C: produc1996495991257130529ist_fm > rule > $o] :
      ( ! [A5: produc1996495991257130529ist_fm,B4: rule] :
          ( ( P2
            = ( produc491467635432902671m_rule @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc7369492558480804444rule_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_1171_case__prodI,axiom,
    ! [F2: list_fm > list_fm > $o,A2: list_fm,B2: list_fm] :
      ( ( F2 @ A2 @ B2 )
     => ( produc2011568048573023402t_fm_o @ F2 @ ( produc381145313068854617ist_fm @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1172_case__prodI,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule > $o,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( F2 @ A2 @ B2 )
     => ( produc7369492558480804444rule_o @ F2 @ ( produc491467635432902671m_rule @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1173_Eps__case__prod__eq,axiom,
    ! [X2: list_fm,Y: list_fm] :
      ( ( fChoic1980167584695909237ist_fm
        @ ( produc2011568048573023402t_fm_o
          @ ^ [X8: list_fm,Y7: list_fm] :
              ( ( X2 = X8 )
              & ( Y = Y7 ) ) ) )
      = ( produc381145313068854617ist_fm @ X2 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_1174_Eps__case__prod__eq,axiom,
    ! [X2: produc1996495991257130529ist_fm,Y: rule] :
      ( ( fChoic9196029409387999681m_rule
        @ ( produc7369492558480804444rule_o
          @ ^ [X8: produc1996495991257130529ist_fm,Y7: rule] :
              ( ( X2 = X8 )
              & ( Y = Y7 ) ) ) )
      = ( produc491467635432902671m_rule @ X2 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_1175_mem__case__prodI2,axiom,
    ! [P2: produc1996495991257130529ist_fm,Z3: rule,C: list_fm > list_fm > set_rule] :
      ( ! [A5: list_fm,B4: list_fm] :
          ( ( P2
            = ( produc381145313068854617ist_fm @ A5 @ B4 ) )
         => ( member_rule @ Z3 @ ( C @ A5 @ B4 ) ) )
     => ( member_rule @ Z3 @ ( produc9016151371058038505t_rule @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1176_mem__case__prodI2,axiom,
    ! [P2: produc1996495991257130529ist_fm,Z3: produc1996495991257130529ist_fm,C: list_fm > list_fm > set_Pr7058068377845519745ist_fm] :
      ( ! [A5: list_fm,B4: list_fm] :
          ( ( P2
            = ( produc381145313068854617ist_fm @ A5 @ B4 ) )
         => ( member8102475879199740618ist_fm @ Z3 @ ( C @ A5 @ B4 ) ) )
     => ( member8102475879199740618ist_fm @ Z3 @ ( produc3038944318576466067ist_fm @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1177_mem__case__prodI2,axiom,
    ! [P2: produc1996495991257130529ist_fm,Z3: fm,C: list_fm > list_fm > set_fm] :
      ( ! [A5: list_fm,B4: list_fm] :
          ( ( P2
            = ( produc381145313068854617ist_fm @ A5 @ B4 ) )
         => ( member_fm @ Z3 @ ( C @ A5 @ B4 ) ) )
     => ( member_fm @ Z3 @ ( produc4802205653939627508set_fm @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1178_mem__case__prodI2,axiom,
    ! [P2: produc1996495991257130529ist_fm,Z3: nat,C: list_fm > list_fm > set_nat] :
      ( ! [A5: list_fm,B4: list_fm] :
          ( ( P2
            = ( produc381145313068854617ist_fm @ A5 @ B4 ) )
         => ( member_nat @ Z3 @ ( C @ A5 @ B4 ) ) )
     => ( member_nat @ Z3 @ ( produc1271363944939325172et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1179_mem__case__prodI2,axiom,
    ! [P2: produc164195504107695125m_rule,Z3: rule,C: produc1996495991257130529ist_fm > rule > set_rule] :
      ( ! [A5: produc1996495991257130529ist_fm,B4: rule] :
          ( ( P2
            = ( produc491467635432902671m_rule @ A5 @ B4 ) )
         => ( member_rule @ Z3 @ ( C @ A5 @ B4 ) ) )
     => ( member_rule @ Z3 @ ( produc6930014626940046263t_rule @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1180_mem__case__prodI2,axiom,
    ! [P2: produc164195504107695125m_rule,Z3: produc1996495991257130529ist_fm,C: produc1996495991257130529ist_fm > rule > set_Pr7058068377845519745ist_fm] :
      ( ! [A5: produc1996495991257130529ist_fm,B4: rule] :
          ( ( P2
            = ( produc491467635432902671m_rule @ A5 @ B4 ) )
         => ( member8102475879199740618ist_fm @ Z3 @ ( C @ A5 @ B4 ) ) )
     => ( member8102475879199740618ist_fm @ Z3 @ ( produc2666912249037264965ist_fm @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1181_mem__case__prodI2,axiom,
    ! [P2: produc164195504107695125m_rule,Z3: fm,C: produc1996495991257130529ist_fm > rule > set_fm] :
      ( ! [A5: produc1996495991257130529ist_fm,B4: rule] :
          ( ( P2
            = ( produc491467635432902671m_rule @ A5 @ B4 ) )
         => ( member_fm @ Z3 @ ( C @ A5 @ B4 ) ) )
     => ( member_fm @ Z3 @ ( produc966423478193074626set_fm @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1182_mem__case__prodI2,axiom,
    ! [P2: produc164195504107695125m_rule,Z3: nat,C: produc1996495991257130529ist_fm > rule > set_nat] :
      ( ! [A5: produc1996495991257130529ist_fm,B4: rule] :
          ( ( P2
            = ( produc491467635432902671m_rule @ A5 @ B4 ) )
         => ( member_nat @ Z3 @ ( C @ A5 @ B4 ) ) )
     => ( member_nat @ Z3 @ ( produc1581039335357782466et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_1183_mem__case__prodI,axiom,
    ! [Z3: rule,C: list_fm > list_fm > set_rule,A2: list_fm,B2: list_fm] :
      ( ( member_rule @ Z3 @ ( C @ A2 @ B2 ) )
     => ( member_rule @ Z3 @ ( produc9016151371058038505t_rule @ C @ ( produc381145313068854617ist_fm @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1184_mem__case__prodI,axiom,
    ! [Z3: produc1996495991257130529ist_fm,C: list_fm > list_fm > set_Pr7058068377845519745ist_fm,A2: list_fm,B2: list_fm] :
      ( ( member8102475879199740618ist_fm @ Z3 @ ( C @ A2 @ B2 ) )
     => ( member8102475879199740618ist_fm @ Z3 @ ( produc3038944318576466067ist_fm @ C @ ( produc381145313068854617ist_fm @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1185_mem__case__prodI,axiom,
    ! [Z3: fm,C: list_fm > list_fm > set_fm,A2: list_fm,B2: list_fm] :
      ( ( member_fm @ Z3 @ ( C @ A2 @ B2 ) )
     => ( member_fm @ Z3 @ ( produc4802205653939627508set_fm @ C @ ( produc381145313068854617ist_fm @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1186_mem__case__prodI,axiom,
    ! [Z3: nat,C: list_fm > list_fm > set_nat,A2: list_fm,B2: list_fm] :
      ( ( member_nat @ Z3 @ ( C @ A2 @ B2 ) )
     => ( member_nat @ Z3 @ ( produc1271363944939325172et_nat @ C @ ( produc381145313068854617ist_fm @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1187_mem__case__prodI,axiom,
    ! [Z3: rule,C: produc1996495991257130529ist_fm > rule > set_rule,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( member_rule @ Z3 @ ( C @ A2 @ B2 ) )
     => ( member_rule @ Z3 @ ( produc6930014626940046263t_rule @ C @ ( produc491467635432902671m_rule @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1188_mem__case__prodI,axiom,
    ! [Z3: produc1996495991257130529ist_fm,C: produc1996495991257130529ist_fm > rule > set_Pr7058068377845519745ist_fm,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( member8102475879199740618ist_fm @ Z3 @ ( C @ A2 @ B2 ) )
     => ( member8102475879199740618ist_fm @ Z3 @ ( produc2666912249037264965ist_fm @ C @ ( produc491467635432902671m_rule @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1189_mem__case__prodI,axiom,
    ! [Z3: fm,C: produc1996495991257130529ist_fm > rule > set_fm,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( member_fm @ Z3 @ ( C @ A2 @ B2 ) )
     => ( member_fm @ Z3 @ ( produc966423478193074626set_fm @ C @ ( produc491467635432902671m_rule @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1190_mem__case__prodI,axiom,
    ! [Z3: nat,C: produc1996495991257130529ist_fm > rule > set_nat,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( member_nat @ Z3 @ ( C @ A2 @ B2 ) )
     => ( member_nat @ Z3 @ ( produc1581039335357782466et_nat @ C @ ( produc491467635432902671m_rule @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1191_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
          ( ( ord_less_set_rule @ A7 @ B8 )
          | ( A7 = B8 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1192_subset__psubset__trans,axiom,
    ! [A: set_rule,B: set_rule,C2: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_set_rule @ B @ C2 )
       => ( ord_less_set_rule @ A @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_1193_subset__not__subset__eq,axiom,
    ( ord_less_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
          ( ( ord_less_eq_set_rule @ A7 @ B8 )
          & ~ ( ord_less_eq_set_rule @ B8 @ A7 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1194_psubset__subset__trans,axiom,
    ! [A: set_rule,B: set_rule,C2: set_rule] :
      ( ( ord_less_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ B @ C2 )
       => ( ord_less_set_rule @ A @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_1195_psubset__imp__subset,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( ord_less_set_rule @ A @ B )
     => ( ord_less_eq_set_rule @ A @ B ) ) ).

% psubset_imp_subset
thf(fact_1196_psubset__eq,axiom,
    ( ord_less_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
          ( ( ord_less_eq_set_rule @ A7 @ B8 )
          & ( A7 != B8 ) ) ) ) ).

% psubset_eq
thf(fact_1197_psubsetE,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( ord_less_set_rule @ A @ B )
     => ~ ( ( ord_less_eq_set_rule @ A @ B )
         => ( ord_less_eq_set_rule @ B @ A ) ) ) ).

% psubsetE
thf(fact_1198_pfsubsetD,axiom,
    ! [A: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le8344328325585982387ist_fm @ A @ B )
     => ( ( fmembe3381613331217039976ist_fm @ C @ A )
       => ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ).

% pfsubsetD
thf(fact_1199_pfsubsetD,axiom,
    ! [A: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( ord_le4463057631884877013m_rule @ A @ B )
     => ( ( fmembe1635983029874454474m_rule @ C @ A )
       => ( fmembe1635983029874454474m_rule @ C @ B ) ) ) ).

% pfsubsetD
thf(fact_1200_mem__case__prodE,axiom,
    ! [Z3: rule,C: list_fm > list_fm > set_rule,P2: produc1996495991257130529ist_fm] :
      ( ( member_rule @ Z3 @ ( produc9016151371058038505t_rule @ C @ P2 ) )
     => ~ ! [X: list_fm,Y4: list_fm] :
            ( ( P2
              = ( produc381145313068854617ist_fm @ X @ Y4 ) )
           => ~ ( member_rule @ Z3 @ ( C @ X @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1201_mem__case__prodE,axiom,
    ! [Z3: produc1996495991257130529ist_fm,C: list_fm > list_fm > set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ Z3 @ ( produc3038944318576466067ist_fm @ C @ P2 ) )
     => ~ ! [X: list_fm,Y4: list_fm] :
            ( ( P2
              = ( produc381145313068854617ist_fm @ X @ Y4 ) )
           => ~ ( member8102475879199740618ist_fm @ Z3 @ ( C @ X @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1202_mem__case__prodE,axiom,
    ! [Z3: fm,C: list_fm > list_fm > set_fm,P2: produc1996495991257130529ist_fm] :
      ( ( member_fm @ Z3 @ ( produc4802205653939627508set_fm @ C @ P2 ) )
     => ~ ! [X: list_fm,Y4: list_fm] :
            ( ( P2
              = ( produc381145313068854617ist_fm @ X @ Y4 ) )
           => ~ ( member_fm @ Z3 @ ( C @ X @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1203_mem__case__prodE,axiom,
    ! [Z3: nat,C: list_fm > list_fm > set_nat,P2: produc1996495991257130529ist_fm] :
      ( ( member_nat @ Z3 @ ( produc1271363944939325172et_nat @ C @ P2 ) )
     => ~ ! [X: list_fm,Y4: list_fm] :
            ( ( P2
              = ( produc381145313068854617ist_fm @ X @ Y4 ) )
           => ~ ( member_nat @ Z3 @ ( C @ X @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1204_mem__case__prodE,axiom,
    ! [Z3: rule,C: produc1996495991257130529ist_fm > rule > set_rule,P2: produc164195504107695125m_rule] :
      ( ( member_rule @ Z3 @ ( produc6930014626940046263t_rule @ C @ P2 ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y4: rule] :
            ( ( P2
              = ( produc491467635432902671m_rule @ X @ Y4 ) )
           => ~ ( member_rule @ Z3 @ ( C @ X @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1205_mem__case__prodE,axiom,
    ! [Z3: produc1996495991257130529ist_fm,C: produc1996495991257130529ist_fm > rule > set_Pr7058068377845519745ist_fm,P2: produc164195504107695125m_rule] :
      ( ( member8102475879199740618ist_fm @ Z3 @ ( produc2666912249037264965ist_fm @ C @ P2 ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y4: rule] :
            ( ( P2
              = ( produc491467635432902671m_rule @ X @ Y4 ) )
           => ~ ( member8102475879199740618ist_fm @ Z3 @ ( C @ X @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1206_mem__case__prodE,axiom,
    ! [Z3: fm,C: produc1996495991257130529ist_fm > rule > set_fm,P2: produc164195504107695125m_rule] :
      ( ( member_fm @ Z3 @ ( produc966423478193074626set_fm @ C @ P2 ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y4: rule] :
            ( ( P2
              = ( produc491467635432902671m_rule @ X @ Y4 ) )
           => ~ ( member_fm @ Z3 @ ( C @ X @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1207_mem__case__prodE,axiom,
    ! [Z3: nat,C: produc1996495991257130529ist_fm > rule > set_nat,P2: produc164195504107695125m_rule] :
      ( ( member_nat @ Z3 @ ( produc1581039335357782466et_nat @ C @ P2 ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y4: rule] :
            ( ( P2
              = ( produc491467635432902671m_rule @ X @ Y4 ) )
           => ~ ( member_nat @ Z3 @ ( C @ X @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_1208_case__prodE,axiom,
    ! [C: list_fm > list_fm > $o,P2: produc1996495991257130529ist_fm] :
      ( ( produc2011568048573023402t_fm_o @ C @ P2 )
     => ~ ! [X: list_fm,Y4: list_fm] :
            ( ( P2
              = ( produc381145313068854617ist_fm @ X @ Y4 ) )
           => ~ ( C @ X @ Y4 ) ) ) ).

% case_prodE
thf(fact_1209_case__prodE,axiom,
    ! [C: produc1996495991257130529ist_fm > rule > $o,P2: produc164195504107695125m_rule] :
      ( ( produc7369492558480804444rule_o @ C @ P2 )
     => ~ ! [X: produc1996495991257130529ist_fm,Y4: rule] :
            ( ( P2
              = ( produc491467635432902671m_rule @ X @ Y4 ) )
           => ~ ( C @ X @ Y4 ) ) ) ).

% case_prodE
thf(fact_1210_case__prodD,axiom,
    ! [F2: list_fm > list_fm > $o,A2: list_fm,B2: list_fm] :
      ( ( produc2011568048573023402t_fm_o @ F2 @ ( produc381145313068854617ist_fm @ A2 @ B2 ) )
     => ( F2 @ A2 @ B2 ) ) ).

% case_prodD
thf(fact_1211_case__prodD,axiom,
    ! [F2: produc1996495991257130529ist_fm > rule > $o,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( produc7369492558480804444rule_o @ F2 @ ( produc491467635432902671m_rule @ A2 @ B2 ) )
     => ( F2 @ A2 @ B2 ) ) ).

% case_prodD
thf(fact_1212_split__paired__Eps,axiom,
    ( fChoic1980167584695909237ist_fm
    = ( ^ [P3: produc1996495991257130529ist_fm > $o] :
          ( fChoic1980167584695909237ist_fm
          @ ( produc2011568048573023402t_fm_o
            @ ^ [A3: list_fm,B5: list_fm] : ( P3 @ ( produc381145313068854617ist_fm @ A3 @ B5 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_1213_split__paired__Eps,axiom,
    ( fChoic9196029409387999681m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o] :
          ( fChoic9196029409387999681m_rule
          @ ( produc7369492558480804444rule_o
            @ ^ [A3: produc1996495991257130529ist_fm,B5: rule] : ( P3 @ ( produc491467635432902671m_rule @ A3 @ B5 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_1214_suntil_Ostep,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( Phi2 @ Omega )
     => ( ( linear6209517582250783242m_rule @ Phi2 @ Psi @ ( stl_Pr950425576149878629m_rule @ Omega ) )
       => ( linear6209517582250783242m_rule @ Phi2 @ Psi @ Omega ) ) ) ).

% suntil.step
thf(fact_1215_suntil_Ostep,axiom,
    ! [Phi2: stream_rule > $o,Omega: stream_rule,Psi: stream_rule > $o] :
      ( ( Phi2 @ Omega )
     => ( ( linear8374973388053238458l_rule @ Phi2 @ Psi @ ( stl_rule @ Omega ) )
       => ( linear8374973388053238458l_rule @ Phi2 @ Psi @ Omega ) ) ) ).

% suntil.step
thf(fact_1216_suntil_Ocases,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o,A2: stream727092118206550309m_rule] :
      ( ( linear6209517582250783242m_rule @ Phi2 @ Psi @ A2 )
     => ( ~ ( Psi @ A2 )
       => ~ ( ( Phi2 @ A2 )
           => ~ ( linear6209517582250783242m_rule @ Phi2 @ Psi @ ( stl_Pr950425576149878629m_rule @ A2 ) ) ) ) ) ).

% suntil.cases
thf(fact_1217_suntil_Ocases,axiom,
    ! [Phi2: stream_rule > $o,Psi: stream_rule > $o,A2: stream_rule] :
      ( ( linear8374973388053238458l_rule @ Phi2 @ Psi @ A2 )
     => ( ~ ( Psi @ A2 )
       => ~ ( ( Phi2 @ A2 )
           => ~ ( linear8374973388053238458l_rule @ Phi2 @ Psi @ ( stl_rule @ A2 ) ) ) ) ) ).

% suntil.cases
thf(fact_1218_suntil_Osimps,axiom,
    ( linear6209517582250783242m_rule
    = ( ^ [Phi: stream727092118206550309m_rule > $o,Psi2: stream727092118206550309m_rule > $o,A3: stream727092118206550309m_rule] :
          ( ? [Omega3: stream727092118206550309m_rule] :
              ( ( A3 = Omega3 )
              & ( Psi2 @ Omega3 ) )
          | ? [Omega3: stream727092118206550309m_rule] :
              ( ( A3 = Omega3 )
              & ( Phi @ Omega3 )
              & ( linear6209517582250783242m_rule @ Phi @ Psi2 @ ( stl_Pr950425576149878629m_rule @ Omega3 ) ) ) ) ) ) ).

% suntil.simps
thf(fact_1219_suntil_Osimps,axiom,
    ( linear8374973388053238458l_rule
    = ( ^ [Phi: stream_rule > $o,Psi2: stream_rule > $o,A3: stream_rule] :
          ( ? [Omega3: stream_rule] :
              ( ( A3 = Omega3 )
              & ( Psi2 @ Omega3 ) )
          | ? [Omega3: stream_rule] :
              ( ( A3 = Omega3 )
              & ( Phi @ Omega3 )
              & ( linear8374973388053238458l_rule @ Phi @ Psi2 @ ( stl_rule @ Omega3 ) ) ) ) ) ) ).

% suntil.simps
thf(fact_1220_suntil__induct__strong,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o,X2: stream727092118206550309m_rule,P: stream727092118206550309m_rule > $o] :
      ( ( linear6209517582250783242m_rule @ Phi2 @ Psi @ X2 )
     => ( ! [Omega2: stream727092118206550309m_rule] :
            ( ( Psi @ Omega2 )
           => ( P @ Omega2 ) )
       => ( ! [Omega2: stream727092118206550309m_rule] :
              ( ( Phi2 @ Omega2 )
             => ( ~ ( Psi @ Omega2 )
               => ( ( linear6209517582250783242m_rule @ Phi2 @ Psi @ ( stl_Pr950425576149878629m_rule @ Omega2 ) )
                 => ( ( P @ ( stl_Pr950425576149878629m_rule @ Omega2 ) )
                   => ( P @ Omega2 ) ) ) ) )
         => ( P @ X2 ) ) ) ) ).

% suntil_induct_strong
thf(fact_1221_suntil__induct__strong,axiom,
    ! [Phi2: stream_rule > $o,Psi: stream_rule > $o,X2: stream_rule,P: stream_rule > $o] :
      ( ( linear8374973388053238458l_rule @ Phi2 @ Psi @ X2 )
     => ( ! [Omega2: stream_rule] :
            ( ( Psi @ Omega2 )
           => ( P @ Omega2 ) )
       => ( ! [Omega2: stream_rule] :
              ( ( Phi2 @ Omega2 )
             => ( ~ ( Psi @ Omega2 )
               => ( ( linear8374973388053238458l_rule @ Phi2 @ Psi @ ( stl_rule @ Omega2 ) )
                 => ( ( P @ ( stl_rule @ Omega2 ) )
                   => ( P @ Omega2 ) ) ) ) )
         => ( P @ X2 ) ) ) ) ).

% suntil_induct_strong
thf(fact_1222_ev__suntil,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ( linear6209517582250783242m_rule @ Phi2 @ Psi @ Omega )
     => ( linear2722748333883190224m_rule @ Psi @ Omega ) ) ).

% ev_suntil
thf(fact_1223_ev__suntil,axiom,
    ! [Phi2: stream_rule > $o,Psi: stream_rule > $o,Omega: stream_rule] :
      ( ( linear8374973388053238458l_rule @ Phi2 @ Psi @ Omega )
     => ( linear6831045468625226112v_rule @ Psi @ Omega ) ) ).

% ev_suntil
thf(fact_1224_ev__suntil__iff,axiom,
    ! [P: stream727092118206550309m_rule > $o,Q: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ ( linear6209517582250783242m_rule @ P @ Q ) @ Omega )
      = ( linear2722748333883190224m_rule @ Q @ Omega ) ) ).

% ev_suntil_iff
thf(fact_1225_ev__suntil__iff,axiom,
    ! [P: stream_rule > $o,Q: stream_rule > $o,Omega: stream_rule] :
      ( ( linear6831045468625226112v_rule @ ( linear8374973388053238458l_rule @ P @ Q ) @ Omega )
      = ( linear6831045468625226112v_rule @ Q @ Omega ) ) ).

% ev_suntil_iff
thf(fact_1226_suntil__cong,axiom,
    ! [P: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule,Q1: stream727092118206550309m_rule > $o,Q22: stream727092118206550309m_rule > $o,R1: stream727092118206550309m_rule > $o,R22: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ P @ Omega )
     => ( ! [Omega2: stream727092118206550309m_rule] :
            ( ( P @ Omega2 )
           => ( ( Q1 @ Omega2 )
              = ( Q22 @ Omega2 ) ) )
       => ( ! [Omega2: stream727092118206550309m_rule] :
              ( ( P @ Omega2 )
             => ( ( R1 @ Omega2 )
                = ( R22 @ Omega2 ) ) )
         => ( ( linear6209517582250783242m_rule @ Q1 @ R1 @ Omega )
            = ( linear6209517582250783242m_rule @ Q22 @ R22 @ Omega ) ) ) ) ) ).

% suntil_cong
thf(fact_1227_suntil__cong,axiom,
    ! [P: stream_rule > $o,Omega: stream_rule,Q1: stream_rule > $o,Q22: stream_rule > $o,R1: stream_rule > $o,R22: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ P @ Omega )
     => ( ! [Omega2: stream_rule] :
            ( ( P @ Omega2 )
           => ( ( Q1 @ Omega2 )
              = ( Q22 @ Omega2 ) ) )
       => ( ! [Omega2: stream_rule] :
              ( ( P @ Omega2 )
             => ( ( R1 @ Omega2 )
                = ( R22 @ Omega2 ) ) )
         => ( ( linear8374973388053238458l_rule @ Q1 @ R1 @ Omega )
            = ( linear8374973388053238458l_rule @ Q22 @ R22 @ Omega ) ) ) ) ) ).

% suntil_cong
thf(fact_1228_suntil__mono,axiom,
    ! [P: stream727092118206550309m_rule > $o,Q1: stream727092118206550309m_rule > $o,Q22: stream727092118206550309m_rule > $o,R1: stream727092118206550309m_rule > $o,R22: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ! [Omega2: stream727092118206550309m_rule] :
          ( ( P @ Omega2 )
         => ( ( Q1 @ Omega2 )
           => ( Q22 @ Omega2 ) ) )
     => ( ! [Omega2: stream727092118206550309m_rule] :
            ( ( P @ Omega2 )
           => ( ( R1 @ Omega2 )
             => ( R22 @ Omega2 ) ) )
       => ( ( linear6209517582250783242m_rule @ Q1 @ R1 @ Omega )
         => ( ( linear5684299538428340377m_rule @ P @ Omega )
           => ( linear6209517582250783242m_rule @ Q22 @ R22 @ Omega ) ) ) ) ) ).

% suntil_mono
thf(fact_1229_suntil__mono,axiom,
    ! [P: stream_rule > $o,Q1: stream_rule > $o,Q22: stream_rule > $o,R1: stream_rule > $o,R22: stream_rule > $o,Omega: stream_rule] :
      ( ! [Omega2: stream_rule] :
          ( ( P @ Omega2 )
         => ( ( Q1 @ Omega2 )
           => ( Q22 @ Omega2 ) ) )
     => ( ! [Omega2: stream_rule] :
            ( ( P @ Omega2 )
           => ( ( R1 @ Omega2 )
             => ( R22 @ Omega2 ) ) )
       => ( ( linear8374973388053238458l_rule @ Q1 @ R1 @ Omega )
         => ( ( linear7625967296368300489w_rule @ P @ Omega )
           => ( linear8374973388053238458l_rule @ Q22 @ R22 @ Omega ) ) ) ) ) ).

% suntil_mono
thf(fact_1230_suntil__inv,axiom,
    ! [F2: stream727092118206550309m_rule > stream727092118206550309m_rule,P: stream727092118206550309m_rule > $o,Q: stream727092118206550309m_rule > $o,S3: stream727092118206550309m_rule] :
      ( ! [S4: stream727092118206550309m_rule] :
          ( ( F2 @ ( stl_Pr950425576149878629m_rule @ S4 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F2 @ S4 ) ) )
     => ( ( linear6209517582250783242m_rule @ P @ Q @ ( F2 @ S3 ) )
        = ( linear6209517582250783242m_rule
          @ ^ [X4: stream727092118206550309m_rule] : ( P @ ( F2 @ X4 ) )
          @ ^ [X4: stream727092118206550309m_rule] : ( Q @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% suntil_inv
thf(fact_1231_suntil__inv,axiom,
    ! [F2: stream727092118206550309m_rule > stream_rule,P: stream_rule > $o,Q: stream_rule > $o,S3: stream727092118206550309m_rule] :
      ( ! [S4: stream727092118206550309m_rule] :
          ( ( F2 @ ( stl_Pr950425576149878629m_rule @ S4 ) )
          = ( stl_rule @ ( F2 @ S4 ) ) )
     => ( ( linear8374973388053238458l_rule @ P @ Q @ ( F2 @ S3 ) )
        = ( linear6209517582250783242m_rule
          @ ^ [X4: stream727092118206550309m_rule] : ( P @ ( F2 @ X4 ) )
          @ ^ [X4: stream727092118206550309m_rule] : ( Q @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% suntil_inv
thf(fact_1232_suntil__inv,axiom,
    ! [F2: stream_rule > stream727092118206550309m_rule,P: stream727092118206550309m_rule > $o,Q: stream727092118206550309m_rule > $o,S3: stream_rule] :
      ( ! [S4: stream_rule] :
          ( ( F2 @ ( stl_rule @ S4 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F2 @ S4 ) ) )
     => ( ( linear6209517582250783242m_rule @ P @ Q @ ( F2 @ S3 ) )
        = ( linear8374973388053238458l_rule
          @ ^ [X4: stream_rule] : ( P @ ( F2 @ X4 ) )
          @ ^ [X4: stream_rule] : ( Q @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% suntil_inv
thf(fact_1233_suntil__inv,axiom,
    ! [F2: stream_rule > stream_rule,P: stream_rule > $o,Q: stream_rule > $o,S3: stream_rule] :
      ( ! [S4: stream_rule] :
          ( ( F2 @ ( stl_rule @ S4 ) )
          = ( stl_rule @ ( F2 @ S4 ) ) )
     => ( ( linear8374973388053238458l_rule @ P @ Q @ ( F2 @ S3 ) )
        = ( linear8374973388053238458l_rule
          @ ^ [X4: stream_rule] : ( P @ ( F2 @ X4 ) )
          @ ^ [X4: stream_rule] : ( Q @ ( F2 @ X4 ) )
          @ S3 ) ) ) ).

% suntil_inv
thf(fact_1234_true__suntil,axiom,
    ! [P: stream727092118206550309m_rule > $o] :
      ( ( linear6209517582250783242m_rule
        @ ^ [Uu: stream727092118206550309m_rule] : $true
        @ P )
      = ( linear2722748333883190224m_rule @ P ) ) ).

% true_suntil
thf(fact_1235_true__suntil,axiom,
    ! [P: stream_rule > $o] :
      ( ( linear8374973388053238458l_rule
        @ ^ [Uu: stream_rule] : $true
        @ P )
      = ( linear6831045468625226112v_rule @ P ) ) ).

% true_suntil
thf(fact_1236_ev__eq__suntil,axiom,
    ( linear2722748333883190224m_rule
    = ( ^ [P3: stream727092118206550309m_rule > $o] :
          ( linear6209517582250783242m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ~ ( P3 @ Xs )
          @ P3 ) ) ) ).

% ev_eq_suntil
thf(fact_1237_ev__eq__suntil,axiom,
    ( linear6831045468625226112v_rule
    = ( ^ [P3: stream_rule > $o] :
          ( linear8374973388053238458l_rule
          @ ^ [Xs: stream_rule] :
              ~ ( P3 @ Xs )
          @ P3 ) ) ) ).

% ev_eq_suntil
thf(fact_1238_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc1996495991257130529ist_fm,A: list_fm > list_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X2 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ A ) ) )
     => ( A @ ( produc1501393135466168645ist_fm @ X2 ) @ ( produc4588648349897876871ist_fm @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_1239_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc164195504107695125m_rule,A: produc1996495991257130529ist_fm > rule > $o] :
      ( ( member4220325220686508332m_rule @ X2 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ A ) ) )
     => ( A @ ( produc6879501374131015971m_rule @ X2 ) @ ( produc7165828336582415457m_rule @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_1240_suntil__aand__nxt,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ( linear6209517582250783242m_rule @ Phi2
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( Phi2 @ Xs )
            & ( linear7357459635800698592rule_o @ Psi @ Xs ) )
        @ Omega )
      = ( ( Phi2 @ Omega )
        & ( linear7357459635800698592rule_o @ ( linear6209517582250783242m_rule @ Phi2 @ Psi ) @ Omega ) ) ) ).

% suntil_aand_nxt
thf(fact_1241_Eps__case__prod,axiom,
    ! [P: produc1996495991257130529ist_fm > rule > $o] :
      ( ( fChoic9196029409387999681m_rule @ ( produc7369492558480804444rule_o @ P ) )
      = ( fChoic9196029409387999681m_rule
        @ ^ [Xy: produc164195504107695125m_rule] : ( P @ ( produc6879501374131015971m_rule @ Xy ) @ ( produc7165828336582415457m_rule @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_1242_suntil__iff__sdrop,axiom,
    ( linear6209517582250783242m_rule
    = ( ^ [P4: stream727092118206550309m_rule > $o,Q4: stream727092118206550309m_rule > $o,Omega3: stream727092118206550309m_rule] :
        ? [J: nat] :
          ( ( Q4 @ ( sdrop_7224736112439592940m_rule @ J @ Omega3 ) )
          & ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ J )
             => ( P4 @ ( sdrop_7224736112439592940m_rule @ K2 @ Omega3 ) ) ) ) ) ) ).

% suntil_iff_sdrop
thf(fact_1243_suntil__iff__sdrop,axiom,
    ( linear8374973388053238458l_rule
    = ( ^ [P4: stream_rule > $o,Q4: stream_rule > $o,Omega3: stream_rule] :
        ? [J: nat] :
          ( ( Q4 @ ( sdrop_rule @ J @ Omega3 ) )
          & ! [K2: nat] :
              ( ( ord_less_nat @ K2 @ J )
             => ( P4 @ ( sdrop_rule @ K2 @ Omega3 ) ) ) ) ) ) ).

% suntil_iff_sdrop
thf(fact_1244_sdrop__if__suntil,axiom,
    ! [P2: stream727092118206550309m_rule > $o,Q2: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
      ( ( linear6209517582250783242m_rule @ P2 @ Q2 @ Omega )
     => ? [J2: nat] :
          ( ( Q2 @ ( sdrop_7224736112439592940m_rule @ J2 @ Omega ) )
          & ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ J2 )
             => ( P2 @ ( sdrop_7224736112439592940m_rule @ K3 @ Omega ) ) ) ) ) ).

% sdrop_if_suntil
thf(fact_1245_sdrop__if__suntil,axiom,
    ! [P2: stream_rule > $o,Q2: stream_rule > $o,Omega: stream_rule] :
      ( ( linear8374973388053238458l_rule @ P2 @ Q2 @ Omega )
     => ? [J2: nat] :
          ( ( Q2 @ ( sdrop_rule @ J2 @ Omega ) )
          & ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ J2 )
             => ( P2 @ ( sdrop_rule @ K3 @ Omega ) ) ) ) ) ).

% sdrop_if_suntil
thf(fact_1246_sdrop__suntil,axiom,
    ! [Q2: stream727092118206550309m_rule > $o,J3: nat,Omega: stream727092118206550309m_rule,P2: stream727092118206550309m_rule > $o] :
      ( ( Q2 @ ( sdrop_7224736112439592940m_rule @ J3 @ Omega ) )
     => ( ! [K4: nat] :
            ( ( ord_less_nat @ K4 @ J3 )
           => ( P2 @ ( sdrop_7224736112439592940m_rule @ K4 @ Omega ) ) )
       => ( linear6209517582250783242m_rule @ P2 @ Q2 @ Omega ) ) ) ).

% sdrop_suntil
thf(fact_1247_sdrop__suntil,axiom,
    ! [Q2: stream_rule > $o,J3: nat,Omega: stream_rule,P2: stream_rule > $o] :
      ( ( Q2 @ ( sdrop_rule @ J3 @ Omega ) )
     => ( ! [K4: nat] :
            ( ( ord_less_nat @ K4 @ J3 )
           => ( P2 @ ( sdrop_rule @ K4 @ Omega ) ) )
       => ( linear8374973388053238458l_rule @ P2 @ Q2 @ Omega ) ) ) ).

% sdrop_suntil
thf(fact_1248_snd__comp__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc7680103964935106114e_rule @ F2 @ G ) )
      = ( comp_r641804198332660996m_rule @ G @ produc7165828336582415457m_rule ) ) ).

% snd_comp_map_prod
thf(fact_1249_fst__comp__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc7680103964935106114e_rule @ F2 @ G ) )
      = ( comp_P8245093624798776376m_rule @ F2 @ produc6879501374131015971m_rule ) ) ).

% fst_comp_map_prod
thf(fact_1250_nxt__wait__least,axiom,
    ! [Phi2: stream_rule > $o,Xs2: stream_rule,N: nat] :
      ( ( linear6831045468625226112v_rule @ Phi2 @ Xs2 )
     => ( ( compow6114205906206174322rule_o @ N @ linear860818938954496176rule_o @ Phi2 @ Xs2 )
       => ( ord_less_eq_nat @ ( linear25361906725448356t_rule @ Phi2 @ Xs2 ) @ N ) ) ) ).

% nxt_wait_least
thf(fact_1251_nxt__wait__least,axiom,
    ! [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,N: nat] :
      ( ( linear2722748333883190224m_rule @ Phi2 @ Xs2 )
     => ( ( compow3961076492690163954rule_o @ N @ linear7357459635800698592rule_o @ Phi2 @ Xs2 )
       => ( ord_less_eq_nat @ ( linear2497734134072303348m_rule @ Phi2 @ Xs2 ) @ N ) ) ) ).

% nxt_wait_least
thf(fact_1252_map__prod__simp,axiom,
    ! [F2: list_fm > list_fm,G: list_fm > list_fm,A2: list_fm,B2: list_fm] :
      ( ( produc3421166768339006330ist_fm @ F2 @ G @ ( produc381145313068854617ist_fm @ A2 @ B2 ) )
      = ( produc381145313068854617ist_fm @ ( F2 @ A2 ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_1253_map__prod__simp,axiom,
    ! [F2: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,A2: list_fm,B2: list_fm] :
      ( ( produc8124276167727720944m_rule @ F2 @ G @ ( produc381145313068854617ist_fm @ A2 @ B2 ) )
      = ( produc491467635432902671m_rule @ ( F2 @ A2 ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_1254_map__prod__simp,axiom,
    ! [F2: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( produc1965614570530558770ist_fm @ F2 @ G @ ( produc491467635432902671m_rule @ A2 @ B2 ) )
      = ( produc381145313068854617ist_fm @ ( F2 @ A2 ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_1255_map__prod__simp,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,A2: produc1996495991257130529ist_fm,B2: rule] :
      ( ( produc7680103964935106114e_rule @ F2 @ G @ ( produc491467635432902671m_rule @ A2 @ B2 ) )
      = ( produc491467635432902671m_rule @ ( F2 @ A2 ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_1256_fst__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc7680103964935106114e_rule @ F2 @ G @ X2 ) )
      = ( F2 @ ( produc6879501374131015971m_rule @ X2 ) ) ) ).

% fst_map_prod
thf(fact_1257_snd__map__prod,axiom,
    ! [F2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X2: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc7680103964935106114e_rule @ F2 @ G @ X2 ) )
      = ( G @ ( produc7165828336582415457m_rule @ X2 ) ) ) ).

% snd_map_prod
thf(fact_1258_less__set__def,axiom,
    ( ord_less_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
          ( ord_less_rule_o
          @ ^ [X4: rule] : ( member_rule @ X4 @ A7 )
          @ ^ [X4: rule] : ( member_rule @ X4 @ B8 ) ) ) ) ).

% less_set_def
thf(fact_1259_less__set__def,axiom,
    ( ord_le5784518313675440685ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B8: set_Pr7058068377845519745ist_fm] :
          ( ord_le8413403837543126768t_fm_o
          @ ^ [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ A7 )
          @ ^ [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ B8 ) ) ) ) ).

% less_set_def
thf(fact_1260_less__set__def,axiom,
    ( ord_less_set_fm
    = ( ^ [A7: set_fm,B8: set_fm] :
          ( ord_less_fm_o
          @ ^ [X4: fm] : ( member_fm @ X4 @ A7 )
          @ ^ [X4: fm] : ( member_fm @ X4 @ B8 ) ) ) ) ).

% less_set_def
thf(fact_1261_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( ord_less_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A7 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B8 ) ) ) ) ).

% less_set_def
thf(fact_1262_psubsetD,axiom,
    ! [A: set_rule,B: set_rule,C: rule] :
      ( ( ord_less_set_rule @ A @ B )
     => ( ( member_rule @ C @ A )
       => ( member_rule @ C @ B ) ) ) ).

% psubsetD
thf(fact_1263_psubsetD,axiom,
    ! [A: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le5784518313675440685ist_fm @ A @ B )
     => ( ( member8102475879199740618ist_fm @ C @ A )
       => ( member8102475879199740618ist_fm @ C @ B ) ) ) ).

% psubsetD
thf(fact_1264_psubsetD,axiom,
    ! [A: set_fm,B: set_fm,C: fm] :
      ( ( ord_less_set_fm @ A @ B )
     => ( ( member_fm @ C @ A )
       => ( member_fm @ C @ B ) ) ) ).

% psubsetD
thf(fact_1265_psubsetD,axiom,
    ! [A: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B ) ) ) ).

% psubsetD
thf(fact_1266_map__prod_Ocomp,axiom,
    ! [F2: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,I: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P8819851879376895428m_rule @ ( produc602106178381476086ist_fm @ F2 @ G ) @ ( produc6213034867950506950m_rule @ H @ I ) )
      = ( produc4449957851134572790ist_fm @ ( comp_P4378417847848123130m_rule @ F2 @ H ) @ ( comp_P4378417847848123130m_rule @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_1267_map__prod_Ocompositionality,axiom,
    ! [F2: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,I: abstra4572470701423150929m_rule > produc164195504107695125m_rule,Prod: produc2413811689340339075m_rule] :
      ( ( produc602106178381476086ist_fm @ F2 @ G @ ( produc6213034867950506950m_rule @ H @ I @ Prod ) )
      = ( produc4449957851134572790ist_fm @ ( comp_P4378417847848123130m_rule @ F2 @ H ) @ ( comp_P4378417847848123130m_rule @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_1268_per__def,axiom,
    ! [R3: rule] :
      ( ( abstra4153889242452539894ist_fm
        @ ^ [R2: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R3 )
      = ( ! [S: produc1996495991257130529ist_fm,R12: rule,Sl6: fset_P661503646757059847ist_fm,S9: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
              & ( abstra3055540779341473792ist_fm
                @ ^ [R2: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R2 @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R3
                @ S )
              & ( member_rule @ R12 @ ( minus_minus_set_rule @ ( sset_rule @ rules ) @ ( insert_rule @ R3 @ bot_bot_set_rule ) ) )
              & ( ( eff @ R12 @ S )
                = ( some_f2162114139285431020ist_fm @ Sl6 ) )
              & ( fmembe3381613331217039976ist_fm @ S9 @ Sl6 ) )
           => ( abstra3055540779341473792ist_fm
              @ ^ [R2: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R2 @ T )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R3
              @ S9 ) ) ) ) ).

% per_def
thf(fact_1269_NE__R,axiom,
    ( ( sset_rule @ rules )
   != bot_bot_set_rule ) ).

% NE_R
thf(fact_1270_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_1271_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1272_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_1273_finite__less__ub,axiom,
    ! [F2: nat > nat,U: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F2 @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_1274_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_1275_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_1276_infinite__enumerate,axiom,
    ! [S6: set_nat] :
      ( ~ ( finite_finite_nat @ S6 )
     => ? [R4: nat > nat] :
          ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R4 )
          & ! [N5: nat] : ( member_nat @ ( R4 @ N5 ) @ S6 ) ) ) ).

% infinite_enumerate

% Helper facts (20)
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X6: nat] : ( P @ X6 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Syntax__Ofm_T,axiom,
    ! [P: fm > $o] :
      ( ( P @ ( fChoice_fm @ P ) )
      = ( ? [X6: fm] : ( P @ X6 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Syntax__Orule_T,axiom,
    ! [P: rule > $o] :
      ( ( P @ ( fChoice_rule @ P ) )
      = ( ? [X6: rule] : ( P @ X6 ) ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X2: stream_rule,Y: stream_rule] :
      ( ( if_stream_rule @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X2: stream_rule,Y: stream_rule] :
      ( ( if_stream_rule @ $true @ X2 @ Y )
      = X2 ) ).

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 ) )
      = ( ? [X6: abstract_tree_rule] : ( P @ X6 ) ) ) ).

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,
    ! [X2: produc1996495991257130529ist_fm,Y: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $false @ X2 @ Y )
      = Y ) ).

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,
    ! [X2: produc1996495991257130529ist_fm,Y: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $true @ X2 @ Y )
      = X2 ) ).

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 ) )
      = ( ? [X6: produc1996495991257130529ist_fm] : ( P @ X6 ) ) ) ).

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,
    ! [X2: fset_P661503646757059847ist_fm,Y: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $false @ X2 @ Y )
      = Y ) ).

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,
    ! [X2: fset_P661503646757059847ist_fm,Y: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_fChoice_1_1_fChoice_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_T,axiom,
    ! [P: produc164195504107695125m_rule > $o] :
      ( ( P @ ( fChoic9196029409387999681m_rule @ P ) )
      = ( ? [X6: produc164195504107695125m_rule] : ( P @ X6 ) ) ) ).

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,
    ! [X2: stream727092118206550309m_rule,Y: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $false @ X2 @ Y )
      = Y ) ).

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,
    ! [X2: stream727092118206550309m_rule,Y: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $true @ X2 @ Y )
      = X2 ) ).

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,
    ! [X2: abstra4572470701423150929m_rule,Y: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $false @ X2 @ Y )
      = Y ) ).

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,
    ! [X2: abstra4572470701423150929m_rule,Y: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $true @ X2 @ Y )
      = X2 ) ).

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 ) )
      = ( ? [X6: abstra4572470701423150929m_rule] : ( P @ X6 ) ) ) ).

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,
    ! [X2: fset_A5396818216090053761m_rule,Y: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Abstract____Completeness__Otree_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_J_T,axiom,
    ! [X2: fset_A5396818216090053761m_rule,Y: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ steps ) ) ) @ ss ).

%------------------------------------------------------------------------------