TPTP Problem File: SLH0700^1.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_00061_001851__12052798_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2071 ( 629 unt; 771 typ;   0 def)
%            Number of atoms       : 3918 (1199 equ;  15 cnn)
%            Maximal formula atoms :   15 (   3 avg)
%            Number of connectives : 15106 ( 228   ~;  54   |; 426   &;12581   @)
%                                         (   0 <=>;1817  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :  136 ( 135 usr)
%            Number of type conns  : 3283 (3283   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  640 ( 636 usr;  22 con; 0-4 aty)
%            Number of variables   : 4833 ( 849   ^;3694   !; 290   ?;4833   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:27:52.691
%------------------------------------------------------------------------------
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    fChoic6673916560228149179at_nat: ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    fChoic2372384411376717755at_nat: ( produc3843707927480180839at_nat > $o ) > produc3843707927480180839at_nat ).

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

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

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

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

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).

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

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_steps,type,
    steps: stream727092118206550309m_rule ).

% Relevant facts (1275)
thf(fact_0_option_Oinject,axiom,
    ! [X2: fset_P661503646757059847ist_fm,Y2: fset_P661503646757059847ist_fm] :
      ( ( ( some_f2162114139285431020ist_fm @ X2 )
        = ( some_f2162114139285431020ist_fm @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_1_RuleSystem__Defs_Oepath_Ocong,axiom,
    abstra5415202973855822231ist_fm = abstra5415202973855822231ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_2_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_3_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_4_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_5_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_6_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_7_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_3185404920971892658e_rule @ N @ ( szip_r730181393530711434e_rule @ S1 @ S2 ) )
      = ( szip_r730181393530711434e_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_8_sdrop__szip,axiom,
    ! [N: nat,S1: stream3588992565182678257e_rule,S2: stream_rule] :
      ( ( sdrop_2433707926279816502e_rule @ N @ ( szip_P3067794746305893590e_rule @ S1 @ S2 ) )
      = ( szip_P3067794746305893590e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_9_sdrop__szip,axiom,
    ! [N: nat,S1: stream_list_fm,S2: stream_list_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( szip_list_fm_list_fm @ S1 @ S2 ) )
      = ( szip_list_fm_list_fm @ ( sdrop_list_fm @ N @ S1 ) @ ( sdrop_list_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_10_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_2468034597186633064ist_fm @ N @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) )
      = ( szip_r8549736553299883584ist_fm @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_11_sdrop__szip,axiom,
    ! [N: nat,S1: stream3588992565182678257e_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_7075696550712280400e_rule @ N @ ( szip_P4443206486938582938e_rule @ S1 @ S2 ) )
      = ( szip_P4443206486938582938e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_szip
thf(fact_12_prover__def,axiom,
    ( prover
    = ( abstra178918799166095863ist_fm
      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R @ S )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ rules ) ) ).

% prover_def
thf(fact_13_wf__ipath__epath,axiom,
    ! [T: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ T )
     => ( ( abstra2285467793620430502m_rule @ T @ Steps )
       => ( abstra5415202973855822231ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ Steps ) ) ) ).

% wf_ipath_epath
thf(fact_14_enabled__def,axiom,
    ! [R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ S3 )
      = ( ? [Sl: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S3 )
            = ( some_f2162114139285431020ist_fm @ Sl ) ) ) ) ).

% enabled_def
thf(fact_15_PersistentRuleSystem__axioms,axiom,
    ( abstra7887161495092839066ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ top_to728902761071950545ist_fm ) ).

% PersistentRuleSystem_axioms
thf(fact_16_per__rules,axiom,
    ! [R2: rule] :
      ( abstra4153889242452539894ist_fm
      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
          ( ( eff @ R @ S )
          = ( some_f2162114139285431020ist_fm @ Ss ) )
      @ rules
      @ top_to728902761071950545ist_fm
      @ R2 ) ).

% per_rules
thf(fact_17_RuleSystem__axioms,axiom,
    ( abstra6455275554694158129ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ top_to728902761071950545ist_fm ) ).

% RuleSystem_axioms
thf(fact_18_enabled__R,axiom,
    ! [S3: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ? [X: rule] :
          ( ( member_rule @ X @ ( sset_rule @ rules ) )
          & ? [Sl2: fset_P661503646757059847ist_fm] :
              ( ( eff @ X @ S3 )
              = ( some_f2162114139285431020ist_fm @ Sl2 ) ) ) ) ).

% enabled_R
thf(fact_19_Saturated__def,axiom,
    ! [Steps: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
      = ( ! [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ X3
              @ Steps ) ) ) ) ).

% Saturated_def
thf(fact_20_PersistentRuleSystem_Oaxioms_I1_J,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S4 )
     => ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 ) ) ).

% PersistentRuleSystem.axioms(1)
thf(fact_21_RuleSystem_Oper_Ocong,axiom,
    abstra4153889242452539894ist_fm = abstra4153889242452539894ist_fm ).

% RuleSystem.per.cong
thf(fact_22_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_rule] : ( Eff @ X @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_23_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat] :
      ( ( abstra5655617845878133111at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P3619101297660450253at_nat] : ( Eff @ X @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_24_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P5670320511379867111at_nat] : ( Eff @ X @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_25_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > produc3843707927480180839at_nat > fset_P2467486768528901581at_nat > $o,Rules: stream_rule,S4: set_Pr4329608150637261639at_nat,S3: produc3843707927480180839at_nat] :
      ( ( abstra5319030069223283063at_nat @ Eff @ Rules @ S4 )
     => ( ( member8757157785044589968at_nat @ S3 @ S4 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P2467486768528901581at_nat] : ( Eff @ X @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_26_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P661503646757059847ist_fm] : ( Eff @ X @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_27_RuleSystem__Defs_Owf_Ocong,axiom,
    abstra4476735159789068068ist_fm = abstra4476735159789068068ist_fm ).

% RuleSystem_Defs.wf.cong
thf(fact_28_PersistentRuleSystem_Oper,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,R2: rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S4 )
     => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
       => ( abstra4153889242452539894ist_fm @ Eff @ Rules @ S4 @ R2 ) ) ) ).

% PersistentRuleSystem.per
thf(fact_29_RuleSystem__Defs_Oenabled__def,axiom,
    ( abstra3055540779341473792ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R: rule,S: produc1996495991257130529ist_fm] :
        ? [X4: fset_P661503646757059847ist_fm] : ( Eff2 @ R @ S @ X4 ) ) ) ).

% RuleSystem_Defs.enabled_def
thf(fact_30_RuleSystem__Defs_OmkTree_Ocong,axiom,
    abstra178918799166095863ist_fm = abstra178918799166095863ist_fm ).

% RuleSystem_Defs.mkTree.cong
thf(fact_31_RuleSystem__Defs_Oenabled_Ocong,axiom,
    abstra3055540779341473792ist_fm = abstra3055540779341473792ist_fm ).

% RuleSystem_Defs.enabled.cong
thf(fact_32_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra8647779336736900908ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,Steps2: stream727092118206550309m_rule] :
        ! [X3: rule] :
          ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
         => ( abstra7252176601000738572ist_fm @ Eff2 @ X3 @ Steps2 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_33_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra8647779336736900908ist_fm = abstra8647779336736900908ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_34_RuleSystem__Defs_Osaturated_Ocong,axiom,
    abstra7252176601000738572ist_fm = abstra7252176601000738572ist_fm ).

% RuleSystem_Defs.saturated.cong
thf(fact_35_RuleSystem__Defs_Owf__ipath__epath,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,T: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff @ Rules @ T )
     => ( ( abstra2285467793620430502m_rule @ T @ Steps )
       => ( abstra5415202973855822231ist_fm @ Eff @ Rules @ Steps ) ) ) ).

% RuleSystem_Defs.wf_ipath_epath
thf(fact_36_per,axiom,
    ! [R2: rule] :
      ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
     => ( abstra4153889242452539894ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R2 ) ) ).

% per
thf(fact_37_ipath__mkTree__saturated,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps )
         => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R2
              @ Steps ) ) ) ) ) ).

% ipath_mkTree_saturated
thf(fact_38_ipath__mkTree__sdrop,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,M: nat] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps )
         => ? [N2: nat,S5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S5 @ top_to728902761071950545ist_fm )
              & ( abstra2285467793620430502m_rule
                @ ( abstra178918799166095863ist_fm
                  @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ ( sdrop_rule @ N2 @ Rs )
                  @ S5 )
                @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_sdrop
thf(fact_39_ipath__mkTree__Saturated,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps )
         => ( abstra8647779336736900908ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ Steps ) ) ) ) ).

% ipath_mkTree_Saturated
thf(fact_40_UNIV__I,axiom,
    ! [X5: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X5 @ top_to8454300755403694231at_nat ) ).

% UNIV_I
thf(fact_41_UNIV__I,axiom,
    ! [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ top_to4669805908274784177at_nat ) ).

% UNIV_I
thf(fact_42_UNIV__I,axiom,
    ! [X5: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X5 @ top_to6833984726390702231at_nat ) ).

% UNIV_I
thf(fact_43_UNIV__I,axiom,
    ! [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ top_to728902761071950545ist_fm ) ).

% UNIV_I
thf(fact_44_UNIV__I,axiom,
    ! [X5: rule] : ( member_rule @ X5 @ top_top_set_rule ) ).

% UNIV_I
thf(fact_45_iso__tuple__UNIV__I,axiom,
    ! [X5: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X5 @ top_to8454300755403694231at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_46_iso__tuple__UNIV__I,axiom,
    ! [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ top_to4669805908274784177at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_47_iso__tuple__UNIV__I,axiom,
    ! [X5: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X5 @ top_to6833984726390702231at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_48_iso__tuple__UNIV__I,axiom,
    ! [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ top_to728902761071950545ist_fm ) ).

% iso_tuple_UNIV_I
thf(fact_49_iso__tuple__UNIV__I,axiom,
    ! [X5: rule] : ( member_rule @ X5 @ top_top_set_rule ) ).

% iso_tuple_UNIV_I
thf(fact_50_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule,M: nat] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: rule] :
                ( ( member_rule @ S5 @ S4 )
                & ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_2199339672054592340e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_51_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,S4: set_list_fm,S3: list_fm,Rs: stream_list_fm,Steps: stream8299795917829157543ist_fm,M: nat] :
      ( ( abstra12045985430712427ist_fm @ Eff @ Rules @ S4 )
     => ( ( member_list_fm @ S3 @ S4 )
       => ( ( abstra274702546776099020ist_fm @ Rules @ Rs )
         => ( ( abstra5863983821914536912ist_fm @ ( abstra3846921640779263781ist_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: list_fm] :
                ( ( member_list_fm @ S5 @ S4 )
                & ( abstra5863983821914536912ist_fm @ ( abstra3846921640779263781ist_fm @ Eff @ ( sdrop_list_fm @ N2 @ Rs ) @ S5 ) @ ( sdrop_4442373711808556042ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_52_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc780299800278082923e_rule > rule > fset_rule > $o,Rules: stream3588992565182678257e_rule,S4: set_rule,S3: rule,Rs: stream3588992565182678257e_rule,Steps: stream4003613590975974891e_rule,M: nat] :
      ( ( abstra1742398642700549575e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra4067143171202666092e_rule @ ( abstra4338455777355667597e_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: rule] :
                ( ( member_rule @ S5 @ S4 )
                & ( abstra4067143171202666092e_rule @ ( abstra4338455777355667597e_rule @ Eff @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_3185404920971892658e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_53_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule,Steps: stream1780317496394867717t_rule,M: nat] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3533331058055637382t_rule @ ( abstra3105372317759539415at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S5 @ S4 )
                & ( abstra3533331058055637382t_rule @ ( abstra3105372317759539415at_nat @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_4214913930985848524t_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_54_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm,Steps: stream1960312990768957601ist_fm,M: nat] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: rule] :
                ( ( member_rule @ S5 @ S4 )
                & ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) @ S5 ) @ ( sdrop_2468034597186633064ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_55_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc780299800278082923e_rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream3588992565182678257e_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream3588992565182678257e_rule,Steps: stream7224824493335796183e_rule,M: nat] :
      ( ( abstra3632413238247259647at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra2308351064978115840e_rule @ ( abstra3792383586331528633at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S5 @ S4 )
                & ( abstra2308351064978115840e_rule @ ( abstra3792383586331528633at_nat @ Eff @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_7505857448455835194e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_56_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,M: nat] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S5 @ S4 )
                & ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_57_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule,Steps: stream1878305877988316353m_rule,M: nat] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: rule] :
                ( ( member_rule @ S5 @ S4 )
                & ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_1764812713384251940m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_58_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc1996495991257130529ist_fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream8299795917829157543ist_fm,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream8299795917829157543ist_fm,Steps: stream1864985929934468237ist_fm,M: nat] :
      ( ( abstra2697211723746494921at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra3364502726910231222ist_fm @ ( abstra1231203810028718467at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S5 @ S4 )
                & ( abstra3364502726910231222ist_fm @ ( abstra1231203810028718467at_nat @ Eff @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) @ S5 ) @ ( sdrop_6343787060873212912ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_59_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc780299800278082923e_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream3588992565182678257e_rule,Steps: stream54883477001725111e_rule,M: nat] :
      ( ( abstra2302509236375889695ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra7728999744517701600e_rule @ ( abstra8979215030598005465ist_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S5 @ S4 )
                & ( abstra7728999744517701600e_rule @ ( abstra8979215030598005465ist_fm @ Eff @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S5 ) @ ( sdrop_8506487140862065946e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_60_wf__mkTree,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra4476735159789068068ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ ( abstra178918799166095863ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 ) ) ) ) ).

% wf_mkTree
thf(fact_61_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule,R2: rule] :
      ( ( abstra8115150628584006924e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra6372423936129899802e_rule @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_62_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: product_prod_nat_nat > rule > fset_rule > $o,Rules: stream6724221391990029191at_nat,S4: set_rule,S3: rule,Rs: stream6724221391990029191at_nat,Steps: stream2767582176000522113at_nat,R2: product_prod_nat_nat] :
      ( ( abstra5596191532008451270t_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1870481761539534101at_nat @ Rules @ Rs )
         => ( ( abstra7582228187744779010at_nat @ ( abstra5213681959199510051t_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member8440522571783428010at_nat @ R2 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ( abstra530515779721846072t_rule @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_63_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule,Steps: stream1780317496394867717t_rule,R2: rule] :
      ( ( abstra3487881890568480634at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3533331058055637382t_rule @ ( abstra3105372317759539415at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra7645578175136651244at_nat @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_64_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm,Steps: stream1960312990768957601ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra1314873687758482406m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( abstra679888793666381912m_rule @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_65_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream6724221391990029191at_nat,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream6724221391990029191at_nat,Steps: stream8372878641218411373at_nat,R2: product_prod_nat_nat] :
      ( ( abstra3727192663846114624at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1870481761539534101at_nat @ Rules @ Rs )
         => ( ( abstra1157841331345613206at_nat @ ( abstra8972947971889770147at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member8440522571783428010at_nat @ R2 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ( abstra951271245706845262at_nat @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_66_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra7252176601000738572ist_fm @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_67_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream6724221391990029191at_nat,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream6724221391990029191at_nat,Steps: stream5916762827556431437at_nat,R2: product_prod_nat_nat] :
      ( ( abstra1630140784663835744ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1870481761539534101at_nat @ Rules @ Rs )
         => ( ( abstra5495853630007432310at_nat @ ( abstra7362676963793170371ist_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member8440522571783428010at_nat @ R2 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ( abstra7290900729481441134ist_fm @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_68_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: produc859450856879609959at_nat > rule > fset_rule > $o,Rules: stream8372878641218411373at_nat,S4: set_rule,S3: rule,Rs: stream8372878641218411373at_nat,Steps: stream7497242749156405735at_nat,R2: produc859450856879609959at_nat] :
      ( ( abstra7673653355469228972t_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules @ Rs )
         => ( ( abstra3018042991430833768at_nat @ ( abstra6341243307501620617t_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member8206827879206165904at_nat @ R2 @ ( sset_P9164763103296375419at_nat @ Rules ) )
             => ( abstra436705010757945630t_rule @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_69_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: produc1996495991257130529ist_fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream8299795917829157543ist_fm,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream8299795917829157543ist_fm,Steps: stream1864985929934468237ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra4722039667754159648at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra3364502726910231222ist_fm @ ( abstra1231203810028718467at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( abstra1159427575716989230at_nat @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_70_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule,Steps: stream1370203160359889003t_rule,R2: rule] :
      ( ( abstra1526291686923892064at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra4530403282039525100t_rule @ ( abstra193881638956283709at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra3512715379067384530at_nat @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_71_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule] :
      ( ( abstra8115150628584006924e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ( abstra4228102177642093818e_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_72_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule,Steps: stream1370203160359889003t_rule] :
      ( ( abstra1526291686923892064at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra4530403282039525100t_rule @ ( abstra193881638956283709at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ( abstra7052252741274667762at_nat @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_73_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule,Steps: stream1780317496394867717t_rule] :
      ( ( abstra3487881890568480634at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3533331058055637382t_rule @ ( abstra3105372317759539415at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ( abstra248010401270669324at_nat @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_74_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > produc3843707927480180839at_nat > fset_P2467486768528901581at_nat > $o,Rules: stream_rule,S4: set_Pr4329608150637261639at_nat,S3: produc3843707927480180839at_nat,Rs: stream_rule,Steps: stream7984548576854626411t_rule] :
      ( ( abstra6634888898293167456at_nat @ Eff @ Rules @ S4 )
     => ( ( member8757157785044589968at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2686194991016466156t_rule @ ( abstra2952254667104538429at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ( abstra6469075505003657970at_nat @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_75_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ( abstra8647779336736900908ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_76_PersistentRuleSystem_Ointro,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstra2778581409224112183ist_fm @ Eff @ Rules @ S4 )
       => ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S4 ) ) ) ).

% PersistentRuleSystem.intro
thf(fact_77_PersistentRuleSystem__def,axiom,
    ( abstra7887161495092839066ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
          ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S6 )
          & ( abstra2778581409224112183ist_fm @ Eff2 @ Rules2 @ S6 ) ) ) ) ).

% PersistentRuleSystem_def
thf(fact_78_RuleSystem__Defs_Ofair_Ocong,axiom,
    abstra1114877437032970289r_rule = abstra1114877437032970289r_rule ).

% RuleSystem_Defs.fair.cong
thf(fact_79_top__set__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm @ top_to6798613745154499404t_fm_o ) ) ).

% top_set_def
thf(fact_80_top__set__def,axiom,
    ( top_top_set_rule
    = ( collect_rule @ top_top_rule_o ) ) ).

% top_set_def
thf(fact_81_sdrop__fair,axiom,
    ! [Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% sdrop_fair
thf(fact_82_i_Osdrop__fair,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( sdrop_7224736112439592940m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_83_i_Osdrop__fair,axiom,
    ! [Rules: stream8299795917829157543ist_fm,Rs: stream8299795917829157543ist_fm,M: nat] :
      ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
     => ( abstra7454370078233524789ist_fm @ Rules @ ( sdrop_4442373711808556042ist_fm @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_84_i_Osdrop__fair,axiom,
    ! [Rules: stream3588992565182678257e_rule,Rs: stream3588992565182678257e_rule,M: nat] :
      ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
     => ( abstra4837122928373946751e_rule @ Rules @ ( sdrop_2199339672054592340e_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_85_i_Osdrop__fair,axiom,
    ! [Rules: stream1878305877988316353m_rule,Rs: stream1878305877988316353m_rule,M: nat] :
      ( ( abstra5219219999209536975m_rule @ Rules @ Rs )
     => ( abstra5219219999209536975m_rule @ Rules @ ( sdrop_1764812713384251940m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_86_i_Osdrop__fair,axiom,
    ! [Rules: stream2312013117288958913e_rule,Rs: stream2312013117288958913e_rule,M: nat] :
      ( ( abstra623789749151397071e_rule @ Rules @ Rs )
     => ( abstra623789749151397071e_rule @ Rules @ ( sdrop_6392754500180887844e_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_87_i_Osdrop__fair,axiom,
    ! [Rules: stream6494289010434245521m_rule,Rs: stream6494289010434245521m_rule,M: nat] :
      ( ( abstra7961590535338393887m_rule @ Rules @ Rs )
     => ( abstra7961590535338393887m_rule @ Rules @ ( sdrop_7373388980841566196m_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_88_i_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_89_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( sdrop_7224736112439592940m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_90_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream8299795917829157543ist_fm,Rs: stream8299795917829157543ist_fm,M: nat] :
      ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
     => ( abstra7454370078233524789ist_fm @ Rules @ ( sdrop_4442373711808556042ist_fm @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_91_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream3588992565182678257e_rule,Rs: stream3588992565182678257e_rule,M: nat] :
      ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
     => ( abstra4837122928373946751e_rule @ Rules @ ( sdrop_2199339672054592340e_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_92_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream1878305877988316353m_rule,Rs: stream1878305877988316353m_rule,M: nat] :
      ( ( abstra5219219999209536975m_rule @ Rules @ Rs )
     => ( abstra5219219999209536975m_rule @ Rules @ ( sdrop_1764812713384251940m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_93_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream2312013117288958913e_rule,Rs: stream2312013117288958913e_rule,M: nat] :
      ( ( abstra623789749151397071e_rule @ Rules @ Rs )
     => ( abstra623789749151397071e_rule @ Rules @ ( sdrop_6392754500180887844e_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_94_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream6494289010434245521m_rule,Rs: stream6494289010434245521m_rule,M: nat] :
      ( ( abstra7961590535338393887m_rule @ Rules @ Rs )
     => ( abstra7961590535338393887m_rule @ Rules @ ( sdrop_7373388980841566196m_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_95_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_96_UNIV__rules,axiom,
    ( ( sset_rule @ rules )
    = top_top_set_rule ) ).

% UNIV_rules
thf(fact_97_UNIV__witness,axiom,
    ? [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ top_to8454300755403694231at_nat ) ).

% UNIV_witness
thf(fact_98_UNIV__witness,axiom,
    ? [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ top_to4669805908274784177at_nat ) ).

% UNIV_witness
thf(fact_99_UNIV__witness,axiom,
    ? [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ top_to6833984726390702231at_nat ) ).

% UNIV_witness
thf(fact_100_UNIV__witness,axiom,
    ? [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ top_to728902761071950545ist_fm ) ).

% UNIV_witness
thf(fact_101_UNIV__witness,axiom,
    ? [X: rule] : ( member_rule @ X @ top_top_set_rule ) ).

% UNIV_witness
thf(fact_102_mem__Collect__eq,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ A @ ( collec7231575776666803724ist_fm @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
    ! [A: rule,P: rule > $o] :
      ( ( member_rule @ A @ ( collect_rule @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_107_Collect__mem__eq,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] :
      ( ( collec7231575776666803724ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_108_Collect__mem__eq,axiom,
    ! [A2: set_rule] :
      ( ( collect_rule
        @ ^ [X3: rule] : ( member_rule @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_112_UNIV__eq__I,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ! [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A2 )
     => ( top_to8454300755403694231at_nat = A2 ) ) ).

% UNIV_eq_I
thf(fact_113_UNIV__eq__I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 )
     => ( top_to4669805908274784177at_nat = A2 ) ) ).

% UNIV_eq_I
thf(fact_114_UNIV__eq__I,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ! [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ A2 )
     => ( top_to6833984726390702231at_nat = A2 ) ) ).

% UNIV_eq_I
thf(fact_115_UNIV__eq__I,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A2 )
     => ( top_to728902761071950545ist_fm = A2 ) ) ).

% UNIV_eq_I
thf(fact_116_UNIV__eq__I,axiom,
    ! [A2: set_rule] :
      ( ! [X: rule] : ( member_rule @ X @ A2 )
     => ( top_top_set_rule = A2 ) ) ).

% UNIV_eq_I
thf(fact_117_PersistentRuleSystem_Oaxioms_I2_J,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S4 )
     => ( abstra2778581409224112183ist_fm @ Eff @ Rules @ S4 ) ) ).

% PersistentRuleSystem.axioms(2)
thf(fact_118_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra5602818997816140546e_rule @ Eff @ Rules @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_119_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1446413340331638506at_nat @ Eff @ Rules @ ( abstra193881638956283709at_nat @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_120_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra8675576244713875460at_nat @ Eff @ Rules @ ( abstra3105372317759539415at_nat @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_121_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > produc3843707927480180839at_nat > fset_P2467486768528901581at_nat > $o,Rules: stream_rule,S4: set_Pr4329608150637261639at_nat,S3: produc3843707927480180839at_nat,Rs: stream_rule] :
      ( ( abstra5319030069223283063at_nat @ Eff @ Rules @ S4 )
     => ( ( member8757157785044589968at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra5755226028053868266at_nat @ Eff @ Rules @ ( abstra2952254667104538429at_nat @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_122_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_123_UNIV__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm
      @ ^ [X3: produc1996495991257130529ist_fm] : $true ) ) ).

% UNIV_def
thf(fact_124_UNIV__def,axiom,
    ( top_top_set_rule
    = ( collect_rule
      @ ^ [X3: rule] : $true ) ) ).

% UNIV_def
thf(fact_125_PersistentRuleSystem__axioms__def,axiom,
    ( abstra2778581409224112183ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
        ! [R: rule] :
          ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( abstra4153889242452539894ist_fm @ Eff2 @ Rules2 @ S6 @ R ) ) ) ) ).

% PersistentRuleSystem_axioms_def
thf(fact_126_PersistentRuleSystem__axioms_Ointro,axiom,
    ! [Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,S4: set_Pr7058068377845519745ist_fm] :
      ( ! [R3: rule] :
          ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
         => ( abstra4153889242452539894ist_fm @ Eff @ Rules @ S4 @ R3 ) )
     => ( abstra2778581409224112183ist_fm @ Eff @ Rules @ S4 ) ) ).

% PersistentRuleSystem_axioms.intro
thf(fact_127_fair__uu__def,axiom,
    abstra1913438439042302728u_rule = abstra1114877437032970289r_rule ).

% fair_uu_def
thf(fact_128_minWait__ex,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ? [N2: nat] :
            ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) )
            @ S3 ) ) ) ).

% minWait_ex
thf(fact_129_trim__fair,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra1114877437032970289r_rule @ rules
          @ ( abstra2363873759530810705ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 ) ) ) ) ).

% trim_fair
thf(fact_130_eff__S,axiom,
    ! [S3: produc1996495991257130529ist_fm,R2: rule,Sl3: fset_P661503646757059847ist_fm,S7: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
       => ( ( ( eff @ R2 @ S3 )
            = ( some_f2162114139285431020ist_fm @ Sl3 ) )
         => ( ( fmembe3381613331217039976ist_fm @ S7 @ Sl3 )
           => ( member8102475879199740618ist_fm @ S7 @ top_to728902761071950545ist_fm ) ) ) ) ) ).

% eff_S
thf(fact_131_pickEff,axiom,
    ! [R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ S3 )
     => ( ( eff @ R2 @ S3 )
        = ( some_f2162114139285431020ist_fm
          @ ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 ) ) ) ) ).

% pickEff
thf(fact_132_trim__alt,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2363873759530810705ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 )
          = ( sdrop_rule
            @ ( abstra738319790293016300ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Rs ) ) ) ) ).

% trim_alt
thf(fact_133_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3154316310710804774e_rule @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_134_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc780299800278082923e_rule > rule > fset_rule > $o,Rules: stream3588992565182678257e_rule,S4: set_rule,S3: rule,Rs: stream3588992565182678257e_rule] :
      ( ( abstra1742398642700549575e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra5553215051456775830e_rule @ Eff @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_135_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3147191065466128608at_nat @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_136_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ? [N2: nat] : ( abstra5706625008861892940m_rule @ Eff @ ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_137_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc780299800278082923e_rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream3588992565182678257e_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream3588992565182678257e_rule] :
      ( ( abstra3632413238247259647at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3867684735323121520at_nat @ Eff @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_138_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3055540779341473792ist_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_139_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc1996495991257130529ist_fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream8299795917829157543ist_fm,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2697211723746494921at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ? [N2: nat] : ( abstra2321963264980577082at_nat @ Eff @ ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_140_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc780299800278082923e_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra2302509236375889695ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra1049073205711939216ist_fm @ Eff @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_141_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra2524698237922071414e_rule @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_142_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra248173851283269062at_nat @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_143_fair__fenum,axiom,
    abstra1114877437032970289r_rule @ rules @ ( abstra7284221463285775110m_rule @ rules ) ).

% fair_fenum
thf(fact_144_sset__fenum,axiom,
    ( ( sset_rule @ ( abstra7284221463285775110m_rule @ rules ) )
    = ( sset_rule @ rules ) ) ).

% sset_fenum
thf(fact_145_Stream_Osmember__def,axiom,
    ( smembe2449974647026456671ist_fm
    = ( ^ [X3: produc1996495991257130529ist_fm,S: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X3 @ ( sset_P3431374123360452149ist_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_146_Stream_Osmember__def,axiom,
    ( smembe1741262129464730021at_nat
    = ( ^ [X3: produc859450856879609959at_nat,S: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ X3 @ ( sset_P9164763103296375419at_nat @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_147_Stream_Osmember__def,axiom,
    ( smembe7612037963498647359at_nat
    = ( ^ [X3: product_prod_nat_nat,S: stream6724221391990029191at_nat] : ( member8440522571783428010at_nat @ X3 @ ( sset_P8584168253256249621at_nat @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_148_Stream_Osmember__def,axiom,
    ( smembe7284261217439725989at_nat
    = ( ^ [X3: produc3843707927480180839at_nat,S: stream8032061917732916077at_nat] : ( member8757157785044589968at_nat @ X3 @ ( sset_P1195531240007660155at_nat @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_149_Stream_Osmember__def,axiom,
    ( smember_rule
    = ( ^ [X3: rule,S: stream_rule] : ( member_rule @ X3 @ ( sset_rule @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_150_i_OSaturated__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps )
      = ( ! [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
           => ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ X3 @ Steps ) ) ) ) ).

% i.Saturated_def
thf(fact_151_RuleSystem__Defs_OpickEff_Ocong,axiom,
    abstra458490714785879965ist_fm = abstra458490714785879965ist_fm ).

% RuleSystem_Defs.pickEff.cong
thf(fact_152_RuleSystem__Defs_Otrim_Ocong,axiom,
    abstra2363873759530810705ist_fm = abstra2363873759530810705ist_fm ).

% RuleSystem_Defs.trim.cong
thf(fact_153_RuleSystem_OminWait_Ocong,axiom,
    abstra738319790293016300ist_fm = abstra738319790293016300ist_fm ).

% RuleSystem.minWait.cong
thf(fact_154_RuleSystem__Defs_Ofenum_Ocong,axiom,
    abstra7284221463285775110m_rule = abstra7284221463285775110m_rule ).

% RuleSystem_Defs.fenum.cong
thf(fact_155_i_OpickEff,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S3 )
     => ( abstra5459341389059251418ist_fm @ Eff3 @ R2 @ S3 @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S3 ) ) ) ).

% i.pickEff
thf(fact_156_shd__sset,axiom,
    ! [A: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) @ ( sset_P3431374123360452149ist_fm @ A ) ) ).

% shd_sset
thf(fact_157_shd__sset,axiom,
    ! [A: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ A ) @ ( sset_P9164763103296375419at_nat @ A ) ) ).

% shd_sset
thf(fact_158_shd__sset,axiom,
    ! [A: stream6724221391990029191at_nat] : ( member8440522571783428010at_nat @ ( shd_Pr4260400998323988397at_nat @ A ) @ ( sset_P8584168253256249621at_nat @ A ) ) ).

% shd_sset
thf(fact_159_shd__sset,axiom,
    ! [A: stream8032061917732916077at_nat] : ( member8757157785044589968at_nat @ ( shd_Pr4412363897699689747at_nat @ A ) @ ( sset_P1195531240007660155at_nat @ A ) ) ).

% shd_sset
thf(fact_160_shd__sset,axiom,
    ! [A: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) @ ( sset_P4979403222752902913m_rule @ A ) ) ).

% shd_sset
thf(fact_161_shd__sset,axiom,
    ! [A: stream_rule] : ( member_rule @ ( shd_rule @ A ) @ ( sset_rule @ A ) ) ).

% shd_sset
thf(fact_162_i_Oenabled__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S3 )
      = ( ? [X4: fset_P661503646757059847ist_fm] : ( abstra5459341389059251418ist_fm @ Eff3 @ R2 @ S3 @ X4 ) ) ) ).

% i.enabled_def
thf(fact_163_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_164_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: product_prod_nat_nat > rule > fset_rule > $o,Rules: stream6724221391990029191at_nat,S4: set_rule,S3: rule,Rs: stream6724221391990029191at_nat] :
      ( ( abstra1542008646636103517t_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1870481761539534101at_nat @ Rules @ Rs )
         => ( member8440522571783428010at_nat @ ( shd_Pr4260400998323988397at_nat @ ( abstra596093905173040509t_rule @ Eff @ Rs @ S3 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_165_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra7711156300587845681at_nat @ Eff @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_166_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs @ S3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_167_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream6724221391990029191at_nat,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream6724221391990029191at_nat] :
      ( ( abstra7719573905946618601at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1870481761539534101at_nat @ Rules @ Rs )
         => ( member8440522571783428010at_nat @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff @ Rs @ S3 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_168_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_169_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream6724221391990029191at_nat,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream6724221391990029191at_nat] :
      ( ( abstra8828684877510946825ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1870481761539534101at_nat @ Rules @ Rs )
         => ( member8440522571783428010at_nat @ ( shd_Pr4260400998323988397at_nat @ ( abstra3441982087891796457ist_fm @ Eff @ Rs @ S3 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_170_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: produc859450856879609959at_nat > rule > fset_rule > $o,Rules: stream8372878641218411373at_nat,S4: set_rule,S3: rule,Rs: stream8372878641218411373at_nat] :
      ( ( abstra2579607477568694211t_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5618069685180253819at_nat @ Rules @ Rs )
         => ( member8206827879206165904at_nat @ ( shd_Pr8412153233960533267at_nat @ ( abstra361488207848709603t_rule @ Eff @ Rs @ S3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_171_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: produc1996495991257130529ist_fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream8299795917829157543ist_fm,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2697211723746494921at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( abstra6533880970982120361at_nat @ Eff @ Rs @ S3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_172_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_173_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra7200325324992684464ist_fm @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_174_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra2524698237922071414e_rule @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_175_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream727092118206550309m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream727092118206550309m_rule] :
      ( ( abstra8123243527896255399at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra1800717694132746102at_nat @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra276577985667043783at_nat @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_176_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream727092118206550309m_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream727092118206550309m_rule] :
      ( ( abstra1119292201781699649at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra4483692673511907984at_nat @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra1939759854238243425at_nat @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_177_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > produc3843707927480180839at_nat > fset_P2467486768528901581at_nat > $o,Rules: stream727092118206550309m_rule,S4: set_Pr4329608150637261639at_nat,S3: produc3843707927480180839at_nat,Rs: stream727092118206550309m_rule] :
      ( ( abstra1694321584795128743at_nat @ Eff @ Rules @ S4 )
     => ( ( member8757157785044589968at_nat @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra63700206635173750at_nat @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra9104697165897128391at_nat @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_178_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra3154316310710804774e_rule @ Eff @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_179_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra248173851283269062at_nat @ Eff @ ( shd_rule @ ( abstra3437498576158148503at_nat @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_180_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra3147191065466128608at_nat @ Eff @ ( shd_rule @ ( abstra7711156300587845681at_nat @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_181_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > produc3843707927480180839at_nat > fset_P2467486768528901581at_nat > $o,Rules: stream_rule,S4: set_Pr4329608150637261639at_nat,S3: produc3843707927480180839at_nat,Rs: stream_rule] :
      ( ( abstra5319030069223283063at_nat @ Eff @ Rules @ S4 )
     => ( ( member8757157785044589968at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra9154741779836564934at_nat @ Eff @ ( shd_rule @ ( abstra1101146469844389783at_nat @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_182_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra3055540779341473792ist_fm @ Eff @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_183_RuleSystem__Defs_OpickEff,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ Eff @ R2 @ S3 )
     => ( Eff @ R2 @ S3 @ ( abstra458490714785879965ist_fm @ Eff @ R2 @ S3 ) ) ) ).

% RuleSystem_Defs.pickEff
thf(fact_184_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,R2: rule,Sl3: fset_rule,S7: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S7 @ Sl3 )
             => ( member_rule @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_185_RuleSystem_Oeff__S,axiom,
    ! [Eff: product_prod_nat_nat > rule > fset_rule > $o,Rules: stream6724221391990029191at_nat,S4: set_rule,S3: rule,R2: product_prod_nat_nat,Sl3: fset_rule,S7: rule] :
      ( ( abstra1542008646636103517t_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member8440522571783428010at_nat @ R2 @ ( sset_P8584168253256249621at_nat @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S7 @ Sl3 )
             => ( member_rule @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_186_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,R2: rule,Sl3: fset_P5670320511379867111at_nat,S7: product_prod_nat_nat] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmembe1449479052724974408at_nat @ S7 @ Sl3 )
             => ( member8440522571783428010at_nat @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_187_RuleSystem_Oeff__S,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,R2: produc1996495991257130529ist_fm,Sl3: fset_rule,S7: rule] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S7 @ Sl3 )
             => ( member_rule @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_188_RuleSystem_Oeff__S,axiom,
    ! [Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream6724221391990029191at_nat,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,R2: product_prod_nat_nat,Sl3: fset_P5670320511379867111at_nat,S7: product_prod_nat_nat] :
      ( ( abstra7719573905946618601at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( member8440522571783428010at_nat @ R2 @ ( sset_P8584168253256249621at_nat @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmembe1449479052724974408at_nat @ S7 @ Sl3 )
             => ( member8440522571783428010at_nat @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_189_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,R2: rule,Sl3: fset_P661503646757059847ist_fm,S7: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmembe3381613331217039976ist_fm @ S7 @ Sl3 )
             => ( member8102475879199740618ist_fm @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_190_RuleSystem_Oeff__S,axiom,
    ! [Eff: produc859450856879609959at_nat > rule > fset_rule > $o,Rules: stream8372878641218411373at_nat,S4: set_rule,S3: rule,R2: produc859450856879609959at_nat,Sl3: fset_rule,S7: rule] :
      ( ( abstra2579607477568694211t_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member8206827879206165904at_nat @ R2 @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S7 @ Sl3 )
             => ( member_rule @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_191_RuleSystem_Oeff__S,axiom,
    ! [Eff: produc1996495991257130529ist_fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream8299795917829157543ist_fm,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,R2: produc1996495991257130529ist_fm,Sl3: fset_P5670320511379867111at_nat,S7: product_prod_nat_nat] :
      ( ( abstra2697211723746494921at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmembe1449479052724974408at_nat @ S7 @ Sl3 )
             => ( member8440522571783428010at_nat @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_192_RuleSystem_Oeff__S,axiom,
    ! [Eff: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream6724221391990029191at_nat,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,R2: product_prod_nat_nat,Sl3: fset_P661503646757059847ist_fm,S7: produc1996495991257130529ist_fm] :
      ( ( abstra8828684877510946825ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( member8440522571783428010at_nat @ R2 @ ( sset_P8584168253256249621at_nat @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmembe3381613331217039976ist_fm @ S7 @ Sl3 )
             => ( member8102475879199740618ist_fm @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_193_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,R2: rule,Sl3: fset_P3619101297660450253at_nat,S7: produc859450856879609959at_nat] :
      ( ( abstra5655617845878133111at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmembe6431293532144391214at_nat @ S7 @ Sl3 )
             => ( member8206827879206165904at_nat @ S7 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_194_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream_rule,Eff: rule > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule @ S8 @ S4 )
         => ! [R3: rule] :
              ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: rule] :
                      ( ( fmember_rule @ S5 @ Sl2 )
                     => ( member_rule @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule @ S8 @ S4 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_195_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream6724221391990029191at_nat,Eff: product_prod_nat_nat > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule @ S8 @ S4 )
         => ! [R3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ R3 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: rule] :
                      ( ( fmember_rule @ S5 @ Sl2 )
                     => ( member_rule @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule @ S8 @ S4 )
           => ? [X6: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X6 @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra1542008646636103517t_rule @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_196_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr1261947904930325089at_nat,Rules: stream_rule,Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o] :
      ( ! [S8: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ S8 @ S4 )
         => ! [R3: rule] :
              ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_P5670320511379867111at_nat] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: product_prod_nat_nat] :
                      ( ( fmembe1449479052724974408at_nat @ S5 @ Sl2 )
                     => ( member8440522571783428010at_nat @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ S8 @ S4 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_P5670320511379867111at_nat] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_197_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule @ S8 @ S4 )
         => ! [R3: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: rule] :
                      ( ( fmember_rule @ S5 @ Sl2 )
                     => ( member_rule @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule @ S8 @ S4 )
           => ? [X6: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X6 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra9106359784214577277m_rule @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_198_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr1261947904930325089at_nat,Rules: stream6724221391990029191at_nat,Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o] :
      ( ! [S8: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ S8 @ S4 )
         => ! [R3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ R3 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ! [Sl2: fset_P5670320511379867111at_nat] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: product_prod_nat_nat] :
                      ( ( fmembe1449479052724974408at_nat @ S5 @ Sl2 )
                     => ( member8440522571783428010at_nat @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ S8 @ S4 )
           => ? [X6: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X6 @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ? [X_12: fset_P5670320511379867111at_nat] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra7719573905946618601at_nat @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_199_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr7058068377845519745ist_fm,Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ! [S8: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ S8 @ S4 )
         => ! [R3: rule] :
              ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_P661503646757059847ist_fm] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: produc1996495991257130529ist_fm] :
                      ( ( fmembe3381613331217039976ist_fm @ S5 @ Sl2 )
                     => ( member8102475879199740618ist_fm @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ S8 @ S4 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_P661503646757059847ist_fm] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_200_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream8372878641218411373at_nat,Eff: produc859450856879609959at_nat > rule > fset_rule > $o] :
      ( ! [S8: rule] :
          ( ( member_rule @ S8 @ S4 )
         => ! [R3: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ R3 @ ( sset_P9164763103296375419at_nat @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: rule] :
                      ( ( fmember_rule @ S5 @ Sl2 )
                     => ( member_rule @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: rule] :
            ( ( member_rule @ S8 @ S4 )
           => ? [X6: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X6 @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra2579607477568694211t_rule @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_201_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr1261947904930325089at_nat,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o] :
      ( ! [S8: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ S8 @ S4 )
         => ! [R3: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ! [Sl2: fset_P5670320511379867111at_nat] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: product_prod_nat_nat] :
                      ( ( fmembe1449479052724974408at_nat @ S5 @ Sl2 )
                     => ( member8440522571783428010at_nat @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ S8 @ S4 )
           => ? [X6: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X6 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ? [X_12: fset_P5670320511379867111at_nat] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra2697211723746494921at_nat @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_202_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr7058068377845519745ist_fm,Rules: stream6724221391990029191at_nat,Eff: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ! [S8: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ S8 @ S4 )
         => ! [R3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ R3 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ! [Sl2: fset_P661503646757059847ist_fm] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: produc1996495991257130529ist_fm] :
                      ( ( fmembe3381613331217039976ist_fm @ S5 @ Sl2 )
                     => ( member8102475879199740618ist_fm @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ S8 @ S4 )
           => ? [X6: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X6 @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ? [X_12: fset_P661503646757059847ist_fm] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra8828684877510946825ist_fm @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_203_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr8693737435421807431at_nat,Rules: stream_rule,Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o] :
      ( ! [S8: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ S8 @ S4 )
         => ! [R3: rule] :
              ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_P3619101297660450253at_nat] :
                  ( ( Eff @ R3 @ S8 @ Sl2 )
                 => ! [S5: produc859450856879609959at_nat] :
                      ( ( fmembe6431293532144391214at_nat @ S5 @ Sl2 )
                     => ( member8206827879206165904at_nat @ S5 @ S4 ) ) ) ) )
     => ( ! [S8: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ S8 @ S4 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_P3619101297660450253at_nat] : ( Eff @ X6 @ S8 @ X_12 ) ) )
       => ( abstra5655617845878133111at_nat @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_204_RuleSystem__def,axiom,
    ( abstra2725590884229999157e_rule
    = ( ^ [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl )
                         => ( member_rule @ S9 @ S6 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ? [X3: rule] :
                  ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
                  & ? [X4: fset_rule] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_205_RuleSystem__def,axiom,
    ( abstra1542008646636103517t_rule
    = ( ^ [Eff2: product_prod_nat_nat > rule > fset_rule > $o,Rules2: stream6724221391990029191at_nat,S6: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ! [R: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ R @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl )
                         => ( member_rule @ S9 @ S6 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ? [X3: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X3 @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                  & ? [X4: fset_rule] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_206_RuleSystem__def,axiom,
    ( abstra8657071042050908689at_nat
    = ( ^ [Eff2: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules2: stream_rule,S6: set_Pr1261947904930325089at_nat] :
          ( ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S6 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_P5670320511379867111at_nat] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: product_prod_nat_nat] :
                          ( ( fmembe1449479052724974408at_nat @ S9 @ Sl )
                         => ( member8440522571783428010at_nat @ S9 @ S6 ) ) ) ) )
          & ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S6 )
             => ? [X3: rule] :
                  ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
                  & ? [X4: fset_P5670320511379867111at_nat] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_207_RuleSystem__def,axiom,
    ( abstra9106359784214577277m_rule
    = ( ^ [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S6: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ! [R: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl )
                         => ( member_rule @ S9 @ S6 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ? [X3: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X3 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                  & ? [X4: fset_rule] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_208_RuleSystem__def,axiom,
    ( abstra7719573905946618601at_nat
    = ( ^ [Eff2: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules2: stream6724221391990029191at_nat,S6: set_Pr1261947904930325089at_nat] :
          ( ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S6 )
             => ! [R: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ R @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                 => ! [Sl: fset_P5670320511379867111at_nat] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: product_prod_nat_nat] :
                          ( ( fmembe1449479052724974408at_nat @ S9 @ Sl )
                         => ( member8440522571783428010at_nat @ S9 @ S6 ) ) ) ) )
          & ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S6 )
             => ? [X3: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X3 @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                  & ? [X4: fset_P5670320511379867111at_nat] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_209_RuleSystem__def,axiom,
    ( abstra6455275554694158129ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
          ( ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S6 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_P661503646757059847ist_fm] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: produc1996495991257130529ist_fm] :
                          ( ( fmembe3381613331217039976ist_fm @ S9 @ Sl )
                         => ( member8102475879199740618ist_fm @ S9 @ S6 ) ) ) ) )
          & ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S6 )
             => ? [X3: rule] :
                  ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
                  & ? [X4: fset_P661503646757059847ist_fm] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_210_RuleSystem__def,axiom,
    ( abstra2579607477568694211t_rule
    = ( ^ [Eff2: produc859450856879609959at_nat > rule > fset_rule > $o,Rules2: stream8372878641218411373at_nat,S6: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ! [R: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ R @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: rule] :
                          ( ( fmember_rule @ S9 @ Sl )
                         => ( member_rule @ S9 @ S6 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ? [X3: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X3 @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
                  & ? [X4: fset_rule] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_211_RuleSystem__def,axiom,
    ( abstra2697211723746494921at_nat
    = ( ^ [Eff2: produc1996495991257130529ist_fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules2: stream8299795917829157543ist_fm,S6: set_Pr1261947904930325089at_nat] :
          ( ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S6 )
             => ! [R: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                 => ! [Sl: fset_P5670320511379867111at_nat] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: product_prod_nat_nat] :
                          ( ( fmembe1449479052724974408at_nat @ S9 @ Sl )
                         => ( member8440522571783428010at_nat @ S9 @ S6 ) ) ) ) )
          & ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S6 )
             => ? [X3: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X3 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                  & ? [X4: fset_P5670320511379867111at_nat] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_212_RuleSystem__def,axiom,
    ( abstra8828684877510946825ist_fm
    = ( ^ [Eff2: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream6724221391990029191at_nat,S6: set_Pr7058068377845519745ist_fm] :
          ( ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S6 )
             => ! [R: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ R @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                 => ! [Sl: fset_P661503646757059847ist_fm] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: produc1996495991257130529ist_fm] :
                          ( ( fmembe3381613331217039976ist_fm @ S9 @ Sl )
                         => ( member8102475879199740618ist_fm @ S9 @ S6 ) ) ) ) )
          & ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S6 )
             => ? [X3: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X3 @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                  & ? [X4: fset_P661503646757059847ist_fm] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_213_RuleSystem__def,axiom,
    ( abstra5655617845878133111at_nat
    = ( ^ [Eff2: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules2: stream_rule,S6: set_Pr8693737435421807431at_nat] :
          ( ! [S: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ S @ S6 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_P3619101297660450253at_nat] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S9: produc859450856879609959at_nat] :
                          ( ( fmembe6431293532144391214at_nat @ S9 @ Sl )
                         => ( member8206827879206165904at_nat @ S9 @ S6 ) ) ) ) )
          & ! [S: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ S @ S6 )
             => ? [X3: rule] :
                  ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
                  & ? [X4: fset_P3619101297660450253at_nat] : ( Eff2 @ X3 @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_214_i_Osset__fenum,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
      = ( sset_rule @ Rules ) ) ).

% i.sset_fenum
thf(fact_215_RuleSystem__Defs_Osset__fenum,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
      = ( sset_rule @ Rules ) ) ).

% RuleSystem_Defs.sset_fenum
thf(fact_216_RuleSystem__Defs_Ofair__fenum,axiom,
    ! [Rules: stream_rule] : ( abstra1114877437032970289r_rule @ Rules @ ( abstra7284221463285775110m_rule @ Rules ) ) ).

% RuleSystem_Defs.fair_fenum
thf(fact_217_i_Ofair__fenum,axiom,
    ! [Rules: stream_rule] : ( abstra1114877437032970289r_rule @ Rules @ ( abstra7284221463285775110m_rule @ Rules ) ) ).

% i.fair_fenum
thf(fact_218_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra5682536783049438522e_rule @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_219_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc780299800278082923e_rule > rule > fset_rule > $o,Rules: stream3588992565182678257e_rule,S4: set_rule,S3: rule,Rs: stream3588992565182678257e_rule] :
      ( ( abstra1742398642700549575e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra5027552996074955751e_rule @ Eff @ Rs @ S3 )
            = ( sdrop_2199339672054592340e_rule @ ( abstra5810752615369301378e_rule @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_220_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7711156300587845681at_nat @ Eff @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra4889254606063302604at_nat @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_221_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S4: set_rule,S3: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra5014957989051229853m_rule @ Eff @ Rs @ S3 )
            = ( sdrop_4442373711808556042ist_fm @ ( abstra3389404019813435448m_rule @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_222_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc780299800278082923e_rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream3588992565182678257e_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream3588992565182678257e_rule] :
      ( ( abstra3632413238247259647at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra4280238386875961311at_nat @ Eff @ Rs @ S3 )
            = ( sdrop_2199339672054592340e_rule @ ( abstra3878188824996679044at_nat @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_223_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra738319790293016300ist_fm @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_224_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra6412047235177443722e_rule @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_225_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc1996495991257130529ist_fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream8299795917829157543ist_fm,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2697211723746494921at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra6533880970982120361at_nat @ Eff @ Rs @ S3 )
            = ( sdrop_4442373711808556042ist_fm @ ( abstra7140573849050296654at_nat @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_226_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc780299800278082923e_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra2302509236375889695ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra3813788296048736511ist_fm @ Eff @ Rs @ S3 )
            = ( sdrop_2199339672054592340e_rule @ ( abstra4936422202967311524ist_fm @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_227_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3437498576158148503at_nat @ Eff @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra4176240314836206258at_nat @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_228_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_229_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra3437498576158148503at_nat @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_230_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra7711156300587845681at_nat @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_231_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > produc3843707927480180839at_nat > fset_P2467486768528901581at_nat > $o,Rules: stream_rule,S4: set_Pr4329608150637261639at_nat,S3: produc3843707927480180839at_nat,Rs: stream_rule] :
      ( ( abstra5319030069223283063at_nat @ Eff @ Rules @ S4 )
     => ( ( member8757157785044589968at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra1101146469844389783at_nat @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_232_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_233_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
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) )
          @ ( sset_rule @ rules ) ) ) ) ).

% trim_in_R
thf(fact_234_trim__enabled,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( abstra3055540779341473792ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) )
          @ S3 ) ) ) ).

% trim_enabled
thf(fact_235_i_Owf__ipath__epath,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,T: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T )
     => ( ( abstra2285467793620430502m_rule @ T @ Steps )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ).

% i.wf_ipath_epath
thf(fact_236_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_237_saturated__effG__uu__def,axiom,
    ( abstra7873564507807625473ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% saturated_effG_uu_def
thf(fact_238_mkTree__effG__uu__def,axiom,
    ( abstra3917079402916513406ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% mkTree_effG_uu_def
thf(fact_239_RuleSystem_Opos,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R2 ) @ Rs ) )
            = R2 ) ) ) ) ).

% RuleSystem.pos
thf(fact_240_pos,axiom,
    ! [Rs: stream_rule,R2: rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
       => ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R2 ) @ Rs ) )
          = R2 ) ) ) ).

% pos
thf(fact_241_fenum__uu__def,axiom,
    abstra8121460317444261103u_rule = abstra7284221463285775110m_rule ).

% fenum_uu_def
thf(fact_242_minWait__def,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra738319790293016300ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( ord_Least_nat
        @ ^ [N3: nat] :
            ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
            @ S3 ) ) ) ).

% minWait_def
thf(fact_243_minWait__least,axiom,
    ! [N: nat,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
        @ S3 )
     => ( ord_less_eq_nat
        @ ( abstra738319790293016300ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 )
        @ N ) ) ).

% minWait_least
thf(fact_244_top__empty__eq,axiom,
    ( top_to1313005431924226822_nat_o
    = ( ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq
thf(fact_245_top__empty__eq,axiom,
    ( top_to3137496036531963500_nat_o
    = ( ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq
thf(fact_246_top__empty__eq,axiom,
    ( top_to1113057434437417734_nat_o
    = ( ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ top_to6833984726390702231at_nat ) ) ) ).

% top_empty_eq
thf(fact_247_top__empty__eq,axiom,
    ( top_to6798613745154499404t_fm_o
    = ( ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq
thf(fact_248_top__empty__eq,axiom,
    ( top_top_rule_o
    = ( ^ [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ) ) ).

% top_empty_eq
thf(fact_249_trim__def,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra2363873759530810705ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( sdrop_while_rule
        @ ^ [R: rule] :
            ~ ( abstra3055540779341473792ist_fm
              @ ^ [S: rule,T2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ S @ T2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R
              @ S3 )
        @ Rs ) ) ).

% trim_def
thf(fact_250_order__refl,axiom,
    ! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).

% order_refl
thf(fact_251_order__refl,axiom,
    ! [X5: set_rule] : ( ord_less_eq_set_rule @ X5 @ X5 ) ).

% order_refl
thf(fact_252_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_253_dual__order_Orefl,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ A ) ).

% dual_order.refl
thf(fact_254_Least1I,axiom,
    ! [P: nat > $o] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y: nat] :
              ( ( P @ Y )
             => ( ord_less_eq_nat @ X6 @ Y ) )
          & ! [Y: nat] :
              ( ( ( P @ Y )
                & ! [Ya: nat] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_nat @ Y @ Ya ) ) )
             => ( Y = X6 ) ) )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% Least1I
thf(fact_255_Least1I,axiom,
    ! [P: set_rule > $o] :
      ( ? [X6: set_rule] :
          ( ( P @ X6 )
          & ! [Y: set_rule] :
              ( ( P @ Y )
             => ( ord_less_eq_set_rule @ X6 @ Y ) )
          & ! [Y: set_rule] :
              ( ( ( P @ Y )
                & ! [Ya: set_rule] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_set_rule @ Y @ Ya ) ) )
             => ( Y = X6 ) ) )
     => ( P @ ( ord_Least_set_rule @ P ) ) ) ).

% Least1I
thf(fact_256_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_257_Least1__le,axiom,
    ! [P: nat > $o,Z: nat] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y: nat] :
              ( ( P @ Y )
             => ( ord_less_eq_nat @ X6 @ Y ) )
          & ! [Y: nat] :
              ( ( ( P @ Y )
                & ! [Ya: nat] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_nat @ Y @ Ya ) ) )
             => ( Y = X6 ) ) )
     => ( ( P @ Z )
       => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ Z ) ) ) ).

% Least1_le
thf(fact_258_Least1__le,axiom,
    ! [P: set_rule > $o,Z: set_rule] :
      ( ? [X6: set_rule] :
          ( ( P @ X6 )
          & ! [Y: set_rule] :
              ( ( P @ Y )
             => ( ord_less_eq_set_rule @ X6 @ Y ) )
          & ! [Y: set_rule] :
              ( ( ( P @ Y )
                & ! [Ya: set_rule] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_set_rule @ Y @ Ya ) ) )
             => ( Y = X6 ) ) )
     => ( ( P @ Z )
       => ( ord_less_eq_set_rule @ ( ord_Least_set_rule @ P ) @ Z ) ) ) ).

% Least1_le
thf(fact_259_LeastI2,axiom,
    ! [P: nat > $o,A: nat,Q: nat > $o] :
      ( ( P @ A )
     => ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2
thf(fact_260_le__cases3,axiom,
    ! [X5: nat,Y3: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X5 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X5 )
         => ~ ( ord_less_eq_nat @ X5 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X5 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X5 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X5 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X5 )
                 => ~ ( ord_less_eq_nat @ X5 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_261_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [X3: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_262_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_rule,Z2: set_rule] : ( Y4 = Z2 ) )
    = ( ^ [X3: set_rule,Y5: set_rule] :
          ( ( ord_less_eq_set_rule @ X3 @ Y5 )
          & ( ord_less_eq_set_rule @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_263_Least__le,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ K ) ) ).

% Least_le
thf(fact_264_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_265_ord__eq__le__trans,axiom,
    ! [A: set_rule,B: set_rule,C: set_rule] :
      ( ( A = B )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ord_less_eq_set_rule @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_266_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_267_ord__le__eq__trans,axiom,
    ! [A: set_rule,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_rule @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_268_LeastI2__order,axiom,
    ! [P: nat > $o,X5: nat,Q: nat > $o] :
      ( ( P @ X5 )
     => ( ! [Y: nat] :
            ( ( P @ Y )
           => ( ord_less_eq_nat @ X5 @ Y ) )
       => ( ! [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_269_LeastI2__order,axiom,
    ! [P: set_rule > $o,X5: set_rule,Q: set_rule > $o] :
      ( ( P @ X5 )
     => ( ! [Y: set_rule] :
            ( ( P @ Y )
           => ( ord_less_eq_set_rule @ X5 @ Y ) )
       => ( ! [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_270_order__antisym,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X5 )
       => ( X5 = Y3 ) ) ) ).

% order_antisym
thf(fact_271_order__antisym,axiom,
    ! [X5: set_rule,Y3: set_rule] :
      ( ( ord_less_eq_set_rule @ X5 @ Y3 )
     => ( ( ord_less_eq_set_rule @ Y3 @ X5 )
       => ( X5 = Y3 ) ) ) ).

% order_antisym
thf(fact_272_LeastI__ex,axiom,
    ! [P: nat > $o] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI_ex
thf(fact_273_Least__equality,axiom,
    ! [P: nat > $o,X5: nat] :
      ( ( P @ X5 )
     => ( ! [Y: nat] :
            ( ( P @ Y )
           => ( ord_less_eq_nat @ X5 @ Y ) )
       => ( ( ord_Least_nat @ P )
          = X5 ) ) ) ).

% Least_equality
thf(fact_274_Least__equality,axiom,
    ! [P: set_rule > $o,X5: set_rule] :
      ( ( P @ X5 )
     => ( ! [Y: set_rule] :
            ( ( P @ Y )
           => ( ord_less_eq_set_rule @ X5 @ Y ) )
       => ( ( ord_Least_set_rule @ P )
          = X5 ) ) ) ).

% Least_equality
thf(fact_275_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_276_order_Otrans,axiom,
    ! [A: set_rule,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ord_less_eq_set_rule @ A @ C ) ) ) ).

% order.trans
thf(fact_277_order__trans,axiom,
    ! [X5: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_eq_nat @ X5 @ Z ) ) ) ).

% order_trans
thf(fact_278_order__trans,axiom,
    ! [X5: set_rule,Y3: set_rule,Z: set_rule] :
      ( ( ord_less_eq_set_rule @ X5 @ Y3 )
     => ( ( ord_less_eq_set_rule @ Y3 @ Z )
       => ( ord_less_eq_set_rule @ X5 @ Z ) ) ) ).

% order_trans
thf(fact_279_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_280_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: nat,B2: nat] :
            ( ( P @ B2 @ A3 )
           => ( P @ A3 @ B2 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_281_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_282_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_rule,Z2: set_rule] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( ( ord_less_eq_set_rule @ B3 @ A4 )
          & ( ord_less_eq_set_rule @ A4 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_283_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_284_dual__order_Oantisym,axiom,
    ! [B: set_rule,A: set_rule] :
      ( ( ord_less_eq_set_rule @ B @ A )
     => ( ( ord_less_eq_set_rule @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_285_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_286_dual__order_Otrans,axiom,
    ! [B: set_rule,A: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ B @ A )
     => ( ( ord_less_eq_set_rule @ C @ B )
       => ( ord_less_eq_set_rule @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_287_LeastI2__wellorder,axiom,
    ! [P: nat > $o,A: nat,Q: nat > $o] :
      ( ( P @ A )
     => ( ! [A3: nat] :
            ( ( P @ A3 )
           => ( ! [B4: nat] :
                  ( ( P @ B4 )
                 => ( ord_less_eq_nat @ A3 @ B4 ) )
             => ( Q @ A3 ) ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_wellorder
thf(fact_288_LeastI2__wellorder__ex,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [A3: nat] :
            ( ( P @ A3 )
           => ( ! [B4: nat] :
                  ( ( P @ B4 )
                 => ( ord_less_eq_nat @ A3 @ B4 ) )
             => ( Q @ A3 ) ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_wellorder_ex
thf(fact_289_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_290_antisym,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_291_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_292_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_rule,Z2: set_rule] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( ( ord_less_eq_set_rule @ A4 @ B3 )
          & ( ord_less_eq_set_rule @ B3 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_293_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_294_order__subst1,axiom,
    ! [A: nat,F: set_rule > nat,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_295_order__subst1,axiom,
    ! [A: set_rule,F: nat > set_rule,B: nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_296_order__subst1,axiom,
    ! [A: set_rule,F: set_rule > set_rule,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_297_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_298_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_set_rule @ ( F @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_299_order__subst2,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_300_order__subst2,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ ( F @ B ) @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_301_order__eq__refl,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( X5 = Y3 )
     => ( ord_less_eq_nat @ X5 @ Y3 ) ) ).

% order_eq_refl
thf(fact_302_order__eq__refl,axiom,
    ! [X5: set_rule,Y3: set_rule] :
      ( ( X5 = Y3 )
     => ( ord_less_eq_set_rule @ X5 @ Y3 ) ) ).

% order_eq_refl
thf(fact_303_linorder__linear,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X5 ) ) ).

% linorder_linear
thf(fact_304_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_305_ord__eq__le__subst,axiom,
    ! [A: set_rule,F: nat > set_rule,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_306_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_rule > nat,B: set_rule,C: set_rule] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_307_ord__eq__le__subst,axiom,
    ! [A: set_rule,F: set_rule > set_rule,B: set_rule,C: set_rule] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_308_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_309_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_310_ord__le__eq__subst,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_311_ord__le__eq__subst,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_312_linorder__le__cases,axiom,
    ! [X5: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X5 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X5 ) ) ).

% linorder_le_cases
thf(fact_313_order__antisym__conv,axiom,
    ! [Y3: nat,X5: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X5 )
     => ( ( ord_less_eq_nat @ X5 @ Y3 )
        = ( X5 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_314_order__antisym__conv,axiom,
    ! [Y3: set_rule,X5: set_rule] :
      ( ( ord_less_eq_set_rule @ Y3 @ X5 )
     => ( ( ord_less_eq_set_rule @ X5 @ Y3 )
        = ( X5 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_315_LeastI,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI
thf(fact_316_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
     => ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_uniqueI
thf(fact_317_top_Oextremum__uniqueI,axiom,
    ! [A: set_rule] :
      ( ( ord_less_eq_set_rule @ top_top_set_rule @ A )
     => ( A = top_top_set_rule ) ) ).

% top.extremum_uniqueI
thf(fact_318_top_Oextremum__unique,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
      = ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_unique
thf(fact_319_top_Oextremum__unique,axiom,
    ! [A: set_rule] :
      ( ( ord_less_eq_set_rule @ top_top_set_rule @ A )
      = ( A = top_top_set_rule ) ) ).

% top.extremum_unique
thf(fact_320_top__greatest,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A @ top_to728902761071950545ist_fm ) ).

% top_greatest
thf(fact_321_top__greatest,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ top_top_set_rule ) ).

% top_greatest
thf(fact_322_pos__least,axiom,
    ! [N: nat,Rs: stream8299795917829157543ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra3513618393749879589ist_fm @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_323_pos__least,axiom,
    ! [N: nat,Rs: stream3588992565182678257e_rule,R2: produc780299800278082923e_rule] :
      ( ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra883788664900704879e_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_324_pos__least,axiom,
    ! [N: nat,Rs: stream1878305877988316353m_rule,R2: produc1660549859252256699m_rule] :
      ( ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra7431954940529686207m_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_325_pos__least,axiom,
    ! [N: nat,Rs: stream2312013117288958913e_rule,R2: produc6288491646048892603e_rule] :
      ( ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra2836524690471546303e_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_326_pos__least,axiom,
    ! [N: nat,Rs: stream6494289010434245521m_rule,R2: produc3793571666044074251m_rule] :
      ( ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra3193765552906301967m_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_327_pos__least,axiom,
    ! [N: nat,Rs: stream727092118206550309m_rule,R2: produc164195504107695125m_rule] :
      ( ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstra1882453022147397137m_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_328_pos__least,axiom,
    ! [N: nat,Rs: stream_rule,R2: rule] :
      ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
        = R2 )
     => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R2 ) @ N ) ) ).

% pos_least
thf(fact_329_pos__def,axiom,
    ( abstra3513618393749879589ist_fm
    = ( ^ [Rs2: stream8299795917829157543ist_fm,R: produc1996495991257130529ist_fm] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_330_pos__def,axiom,
    ( abstra883788664900704879e_rule
    = ( ^ [Rs2: stream3588992565182678257e_rule,R: produc780299800278082923e_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_331_pos__def,axiom,
    ( abstra7431954940529686207m_rule
    = ( ^ [Rs2: stream1878305877988316353m_rule,R: produc1660549859252256699m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_332_pos__def,axiom,
    ( abstra2836524690471546303e_rule
    = ( ^ [Rs2: stream2312013117288958913e_rule,R: produc6288491646048892603e_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_333_pos__def,axiom,
    ( abstra3193765552906301967m_rule
    = ( ^ [Rs2: stream6494289010434245521m_rule,R: produc3793571666044074251m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_334_pos__def,axiom,
    ( abstra1882453022147397137m_rule
    = ( ^ [Rs2: stream727092118206550309m_rule,R: produc164195504107695125m_rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_335_pos__def,axiom,
    ( abstract_pos_rule
    = ( ^ [Rs2: stream_rule,R: rule] :
          ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs2 ) )
              = R ) ) ) ) ).

% pos_def
thf(fact_336_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream8299795917829157543ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra3513618393749879589ist_fm @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_337_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream3588992565182678257e_rule,R2: produc780299800278082923e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra883788664900704879e_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_338_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream1878305877988316353m_rule,R2: produc1660549859252256699m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra7431954940529686207m_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_339_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream2312013117288958913e_rule,R2: produc6288491646048892603e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra2836524690471546303e_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_340_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream6494289010434245521m_rule,R2: produc3793571666044074251m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra3193765552906301967m_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_341_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream727092118206550309m_rule,R2: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra1882453022147397137m_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_342_RuleSystem_Opos__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_343_RuleSystem_Opos__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream8299795917829157543ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstra3513618393749879589ist_fm @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_344_RuleSystem_Opos__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream3588992565182678257e_rule,R2: produc780299800278082923e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstra883788664900704879e_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_345_RuleSystem_Opos__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream1878305877988316353m_rule,R2: produc1660549859252256699m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstra7431954940529686207m_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_346_RuleSystem_Opos__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream2312013117288958913e_rule,R2: produc6288491646048892603e_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstra2836524690471546303e_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_347_RuleSystem_Opos__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream6494289010434245521m_rule,R2: produc3793571666044074251m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstra3193765552906301967m_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_348_RuleSystem_Opos__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream727092118206550309m_rule,R2: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstra1882453022147397137m_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_349_RuleSystem_Opos__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstract_pos_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_350_RuleSystem_OminWait__le__pos,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( abstra3055540779341473792ist_fm @ Eff @ R2 @ S3 )
           => ( ord_less_eq_nat @ ( abstra738319790293016300ist_fm @ Eff @ Rs @ S3 ) @ ( abstract_pos_rule @ Rs @ R2 ) ) ) ) ) ) ).

% RuleSystem.minWait_le_pos
thf(fact_351_RuleSystem__Defs_Otrim__def,axiom,
    ( abstra2363873759530810705ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] :
          ( sdrop_while_rule
          @ ^ [R: rule] :
              ~ ( abstra3055540779341473792ist_fm @ Eff2 @ R @ S )
          @ Rs2 ) ) ) ).

% RuleSystem_Defs.trim_def
thf(fact_352_minWait__le__pos,axiom,
    ! [Rs: stream_rule,R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
       => ( ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 )
         => ( ord_less_eq_nat
            @ ( abstra738319790293016300ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ ( abstract_pos_rule @ Rs @ R2 ) ) ) ) ) ).

% minWait_le_pos
thf(fact_353_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
        @ ^ [R: rule] :
            ~ ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S3 )
        @ Rs ) ) ).

% i.trim_def
thf(fact_354_RuleSystem_OminWait__least,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( 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_355_RuleSystem_OminWait__def,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( 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_356_per__rules_H,axiom,
    ! [R2: rule,A2: list_fm,B5: list_fm,A5: list_fm,B6: list_fm,R4: rule,Ss2: fset_P661503646757059847ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ ( produc381145313068854617ist_fm @ A2 @ B5 ) )
     => ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ ( produc381145313068854617ist_fm @ A5 @ B6 ) )
       => ( ( ( eff @ R4 @ ( produc381145313068854617ist_fm @ A2 @ B5 ) )
            = ( some_f2162114139285431020ist_fm @ Ss2 ) )
         => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A5 @ B6 ) @ Ss2 )
           => ( R4 = R2 ) ) ) ) ) ).

% per_rules'
thf(fact_357_in__cont__mkTree,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,T3: abstra4572470701423150929m_rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( fmembe1635983029874454474m_rule @ T3
            @ ( abstra4962253831105676628m_rule
              @ ( abstra178918799166095863ist_fm
                @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S3 ) ) )
         => ? [Sl4: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S5 @ top_to728902761071950545ist_fm )
              & ( ( eff
                  @ ( shd_rule
                    @ ( abstra2363873759530810705ist_fm
                      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R @ S )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ Rs
                      @ S3 ) )
                  @ S3 )
                = ( some_f2162114139285431020ist_fm @ Sl4 ) )
              & ( fmembe3381613331217039976ist_fm @ S5 @ Sl4 )
              & ( T3
                = ( abstra178918799166095863ist_fm
                  @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ ( stl_rule
                    @ ( abstra2363873759530810705ist_fm
                      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R @ S )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ Rs
                      @ S3 ) )
                  @ S5 ) ) ) ) ) ) ).

% in_cont_mkTree
thf(fact_358_GreatestI2__order,axiom,
    ! [P: set_rule > $o,X5: set_rule,Q: set_rule > $o] :
      ( ( P @ X5 )
     => ( ! [Y: set_rule] :
            ( ( P @ Y )
           => ( ord_less_eq_set_rule @ Y @ X5 ) )
       => ( ! [X: set_rule] :
              ( ( P @ X )
             => ( ! [Y6: set_rule] :
                    ( ( P @ Y6 )
                   => ( ord_less_eq_set_rule @ Y6 @ X ) )
               => ( Q @ X ) ) )
         => ( Q @ ( order_8905881487648567394t_rule @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_359_GreatestI2__order,axiom,
    ! [P: nat > $o,X5: nat,Q: nat > $o] :
      ( ( P @ X5 )
     => ( ! [Y: nat] :
            ( ( P @ Y )
           => ( ord_less_eq_nat @ Y @ X5 ) )
       => ( ! [X: nat] :
              ( ( P @ X )
             => ( ! [Y6: nat] :
                    ( ( P @ Y6 )
                   => ( ord_less_eq_nat @ Y6 @ X ) )
               => ( Q @ X ) ) )
         => ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_360_Greatest__equality,axiom,
    ! [P: set_rule > $o,X5: set_rule] :
      ( ( P @ X5 )
     => ( ! [Y: set_rule] :
            ( ( P @ Y )
           => ( ord_less_eq_set_rule @ Y @ X5 ) )
       => ( ( order_8905881487648567394t_rule @ P )
          = X5 ) ) ) ).

% Greatest_equality
thf(fact_361_Greatest__equality,axiom,
    ! [P: nat > $o,X5: nat] :
      ( ( P @ X5 )
     => ( ! [Y: nat] :
            ( ( P @ Y )
           => ( ord_less_eq_nat @ Y @ X5 ) )
       => ( ( order_Greatest_nat @ P )
          = X5 ) ) ) ).

% Greatest_equality
thf(fact_362_eqfelem__imp__iff,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( X5 = Y3 )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
        = ( fmembe3381613331217039976ist_fm @ Y3 @ A2 ) ) ) ).

% eqfelem_imp_iff
thf(fact_363_eqfelem__imp__iff,axiom,
    ! [X5: abstra4572470701423150929m_rule,Y3: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( X5 = Y3 )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
        = ( fmembe1635983029874454474m_rule @ Y3 @ A2 ) ) ) ).

% eqfelem_imp_iff
thf(fact_364_if__split__fmem2,axiom,
    ! [A: produc1996495991257130529ist_fm,Q: $o,X5: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A @ ( if_fse3714977293194272717ist_fm @ Q @ X5 @ Y3 ) )
      = ( ( Q
         => ( fmembe3381613331217039976ist_fm @ A @ X5 ) )
        & ( ~ Q
         => ( fmembe3381613331217039976ist_fm @ A @ Y3 ) ) ) ) ).

% if_split_fmem2
thf(fact_365_if__split__fmem2,axiom,
    ! [A: abstra4572470701423150929m_rule,Q: $o,X5: fset_A5396818216090053761m_rule,Y3: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ A @ ( if_fse3370020817956878651m_rule @ Q @ X5 @ Y3 ) )
      = ( ( Q
         => ( fmembe1635983029874454474m_rule @ A @ X5 ) )
        & ( ~ Q
         => ( fmembe1635983029874454474m_rule @ A @ Y3 ) ) ) ) ).

% if_split_fmem2
thf(fact_366_if__split__fmem1,axiom,
    ! [Q: $o,X5: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ ( if_Pro3930376587665744871ist_fm @ Q @ X5 @ Y3 ) @ B )
      = ( ( Q
         => ( fmembe3381613331217039976ist_fm @ X5 @ B ) )
        & ( ~ Q
         => ( fmembe3381613331217039976ist_fm @ Y3 @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_367_if__split__fmem1,axiom,
    ! [Q: $o,X5: abstra4572470701423150929m_rule,Y3: abstra4572470701423150929m_rule,B: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ ( if_Abs4566652138351786251m_rule @ Q @ X5 @ Y3 ) @ B )
      = ( ( Q
         => ( fmembe1635983029874454474m_rule @ X5 @ B ) )
        & ( ~ Q
         => ( fmembe1635983029874454474m_rule @ Y3 @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_368_eqfset__imp__iff,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,X5: produc1996495991257130529ist_fm] :
      ( ( A2 = B5 )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
        = ( fmembe3381613331217039976ist_fm @ X5 @ B5 ) ) ) ).

% eqfset_imp_iff
thf(fact_369_eqfset__imp__iff,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,X5: abstra4572470701423150929m_rule] :
      ( ( A2 = B5 )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
        = ( fmembe1635983029874454474m_rule @ X5 @ B5 ) ) ) ).

% eqfset_imp_iff
thf(fact_370_fsubsetI,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A2 )
         => ( fmembe3381613331217039976ist_fm @ X @ B5 ) )
     => ( ord_le2064643713053750439ist_fm @ A2 @ B5 ) ) ).

% fsubsetI
thf(fact_371_fsubsetI,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A2 )
         => ( fmembe1635983029874454474m_rule @ X @ B5 ) )
     => ( ord_le6951374404268211425m_rule @ A2 @ B5 ) ) ).

% fsubsetI
thf(fact_372_pred__subset__eq2,axiom,
    ! [R5: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R5 )
        @ ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ S4 ) )
      = ( ord_le3000389064537975527at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_373_pred__subset__eq2,axiom,
    ! [R5: set_Pr7058068377845519745ist_fm,S4: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le1774903256965159270t_fm_o
        @ ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ R5 )
        @ ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ S4 ) )
      = ( ord_le2055072402069232929ist_fm @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_374_pred__subset__eq2,axiom,
    ! [R5: set_Pr7459493094073627847at_nat,S4: set_Pr7459493094073627847at_nat] :
      ( ( ord_le3072208448688395470_nat_o
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R5 )
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ S4 ) )
      = ( ord_le5997549366648089703at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_375_pred__subset__eq2,axiom,
    ! [R5: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R5 )
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ S4 ) )
      = ( ord_le1268244103169919719at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_376_pred__subset__eq2,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R5 )
        @ ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_377_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ S3 ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_378_subrelI,axiom,
    ! [R2: set_Pr7058068377845519745ist_fm,S3: set_Pr7058068377845519745ist_fm] :
      ( ! [X: list_fm,Y: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ R2 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ S3 ) )
     => ( ord_le2055072402069232929ist_fm @ R2 @ S3 ) ) ).

% subrelI
thf(fact_379_subrelI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S3: set_Pr7459493094073627847at_nat] :
      ( ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ S3 ) )
     => ( ord_le5997549366648089703at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_380_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ S3 ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_381_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_382_pred__equals__eq2,axiom,
    ! [R5: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ R5 ) )
        = ( ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_383_pred__equals__eq2,axiom,
    ! [R5: set_Pr7058068377845519745ist_fm,S4: set_Pr7058068377845519745ist_fm] :
      ( ( ( ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ R5 ) )
        = ( ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_384_pred__equals__eq2,axiom,
    ! [R5: set_Pr7459493094073627847at_nat,S4: set_Pr7459493094073627847at_nat] :
      ( ( ( ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ R5 ) )
        = ( ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_385_pred__equals__eq2,axiom,
    ! [R5: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ R5 ) )
        = ( ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_386_pred__equals__eq2,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ R5 ) )
        = ( ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_387_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_388_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_389_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_390_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_391_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_392_fin__mono,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,X5: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A2 @ B5 )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
       => ( fmembe3381613331217039976ist_fm @ X5 @ B5 ) ) ) ).

% fin_mono
thf(fact_393_fin__mono,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,X5: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A2 @ B5 )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
       => ( fmembe1635983029874454474m_rule @ X5 @ B5 ) ) ) ).

% fin_mono
thf(fact_394_fsubsetD,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A2 @ B5 )
     => ( ( fmembe3381613331217039976ist_fm @ C @ A2 )
       => ( fmembe3381613331217039976ist_fm @ C @ B5 ) ) ) ).

% fsubsetD
thf(fact_395_fsubsetD,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A2 @ B5 )
     => ( ( fmembe1635983029874454474m_rule @ C @ A2 )
       => ( fmembe1635983029874454474m_rule @ C @ B5 ) ) ) ).

% fsubsetD
thf(fact_396_stream_Ocoinduct__strong,axiom,
    ! [R5: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R5 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R5 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( ( R5 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) )
                | ( ( stl_rule @ Stream3 )
                  = ( stl_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_397_stream_Ocoinduct__strong,axiom,
    ! [R5: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R5 @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R5 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( ( R5 @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) )
                | ( ( stl_Pr950425576149878629m_rule @ Stream3 )
                  = ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_398_stream_Ocoinduct,axiom,
    ! [R5: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R5 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R5 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( R5 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_399_stream_Ocoinduct,axiom,
    ! [R5: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R5 @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R5 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( R5 @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_400_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_401_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_402_stl__sset,axiom,
    ! [X5: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm] :
      ( ( member8102475879199740618ist_fm @ X5 @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) )
     => ( member8102475879199740618ist_fm @ X5 @ ( sset_P3431374123360452149ist_fm @ A ) ) ) ).

% stl_sset
thf(fact_403_stl__sset,axiom,
    ! [X5: produc859450856879609959at_nat,A: stream8372878641218411373at_nat] :
      ( ( member8206827879206165904at_nat @ X5 @ ( sset_P9164763103296375419at_nat @ ( stl_Pr5010465336888994071at_nat @ A ) ) )
     => ( member8206827879206165904at_nat @ X5 @ ( sset_P9164763103296375419at_nat @ A ) ) ) ).

% stl_sset
thf(fact_404_stl__sset,axiom,
    ! [X5: product_prod_nat_nat,A: stream6724221391990029191at_nat] :
      ( ( member8440522571783428010at_nat @ X5 @ ( sset_P8584168253256249621at_nat @ ( stl_Pr5027827701538482609at_nat @ A ) ) )
     => ( member8440522571783428010at_nat @ X5 @ ( sset_P8584168253256249621at_nat @ A ) ) ) ).

% stl_sset
thf(fact_405_stl__sset,axiom,
    ! [X5: produc3843707927480180839at_nat,A: stream8032061917732916077at_nat] :
      ( ( member8757157785044589968at_nat @ X5 @ ( sset_P1195531240007660155at_nat @ ( stl_Pr6836137205557096727at_nat @ A ) ) )
     => ( member8757157785044589968at_nat @ X5 @ ( sset_P1195531240007660155at_nat @ A ) ) ) ).

% stl_sset
thf(fact_406_stl__sset,axiom,
    ! [X5: produc164195504107695125m_rule,A: stream727092118206550309m_rule] :
      ( ( member4220325220686508332m_rule @ X5 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) )
     => ( member4220325220686508332m_rule @ X5 @ ( sset_P4979403222752902913m_rule @ A ) ) ) ).

% stl_sset
thf(fact_407_stl__sset,axiom,
    ! [X5: rule,A: stream_rule] :
      ( ( member_rule @ X5 @ ( sset_rule @ ( stl_rule @ A ) ) )
     => ( member_rule @ X5 @ ( sset_rule @ A ) ) ) ).

% stl_sset
thf(fact_408_sdrop__stl,axiom,
    ! [N: nat,S3: stream8299795917829157543ist_fm] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( stl_Pr5027778045650968273ist_fm @ S3 ) )
      = ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_409_sdrop__stl,axiom,
    ! [N: nat,S3: stream3588992565182678257e_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( stl_Pr4904654673476115995e_rule @ S3 ) )
      = ( stl_Pr4904654673476115995e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_410_sdrop__stl,axiom,
    ! [N: nat,S3: stream1878305877988316353m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N @ ( stl_Pr5425948690297933931m_rule @ S3 ) )
      = ( stl_Pr5425948690297933931m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_411_sdrop__stl,axiom,
    ! [N: nat,S3: stream2312013117288958913e_rule] :
      ( ( sdrop_6392754500180887844e_rule @ N @ ( stl_Pr830518440239794027e_rule @ S3 ) )
      = ( stl_Pr830518440239794027e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_412_sdrop__stl,axiom,
    ! [N: nat,S3: stream6494289010434245521m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( stl_Pr2200982372225425851m_rule @ S3 ) )
      = ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_413_sdrop__stl,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S3 ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_414_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_415_subset__UNIV,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A2 @ top_to728902761071950545ist_fm ) ).

% subset_UNIV
thf(fact_416_subset__UNIV,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ top_top_set_rule ) ).

% subset_UNIV
thf(fact_417_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_418_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_419_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_420_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_421_top__empty__eq2,axiom,
    ( top_to2659251575160290430_nat_o
    = ( ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y5 ) @ top_to8454300755403694231at_nat ) ) ) ).

% top_empty_eq2
thf(fact_422_top__empty__eq2,axiom,
    ( top_to1326704617074743934_nat_o
    = ( ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y5 ) @ top_to5281548843312042519at_nat ) ) ) ).

% top_empty_eq2
thf(fact_423_top__empty__eq2,axiom,
    ( top_to8903224817164384382_nat_o
    = ( ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y5 ) @ top_to6833984726390702231at_nat ) ) ) ).

% top_empty_eq2
thf(fact_424_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X3: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y5 ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_425_top__empty__eq2,axiom,
    ( top_to7326020031456794390t_fm_o
    = ( ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_426_sset__induct,axiom,
    ! [Y3: produc1996495991257130529ist_fm,S3: stream8299795917829157543ist_fm,P: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Y3 @ ( sset_P3431374123360452149ist_fm @ S3 ) )
     => ( ! [S8: stream8299795917829157543ist_fm] : ( P @ ( shd_Pr772355297128350925ist_fm @ S8 ) @ S8 )
       => ( ! [S8: stream8299795917829157543ist_fm,Y: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Y @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ S8 ) ) )
             => ( ( P @ Y @ ( stl_Pr5027778045650968273ist_fm @ S8 ) )
               => ( P @ Y @ S8 ) ) )
         => ( P @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_427_sset__induct,axiom,
    ! [Y3: produc859450856879609959at_nat,S3: stream8372878641218411373at_nat,P: produc859450856879609959at_nat > stream8372878641218411373at_nat > $o] :
      ( ( member8206827879206165904at_nat @ Y3 @ ( sset_P9164763103296375419at_nat @ S3 ) )
     => ( ! [S8: stream8372878641218411373at_nat] : ( P @ ( shd_Pr8412153233960533267at_nat @ S8 ) @ S8 )
       => ( ! [S8: stream8372878641218411373at_nat,Y: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Y @ ( sset_P9164763103296375419at_nat @ ( stl_Pr5010465336888994071at_nat @ S8 ) ) )
             => ( ( P @ Y @ ( stl_Pr5010465336888994071at_nat @ S8 ) )
               => ( P @ Y @ S8 ) ) )
         => ( P @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_428_sset__induct,axiom,
    ! [Y3: product_prod_nat_nat,S3: stream6724221391990029191at_nat,P: product_prod_nat_nat > stream6724221391990029191at_nat > $o] :
      ( ( member8440522571783428010at_nat @ Y3 @ ( sset_P8584168253256249621at_nat @ S3 ) )
     => ( ! [S8: stream6724221391990029191at_nat] : ( P @ ( shd_Pr4260400998323988397at_nat @ S8 ) @ S8 )
       => ( ! [S8: stream6724221391990029191at_nat,Y: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y @ ( sset_P8584168253256249621at_nat @ ( stl_Pr5027827701538482609at_nat @ S8 ) ) )
             => ( ( P @ Y @ ( stl_Pr5027827701538482609at_nat @ S8 ) )
               => ( P @ Y @ S8 ) ) )
         => ( P @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_429_sset__induct,axiom,
    ! [Y3: produc3843707927480180839at_nat,S3: stream8032061917732916077at_nat,P: produc3843707927480180839at_nat > stream8032061917732916077at_nat > $o] :
      ( ( member8757157785044589968at_nat @ Y3 @ ( sset_P1195531240007660155at_nat @ S3 ) )
     => ( ! [S8: stream8032061917732916077at_nat] : ( P @ ( shd_Pr4412363897699689747at_nat @ S8 ) @ S8 )
       => ( ! [S8: stream8032061917732916077at_nat,Y: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Y @ ( sset_P1195531240007660155at_nat @ ( stl_Pr6836137205557096727at_nat @ S8 ) ) )
             => ( ( P @ Y @ ( stl_Pr6836137205557096727at_nat @ S8 ) )
               => ( P @ Y @ S8 ) ) )
         => ( P @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_430_sset__induct,axiom,
    ! [Y3: produc164195504107695125m_rule,S3: stream727092118206550309m_rule,P: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ S3 ) )
     => ( ! [S8: stream727092118206550309m_rule] : ( P @ ( shd_Pr7235097944458474089m_rule @ S8 ) @ S8 )
       => ( ! [S8: stream727092118206550309m_rule,Y: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Y @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ S8 ) ) )
             => ( ( P @ Y @ ( stl_Pr950425576149878629m_rule @ S8 ) )
               => ( P @ Y @ S8 ) ) )
         => ( P @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_431_sset__induct,axiom,
    ! [Y3: rule,S3: stream_rule,P: rule > stream_rule > $o] :
      ( ( member_rule @ Y3 @ ( sset_rule @ S3 ) )
     => ( ! [S8: stream_rule] : ( P @ ( shd_rule @ S8 ) @ S8 )
       => ( ! [S8: stream_rule,Y: rule] :
              ( ( member_rule @ Y @ ( sset_rule @ ( stl_rule @ S8 ) ) )
             => ( ( P @ Y @ ( stl_rule @ S8 ) )
               => ( P @ Y @ S8 ) ) )
         => ( P @ Y3 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_432_fair__stl,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).

% fair_stl
thf(fact_433_sdrop__while_Osimps,axiom,
    ( sdrop_5541336384294084785m_rule
    = ( ^ [P2: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( if_str8948254419368749791m_rule @ ( P2 @ ( shd_Pr7235097944458474089m_rule @ S ) ) @ ( sdrop_5541336384294084785m_rule @ P2 @ ( stl_Pr950425576149878629m_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_434_sdrop__while_Osimps,axiom,
    ( sdrop_while_rule
    = ( ^ [P2: rule > $o,S: stream_rule] : ( if_stream_rule @ ( P2 @ ( shd_rule @ S ) ) @ ( sdrop_while_rule @ P2 @ ( stl_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_435_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_436_sset__sdrop,axiom,
    ! [N: nat,S3: stream8299795917829157543ist_fm] : ( ord_le2055072402069232929ist_fm @ ( sset_P3431374123360452149ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S3 ) ) @ ( sset_P3431374123360452149ist_fm @ S3 ) ) ).

% sset_sdrop
thf(fact_437_sset__sdrop,axiom,
    ! [N: nat,S3: stream3588992565182678257e_rule] : ( ord_le4515388646595166059e_rule @ ( sset_P4203794550272384383e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S3 ) ) @ ( sset_P4203794550272384383e_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_438_sset__sdrop,axiom,
    ! [N: nat,S3: stream1878305877988316353m_rule] : ( ord_le6218097786609347643m_rule @ ( sset_P8020534884870570447m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S3 ) ) @ ( sset_P8020534884870570447m_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_439_sset__sdrop,axiom,
    ! [N: nat,S3: stream2312013117288958913e_rule] : ( ord_le6651805025909990203e_rule @ ( sset_P3425104634812430543e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S3 ) ) @ ( sset_P3425104634812430543e_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_440_sset__sdrop,axiom,
    ! [N: nat,S3: stream6494289010434245521m_rule] : ( ord_le3684904531561281547m_rule @ ( sset_P270791984657859871m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S3 ) ) @ ( sset_P270791984657859871m_rule @ S3 ) ) ).

% sset_sdrop
thf(fact_441_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_442_szip_Osimps_I1_J,axiom,
    ! [S1: stream6724221391990029191at_nat,S2: stream6724221391990029191at_nat] :
      ( ( shd_Pr8412153233960533267at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) )
      = ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S1 ) @ ( shd_Pr4260400998323988397at_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_443_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_444_szip_Osimps_I1_J,axiom,
    ! [S1: stream3627166509194609229at_nat,S2: stream3627166509194609229at_nat] :
      ( ( shd_Pr943172039038359955at_nat @ ( szip_s1528810966652963738at_nat @ S1 @ S2 ) )
      = ( produc9060074326276436823at_nat @ ( shd_se2672623163517224947at_nat @ S1 ) @ ( shd_se2672623163517224947at_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_445_szip_Osimps_I1_J,axiom,
    ! [S1: stream7255200869668761831at_nat,S2: stream7255200869668761831at_nat] :
      ( ( shd_Pr4412363897699689747at_nat @ ( szip_s6495482949952704538at_nat @ S1 @ S2 ) )
      = ( produc2922128104949294807at_nat @ ( shd_se6818656645295851789at_nat @ S1 ) @ ( shd_se6818656645295851789at_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_446_szip_Osimps_I1_J,axiom,
    ! [S1: stream_nat,S2: stream_nat] :
      ( ( shd_Pr4260400998323988397at_nat @ ( szip_nat_nat @ S1 @ S2 ) )
      = ( product_Pair_nat_nat @ ( shd_nat @ S1 ) @ ( shd_nat @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_447_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_448_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_449_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_450_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_451_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_452_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,Rs: stream_rule,T3: abstra1071693965251717573e_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe5852619498312149900e_rule @ T3 @ ( abstra2651384284180525036e_rule @ ( abstra7443870075974203631e_rule @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_rule,S5: rule] :
                ( ( member_rule @ S5 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmember_rule @ S5 @ Sl4 )
                & ( T3
                  = ( abstra7443870075974203631e_rule @ Eff @ ( stl_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_453_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_rule,T3: abstra9001827361428689457t_rule] :
      ( ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe527247652794266794t_rule @ T3 @ ( abstra4795472920223927348t_rule @ ( abstra3105372317759539415at_nat @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_P5670320511379867111at_nat,S5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S5 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra7711156300587845681at_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe1449479052724974408at_nat @ S5 @ Sl4 )
                & ( T3
                  = ( abstra3105372317759539415at_nat @ Eff @ ( stl_rule @ ( abstra7711156300587845681at_nat @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_454_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_rule,T3: abstra4572470701423150929m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe1635983029874454474m_rule @ T3 @ ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S5 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe3381613331217039976ist_fm @ S5 @ Sl4 )
                & ( T3
                  = ( abstra178918799166095863ist_fm @ Eff @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_455_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S4: set_rule,S3: rule,Rs: stream727092118206550309m_rule,T3: abstra4791331342562632597m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe1981849019698968796m_rule @ T3 @ ( abstra5403976286072314556m_rule @ ( abstra1231468025953361855e_rule @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_rule,S5: rule] :
                ( ( member_rule @ S5 @ S4 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmember_rule @ S5 @ Sl4 )
                & ( T3
                  = ( abstra1231468025953361855e_rule @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_456_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream_rule,T3: abstra5175790144536110231t_rule] :
      ( ( abstra5655617845878133111at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe4064145320595246480t_rule @ T3 @ ( abstra553533796049368218t_rule @ ( abstra193881638956283709at_nat @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_P3619101297660450253at_nat,S5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S5 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra3437498576158148503at_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe6431293532144391214at_nat @ S5 @ Sl4 )
                & ( T3
                  = ( abstra193881638956283709at_nat @ Eff @ ( stl_rule @ ( abstra3437498576158148503at_nat @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_457_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_rule,S4: set_Ab3744519486624674951m_rule,S3: abstra4572470701423150929m_rule,Rs: stream_rule,T3: abstra4125161011840266193e_rule] :
      ( ( abstra6244190358723783233m_rule @ Eff @ Rules @ S4 )
     => ( ( member1590433669991072232m_rule @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe8010993199532995224e_rule @ T3 @ ( abstra6162512231279892984e_rule @ ( abstra2861652164185304827m_rule @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_A5396818216090053761m_rule,S5: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S5 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra1865002795007470625m_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe1635983029874454474m_rule @ S5 @ Sl4 )
                & ( T3
                  = ( abstra2861652164185304827m_rule @ Eff @ ( stl_rule @ ( abstra1865002795007470625m_rule @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_458_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream727092118206550309m_rule,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream727092118206550309m_rule,T3: abstra4998923703602266625m_rule] :
      ( ( abstra1119292201781699649at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe5371621603494965498m_rule @ T3 @ ( abstra8631692192530684676m_rule @ ( abstra5880731324333133319at_nat @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_P5670320511379867111at_nat,S5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S5 @ S4 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra1939759854238243425at_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe1449479052724974408at_nat @ S5 @ Sl4 )
                & ( T3
                  = ( abstra5880731324333133319at_nat @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra1939759854238243425at_nat @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_459_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > produc3843707927480180839at_nat > fset_P2467486768528901581at_nat > $o,Rules: stream_rule,S4: set_Pr4329608150637261639at_nat,S3: produc3843707927480180839at_nat,Rs: stream_rule,T3: abstra7387796307657911447t_rule] :
      ( ( abstra5319030069223283063at_nat @ Eff @ Rules @ S4 )
     => ( ( member8757157785044589968at_nat @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe1241436755953847696t_rule @ T3 @ ( abstra8250653879840296090t_rule @ ( abstra2952254667104538429at_nat @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_P2467486768528901581at_nat,S5: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ S5 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra1101146469844389783at_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe9188685285568734254at_nat @ S5 @ Sl4 )
                & ( T3
                  = ( abstra2952254667104538429at_nat @ Eff @ ( stl_rule @ ( abstra1101146469844389783at_nat @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_460_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule,T3: abstra4658698708444432161m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe2925761000581315610m_rule @ T3 @ ( abstra5924671108449608740m_rule @ ( abstra2161496378874750759ist_fm @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_P661503646757059847ist_fm,S5: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S5 @ S4 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe3381613331217039976ist_fm @ S5 @ Sl4 )
                & ( T3
                  = ( abstra2161496378874750759ist_fm @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra6201285654312316801ist_fm @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_461_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > produc859450856879609959at_nat > fset_P3619101297660450253at_nat > $o,Rules: stream727092118206550309m_rule,S4: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat,Rs: stream727092118206550309m_rule,T3: abstra8173387307635926887m_rule] :
      ( ( abstra8123243527896255399at_nat @ Eff @ Rules @ S4 )
     => ( ( member8206827879206165904at_nat @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe2908244859456314592m_rule @ T3 @ ( abstra8750048091207030378m_rule @ ( abstra8186850217458087021at_nat @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_P3619101297660450253at_nat,S5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S5 @ S4 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra276577985667043783at_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe6431293532144391214at_nat @ S5 @ Sl4 )
                & ( T3
                  = ( abstra8186850217458087021at_nat @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra276577985667043783at_nat @ Eff @ Rs @ S3 ) ) @ S5 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_462_fset__eqI,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A2 )
          = ( fmembe3381613331217039976ist_fm @ X @ B5 ) )
     => ( A2 = B5 ) ) ).

% fset_eqI
thf(fact_463_fset__eqI,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A2 )
          = ( fmembe1635983029874454474m_rule @ X @ B5 ) )
     => ( A2 = B5 ) ) ).

% fset_eqI
thf(fact_464_fequalityCE,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( A2 = B5 )
     => ( ( ( fmembe3381613331217039976ist_fm @ C @ A2 )
         => ~ ( fmembe3381613331217039976ist_fm @ C @ B5 ) )
       => ~ ( ~ ( fmembe3381613331217039976ist_fm @ C @ A2 )
           => ( fmembe3381613331217039976ist_fm @ C @ B5 ) ) ) ) ).

% fequalityCE
thf(fact_465_fequalityCE,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( A2 = B5 )
     => ( ( ( fmembe1635983029874454474m_rule @ C @ A2 )
         => ~ ( fmembe1635983029874454474m_rule @ C @ B5 ) )
       => ~ ( ~ ( fmembe1635983029874454474m_rule @ C @ A2 )
           => ( fmembe1635983029874454474m_rule @ C @ B5 ) ) ) ) ).

% fequalityCE
thf(fact_466_eq__fmem__trans,axiom,
    ! [A: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( A = B )
     => ( ( fmembe3381613331217039976ist_fm @ B @ A2 )
       => ( fmembe3381613331217039976ist_fm @ A @ A2 ) ) ) ).

% eq_fmem_trans
thf(fact_467_eq__fmem__trans,axiom,
    ! [A: abstra4572470701423150929m_rule,B: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( A = B )
     => ( ( fmembe1635983029874454474m_rule @ B @ A2 )
       => ( fmembe1635983029874454474m_rule @ A @ A2 ) ) ) ).

% eq_fmem_trans
thf(fact_468_mkTree_Osimps_I2_J,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 ) )
      = ( fimage748043113761278389m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( stl_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) ) )
        @ ( abstra458490714785879965ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) )
          @ S3 ) ) ) ).

% mkTree.simps(2)
thf(fact_469_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X2 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_470_prod_Oinject,axiom,
    ! [X1: list_fm,X2: list_fm,Y1: list_fm,Y2: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ X1 @ X2 )
        = ( produc381145313068854617ist_fm @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_471_prod_Oinject,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ X1 @ X2 )
        = ( produc9060074326276436823at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_472_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ X1 @ X2 )
        = ( produc2922128104949294807at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_473_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_474_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A6: product_prod_nat_nat,B7: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A6 @ B7 ) )
      = ( ( A = A6 )
        & ( B = B7 ) ) ) ).

% old.prod.inject
thf(fact_475_old_Oprod_Oinject,axiom,
    ! [A: list_fm,B: list_fm,A6: list_fm,B7: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A6 @ B7 ) )
      = ( ( A = A6 )
        & ( B = B7 ) ) ) ).

% old.prod.inject
thf(fact_476_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A6: set_Pr4329608150637261639at_nat,B7: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B )
        = ( produc9060074326276436823at_nat @ A6 @ B7 ) )
      = ( ( A = A6 )
        & ( B = B7 ) ) ) ).

% old.prod.inject
thf(fact_477_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A6 @ B7 ) )
      = ( ( A = A6 )
        & ( B = B7 ) ) ) ).

% old.prod.inject
thf(fact_478_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A6: nat,B7: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A6 @ B7 ) )
      = ( ( A = A6 )
        & ( B = B7 ) ) ) ).

% old.prod.inject
thf(fact_479_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_480_ipath_Ocases,axiom,
    ! [A1: abstract_tree_rule,A22: stream_rule] :
      ( ( abstract_ipath_rule @ A1 @ A22 )
     => ~ ( ( ( abstract_root_rule @ A1 )
            = ( shd_rule @ A22 ) )
         => ! [T4: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T4 @ ( abstract_cont_rule @ A1 ) )
             => ~ ( abstract_ipath_rule @ T4 @ ( stl_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_481_ipath_Ocases,axiom,
    ! [A1: abstra4572470701423150929m_rule,A22: stream727092118206550309m_rule] :
      ( ( abstra2285467793620430502m_rule @ A1 @ A22 )
     => ~ ( ( ( abstra3503897793822247108m_rule @ A1 )
            = ( shd_Pr7235097944458474089m_rule @ A22 ) )
         => ! [T4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ A1 ) )
             => ~ ( abstra2285467793620430502m_rule @ T4 @ ( stl_Pr950425576149878629m_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_482_ipath_Osimps,axiom,
    ( abstract_ipath_rule
    = ( ^ [A12: abstract_tree_rule,A23: stream_rule] :
        ? [T2: abstract_tree_rule,Steps2: stream_rule,T5: abstract_tree_rule] :
          ( ( A12 = T2 )
          & ( A23 = Steps2 )
          & ( ( abstract_root_rule @ T2 )
            = ( shd_rule @ Steps2 ) )
          & ( fmembe7964111328986570234e_rule @ T5 @ ( abstract_cont_rule @ T2 ) )
          & ( abstract_ipath_rule @ T5 @ ( stl_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_483_ipath_Osimps,axiom,
    ( abstra2285467793620430502m_rule
    = ( ^ [A12: abstra4572470701423150929m_rule,A23: stream727092118206550309m_rule] :
        ? [T2: abstra4572470701423150929m_rule,Steps2: stream727092118206550309m_rule,T5: abstra4572470701423150929m_rule] :
          ( ( A12 = T2 )
          & ( A23 = Steps2 )
          & ( ( abstra3503897793822247108m_rule @ T2 )
            = ( shd_Pr7235097944458474089m_rule @ Steps2 ) )
          & ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T2 ) )
          & ( abstra2285467793620430502m_rule @ T5 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_484_ipath_Ointros,axiom,
    ! [T: abstract_tree_rule,Steps: stream_rule,T3: abstract_tree_rule] :
      ( ( ( abstract_root_rule @ T )
        = ( shd_rule @ Steps ) )
     => ( ( fmembe7964111328986570234e_rule @ T3 @ ( abstract_cont_rule @ T ) )
       => ( ( abstract_ipath_rule @ T3 @ ( stl_rule @ Steps ) )
         => ( abstract_ipath_rule @ T @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_485_ipath_Ointros,axiom,
    ! [T: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule,T3: abstra4572470701423150929m_rule] :
      ( ( ( abstra3503897793822247108m_rule @ T )
        = ( shd_Pr7235097944458474089m_rule @ Steps ) )
     => ( ( fmembe1635983029874454474m_rule @ T3 @ ( abstra4962253831105676628m_rule @ T ) )
       => ( ( abstra2285467793620430502m_rule @ T3 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
         => ( abstra2285467793620430502m_rule @ T @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_486_ipath_Ocoinduct,axiom,
    ! [X7: abstract_tree_rule > stream_rule > $o,X1: abstract_tree_rule,X2: stream_rule] :
      ( ( X7 @ X1 @ X2 )
     => ( ! [X12: abstract_tree_rule,X22: stream_rule] :
            ( ( X7 @ X12 @ X22 )
           => ? [T6: abstract_tree_rule,Steps3: stream_rule,T7: abstract_tree_rule] :
                ( ( X12 = T6 )
                & ( X22 = Steps3 )
                & ( ( abstract_root_rule @ T6 )
                  = ( shd_rule @ Steps3 ) )
                & ( fmembe7964111328986570234e_rule @ T7 @ ( abstract_cont_rule @ T6 ) )
                & ( ( X7 @ T7 @ ( stl_rule @ Steps3 ) )
                  | ( abstract_ipath_rule @ T7 @ ( stl_rule @ Steps3 ) ) ) ) )
       => ( abstract_ipath_rule @ X1 @ X2 ) ) ) ).

% ipath.coinduct
thf(fact_487_ipath_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > stream727092118206550309m_rule > $o,X1: abstra4572470701423150929m_rule,X2: stream727092118206550309m_rule] :
      ( ( X7 @ X1 @ X2 )
     => ( ! [X12: abstra4572470701423150929m_rule,X22: stream727092118206550309m_rule] :
            ( ( X7 @ X12 @ X22 )
           => ? [T6: abstra4572470701423150929m_rule,Steps3: stream727092118206550309m_rule,T7: abstra4572470701423150929m_rule] :
                ( ( X12 = T6 )
                & ( X22 = Steps3 )
                & ( ( abstra3503897793822247108m_rule @ T6 )
                  = ( shd_Pr7235097944458474089m_rule @ Steps3 ) )
                & ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ T6 ) )
                & ( ( X7 @ T7 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra2285467793620430502m_rule @ T7 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra2285467793620430502m_rule @ X1 @ X2 ) ) ) ).

% ipath.coinduct
thf(fact_488_subsetI,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X @ A2 )
         => ( member8102475879199740618ist_fm @ X @ B5 ) )
     => ( ord_le2055072402069232929ist_fm @ A2 @ B5 ) ) ).

% subsetI
thf(fact_489_subsetI,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [X: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ A2 )
         => ( member8206827879206165904at_nat @ X @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_490_subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ A2 )
         => ( member8440522571783428010at_nat @ X @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_491_subsetI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ! [X: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X @ A2 )
         => ( member8757157785044589968at_nat @ X @ B5 ) )
     => ( ord_le1268244103169919719at_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_492_subsetI,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ! [X: rule] :
          ( ( member_rule @ X @ A2 )
         => ( member_rule @ X @ B5 ) )
     => ( ord_less_eq_set_rule @ A2 @ B5 ) ) ).

% subsetI
thf(fact_493_subset__antisym,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ( ord_less_eq_set_rule @ B5 @ A2 )
       => ( A2 = B5 ) ) ) ).

% subset_antisym
thf(fact_494_fimage__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F @ A2 ) ) ) ) ).

% fimage_eqI
thf(fact_495_fimage__eqI,axiom,
    ! [B: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A2 ) ) ) ) ).

% fimage_eqI
thf(fact_496_fimage__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A2 ) ) ) ) ).

% fimage_eqI
thf(fact_497_fimage__eqI,axiom,
    ! [B: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( B
        = ( F @ X5 ) )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F @ A2 ) ) ) ) ).

% fimage_eqI
thf(fact_498_pred__subset__eq,axiom,
    ! [R5: set_Pr7058068377845519745ist_fm,S4: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le8518180823940269308t_fm_o
        @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ R5 )
        @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ S4 ) )
      = ( ord_le2055072402069232929ist_fm @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_499_pred__subset__eq,axiom,
    ! [R5: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le7858099551454983350_nat_o
        @ ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ R5 )
        @ ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ S4 ) )
      = ( ord_le3000389064537975527at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_500_pred__subset__eq,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le704812498762024988_nat_o
        @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ R5 )
        @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_501_pred__subset__eq,axiom,
    ! [R5: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1999416642142172342_nat_o
        @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ R5 )
        @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ S4 ) )
      = ( ord_le1268244103169919719at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_502_pred__subset__eq,axiom,
    ! [R5: set_rule,S4: set_rule] :
      ( ( ord_less_eq_rule_o
        @ ^ [X3: rule] : ( member_rule @ X3 @ R5 )
        @ ^ [X3: rule] : ( member_rule @ X3 @ S4 ) )
      = ( ord_less_eq_set_rule @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_503_subset__fimage__iff,axiom,
    ! [B5: fset_A5396818216090053761m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A2: fset_P661503646757059847ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ B5 @ ( fimage748043113761278389m_rule @ F @ A2 ) )
      = ( ? [AA: fset_P661503646757059847ist_fm] :
            ( ( ord_le2064643713053750439ist_fm @ AA @ A2 )
            & ( B5
              = ( fimage748043113761278389m_rule @ F @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_504_subset__fimage__iff,axiom,
    ! [B5: fset_P661503646757059847ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A2: fset_A5396818216090053761m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ B5 @ ( fimage9164954772359936849ist_fm @ F @ A2 ) )
      = ( ? [AA: fset_A5396818216090053761m_rule] :
            ( ( ord_le6951374404268211425m_rule @ AA @ A2 )
            & ( B5
              = ( fimage9164954772359936849ist_fm @ F @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_505_fimage__mono,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B5: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ A2 @ B5 )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F @ A2 ) @ ( fimage748043113761278389m_rule @ F @ B5 ) ) ) ).

% fimage_mono
thf(fact_506_fimage__mono,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B5: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ A2 @ B5 )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F @ A2 ) @ ( fimage9164954772359936849ist_fm @ F @ B5 ) ) ) ).

% fimage_mono
thf(fact_507_in__mono,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm,X5: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A2 @ B5 )
     => ( ( member8102475879199740618ist_fm @ X5 @ A2 )
       => ( member8102475879199740618ist_fm @ X5 @ B5 ) ) ) ).

% in_mono
thf(fact_508_in__mono,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,X5: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A2 @ B5 )
     => ( ( member8206827879206165904at_nat @ X5 @ A2 )
       => ( member8206827879206165904at_nat @ X5 @ B5 ) ) ) ).

% in_mono
thf(fact_509_in__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B5 )
     => ( ( member8440522571783428010at_nat @ X5 @ A2 )
       => ( member8440522571783428010at_nat @ X5 @ B5 ) ) ) ).

% in_mono
thf(fact_510_in__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,X5: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ B5 )
     => ( ( member8757157785044589968at_nat @ X5 @ A2 )
       => ( member8757157785044589968at_nat @ X5 @ B5 ) ) ) ).

% in_mono
thf(fact_511_in__mono,axiom,
    ! [A2: set_rule,B5: set_rule,X5: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ( member_rule @ X5 @ A2 )
       => ( member_rule @ X5 @ B5 ) ) ) ).

% in_mono
thf(fact_512_subsetD,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,B5: set_Pr7058068377845519745ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A2 @ B5 )
     => ( ( member8102475879199740618ist_fm @ C @ A2 )
       => ( member8102475879199740618ist_fm @ C @ B5 ) ) ) ).

% subsetD
thf(fact_513_subsetD,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A2 @ B5 )
     => ( ( member8206827879206165904at_nat @ C @ A2 )
       => ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_514_subsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B5 )
     => ( ( member8440522571783428010at_nat @ C @ A2 )
       => ( member8440522571783428010at_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_515_subsetD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ B5 )
     => ( ( member8757157785044589968at_nat @ C @ A2 )
       => ( member8757157785044589968at_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_516_subsetD,axiom,
    ! [A2: set_rule,B5: set_rule,C: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ( member_rule @ C @ A2 )
       => ( member_rule @ C @ B5 ) ) ) ).

% subsetD
thf(fact_517_equalityE,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( A2 = B5 )
     => ~ ( ( ord_less_eq_set_rule @ A2 @ B5 )
         => ~ ( ord_less_eq_set_rule @ B5 @ A2 ) ) ) ).

% equalityE
thf(fact_518_subset__eq,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B8: set_Pr7058068377845519745ist_fm] :
        ! [X3: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X3 @ A7 )
         => ( member8102475879199740618ist_fm @ X3 @ B8 ) ) ) ) ).

% subset_eq
thf(fact_519_subset__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B8: set_Pr8693737435421807431at_nat] :
        ! [X3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A7 )
         => ( member8206827879206165904at_nat @ X3 @ B8 ) ) ) ) ).

% subset_eq
thf(fact_520_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
        ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A7 )
         => ( member8440522571783428010at_nat @ X3 @ B8 ) ) ) ) ).

% subset_eq
thf(fact_521_subset__eq,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A7: set_Pr4329608150637261639at_nat,B8: set_Pr4329608150637261639at_nat] :
        ! [X3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A7 )
         => ( member8757157785044589968at_nat @ X3 @ B8 ) ) ) ) ).

% subset_eq
thf(fact_522_subset__eq,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
        ! [X3: rule] :
          ( ( member_rule @ X3 @ A7 )
         => ( member_rule @ X3 @ B8 ) ) ) ) ).

% subset_eq
thf(fact_523_equalityD1,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_rule @ A2 @ B5 ) ) ).

% equalityD1
thf(fact_524_equalityD2,axiom,
    ! [A2: set_rule,B5: set_rule] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_rule @ B5 @ A2 ) ) ).

% equalityD2
thf(fact_525_subset__iff,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B8: set_Pr7058068377845519745ist_fm] :
        ! [T2: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ T2 @ A7 )
         => ( member8102475879199740618ist_fm @ T2 @ B8 ) ) ) ) ).

% subset_iff
thf(fact_526_subset__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B8: set_Pr8693737435421807431at_nat] :
        ! [T2: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ T2 @ A7 )
         => ( member8206827879206165904at_nat @ T2 @ B8 ) ) ) ) ).

% subset_iff
thf(fact_527_subset__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
        ! [T2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ T2 @ A7 )
         => ( member8440522571783428010at_nat @ T2 @ B8 ) ) ) ) ).

% subset_iff
thf(fact_528_subset__iff,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A7: set_Pr4329608150637261639at_nat,B8: set_Pr4329608150637261639at_nat] :
        ! [T2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ T2 @ A7 )
         => ( member8757157785044589968at_nat @ T2 @ B8 ) ) ) ) ).

% subset_iff
thf(fact_529_subset__iff,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
        ! [T2: rule] :
          ( ( member_rule @ T2 @ A7 )
         => ( member_rule @ T2 @ B8 ) ) ) ) ).

% subset_iff
thf(fact_530_subset__refl,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ A2 ) ).

% subset_refl
thf(fact_531_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_532_subset__trans,axiom,
    ! [A2: set_rule,B5: set_rule,C2: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B5 )
     => ( ( ord_less_eq_set_rule @ B5 @ C2 )
       => ( ord_less_eq_set_rule @ A2 @ C2 ) ) ) ).

% subset_trans
thf(fact_533_set__eq__subset,axiom,
    ( ( ^ [Y4: set_rule,Z2: set_rule] : ( Y4 = Z2 ) )
    = ( ^ [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_534_less__eq__set__def,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B8: set_Pr7058068377845519745ist_fm] :
          ( ord_le8518180823940269308t_fm_o
          @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ A7 )
          @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_535_less__eq__set__def,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,B8: set_Pr8693737435421807431at_nat] :
          ( ord_le7858099551454983350_nat_o
          @ ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ A7 )
          @ ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_536_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A7 )
          @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_537_less__eq__set__def,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A7: set_Pr4329608150637261639at_nat,B8: set_Pr4329608150637261639at_nat] :
          ( ord_le1999416642142172342_nat_o
          @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A7 )
          @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_538_less__eq__set__def,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
          ( ord_less_eq_rule_o
          @ ^ [X3: rule] : ( member_rule @ X3 @ A7 )
          @ ^ [X3: rule] : ( member_rule @ X3 @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_539_Collect__mono__iff,axiom,
    ! [P: rule > $o,Q: rule > $o] :
      ( ( ord_less_eq_set_rule @ ( collect_rule @ P ) @ ( collect_rule @ Q ) )
      = ( ! [X3: rule] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_540_fimage__fimage,axiom,
    ! [F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A2: fset_P661503646757059847ist_fm] :
      ( ( fimage3349360726776475169m_rule @ F @ ( fimage748043113761278389m_rule @ G @ A2 ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_541_fimage__fimage,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A2: fset_A5396818216090053761m_rule] :
      ( ( fimage8674584286907012797ist_fm @ F @ ( fimage9164954772359936849ist_fm @ G @ A2 ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X3: abstra4572470701423150929m_rule] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_542_fimage__fimage,axiom,
    ! [F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( fimage748043113761278389m_rule @ F @ ( fimage8674584286907012797ist_fm @ G @ A2 ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_543_fimage__fimage,axiom,
    ! [F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A2: fset_A5396818216090053761m_rule] :
      ( ( fimage748043113761278389m_rule @ F @ ( fimage9164954772359936849ist_fm @ G @ A2 ) )
      = ( fimage3349360726776475169m_rule
        @ ^ [X3: abstra4572470701423150929m_rule] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_544_fimage__fimage,axiom,
    ! [F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( fimage9164954772359936849ist_fm @ F @ ( fimage3349360726776475169m_rule @ G @ A2 ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X3: abstra4572470701423150929m_rule] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_545_fimage__fimage,axiom,
    ! [F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A2: fset_P661503646757059847ist_fm] :
      ( ( fimage9164954772359936849ist_fm @ F @ ( fimage748043113761278389m_rule @ G @ A2 ) )
      = ( fimage8674584286907012797ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_546_Collect__subset,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_547_Collect__subset,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_548_Collect__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_549_Collect__subset,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ord_le1268244103169919719at_nat
      @ ( collec6321179662152712658at_nat
        @ ^ [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_550_Collect__subset,axiom,
    ! [A2: set_rule,P: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X3: rule] :
            ( ( member_rule @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_551_rev__fimage__eqI,axiom,
    ! [X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm,B: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
     => ( ( B
          = ( F @ X5 ) )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F @ A2 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_552_rev__fimage__eqI,axiom,
    ! [X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm,B: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
     => ( ( B
          = ( F @ X5 ) )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A2 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_553_rev__fimage__eqI,axiom,
    ! [X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule,B: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
     => ( ( B
          = ( F @ X5 ) )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A2 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_554_rev__fimage__eqI,axiom,
    ! [X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule,B: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
     => ( ( B
          = ( F @ X5 ) )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F @ A2 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_555_fimage__cong,axiom,
    ! [M2: fset_P661503646757059847ist_fm,N4: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( M2 = N4 )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( fmembe3381613331217039976ist_fm @ X @ N4 )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( fimage748043113761278389m_rule @ F @ M2 )
          = ( fimage748043113761278389m_rule @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_556_fimage__cong,axiom,
    ! [M2: fset_A5396818216090053761m_rule,N4: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( M2 = N4 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( fmembe1635983029874454474m_rule @ X @ N4 )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( fimage9164954772359936849ist_fm @ F @ M2 )
          = ( fimage9164954772359936849ist_fm @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_557_fimageI,axiom,
    ! [X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
     => ( fmembe3381613331217039976ist_fm @ ( F @ X5 ) @ ( fimage8674584286907012797ist_fm @ F @ A2 ) ) ) ).

% fimageI
thf(fact_558_fimageI,axiom,
    ! [X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
     => ( fmembe1635983029874454474m_rule @ ( F @ X5 ) @ ( fimage748043113761278389m_rule @ F @ A2 ) ) ) ).

% fimageI
thf(fact_559_fimageI,axiom,
    ! [X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
     => ( fmembe3381613331217039976ist_fm @ ( F @ X5 ) @ ( fimage9164954772359936849ist_fm @ F @ A2 ) ) ) ).

% fimageI
thf(fact_560_fimageI,axiom,
    ! [X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
     => ( fmembe1635983029874454474m_rule @ ( F @ X5 ) @ ( fimage3349360726776475169m_rule @ F @ A2 ) ) ) ).

% fimageI
thf(fact_561_fimageE,axiom,
    ! [B: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F @ A2 ) )
     => ~ ! [X: produc1996495991257130529ist_fm] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X @ A2 ) ) ) ).

% fimageE
thf(fact_562_fimageE,axiom,
    ! [B: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A2: fset_A5396818216090053761m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A2 ) )
     => ~ ! [X: abstra4572470701423150929m_rule] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe1635983029874454474m_rule @ X @ A2 ) ) ) ).

% fimageE
thf(fact_563_fimageE,axiom,
    ! [B: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A2: fset_P661503646757059847ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A2 ) )
     => ~ ! [X: produc1996495991257130529ist_fm] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X @ A2 ) ) ) ).

% fimageE
thf(fact_564_fimageE,axiom,
    ! [B: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F @ A2 ) )
     => ~ ! [X: abstra4572470701423150929m_rule] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe1635983029874454474m_rule @ X @ A2 ) ) ) ).

% fimageE
thf(fact_565_fimage__fsubsetI,axiom,
    ! [A2: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,B5: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A2 )
         => ( fmembe3381613331217039976ist_fm @ ( F @ X ) @ B5 ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage8674584286907012797ist_fm @ F @ A2 ) @ B5 ) ) ).

% fimage_fsubsetI
thf(fact_566_fimage__fsubsetI,axiom,
    ! [A2: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,B5: fset_A5396818216090053761m_rule] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A2 )
         => ( fmembe1635983029874454474m_rule @ ( F @ X ) @ B5 ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F @ A2 ) @ B5 ) ) ).

% fimage_fsubsetI
thf(fact_567_fimage__fsubsetI,axiom,
    ! [A2: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,B5: fset_P661503646757059847ist_fm] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A2 )
         => ( fmembe3381613331217039976ist_fm @ ( F @ X ) @ B5 ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F @ A2 ) @ B5 ) ) ).

% fimage_fsubsetI
thf(fact_568_fimage__fsubsetI,axiom,
    ! [A2: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,B5: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A2 )
         => ( fmembe1635983029874454474m_rule @ ( F @ X ) @ B5 ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage3349360726776475169m_rule @ F @ A2 ) @ B5 ) ) ).

% fimage_fsubsetI
thf(fact_569_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_570_mkTree_Osimps_I1_J,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 ) )
      = ( produc491467635432902671m_rule @ S3
        @ ( shd_rule
          @ ( abstra2363873759530810705ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 ) ) ) ) ).

% mkTree.simps(1)
thf(fact_571_prod__induct3,axiom,
    ! [P: produc859450856879609959at_nat > $o,X5: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B2: nat,C3: nat] : ( P @ ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B2 @ C3 ) ) )
     => ( P @ X5 ) ) ).

% prod_induct3
thf(fact_572_prod__cases3,axiom,
    ! [Y3: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B2: nat,C3: nat] :
          ( Y3
         != ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_573_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A6: product_prod_nat_nat,B7: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A6 @ B7 ) )
     => ~ ( ( A = A6 )
         => ( B != B7 ) ) ) ).

% Pair_inject
thf(fact_574_Pair__inject,axiom,
    ! [A: list_fm,B: list_fm,A6: list_fm,B7: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A6 @ B7 ) )
     => ~ ( ( A = A6 )
         => ( B != B7 ) ) ) ).

% Pair_inject
thf(fact_575_Pair__inject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A6: set_Pr4329608150637261639at_nat,B7: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B )
        = ( produc9060074326276436823at_nat @ A6 @ B7 ) )
     => ~ ( ( A = A6 )
         => ( B != B7 ) ) ) ).

% Pair_inject
thf(fact_576_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A6 @ B7 ) )
     => ~ ( ( A = A6 )
         => ( B != B7 ) ) ) ).

% Pair_inject
thf(fact_577_Pair__inject,axiom,
    ! [A: nat,B: nat,A6: nat,B7: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A6 @ B7 ) )
     => ~ ( ( A = A6 )
         => ( B != B7 ) ) ) ).

% Pair_inject
thf(fact_578_prod__cases,axiom,
    ! [P: produc859450856879609959at_nat > $o,P3: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A3 @ B2 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_579_prod__cases,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,P3: produc1996495991257130529ist_fm] :
      ( ! [A3: list_fm,B2: list_fm] : ( P @ ( produc381145313068854617ist_fm @ A3 @ B2 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_580_prod__cases,axiom,
    ! [P: produc1319942482725812455at_nat > $o,P3: produc1319942482725812455at_nat] :
      ( ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( P @ ( produc9060074326276436823at_nat @ A3 @ B2 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_581_prod__cases,axiom,
    ! [P: produc3843707927480180839at_nat > $o,P3: produc3843707927480180839at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( P @ ( produc2922128104949294807at_nat @ A3 @ B2 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_582_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
      ( ! [A3: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B2 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_583_surj__pair,axiom,
    ! [P3: produc859450856879609959at_nat] :
    ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( P3
      = ( produc6161850002892822231at_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_584_surj__pair,axiom,
    ! [P3: produc1996495991257130529ist_fm] :
    ? [X: list_fm,Y: list_fm] :
      ( P3
      = ( produc381145313068854617ist_fm @ X @ Y ) ) ).

% surj_pair
thf(fact_585_surj__pair,axiom,
    ! [P3: produc1319942482725812455at_nat] :
    ? [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( P3
      = ( produc9060074326276436823at_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_586_surj__pair,axiom,
    ! [P3: produc3843707927480180839at_nat] :
    ? [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( P3
      = ( produc2922128104949294807at_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_587_surj__pair,axiom,
    ! [P3: product_prod_nat_nat] :
    ? [X: nat,Y: nat] :
      ( P3
      = ( product_Pair_nat_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_588_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
          ( Y3
         != ( produc6161850002892822231at_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_589_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1996495991257130529ist_fm] :
      ~ ! [A3: list_fm,B2: list_fm] :
          ( Y3
         != ( produc381145313068854617ist_fm @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_590_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1319942482725812455at_nat] :
      ~ ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
          ( Y3
         != ( produc9060074326276436823at_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_591_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc3843707927480180839at_nat] :
      ~ ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( Y3
         != ( produc2922128104949294807at_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_592_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ! [A3: nat,B2: nat] :
          ( Y3
         != ( product_Pair_nat_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_593_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ Eff @ Rs @ S3 ) )
      = ( produc6161850002892822231at_nat @ S3 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_594_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_595_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > fset_s466195290552925357at_nat > $o,Rs: stream3627166509194609229at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( abstra106331753906253176at_nat @ ( abstra8108511269830108963at_nat @ Eff @ Rs @ S3 ) )
      = ( produc9060074326276436823at_nat @ S3 @ ( shd_se2672623163517224947at_nat @ ( abstra5441914787988489033at_nat @ Eff @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_596_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > fset_s3385143008492993351at_nat > $o,Rs: stream7255200869668761831at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( abstra973273217209065464at_nat @ ( abstra7709754627474349731at_nat @ Eff @ Rs @ S3 ) )
      = ( produc2922128104949294807at_nat @ S3 @ ( shd_se6818656645295851789at_nat @ ( abstra6541677642950016201at_nat @ Eff @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_597_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: nat > nat > fset_nat > $o,Rs: stream_nat,S3: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ Eff @ Rs @ S3 ) )
      = ( product_Pair_nat_nat @ S3 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ 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_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra35009723297898488at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) )
      = ( produc6161850002892822231at_nat @ S3 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_600_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_601_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > option8009148030513158131at_nat,Rs: stream3627166509194609229at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( abstra106331753906253176at_nat @ ( abstra8108511269830108963at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ Rs @ S3 ) )
      = ( produc9060074326276436823at_nat @ S3 @ ( shd_se2672623163517224947at_nat @ ( abstra5441914787988489033at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_602_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > option5910749414053197325at_nat,Rs: stream7255200869668761831at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( abstra973273217209065464at_nat @ ( abstra7709754627474349731at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ Rs @ S3 ) )
      = ( produc2922128104949294807at_nat @ S3 @ ( shd_se6818656645295851789at_nat @ ( abstra6541677642950016201at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_603_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: nat > nat > option_fset_nat,Rs: stream_nat,S3: nat] :
      ( ( abstra3259451721936639506at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) )
      = ( product_Pair_nat_nat @ S3 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_604_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_605_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_606_mkTree_Ocode,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( abstra7467780926338743076m_rule
        @ ( produc491467635432902671m_rule @ S3
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 ) ) )
        @ ( fimage748043113761278389m_rule
          @ ( abstra178918799166095863ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( stl_rule
              @ ( abstra2363873759530810705ist_fm
                @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S3 ) ) )
          @ ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ ( shd_rule
              @ ( abstra2363873759530810705ist_fm
                @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ Rs
                @ S3 ) )
            @ S3 ) ) ) ) ).

% mkTree.code
thf(fact_607_i_OmkTree_Ocode,axiom,
    ! [Eff3: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 )
      = ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S3 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ ( stl_Pr5027827701538482609at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra673899893360562045at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_608_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_609_i_OmkTree_Ocode,axiom,
    ! [Eff3: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > option8009148030513158131at_nat,Rs: stream3627166509194609229at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( abstra8108511269830108963at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ Rs @ S3 )
      = ( abstra6446703985809605400at_nat @ ( produc9060074326276436823at_nat @ S3 @ ( shd_se2672623163517224947at_nat @ ( abstra5441914787988489033at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage7170464556407549111at_nat @ ( abstra8108511269830108963at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ ( stl_se8095006981682938871at_nat @ ( abstra5441914787988489033at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra5031851239426275837at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ ( shd_se2672623163517224947at_nat @ ( abstra5441914787988489033at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_610_i_OmkTree_Ocode,axiom,
    ! [Eff3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > option5910749414053197325at_nat,Rs: stream7255200869668761831at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( abstra7709754627474349731at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ Rs @ S3 )
      = ( abstra4133525815659287960at_nat @ ( produc2922128104949294807at_nat @ S3 @ ( shd_se6818656645295851789at_nat @ ( abstra6541677642950016201at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage9092832869046556317at_nat @ ( abstra7709754627474349731at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ ( stl_se6007078998265101073at_nat @ ( abstra6541677642950016201at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra6518908551485222781at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ ( shd_se6818656645295851789at_nat @ ( abstra6541677642950016201at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_611_i_OmkTree_Ocode,axiom,
    ! [Eff3: nat > nat > option_fset_nat,Rs: stream_nat,S3: nat] :
      ( ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 )
      = ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S3 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ ( stl_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra969719415724737503at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ ( shd_nat @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_612_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_613_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra8972947971889770147at_nat
    = ( ^ [Eff2: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs2: stream6724221391990029191at_nat,S: product_prod_nat_nat] : ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff2 @ Rs2 @ S ) ) ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ Eff2 @ ( stl_Pr5027827701538482609at_nat @ ( abstra4493250544927931593at_nat @ Eff2 @ Rs2 @ S ) ) ) @ ( abstra673899893360562045at_nat @ Eff2 @ ( shd_Pr4260400998323988397at_nat @ ( abstra4493250544927931593at_nat @ Eff2 @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_614_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_615_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra8108511269830108963at_nat
    = ( ^ [Eff2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > fset_s466195290552925357at_nat > $o,Rs2: stream3627166509194609229at_nat,S: set_Pr4329608150637261639at_nat] : ( abstra6446703985809605400at_nat @ ( produc9060074326276436823at_nat @ S @ ( shd_se2672623163517224947at_nat @ ( abstra5441914787988489033at_nat @ Eff2 @ Rs2 @ S ) ) ) @ ( fimage7170464556407549111at_nat @ ( abstra8108511269830108963at_nat @ Eff2 @ ( stl_se8095006981682938871at_nat @ ( abstra5441914787988489033at_nat @ Eff2 @ Rs2 @ S ) ) ) @ ( abstra5031851239426275837at_nat @ Eff2 @ ( shd_se2672623163517224947at_nat @ ( abstra5441914787988489033at_nat @ Eff2 @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_616_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra7709754627474349731at_nat
    = ( ^ [Eff2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > fset_s3385143008492993351at_nat > $o,Rs2: stream7255200869668761831at_nat,S: set_Pr1261947904930325089at_nat] : ( abstra4133525815659287960at_nat @ ( produc2922128104949294807at_nat @ S @ ( shd_se6818656645295851789at_nat @ ( abstra6541677642950016201at_nat @ Eff2 @ Rs2 @ S ) ) ) @ ( fimage9092832869046556317at_nat @ ( abstra7709754627474349731at_nat @ Eff2 @ ( stl_se6007078998265101073at_nat @ ( abstra6541677642950016201at_nat @ Eff2 @ Rs2 @ S ) ) ) @ ( abstra6518908551485222781at_nat @ Eff2 @ ( shd_se6818656645295851789at_nat @ ( abstra6541677642950016201at_nat @ Eff2 @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_617_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra1192804020863382533at_nat
    = ( ^ [Eff2: nat > nat > fset_nat > $o,Rs2: stream_nat,S: nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff2 @ Rs2 @ S ) ) ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ Eff2 @ ( stl_nat @ ( abstra7573781713853518635at_nat @ Eff2 @ Rs2 @ S ) ) ) @ ( abstra969719415724737503at_nat @ Eff2 @ ( shd_nat @ ( abstra7573781713853518635at_nat @ Eff2 @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_618_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_619_mkTree__unfold,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S3 )
      = ( case_s2494923709252486212m_rule
        @ ^ [R: rule,S9: stream_rule] :
            ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S3 @ R )
            @ ( fimage748043113761278389m_rule
              @ ( abstra178918799166095863ist_fm
                @ ^ [S: rule,T2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S @ T2 )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ S9 )
              @ ( abstra458490714785879965ist_fm
                @ ^ [S: rule,T2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S @ T2 )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R
                @ S3 ) ) )
        @ ( abstra2363873759530810705ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S3 ) ) ) ).

% mkTree_unfold
thf(fact_620_epath__completeness__Saturated,axiom,
    ! [S3: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ? [T8: abstra4572470701423150929m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) )
              = S3 )
            & ( abstra4476735159789068068ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ T8 )
            & ( abstra3507476289341504315m_rule @ T8 ) )
        | ? [Steps4: stream727092118206550309m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps4 ) )
              = S3 )
            & ( abstra5415202973855822231ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps4 )
            & ( abstra8647779336736900908ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps4 ) ) ) ) ).

% epath_completeness_Saturated
thf(fact_621_konig_Osimps_I1_J,axiom,
    ! [T: abstract_tree_rule] :
      ( ( shd_rule @ ( abstract_konig_rule @ T ) )
      = ( abstract_root_rule @ T ) ) ).

% konig.simps(1)
thf(fact_622_konig_Osimps_I1_J,axiom,
    ! [T: abstra4572470701423150929m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( abstra3079604473823497664m_rule @ T ) )
      = ( abstra3503897793822247108m_rule @ T ) ) ).

% konig.simps(1)
thf(fact_623_i_OmkTree__unfold,axiom,
    ! [Eff3: product_prod_nat_nat > product_prod_nat_nat > option7922001270233832621at_nat,Rs: stream6724221391990029191at_nat,S3: product_prod_nat_nat] :
      ( ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 )
      = ( case_s6993487711257930156at_nat
        @ ^ [R: product_prod_nat_nat,S9: stream6724221391990029191at_nat] : ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S3 @ R ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ S9 ) @ ( abstra673899893360562045at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra4493250544927931593at_nat @ ( abstra426094224824445696at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_624_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
        @ ^ [R: list_fm,S9: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S3 @ R ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ S9 ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_625_i_OmkTree__unfold,axiom,
    ! [Eff3: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > option8009148030513158131at_nat,Rs: stream3627166509194609229at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( abstra8108511269830108963at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ Rs @ S3 )
      = ( case_s8583765366284864230at_nat
        @ ^ [R: set_Pr4329608150637261639at_nat,S9: stream3627166509194609229at_nat] : ( abstra6446703985809605400at_nat @ ( produc9060074326276436823at_nat @ S3 @ R ) @ ( fimage7170464556407549111at_nat @ ( abstra8108511269830108963at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ S9 ) @ ( abstra5031851239426275837at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra5441914787988489033at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_626_i_OmkTree__unfold,axiom,
    ! [Eff3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > option5910749414053197325at_nat,Rs: stream7255200869668761831at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( abstra7709754627474349731at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ Rs @ S3 )
      = ( case_s4387612206358168652at_nat
        @ ^ [R: set_Pr1261947904930325089at_nat,S9: stream7255200869668761831at_nat] : ( abstra4133525815659287960at_nat @ ( produc2922128104949294807at_nat @ S3 @ R ) @ ( fimage9092832869046556317at_nat @ ( abstra7709754627474349731at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ S9 ) @ ( abstra6518908551485222781at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra6541677642950016201at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_627_i_OmkTree__unfold,axiom,
    ! [Eff3: nat > nat > option_fset_nat,Rs: stream_nat,S3: nat] :
      ( ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 )
      = ( case_s2280698752336340829at_nat
        @ ^ [R: nat,S9: stream_nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S3 @ R ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ S9 ) @ ( abstra969719415724737503at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_628_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
        @ ^ [R: rule,S9: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S3 @ R ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ S9 ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_629_tree_Oinject,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule,Y1: produc164195504107695125m_rule,Y2: fset_A5396818216090053761m_rule] :
      ( ( ( abstra7467780926338743076m_rule @ X1 @ X2 )
        = ( abstra7467780926338743076m_rule @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% tree.inject
thf(fact_630_tree_Ocollapse,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) )
      = Tree ) ).

% tree.collapse
thf(fact_631_tree_Oexhaust,axiom,
    ! [Y3: abstra4572470701423150929m_rule] :
      ~ ! [X12: produc164195504107695125m_rule,X22: fset_A5396818216090053761m_rule] :
          ( Y3
         != ( abstra7467780926338743076m_rule @ X12 @ X22 ) ) ).

% tree.exhaust
thf(fact_632_fst__conv,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_633_fst__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_634_fst__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
      ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_635_fst__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_636_fst__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_637_fst__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_638_fst__eqD,axiom,
    ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_639_fst__eqD,axiom,
    ! [X5: list_fm,Y3: list_fm,A: list_fm] :
      ( ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y3 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_640_fst__eqD,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y3 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_641_fst__eqD,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y3 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_642_fst__eqD,axiom,
    ! [X5: nat,Y3: nat,A: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_643_fst__eqD,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y3: rule,A: produc1996495991257130529ist_fm] :
      ( ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X5 @ Y3 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_644_stream_Ocase__distrib,axiom,
    ! [H: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,F: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( H @ ( case_s2494923709252486212m_rule @ F @ Stream ) )
      = ( case_s2494923709252486212m_rule
        @ ^ [X13: rule,X23: stream_rule] : ( H @ ( F @ X13 @ X23 ) )
        @ Stream ) ) ).

% stream.case_distrib
thf(fact_645_Konig,axiom,
    ! [T: abstra4572470701423150929m_rule] :
      ( ~ ( abstra3507476289341504315m_rule @ T )
     => ( abstra2285467793620430502m_rule @ T @ ( abstra3079604473823497664m_rule @ T ) ) ) ).

% Konig
thf(fact_646_tree_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra3503897793822247108m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X2 ) )
      = X1 ) ).

% tree.sel(1)
thf(fact_647_tree_Osel_I2_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra4962253831105676628m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X2 ) )
      = X2 ) ).

% tree.sel(2)
thf(fact_648_ftree__no__ipath,axiom,
    ! [T: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra3507476289341504315m_rule @ T )
     => ~ ( abstra2285467793620430502m_rule @ T @ Steps ) ) ).

% ftree_no_ipath
thf(fact_649_shd__def,axiom,
    ( shd_rule
    = ( case_s3980788006751379256e_rule
      @ ^ [X13: rule,X23: stream_rule] : X13 ) ) ).

% shd_def
thf(fact_650_shd__def,axiom,
    ( shd_Pr7235097944458474089m_rule
    = ( case_s4599053733867249880m_rule
      @ ^ [X13: produc164195504107695125m_rule,X23: stream727092118206550309m_rule] : X13 ) ) ).

% shd_def
thf(fact_651_tree_Oexhaust__sel,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( Tree
      = ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) ) ) ).

% tree.exhaust_sel
thf(fact_652_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ? [T8: abstra4572470701423150929m_rule] :
              ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) )
                = S3 )
              & ( abstra4476735159789068068ist_fm @ Eff @ Rules @ T8 )
              & ( abstra3507476289341504315m_rule @ T8 ) )
          | ? [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_653_stream_Ocase__eq__if,axiom,
    ( case_s2494923709252486212m_rule
    = ( ^ [F2: rule > stream_rule > abstra4572470701423150929m_rule,Stream5: stream_rule] : ( F2 @ ( shd_rule @ Stream5 ) @ ( stl_rule @ Stream5 ) ) ) ) ).

% stream.case_eq_if
thf(fact_654_tfinite,axiom,
    ! [T: abstra4572470701423150929m_rule] :
      ( ! [T4: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T ) )
         => ( abstra3507476289341504315m_rule @ T4 ) )
     => ( abstra3507476289341504315m_rule @ T ) ) ).

% tfinite
thf(fact_655_tfinite_Osimps,axiom,
    ( abstra3507476289341504315m_rule
    = ( ^ [A4: abstra4572470701423150929m_rule] :
        ? [T2: abstra4572470701423150929m_rule] :
          ( ( A4 = T2 )
          & ! [X3: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X3 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra3507476289341504315m_rule @ X3 ) ) ) ) ) ).

% tfinite.simps
thf(fact_656_tfinite_Ocases,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra3507476289341504315m_rule @ A )
     => ! [T7: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ A ) )
         => ( abstra3507476289341504315m_rule @ T7 ) ) ) ).

% tfinite.cases
thf(fact_657_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra8972947971889770147at_nat
    = ( ^ [Eff2: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rs2: stream6724221391990029191at_nat,S: product_prod_nat_nat] :
          ( case_s6993487711257930156at_nat
          @ ^ [R: product_prod_nat_nat,S9: stream6724221391990029191at_nat] : ( abstra6467204794154587544at_nat @ ( produc6161850002892822231at_nat @ S @ R ) @ ( fimage9147729261254313981at_nat @ ( abstra8972947971889770147at_nat @ Eff2 @ S9 ) @ ( abstra673899893360562045at_nat @ Eff2 @ R @ S ) ) )
          @ ( abstra4493250544927931593at_nat @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_658_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
          @ ^ [R: list_fm,S9: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S @ R ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ S9 ) @ ( abstra1097029126060538623ist_fm @ Eff2 @ R @ S ) ) )
          @ ( abstra2983386638146376267ist_fm @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_659_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra8108511269830108963at_nat
    = ( ^ [Eff2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > fset_s466195290552925357at_nat > $o,Rs2: stream3627166509194609229at_nat,S: set_Pr4329608150637261639at_nat] :
          ( case_s8583765366284864230at_nat
          @ ^ [R: set_Pr4329608150637261639at_nat,S9: stream3627166509194609229at_nat] : ( abstra6446703985809605400at_nat @ ( produc9060074326276436823at_nat @ S @ R ) @ ( fimage7170464556407549111at_nat @ ( abstra8108511269830108963at_nat @ Eff2 @ S9 ) @ ( abstra5031851239426275837at_nat @ Eff2 @ R @ S ) ) )
          @ ( abstra5441914787988489033at_nat @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_660_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra7709754627474349731at_nat
    = ( ^ [Eff2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > fset_s3385143008492993351at_nat > $o,Rs2: stream7255200869668761831at_nat,S: set_Pr1261947904930325089at_nat] :
          ( case_s4387612206358168652at_nat
          @ ^ [R: set_Pr1261947904930325089at_nat,S9: stream7255200869668761831at_nat] : ( abstra4133525815659287960at_nat @ ( produc2922128104949294807at_nat @ S @ R ) @ ( fimage9092832869046556317at_nat @ ( abstra7709754627474349731at_nat @ Eff2 @ S9 ) @ ( abstra6518908551485222781at_nat @ Eff2 @ R @ S ) ) )
          @ ( abstra6541677642950016201at_nat @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_661_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra1192804020863382533at_nat
    = ( ^ [Eff2: nat > nat > fset_nat > $o,Rs2: stream_nat,S: nat] :
          ( case_s2280698752336340829at_nat
          @ ^ [R: nat,S9: stream_nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S @ R ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ Eff2 @ S9 ) @ ( abstra969719415724737503at_nat @ Eff2 @ R @ S ) ) )
          @ ( abstra7573781713853518635at_nat @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_662_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra178918799166095863ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] :
          ( case_s2494923709252486212m_rule
          @ ^ [R: rule,S9: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ R ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ S9 ) @ ( abstra458490714785879965ist_fm @ Eff2 @ R @ S ) ) )
          @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_663_epath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X5: stream727092118206550309m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X7 @ X )
           => ? [Steps3: stream727092118206550309m_rule,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl5 )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) )
                  = ( some_f2162114139285431020ist_fm @ Sl5 ) )
                & ( ( X7 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra5415202973855822231ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ X5 ) ) ) ).

% epath.coinduct
thf(fact_664_epath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Sl3: fset_P661503646757059847ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
            = ( some_f2162114139285431020ist_fm @ Sl3 ) )
         => ( ( abstra5415202973855822231ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ Steps ) ) ) ) ) ).

% epath.intros
thf(fact_665_epath_Osimps,axiom,
    ! [A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
      = ( ? [Steps2: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl )
            & ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) )
              = ( some_f2162114139285431020ist_fm @ Sl ) )
            & ( abstra5415202973855822231ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% epath.simps
thf(fact_666_epath_Ocases,axiom,
    ! [A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) ) @ Sl2 )
             => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) )
                  = ( some_f2162114139285431020ist_fm @ Sl2 ) )
               => ~ ( abstra5415202973855822231ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% epath.cases
thf(fact_667_i_Oepath_Ocases,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) @ Sl2 )
             => ( ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_668_i_Oepath_Ocases,axiom,
    ! [Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8372878641218411373at_nat,A: stream1622999302952468147at_nat] :
      ( ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_669_i_Oepath_Ocases,axiom,
    ! [Eff3: product_prod_nat_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream6724221391990029191at_nat,A: stream5916762827556431437at_nat] :
      ( ( abstra4041229200784676899ist_fm @ ( abstra5696132678714034720ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8440522571783428010at_nat @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ A ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ ( stl_Pr3810959331343356407at_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra5696132678714034720ist_fm @ Eff3 @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ A ) ) @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra4041229200784676899ist_fm @ ( abstra5696132678714034720ist_fm @ Eff3 ) @ Rules @ ( stl_Pr3810959331343356407at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_670_i_Oepath_Ocases,axiom,
    ! [Eff3: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8032061917732916077at_nat,A: stream7788502890425800371at_nat] :
      ( ( abstra6262887151753371453ist_fm @ ( abstra945866278424861114ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8757157785044589968at_nat @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ A ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ ( stl_Pr8376929985763293661at_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra945866278424861114ist_fm @ Eff3 @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ A ) ) @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra6262887151753371453ist_fm @ ( abstra945866278424861114ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8376929985763293661at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_671_i_Oepath_Ocases,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) @ Sl2 )
             => ( ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_672_i_Oepath_Ocases,axiom,
    ! [Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8372878641218411373at_nat,A: stream1819640557982756571at_nat] :
      ( ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_673_i_Oepath_Ocases,axiom,
    ! [Eff3: product_prod_nat_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream6724221391990029191at_nat,A: stream6060259936882193013at_nat] :
      ( ( abstra9014086451575537615m_rule @ ( abstra8432771076430573074m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8440522571783428010at_nat @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ A ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ ( stl_Pr8881980968988507061at_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra8432771076430573074m_rule @ Eff3 @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ A ) ) @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra9014086451575537615m_rule @ ( abstra8432771076430573074m_rule @ Eff3 ) @ Rules @ ( stl_Pr8881980968988507061at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_674_i_Oepath_Ocases,axiom,
    ! [Eff3: produc3843707927480180839at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8032061917732916077at_nat,A: stream9093217665182992091at_nat] :
      ( ( abstra8385385442391400757m_rule @ ( abstra3500458500780140280m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member8757157785044589968at_nat @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ A ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ ( stl_Pr6233422408842714779at_nat @ A ) ) ) @ Sl2 )
             => ( ( abstra3500458500780140280m_rule @ Eff3 @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ A ) ) @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra8385385442391400757m_rule @ ( abstra3500458500780140280m_rule @ Eff3 ) @ Rules @ ( stl_Pr6233422408842714779at_nat @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_675_i_Oepath_Ocases,axiom,
    ! [Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) @ Sl2 )
             => ( ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ Sl2 )
               => ~ ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_676_i_Oepath_Ocases,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) ) @ Sl2 )
             => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_677_i_Oepath_Osimps,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ Sl )
            & ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_678_i_Oepath_Osimps,axiom,
    ! [Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8372878641218411373at_nat,A: stream1622999302952468147at_nat] :
      ( ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream1622999302952468147at_nat,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) @ Sl )
            & ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ Sl )
            & ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_679_i_Oepath_Osimps,axiom,
    ! [Eff3: product_prod_nat_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream6724221391990029191at_nat,A: stream5916762827556431437at_nat] :
      ( ( abstra4041229200784676899ist_fm @ ( abstra5696132678714034720ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream5916762827556431437at_nat,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member8440522571783428010at_nat @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps2 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ ( stl_Pr3810959331343356407at_nat @ Steps2 ) ) ) @ Sl )
            & ( abstra5696132678714034720ist_fm @ Eff3 @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps2 ) ) @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps2 ) ) @ Sl )
            & ( abstra4041229200784676899ist_fm @ ( abstra5696132678714034720ist_fm @ Eff3 ) @ Rules @ ( stl_Pr3810959331343356407at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_680_i_Oepath_Osimps,axiom,
    ! [Eff3: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream8032061917732916077at_nat,A: stream7788502890425800371at_nat] :
      ( ( abstra6262887151753371453ist_fm @ ( abstra945866278424861114ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream7788502890425800371at_nat,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member8757157785044589968at_nat @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps2 ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ ( stl_Pr8376929985763293661at_nat @ Steps2 ) ) ) @ Sl )
            & ( abstra945866278424861114ist_fm @ Eff3 @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps2 ) ) @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps2 ) ) @ Sl )
            & ( abstra6262887151753371453ist_fm @ ( abstra945866278424861114ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8376929985763293661at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_681_i_Oepath_Osimps,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ Sl )
            & ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_682_i_Oepath_Osimps,axiom,
    ! [Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8372878641218411373at_nat,A: stream1819640557982756571at_nat] :
      ( ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream1819640557982756571at_nat,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) @ Sl )
            & ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ Sl )
            & ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_683_i_Oepath_Osimps,axiom,
    ! [Eff3: product_prod_nat_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream6724221391990029191at_nat,A: stream6060259936882193013at_nat] :
      ( ( abstra9014086451575537615m_rule @ ( abstra8432771076430573074m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream6060259936882193013at_nat,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member8440522571783428010at_nat @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps2 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ ( stl_Pr8881980968988507061at_nat @ Steps2 ) ) ) @ Sl )
            & ( abstra8432771076430573074m_rule @ Eff3 @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps2 ) ) @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps2 ) ) @ Sl )
            & ( abstra9014086451575537615m_rule @ ( abstra8432771076430573074m_rule @ Eff3 ) @ Rules @ ( stl_Pr8881980968988507061at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_684_i_Oepath_Osimps,axiom,
    ! [Eff3: produc3843707927480180839at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream8032061917732916077at_nat,A: stream9093217665182992091at_nat] :
      ( ( abstra8385385442391400757m_rule @ ( abstra3500458500780140280m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream9093217665182992091at_nat,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member8757157785044589968at_nat @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps2 ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ ( stl_Pr6233422408842714779at_nat @ Steps2 ) ) ) @ Sl )
            & ( abstra3500458500780140280m_rule @ Eff3 @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps2 ) ) @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps2 ) ) @ Sl )
            & ( abstra8385385442391400757m_rule @ ( abstra3500458500780140280m_rule @ Eff3 ) @ Rules @ ( stl_Pr6233422408842714779at_nat @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_685_i_Oepath_Osimps,axiom,
    ! [Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) @ Sl )
            & ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ Sl )
            & ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_686_i_Oepath_Osimps,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl )
            & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ Sl )
            & ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_687_i_Oepath_Ointros,axiom,
    ! [Steps: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_P661503646757059847ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps ) )
           => ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_688_i_Oepath_Ointros,axiom,
    ! [Steps: stream1622999302952468147at_nat,Rules: stream8372878641218411373at_nat,Sl3: fset_P661503646757059847ist_fm,Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps ) )
           => ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_689_i_Oepath_Ointros,axiom,
    ! [Steps: stream5916762827556431437at_nat,Rules: stream6724221391990029191at_nat,Sl3: fset_P661503646757059847ist_fm,Eff3: product_prod_nat_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member8440522571783428010at_nat @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ ( stl_Pr3810959331343356407at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra5696132678714034720ist_fm @ Eff3 @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps ) ) @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra4041229200784676899ist_fm @ ( abstra5696132678714034720ist_fm @ Eff3 ) @ Rules @ ( stl_Pr3810959331343356407at_nat @ Steps ) )
           => ( abstra4041229200784676899ist_fm @ ( abstra5696132678714034720ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_690_i_Oepath_Ointros,axiom,
    ! [Steps: stream7788502890425800371at_nat,Rules: stream8032061917732916077at_nat,Sl3: fset_P661503646757059847ist_fm,Eff3: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member8757157785044589968at_nat @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ ( stl_Pr8376929985763293661at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra945866278424861114ist_fm @ Eff3 @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps ) ) @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra6262887151753371453ist_fm @ ( abstra945866278424861114ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8376929985763293661at_nat @ Steps ) )
           => ( abstra6262887151753371453ist_fm @ ( abstra945866278424861114ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_691_i_Oepath_Ointros,axiom,
    ! [Steps: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_A5396818216090053761m_rule,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps ) )
           => ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_692_i_Oepath_Ointros,axiom,
    ! [Steps: stream1819640557982756571at_nat,Rules: stream8372878641218411373at_nat,Sl3: fset_A5396818216090053761m_rule,Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps ) )
           => ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_693_i_Oepath_Ointros,axiom,
    ! [Steps: stream6060259936882193013at_nat,Rules: stream6724221391990029191at_nat,Sl3: fset_A5396818216090053761m_rule,Eff3: product_prod_nat_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member8440522571783428010at_nat @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ ( stl_Pr8881980968988507061at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra8432771076430573074m_rule @ Eff3 @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps ) ) @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra9014086451575537615m_rule @ ( abstra8432771076430573074m_rule @ Eff3 ) @ Rules @ ( stl_Pr8881980968988507061at_nat @ Steps ) )
           => ( abstra9014086451575537615m_rule @ ( abstra8432771076430573074m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_694_i_Oepath_Ointros,axiom,
    ! [Steps: stream9093217665182992091at_nat,Rules: stream8032061917732916077at_nat,Sl3: fset_A5396818216090053761m_rule,Eff3: produc3843707927480180839at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member8757157785044589968at_nat @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ ( stl_Pr6233422408842714779at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( abstra3500458500780140280m_rule @ Eff3 @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps ) ) @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra8385385442391400757m_rule @ ( abstra3500458500780140280m_rule @ Eff3 ) @ Rules @ ( stl_Pr6233422408842714779at_nat @ Steps ) )
           => ( abstra8385385442391400757m_rule @ ( abstra3500458500780140280m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_695_i_Oepath_Ointros,axiom,
    ! [Steps: stream7755889244956737277e_rule,Rules: stream_rule,Sl3: fset_A5396818216090053761m_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) @ Sl3 )
       => ( ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps ) )
           => ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_696_i_Oepath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Rules: stream_rule,Sl3: fset_P661503646757059847ist_fm,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_697_prod_Ocollapse,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_698_prod_Ocollapse,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_699_prod_Ocollapse,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_700_prod_Ocollapse,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_701_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_702_prod_Ocollapse,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_703_snd__eqD,axiom,
    ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_704_snd__eqD,axiom,
    ! [X5: list_fm,Y3: list_fm,A: list_fm] :
      ( ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_705_snd__eqD,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_706_snd__eqD,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_707_snd__eqD,axiom,
    ! [X5: nat,Y3: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_708_snd__eqD,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y3: rule,A: rule] :
      ( ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X5 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_709_snd__conv,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_710_snd__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_711_snd__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
      ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_712_snd__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_713_snd__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_714_snd__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_715_surjective__pairing,axiom,
    ! [T: produc859450856879609959at_nat] :
      ( T
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ T ) @ ( produc6408287024330202629at_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_716_surjective__pairing,axiom,
    ! [T: produc1996495991257130529ist_fm] :
      ( T
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ T ) @ ( produc4588648349897876871ist_fm @ T ) ) ) ).

% surjective_pairing
thf(fact_717_surjective__pairing,axiom,
    ! [T: produc1319942482725812455at_nat] :
      ( T
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ T ) @ ( produc826222791318859909at_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_718_surjective__pairing,axiom,
    ! [T: produc3843707927480180839at_nat] :
      ( T
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ T ) @ ( produc2937545259423399429at_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_719_surjective__pairing,axiom,
    ! [T: product_prod_nat_nat] :
      ( T
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_720_surjective__pairing,axiom,
    ! [T: produc164195504107695125m_rule] :
      ( T
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ T ) @ ( produc7165828336582415457m_rule @ T ) ) ) ).

% surjective_pairing
thf(fact_721_prod_Oexhaust__sel,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( Prod
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_722_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( Prod
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_723_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( Prod
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_724_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( Prod
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_725_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( Prod
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_726_prod_Oexhaust__sel,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( Prod
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_727_stl__def,axiom,
    ( stl_rule
    = ( case_s5295311805034529096m_rule
      @ ^ [X13: rule,X23: stream_rule] : X23 ) ) ).

% stl_def
thf(fact_728_stl__def,axiom,
    ( stl_Pr950425576149878629m_rule
    = ( case_s474603282864825704m_rule
      @ ^ [X13: produc164195504107695125m_rule,X23: stream727092118206550309m_rule] : X23 ) ) ).

% stl_def
thf(fact_729_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream3409308193418444653ist_fm > $o,X5: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream3409308193418444653ist_fm] :
            ( ( X7 @ X )
           => ? [Steps3: stream3409308193418444653ist_fm,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) )
                  | ( abstra5061764961602482947ist_fm @ Eff @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_730_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream1622999302952468147at_nat > $o,X5: stream1622999302952468147at_nat,Rules: stream8372878641218411373at_nat,Eff: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream1622999302952468147at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream1622999302952468147at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) )
                  | ( abstra1311857326046938941ist_fm @ Eff @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) ) )
       => ( abstra1311857326046938941ist_fm @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_731_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream5916762827556431437at_nat > $o,X5: stream5916762827556431437at_nat,Rules: stream6724221391990029191at_nat,Eff: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream5916762827556431437at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream5916762827556431437at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8440522571783428010at_nat @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps3 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ ( stl_Pr3810959331343356407at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps3 ) ) @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr3810959331343356407at_nat @ Steps3 ) )
                  | ( abstra4041229200784676899ist_fm @ Eff @ Rules @ ( stl_Pr3810959331343356407at_nat @ Steps3 ) ) ) ) )
       => ( abstra4041229200784676899ist_fm @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_732_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream7788502890425800371at_nat > $o,X5: stream7788502890425800371at_nat,Rules: stream8032061917732916077at_nat,Eff: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream7788502890425800371at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream7788502890425800371at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8757157785044589968at_nat @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps3 ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ ( stl_Pr8376929985763293661at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps3 ) ) @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr8376929985763293661at_nat @ Steps3 ) )
                  | ( abstra6262887151753371453ist_fm @ Eff @ Rules @ ( stl_Pr8376929985763293661at_nat @ Steps3 ) ) ) ) )
       => ( abstra6262887151753371453ist_fm @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_733_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream7989562319868115861ist_fm > $o,X5: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream7989562319868115861ist_fm] :
            ( ( X7 @ X )
           => ? [Steps3: stream7989562319868115861ist_fm,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) )
                  | ( abstra8066765647771630831m_rule @ Eff @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_734_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream1819640557982756571at_nat > $o,X5: stream1819640557982756571at_nat,Rules: stream8372878641218411373at_nat,Eff: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream1819640557982756571at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream1819640557982756571at_nat,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) )
                  | ( abstra3157339599784851765m_rule @ Eff @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) ) )
       => ( abstra3157339599784851765m_rule @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_735_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream6060259936882193013at_nat > $o,X5: stream6060259936882193013at_nat,Rules: stream6724221391990029191at_nat,Eff: product_prod_nat_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream6060259936882193013at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream6060259936882193013at_nat,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8440522571783428010at_nat @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps3 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ ( stl_Pr8881980968988507061at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps3 ) ) @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr8881980968988507061at_nat @ Steps3 ) )
                  | ( abstra9014086451575537615m_rule @ Eff @ Rules @ ( stl_Pr8881980968988507061at_nat @ Steps3 ) ) ) ) )
       => ( abstra9014086451575537615m_rule @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_736_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream9093217665182992091at_nat > $o,X5: stream9093217665182992091at_nat,Rules: stream8032061917732916077at_nat,Eff: produc3843707927480180839at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream9093217665182992091at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream9093217665182992091at_nat,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8757157785044589968at_nat @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps3 ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ ( stl_Pr6233422408842714779at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps3 ) ) @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr6233422408842714779at_nat @ Steps3 ) )
                  | ( abstra8385385442391400757m_rule @ Eff @ Rules @ ( stl_Pr6233422408842714779at_nat @ Steps3 ) ) ) ) )
       => ( abstra8385385442391400757m_rule @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_737_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream7755889244956737277e_rule > $o,X5: stream7755889244956737277e_rule,Rules: stream_rule,Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream7755889244956737277e_rule] :
            ( ( X7 @ X )
           => ? [Steps3: stream7755889244956737277e_rule,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) )
                  | ( abstra5823804033175701339m_rule @ Eff @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_738_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X5: stream727092118206550309m_rule,Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X7 @ X )
           => ? [Steps3: stream727092118206550309m_rule,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra5415202973855822231ist_fm @ Eff @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_739_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_P661503646757059847ist_fm,Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra5061764961602482947ist_fm @ Eff @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps ) )
           => ( abstra5061764961602482947ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_740_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream1622999302952468147at_nat,Rules: stream8372878641218411373at_nat,Sl3: fset_P661503646757059847ist_fm,Eff: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra1311857326046938941ist_fm @ Eff @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps ) )
           => ( abstra1311857326046938941ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_741_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream5916762827556431437at_nat,Rules: stream6724221391990029191at_nat,Sl3: fset_P661503646757059847ist_fm,Eff: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8440522571783428010at_nat @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ ( stl_Pr3810959331343356407at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps ) ) @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra4041229200784676899ist_fm @ Eff @ Rules @ ( stl_Pr3810959331343356407at_nat @ Steps ) )
           => ( abstra4041229200784676899ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_742_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7788502890425800371at_nat,Rules: stream8032061917732916077at_nat,Sl3: fset_P661503646757059847ist_fm,Eff: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8757157785044589968at_nat @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ ( stl_Pr8376929985763293661at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps ) ) @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra6262887151753371453ist_fm @ Eff @ Rules @ ( stl_Pr8376929985763293661at_nat @ Steps ) )
           => ( abstra6262887151753371453ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_743_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_A5396818216090053761m_rule,Eff: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra8066765647771630831m_rule @ Eff @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps ) )
           => ( abstra8066765647771630831m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_744_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream1819640557982756571at_nat,Rules: stream8372878641218411373at_nat,Sl3: fset_A5396818216090053761m_rule,Eff: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra3157339599784851765m_rule @ Eff @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps ) )
           => ( abstra3157339599784851765m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_745_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream6060259936882193013at_nat,Rules: stream6724221391990029191at_nat,Sl3: fset_A5396818216090053761m_rule,Eff: product_prod_nat_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8440522571783428010at_nat @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ ( stl_Pr8881980968988507061at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps ) ) @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra9014086451575537615m_rule @ Eff @ Rules @ ( stl_Pr8881980968988507061at_nat @ Steps ) )
           => ( abstra9014086451575537615m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_746_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream9093217665182992091at_nat,Rules: stream8032061917732916077at_nat,Sl3: fset_A5396818216090053761m_rule,Eff: produc3843707927480180839at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8757157785044589968at_nat @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ ( stl_Pr6233422408842714779at_nat @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps ) ) @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps ) ) @ Sl3 )
         => ( ( abstra8385385442391400757m_rule @ Eff @ Rules @ ( stl_Pr6233422408842714779at_nat @ Steps ) )
           => ( abstra8385385442391400757m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_747_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7755889244956737277e_rule,Rules: stream_rule,Sl3: fset_A5396818216090053761m_rule,Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5823804033175701339m_rule @ Eff @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps ) )
           => ( abstra5823804033175701339m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_748_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Rules: stream_rule,Sl3: fset_P661503646757059847ist_fm,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( sset_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5415202973855822231ist_fm @ Eff @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_749_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5061764961602482947ist_fm
    = ( ^ [Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8299795917829157543ist_fm,A4: stream3409308193418444653ist_fm] :
        ? [Steps2: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
          ( ( A4 = Steps2 )
          & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ Sl )
          & ( abstra5061764961602482947ist_fm @ Eff2 @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_750_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1311857326046938941ist_fm
    = ( ^ [Eff2: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8372878641218411373at_nat,A4: stream1622999302952468147at_nat] :
        ? [Steps2: stream1622999302952468147at_nat,Sl: fset_P661503646757059847ist_fm] :
          ( ( A4 = Steps2 )
          & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps2 ) ) @ Sl )
          & ( abstra1311857326046938941ist_fm @ Eff2 @ Rules2 @ ( stl_Pr1564049449699658205at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_751_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra4041229200784676899ist_fm
    = ( ^ [Eff2: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream6724221391990029191at_nat,A4: stream5916762827556431437at_nat] :
        ? [Steps2: stream5916762827556431437at_nat,Sl: fset_P661503646757059847ist_fm] :
          ( ( A4 = Steps2 )
          & ( member8440522571783428010at_nat @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps2 ) ) @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ ( stl_Pr3810959331343356407at_nat @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps2 ) ) @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps2 ) ) @ Sl )
          & ( abstra4041229200784676899ist_fm @ Eff2 @ Rules2 @ ( stl_Pr3810959331343356407at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_752_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6262887151753371453ist_fm
    = ( ^ [Eff2: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8032061917732916077at_nat,A4: stream7788502890425800371at_nat] :
        ? [Steps2: stream7788502890425800371at_nat,Sl: fset_P661503646757059847ist_fm] :
          ( ( A4 = Steps2 )
          & ( member8757157785044589968at_nat @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps2 ) ) @ ( sset_P1195531240007660155at_nat @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ ( stl_Pr8376929985763293661at_nat @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps2 ) ) @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps2 ) ) @ Sl )
          & ( abstra6262887151753371453ist_fm @ Eff2 @ Rules2 @ ( stl_Pr8376929985763293661at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_753_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra8066765647771630831m_rule
    = ( ^ [Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8299795917829157543ist_fm,A4: stream7989562319868115861ist_fm] :
        ? [Steps2: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
          ( ( A4 = Steps2 )
          & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ Sl )
          & ( abstra8066765647771630831m_rule @ Eff2 @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_754_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra3157339599784851765m_rule
    = ( ^ [Eff2: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8372878641218411373at_nat,A4: stream1819640557982756571at_nat] :
        ? [Steps2: stream1819640557982756571at_nat,Sl: fset_A5396818216090053761m_rule] :
          ( ( A4 = Steps2 )
          & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( sset_P9164763103296375419at_nat @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps2 ) ) @ Sl )
          & ( abstra3157339599784851765m_rule @ Eff2 @ Rules2 @ ( stl_Pr2369401607900929691at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_755_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra9014086451575537615m_rule
    = ( ^ [Eff2: product_prod_nat_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream6724221391990029191at_nat,A4: stream6060259936882193013at_nat] :
        ? [Steps2: stream6060259936882193013at_nat,Sl: fset_A5396818216090053761m_rule] :
          ( ( A4 = Steps2 )
          & ( member8440522571783428010at_nat @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps2 ) ) @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ ( stl_Pr8881980968988507061at_nat @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps2 ) ) @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps2 ) ) @ Sl )
          & ( abstra9014086451575537615m_rule @ Eff2 @ Rules2 @ ( stl_Pr8881980968988507061at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_756_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra8385385442391400757m_rule
    = ( ^ [Eff2: produc3843707927480180839at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8032061917732916077at_nat,A4: stream9093217665182992091at_nat] :
        ? [Steps2: stream9093217665182992091at_nat,Sl: fset_A5396818216090053761m_rule] :
          ( ( A4 = Steps2 )
          & ( member8757157785044589968at_nat @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps2 ) ) @ ( sset_P1195531240007660155at_nat @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ ( stl_Pr6233422408842714779at_nat @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps2 ) ) @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps2 ) ) @ Sl )
          & ( abstra8385385442391400757m_rule @ Eff2 @ Rules2 @ ( stl_Pr6233422408842714779at_nat @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_757_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5823804033175701339m_rule
    = ( ^ [Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_rule,A4: stream7755889244956737277e_rule] :
        ? [Steps2: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
          ( ( A4 = Steps2 )
          & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ Sl )
          & ( abstra5823804033175701339m_rule @ Eff2 @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_758_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5415202973855822231ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A4: stream727092118206550309m_rule] :
        ? [Steps2: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
          ( ( A4 = Steps2 )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ Sl )
          & ( abstra5415202973855822231ist_fm @ Eff2 @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_759_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ Eff @ Rules @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra5061764961602482947ist_fm @ Eff @ Rules @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_760_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: produc859450856879609959at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8372878641218411373at_nat,A: stream1622999302952468147at_nat] :
      ( ( abstra1311857326046938941ist_fm @ Eff @ Rules @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra1311857326046938941ist_fm @ Eff @ Rules @ ( stl_Pr1564049449699658205at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_761_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream6724221391990029191at_nat,A: stream5916762827556431437at_nat] :
      ( ( abstra4041229200784676899ist_fm @ Eff @ Rules @ A )
     => ~ ( ( member8440522571783428010at_nat @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ A ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ ( stl_Pr3810959331343356407at_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ A ) ) @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra4041229200784676899ist_fm @ Eff @ Rules @ ( stl_Pr3810959331343356407at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_762_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8032061917732916077at_nat,A: stream7788502890425800371at_nat] :
      ( ( abstra6262887151753371453ist_fm @ Eff @ Rules @ A )
     => ~ ( ( member8757157785044589968at_nat @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ A ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ ( stl_Pr8376929985763293661at_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ A ) ) @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra6262887151753371453ist_fm @ Eff @ Rules @ ( stl_Pr8376929985763293661at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_763_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ Eff @ Rules @ A )
     => ~ ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra8066765647771630831m_rule @ Eff @ Rules @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_764_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: produc859450856879609959at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream8372878641218411373at_nat,A: stream1819640557982756571at_nat] :
      ( ( abstra3157339599784851765m_rule @ Eff @ Rules @ A )
     => ~ ( ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra3157339599784851765m_rule @ Eff @ Rules @ ( stl_Pr2369401607900929691at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_765_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: product_prod_nat_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream6724221391990029191at_nat,A: stream6060259936882193013at_nat] :
      ( ( abstra9014086451575537615m_rule @ Eff @ Rules @ A )
     => ~ ( ( member8440522571783428010at_nat @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ A ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ ( stl_Pr8881980968988507061at_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ A ) ) @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra9014086451575537615m_rule @ Eff @ Rules @ ( stl_Pr8881980968988507061at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_766_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: produc3843707927480180839at_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream8032061917732916077at_nat,A: stream9093217665182992091at_nat] :
      ( ( abstra8385385442391400757m_rule @ Eff @ Rules @ A )
     => ~ ( ( member8757157785044589968at_nat @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ A ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ ( stl_Pr6233422408842714779at_nat @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ A ) ) @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ A ) ) @ Sl2 )
               => ~ ( abstra8385385442391400757m_rule @ Eff @ Rules @ ( stl_Pr6233422408842714779at_nat @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_767_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ Eff @ Rules @ A )
     => ~ ( ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ Sl2 )
               => ~ ( abstra5823804033175701339m_rule @ Eff @ Rules @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_768_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ Eff @ Rules @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra5415202973855822231ist_fm @ Eff @ Rules @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_769_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream3409308193418444653ist_fm > $o,X5: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X: stream3409308193418444653ist_fm] :
            ( ( X7 @ X )
           => ? [Steps3: stream3409308193418444653ist_fm,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl5 )
                & ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) )
                  | ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_770_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream1622999302952468147at_nat > $o,X5: stream1622999302952468147at_nat,Rules: stream8372878641218411373at_nat,Eff3: produc859450856879609959at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X: stream1622999302952468147at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream1622999302952468147at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8206827879206165904at_nat @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( abstra6247935586015979962ist_fm @ Eff3 @ ( produc2750861588394444363at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ ( produc2934448030062000137at_nat @ ( shd_Pr5062688428016131033at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) )
                  | ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ ( stl_Pr1564049449699658205at_nat @ Steps3 ) ) ) ) )
       => ( abstra1311857326046938941ist_fm @ ( abstra6247935586015979962ist_fm @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_771_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream5916762827556431437at_nat > $o,X5: stream5916762827556431437at_nat,Rules: stream6724221391990029191at_nat,Eff3: product_prod_nat_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X: stream5916762827556431437at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream5916762827556431437at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8440522571783428010at_nat @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps3 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ ( stl_Pr3810959331343356407at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( abstra5696132678714034720ist_fm @ Eff3 @ ( produc8873327162560832741at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps3 ) ) @ ( produc6380178601224530851at_nat @ ( shd_Pr6154206852037299699at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr3810959331343356407at_nat @ Steps3 ) )
                  | ( abstra4041229200784676899ist_fm @ ( abstra5696132678714034720ist_fm @ Eff3 ) @ Rules @ ( stl_Pr3810959331343356407at_nat @ Steps3 ) ) ) ) )
       => ( abstra4041229200784676899ist_fm @ ( abstra5696132678714034720ist_fm @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_772_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7788502890425800371at_nat > $o,X5: stream7788502890425800371at_nat,Rules: stream8032061917732916077at_nat,Eff3: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X: stream7788502890425800371at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream7788502890425800371at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8757157785044589968at_nat @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps3 ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ ( stl_Pr8376929985763293661at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( abstra945866278424861114ist_fm @ Eff3 @ ( produc8437926920159080011at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps3 ) ) @ ( produc5735555203892637705at_nat @ ( shd_Pr1618269646678715353at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr8376929985763293661at_nat @ Steps3 ) )
                  | ( abstra6262887151753371453ist_fm @ ( abstra945866278424861114ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8376929985763293661at_nat @ Steps3 ) ) ) ) )
       => ( abstra6262887151753371453ist_fm @ ( abstra945866278424861114ist_fm @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_773_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7989562319868115861ist_fm > $o,X5: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X: stream7989562319868115861ist_fm] :
            ( ( X7 @ X )
           => ? [Steps3: stream7989562319868115861ist_fm,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl5 )
                & ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) )
                  | ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_774_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream1819640557982756571at_nat > $o,X5: stream1819640557982756571at_nat,Rules: stream8372878641218411373at_nat,Eff3: produc859450856879609959at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X: stream1819640557982756571at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream1819640557982756571at_nat,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8206827879206165904at_nat @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( sset_P9164763103296375419at_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( abstra3953117623887779576m_rule @ Eff3 @ ( produc601515061966782543at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ ( produc567330280010208273at_nat @ ( shd_Pr618815079712798623at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) )
                  | ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ ( stl_Pr2369401607900929691at_nat @ Steps3 ) ) ) ) )
       => ( abstra3157339599784851765m_rule @ ( abstra3953117623887779576m_rule @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_775_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream6060259936882193013at_nat > $o,X5: stream6060259936882193013at_nat,Rules: stream6724221391990029191at_nat,Eff3: product_prod_nat_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X: stream6060259936882193013at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream6060259936882193013at_nat,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8440522571783428010at_nat @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps3 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ ( stl_Pr8881980968988507061at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( abstra8432771076430573074m_rule @ Eff3 @ ( produc8148265381959644393at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps3 ) ) @ ( produc4617181344899183531at_nat @ ( shd_Pr2385784996796508857at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr8881980968988507061at_nat @ Steps3 ) )
                  | ( abstra9014086451575537615m_rule @ ( abstra8432771076430573074m_rule @ Eff3 ) @ Rules @ ( stl_Pr8881980968988507061at_nat @ Steps3 ) ) ) ) )
       => ( abstra9014086451575537615m_rule @ ( abstra8432771076430573074m_rule @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_776_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream9093217665182992091at_nat > $o,X5: stream9093217665182992091at_nat,Rules: stream8032061917732916077at_nat,Eff3: produc3843707927480180839at_nat > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X: stream9093217665182992091at_nat] :
            ( ( X7 @ X )
           => ? [Steps3: stream9093217665182992091at_nat,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8757157785044589968at_nat @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps3 ) ) @ ( sset_P1195531240007660155at_nat @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ ( stl_Pr6233422408842714779at_nat @ Steps3 ) ) ) @ Sl5 )
                & ( abstra3500458500780140280m_rule @ Eff3 @ ( produc4188740258898430031at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps3 ) ) @ ( produc7875181159794461713at_nat @ ( shd_Pr5894612048456619935at_nat @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr6233422408842714779at_nat @ Steps3 ) )
                  | ( abstra8385385442391400757m_rule @ ( abstra3500458500780140280m_rule @ Eff3 ) @ Rules @ ( stl_Pr6233422408842714779at_nat @ Steps3 ) ) ) ) )
       => ( abstra8385385442391400757m_rule @ ( abstra3500458500780140280m_rule @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_777_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7755889244956737277e_rule > $o,X5: stream7755889244956737277e_rule,Rules: stream_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X: stream7755889244956737277e_rule] :
            ( ( X7 @ X )
           => ? [Steps3: stream7755889244956737277e_rule,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl5 )
                & ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) )
                  | ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_778_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X5: stream727092118206550309m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X7 @ X )
           => ? [Steps3: stream727092118206550309m_rule,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl5 )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_779_wf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X5: abstra4572470701423150929m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X7 @ X )
           => ? [T6: abstra4572470701423150929m_rule] :
                ( ( X = T6 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( sset_rule @ rules ) )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) )
                  = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T6 ) ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra4476735159789068068ist_fm
                        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                            ( ( eff @ R @ S )
                            = ( some_f2162114139285431020ist_fm @ Ss ) )
                        @ rules
                        @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ X5 ) ) ) ).

% wf.coinduct
thf(fact_780_wf_Ointros,axiom,
    ! [T: abstra4572470701423150929m_rule] :
      ( ( 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 ) ) ) )
       => ( ! [T4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra4476735159789068068ist_fm
                @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ rules
                @ T4 ) )
         => ( abstra4476735159789068068ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ T ) ) ) ) ).

% wf.intros
thf(fact_781_wf_Osimps,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
      = ( ? [T2: abstra4572470701423150929m_rule] :
            ( ( A = T2 )
            & ( 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 ) ) ) )
            & ! [X3: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X3 @ ( abstra4962253831105676628m_rule @ T2 ) )
               => ( abstra4476735159789068068ist_fm
                  @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ rules
                  @ X3 ) ) ) ) ) ).

% wf.simps
thf(fact_782_wf_Ocases,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ rules ) )
         => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) )
              = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) ) )
           => ~ ! [T7: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ T7 ) ) ) ) ) ).

% wf.cases
thf(fact_783_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y3: product_prod_nat_nat,A: produc859450856879609959at_nat] :
      ( ( P @ X5 @ Y3 )
     => ( ( A
          = ( produc6161850002892822231at_nat @ X5 @ Y3 ) )
       => ( P @ ( produc3213797794245857475at_nat @ A ) @ ( produc6408287024330202629at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_784_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: list_fm > list_fm > $o,X5: list_fm,Y3: list_fm,A: produc1996495991257130529ist_fm] :
      ( ( P @ X5 @ Y3 )
     => ( ( A
          = ( produc381145313068854617ist_fm @ X5 @ Y3 ) )
       => ( P @ ( produc1501393135466168645ist_fm @ A ) @ ( produc4588648349897876871ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_785_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,A: produc1319942482725812455at_nat] :
      ( ( P @ X5 @ Y3 )
     => ( ( A
          = ( produc9060074326276436823at_nat @ X5 @ Y3 ) )
       => ( P @ ( produc4486455499028916547at_nat @ A ) @ ( produc826222791318859909at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_786_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat] :
      ( ( P @ X5 @ Y3 )
     => ( ( A
          = ( produc2922128104949294807at_nat @ X5 @ Y3 ) )
       => ( P @ ( produc6011863565407846595at_nat @ A ) @ ( produc2937545259423399429at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_787_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: nat > nat > $o,X5: nat,Y3: nat,A: product_prod_nat_nat] :
      ( ( P @ X5 @ Y3 )
     => ( ( A
          = ( product_Pair_nat_nat @ X5 @ Y3 ) )
       => ( P @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_788_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: produc1996495991257130529ist_fm > rule > $o,X5: produc1996495991257130529ist_fm,Y3: rule,A: produc164195504107695125m_rule] :
      ( ( P @ X5 @ Y3 )
     => ( ( A
          = ( produc491467635432902671m_rule @ X5 @ Y3 ) )
       => ( P @ ( produc6879501374131015971m_rule @ A ) @ ( produc7165828336582415457m_rule @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_789_conjI__realizer,axiom,
    ! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat,Q: product_prod_nat_nat > $o,Q2: product_prod_nat_nat] :
      ( ( P @ P3 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ P3 @ Q2 ) ) )
          & ( Q @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ P3 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_790_conjI__realizer,axiom,
    ! [P: list_fm > $o,P3: list_fm,Q: list_fm > $o,Q2: list_fm] :
      ( ( P @ P3 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ P3 @ Q2 ) ) )
          & ( Q @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ P3 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_791_conjI__realizer,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,P3: set_Pr4329608150637261639at_nat,Q: set_Pr4329608150637261639at_nat > $o,Q2: set_Pr4329608150637261639at_nat] :
      ( ( P @ P3 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ P3 @ Q2 ) ) )
          & ( Q @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ P3 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_792_conjI__realizer,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,P3: set_Pr1261947904930325089at_nat,Q: set_Pr1261947904930325089at_nat > $o,Q2: set_Pr1261947904930325089at_nat] :
      ( ( P @ P3 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ P3 @ Q2 ) ) )
          & ( Q @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ P3 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_793_conjI__realizer,axiom,
    ! [P: nat > $o,P3: nat,Q: nat > $o,Q2: nat] :
      ( ( P @ P3 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q2 ) ) )
          & ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_794_conjI__realizer,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,P3: produc1996495991257130529ist_fm,Q: rule > $o,Q2: rule] :
      ( ( P @ P3 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ P3 @ Q2 ) ) )
          & ( Q @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ P3 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_795_fset_Omap__comp,axiom,
    ! [G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc164195504107695125m_rule,V: fset_A5396818216090053761m_rule] :
      ( ( fimage6650965236902554765ist_fm @ G @ ( fimage6814378747598375653m_rule @ F @ V ) )
      = ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ G @ F ) @ V ) ) ).

% fset.map_comp
thf(fact_796_fset_Omap__comp,axiom,
    ! [G: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,V: fset_P661503646757059847ist_fm] :
      ( ( fimage3349360726776475169m_rule @ G @ ( fimage748043113761278389m_rule @ F @ V ) )
      = ( fimage748043113761278389m_rule @ ( comp_A7346078547394085594ist_fm @ G @ F ) @ V ) ) ).

% fset.map_comp
thf(fact_797_fset_Omap__comp,axiom,
    ! [G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,V: fset_A5396818216090053761m_rule] :
      ( ( fimage8674584286907012797ist_fm @ G @ ( fimage9164954772359936849ist_fm @ F @ V ) )
      = ( fimage9164954772359936849ist_fm @ ( comp_P2882693189979712244m_rule @ G @ F ) @ V ) ) ).

% fset.map_comp
thf(fact_798_fset_Omap__comp,axiom,
    ! [G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,V: fset_P661503646757059847ist_fm] :
      ( ( fimage748043113761278389m_rule @ G @ ( fimage8674584286907012797ist_fm @ F @ V ) )
      = ( fimage748043113761278389m_rule @ ( comp_P2076232811723594896ist_fm @ G @ F ) @ V ) ) ).

% fset.map_comp
thf(fact_799_fset_Omap__comp,axiom,
    ! [G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,V: fset_A5396818216090053761m_rule] :
      ( ( fimage748043113761278389m_rule @ G @ ( fimage9164954772359936849ist_fm @ F @ V ) )
      = ( fimage3349360726776475169m_rule @ ( comp_P4439951101548160930m_rule @ G @ F ) @ V ) ) ).

% fset.map_comp
thf(fact_800_fset_Omap__comp,axiom,
    ! [G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,V: fset_A5396818216090053761m_rule] :
      ( ( fimage9164954772359936849ist_fm @ G @ ( fimage3349360726776475169m_rule @ F @ V ) )
      = ( fimage9164954772359936849ist_fm @ ( comp_A8152538925650202942m_rule @ G @ F ) @ V ) ) ).

% fset.map_comp
thf(fact_801_fset_Omap__comp,axiom,
    ! [G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,V: fset_P661503646757059847ist_fm] :
      ( ( fimage9164954772359936849ist_fm @ G @ ( fimage748043113761278389m_rule @ F @ V ) )
      = ( fimage8674584286907012797ist_fm @ ( comp_A2205678127158149108ist_fm @ G @ F ) @ V ) ) ).

% fset.map_comp
thf(fact_802_K__record__comp,axiom,
    ! [C: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P4378417847848123130m_rule
        @ ^ [X3: produc164195504107695125m_rule] : C
        @ F )
      = ( ^ [X3: abstra4572470701423150929m_rule] : C ) ) ).

% K_record_comp
thf(fact_803_rewriteR__comp__comp2,axiom,
    ! [G: abstra4572470701423150929m_rule > produc164195504107695125m_rule,H: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,R1: produc164195504107695125m_rule > produc164195504107695125m_rule,R22: abstra4572470701423150929m_rule > produc164195504107695125m_rule,F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,L: produc164195504107695125m_rule > produc1996495991257130529ist_fm] :
      ( ( ( comp_A7552847697395116572m_rule @ G @ H )
        = ( comp_P8444364099071156576m_rule @ R1 @ R22 ) )
     => ( ( ( comp_P7463969011078018366m_rule @ F @ R1 )
          = L )
       => ( ( comp_A8152538925650202942m_rule @ ( comp_P4378417847848123130m_rule @ F @ G ) @ H )
          = ( comp_P4378417847848123130m_rule @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_804_rewriteR__comp__comp2,axiom,
    ! [G: produc164195504107695125m_rule > produc164195504107695125m_rule,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,R1: abstra4572470701423150929m_rule > produc164195504107695125m_rule,R22: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,L: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ( comp_P8444364099071156576m_rule @ G @ H )
        = ( comp_A7552847697395116572m_rule @ R1 @ R22 ) )
     => ( ( ( comp_P4378417847848123130m_rule @ F @ R1 )
          = L )
       => ( ( comp_P4378417847848123130m_rule @ ( comp_P7463969011078018366m_rule @ F @ G ) @ H )
          = ( comp_A8152538925650202942m_rule @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_805_rewriteR__comp__comp2,axiom,
    ! [G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,R1: produc164195504107695125m_rule > produc1996495991257130529ist_fm,R22: abstra4572470701423150929m_rule > produc164195504107695125m_rule,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,L: produc164195504107695125m_rule > produc1996495991257130529ist_fm] :
      ( ( ( comp_P4378417847848123130m_rule @ G @ H )
        = ( comp_P4378417847848123130m_rule @ R1 @ R22 ) )
     => ( ( ( comp_P8245093624798776376m_rule @ F @ R1 )
          = L )
       => ( ( comp_P4378417847848123130m_rule @ ( comp_P8245093624798776376m_rule @ F @ G ) @ H )
          = ( comp_P4378417847848123130m_rule @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_806_rewriteL__comp__comp2,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,L1: produc164195504107695125m_rule > produc1996495991257130529ist_fm,L2: produc164195504107695125m_rule > produc164195504107695125m_rule,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,R2: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( ( comp_P8245093624798776376m_rule @ F @ G )
        = ( comp_P7463969011078018366m_rule @ L1 @ L2 ) )
     => ( ( ( comp_P8444364099071156576m_rule @ L2 @ H )
          = R2 )
       => ( ( comp_P2882693189979712244m_rule @ F @ ( comp_P4378417847848123130m_rule @ G @ H ) )
          = ( comp_P4378417847848123130m_rule @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_807_rewriteL__comp__comp2,axiom,
    ! [F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: produc164195504107695125m_rule > produc164195504107695125m_rule,L1: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,L2: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,R2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ( comp_P7463969011078018366m_rule @ F @ G )
        = ( comp_P8245093624798776376m_rule @ L1 @ L2 ) )
     => ( ( ( comp_P4378417847848123130m_rule @ L2 @ H )
          = R2 )
       => ( ( comp_P4378417847848123130m_rule @ F @ ( comp_P8444364099071156576m_rule @ G @ H ) )
          = ( comp_P2882693189979712244m_rule @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_808_rewriteL__comp__comp2,axiom,
    ! [F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc164195504107695125m_rule,L1: produc164195504107695125m_rule > produc1996495991257130529ist_fm,L2: abstra4572470701423150929m_rule > produc164195504107695125m_rule,H: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,R2: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( ( comp_P4378417847848123130m_rule @ F @ G )
        = ( comp_P4378417847848123130m_rule @ L1 @ L2 ) )
     => ( ( ( comp_A7552847697395116572m_rule @ L2 @ H )
          = R2 )
       => ( ( comp_P4378417847848123130m_rule @ F @ ( comp_A7552847697395116572m_rule @ G @ H ) )
          = ( comp_P4378417847848123130m_rule @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_809_rewriteR__comp__comp,axiom,
    ! [G: abstra4572470701423150929m_rule > produc164195504107695125m_rule,H: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,R2: abstra4572470701423150929m_rule > produc164195504107695125m_rule,F: produc164195504107695125m_rule > produc1996495991257130529ist_fm] :
      ( ( ( comp_A7552847697395116572m_rule @ G @ H )
        = R2 )
     => ( ( comp_A8152538925650202942m_rule @ ( comp_P4378417847848123130m_rule @ F @ G ) @ H )
        = ( comp_P4378417847848123130m_rule @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_810_rewriteR__comp__comp,axiom,
    ! [G: produc164195504107695125m_rule > produc164195504107695125m_rule,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,R2: abstra4572470701423150929m_rule > produc164195504107695125m_rule,F: produc164195504107695125m_rule > produc1996495991257130529ist_fm] :
      ( ( ( comp_P8444364099071156576m_rule @ G @ H )
        = R2 )
     => ( ( comp_P4378417847848123130m_rule @ ( comp_P7463969011078018366m_rule @ F @ G ) @ H )
        = ( comp_P4378417847848123130m_rule @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_811_rewriteR__comp__comp,axiom,
    ! [G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,R2: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( ( comp_P4378417847848123130m_rule @ G @ H )
        = R2 )
     => ( ( comp_P4378417847848123130m_rule @ ( comp_P8245093624798776376m_rule @ F @ G ) @ H )
        = ( comp_P2882693189979712244m_rule @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_812_rewriteL__comp__comp,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,L: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( ( comp_P8245093624798776376m_rule @ F @ G )
        = L )
     => ( ( comp_P2882693189979712244m_rule @ F @ ( comp_P4378417847848123130m_rule @ G @ H ) )
        = ( comp_P4378417847848123130m_rule @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_813_rewriteL__comp__comp,axiom,
    ! [F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: produc164195504107695125m_rule > produc164195504107695125m_rule,L: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( ( comp_P7463969011078018366m_rule @ F @ G )
        = L )
     => ( ( comp_P4378417847848123130m_rule @ F @ ( comp_P8444364099071156576m_rule @ G @ H ) )
        = ( comp_P4378417847848123130m_rule @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_814_rewriteL__comp__comp,axiom,
    ! [F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc164195504107695125m_rule,L: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( ( comp_P4378417847848123130m_rule @ F @ G )
        = L )
     => ( ( comp_P4378417847848123130m_rule @ F @ ( comp_A7552847697395116572m_rule @ G @ H ) )
        = ( comp_A8152538925650202942m_rule @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_815_subset__CollectI,axiom,
    ! [B5: set_Pr7058068377845519745ist_fm,A2: set_Pr7058068377845519745ist_fm,Q: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B5 @ A2 )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X @ B5 )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le2055072402069232929ist_fm
          @ ( collec7231575776666803724ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ B5 )
                & ( Q @ X3 ) ) )
          @ ( collec7231575776666803724ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_816_subset__CollectI,axiom,
    ! [B5: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,Q: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A2 )
     => ( ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ B5 )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le3000389064537975527at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X3 @ B5 )
                & ( Q @ X3 ) ) )
          @ ( collec7088162979684241874at_nat
            @ ^ [X3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_817_subset__CollectI,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B5 @ A2 )
     => ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ B5 )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ B5 )
                & ( Q @ X3 ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_818_subset__CollectI,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,Q: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ B5 @ A2 )
     => ( ! [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ B5 )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_le1268244103169919719at_nat
          @ ( collec6321179662152712658at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X3 @ B5 )
                & ( Q @ X3 ) ) )
          @ ( collec6321179662152712658at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_819_subset__CollectI,axiom,
    ! [B5: set_rule,A2: set_rule,Q: rule > $o,P: rule > $o] :
      ( ( ord_less_eq_set_rule @ B5 @ A2 )
     => ( ! [X: rule] :
            ( ( member_rule @ X @ B5 )
           => ( ( Q @ X )
             => ( P @ X ) ) )
       => ( ord_less_eq_set_rule
          @ ( collect_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ B5 )
                & ( Q @ X3 ) ) )
          @ ( collect_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_820_subset__Collect__iff,axiom,
    ! [B5: set_Pr7058068377845519745ist_fm,A2: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B5 @ A2 )
     => ( ( ord_le2055072402069232929ist_fm @ B5
          @ ( collec7231575776666803724ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( ! [X3: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ X3 @ B5 )
             => ( P @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_821_subset__Collect__iff,axiom,
    ! [B5: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A2 )
     => ( ( ord_le3000389064537975527at_nat @ B5
          @ ( collec7088162979684241874at_nat
            @ ^ [X3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( ! [X3: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X3 @ B5 )
             => ( P @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_822_subset__Collect__iff,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B5 @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ B5
          @ ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ B5 )
             => ( P @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_823_subset__Collect__iff,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ B5 @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ B5
          @ ( collec6321179662152712658at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( ! [X3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X3 @ B5 )
             => ( P @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_824_subset__Collect__iff,axiom,
    ! [B5: set_rule,A2: set_rule,P: rule > $o] :
      ( ( ord_less_eq_set_rule @ B5 @ A2 )
     => ( ( ord_less_eq_set_rule @ B5
          @ ( collect_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( ! [X3: rule] :
              ( ( member_rule @ X3 @ B5 )
             => ( P @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_825_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff @ Rules @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ( ( Eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) )
           => ~ ! [T7: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ T7 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_826_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra4476735159789068068ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A4: abstra4572470701423150929m_rule] :
        ? [T2: abstra4572470701423150929m_rule] :
          ( ( A4 = T2 )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ Rules2 ) )
          & ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) )
          & ! [X3: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X3 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ X3 ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_827_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T: abstra4572470701423150929m_rule,Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ Rules ) )
     => ( ( Eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) )
       => ( ! [T4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ T4 ) )
         => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ T ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_828_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X5: abstra4572470701423150929m_rule,Rules: stream_rule,Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X5 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X7 @ X )
           => ? [T6: abstra4572470701423150929m_rule] :
                ( ( X = T6 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( sset_rule @ Rules ) )
                & ( Eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T6 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ Eff @ Rules @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_829_i_Owf_Ocases,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) )
           => ~ ! [T7: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T7 ) ) ) ) ) ).

% i.wf.cases
thf(fact_830_i_Owf_Osimps,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [T2: abstra4572470701423150929m_rule] :
            ( ( A = T2 )
            & ( 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 ) ) )
            & ! [X3: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X3 @ ( abstra4962253831105676628m_rule @ T2 ) )
               => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ) ) ).

% i.wf.simps
thf(fact_831_i_Owf_Ointros,axiom,
    ! [T: abstra4572470701423150929m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( 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 ) ) )
       => ( ! [T4: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T4 ) )
         => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T ) ) ) ) ).

% i.wf.intros
thf(fact_832_i_Owf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X5: abstra4572470701423150929m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X7 @ X )
           => ? [T6: abstra4572470701423150929m_rule] :
                ( ( X = T6 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( sset_rule @ Rules ) )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T6 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T6 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.wf.coinduct
thf(fact_833_exI__realizer,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o,Y3: product_prod_nat_nat,X5: product_prod_nat_nat] :
      ( ( P @ Y3 @ X5 )
     => ( P @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) ) @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_834_exI__realizer,axiom,
    ! [P: list_fm > list_fm > $o,Y3: list_fm,X5: list_fm] :
      ( ( P @ Y3 @ X5 )
     => ( P @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y3 ) ) @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_835_exI__realizer,axiom,
    ! [P: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,Y3: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat] :
      ( ( P @ Y3 @ X5 )
     => ( P @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y3 ) ) @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_836_exI__realizer,axiom,
    ! [P: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Y3: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( P @ Y3 @ X5 )
     => ( P @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y3 ) ) @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_837_exI__realizer,axiom,
    ! [P: nat > nat > $o,Y3: nat,X5: nat] :
      ( ( P @ Y3 @ X5 )
     => ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_838_exI__realizer,axiom,
    ! [P: rule > produc1996495991257130529ist_fm > $o,Y3: rule,X5: produc1996495991257130529ist_fm] :
      ( ( P @ Y3 @ X5 )
     => ( P @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X5 @ Y3 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X5 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_839_eq__snd__iff,axiom,
    ! [B: product_prod_nat_nat,P3: produc859450856879609959at_nat] :
      ( ( B
        = ( produc6408287024330202629at_nat @ P3 ) )
      = ( ? [A4: product_prod_nat_nat] :
            ( P3
            = ( produc6161850002892822231at_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_840_eq__snd__iff,axiom,
    ! [B: list_fm,P3: produc1996495991257130529ist_fm] :
      ( ( B
        = ( produc4588648349897876871ist_fm @ P3 ) )
      = ( ? [A4: list_fm] :
            ( P3
            = ( produc381145313068854617ist_fm @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_841_eq__snd__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,P3: produc1319942482725812455at_nat] :
      ( ( B
        = ( produc826222791318859909at_nat @ P3 ) )
      = ( ? [A4: set_Pr4329608150637261639at_nat] :
            ( P3
            = ( produc9060074326276436823at_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_842_eq__snd__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,P3: produc3843707927480180839at_nat] :
      ( ( B
        = ( produc2937545259423399429at_nat @ P3 ) )
      = ( ? [A4: set_Pr1261947904930325089at_nat] :
            ( P3
            = ( produc2922128104949294807at_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_843_eq__snd__iff,axiom,
    ! [B: nat,P3: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P3 ) )
      = ( ? [A4: nat] :
            ( P3
            = ( product_Pair_nat_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_844_eq__snd__iff,axiom,
    ! [B: rule,P3: produc164195504107695125m_rule] :
      ( ( B
        = ( produc7165828336582415457m_rule @ P3 ) )
      = ( ? [A4: produc1996495991257130529ist_fm] :
            ( P3
            = ( produc491467635432902671m_rule @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_845_sndI,axiom,
    ! [X5: produc859450856879609959at_nat,Y3: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( X5
        = ( produc6161850002892822231at_nat @ Y3 @ Z ) )
     => ( ( produc6408287024330202629at_nat @ X5 )
        = Z ) ) ).

% sndI
thf(fact_846_sndI,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y3: list_fm,Z: list_fm] :
      ( ( X5
        = ( produc381145313068854617ist_fm @ Y3 @ Z ) )
     => ( ( produc4588648349897876871ist_fm @ X5 )
        = Z ) ) ).

% sndI
thf(fact_847_sndI,axiom,
    ! [X5: produc1319942482725812455at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( X5
        = ( produc9060074326276436823at_nat @ Y3 @ Z ) )
     => ( ( produc826222791318859909at_nat @ X5 )
        = Z ) ) ).

% sndI
thf(fact_848_sndI,axiom,
    ! [X5: produc3843707927480180839at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( X5
        = ( produc2922128104949294807at_nat @ Y3 @ Z ) )
     => ( ( produc2937545259423399429at_nat @ X5 )
        = Z ) ) ).

% sndI
thf(fact_849_sndI,axiom,
    ! [X5: product_prod_nat_nat,Y3: nat,Z: nat] :
      ( ( X5
        = ( product_Pair_nat_nat @ Y3 @ Z ) )
     => ( ( product_snd_nat_nat @ X5 )
        = Z ) ) ).

% sndI
thf(fact_850_sndI,axiom,
    ! [X5: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z: rule] :
      ( ( X5
        = ( produc491467635432902671m_rule @ Y3 @ Z ) )
     => ( ( produc7165828336582415457m_rule @ X5 )
        = Z ) ) ).

% sndI
thf(fact_851_fstI,axiom,
    ! [X5: produc859450856879609959at_nat,Y3: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( X5
        = ( produc6161850002892822231at_nat @ Y3 @ Z ) )
     => ( ( produc3213797794245857475at_nat @ X5 )
        = Y3 ) ) ).

% fstI
thf(fact_852_fstI,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y3: list_fm,Z: list_fm] :
      ( ( X5
        = ( produc381145313068854617ist_fm @ Y3 @ Z ) )
     => ( ( produc1501393135466168645ist_fm @ X5 )
        = Y3 ) ) ).

% fstI
thf(fact_853_fstI,axiom,
    ! [X5: produc1319942482725812455at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( X5
        = ( produc9060074326276436823at_nat @ Y3 @ Z ) )
     => ( ( produc4486455499028916547at_nat @ X5 )
        = Y3 ) ) ).

% fstI
thf(fact_854_fstI,axiom,
    ! [X5: produc3843707927480180839at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( X5
        = ( produc2922128104949294807at_nat @ Y3 @ Z ) )
     => ( ( produc6011863565407846595at_nat @ X5 )
        = Y3 ) ) ).

% fstI
thf(fact_855_fstI,axiom,
    ! [X5: product_prod_nat_nat,Y3: nat,Z: nat] :
      ( ( X5
        = ( product_Pair_nat_nat @ Y3 @ Z ) )
     => ( ( product_fst_nat_nat @ X5 )
        = Y3 ) ) ).

% fstI
thf(fact_856_fstI,axiom,
    ! [X5: produc164195504107695125m_rule,Y3: produc1996495991257130529ist_fm,Z: rule] :
      ( ( X5
        = ( produc491467635432902671m_rule @ Y3 @ Z ) )
     => ( ( produc6879501374131015971m_rule @ X5 )
        = Y3 ) ) ).

% fstI
thf(fact_857_eq__fst__iff,axiom,
    ! [A: product_prod_nat_nat,P3: produc859450856879609959at_nat] :
      ( ( A
        = ( produc3213797794245857475at_nat @ P3 ) )
      = ( ? [B3: product_prod_nat_nat] :
            ( P3
            = ( produc6161850002892822231at_nat @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_858_eq__fst__iff,axiom,
    ! [A: list_fm,P3: produc1996495991257130529ist_fm] :
      ( ( A
        = ( produc1501393135466168645ist_fm @ P3 ) )
      = ( ? [B3: list_fm] :
            ( P3
            = ( produc381145313068854617ist_fm @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_859_eq__fst__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,P3: produc1319942482725812455at_nat] :
      ( ( A
        = ( produc4486455499028916547at_nat @ P3 ) )
      = ( ? [B3: set_Pr4329608150637261639at_nat] :
            ( P3
            = ( produc9060074326276436823at_nat @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_860_eq__fst__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,P3: produc3843707927480180839at_nat] :
      ( ( A
        = ( produc6011863565407846595at_nat @ P3 ) )
      = ( ? [B3: set_Pr1261947904930325089at_nat] :
            ( P3
            = ( produc2922128104949294807at_nat @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_861_eq__fst__iff,axiom,
    ! [A: nat,P3: product_prod_nat_nat] :
      ( ( A
        = ( product_fst_nat_nat @ P3 ) )
      = ( ? [B3: nat] :
            ( P3
            = ( product_Pair_nat_nat @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_862_eq__fst__iff,axiom,
    ! [A: produc1996495991257130529ist_fm,P3: produc164195504107695125m_rule] :
      ( ( A
        = ( produc6879501374131015971m_rule @ P3 ) )
      = ( ? [B3: rule] :
            ( P3
            = ( produc491467635432902671m_rule @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_863_konig_Osimps_I2_J,axiom,
    ! [T: abstract_tree_rule] :
      ( ( stl_rule @ ( abstract_konig_rule @ T ) )
      = ( abstract_konig_rule
        @ ( fChoic8596484284512669613e_rule
          @ ^ [T5: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T5 @ ( abstract_cont_rule @ T ) )
              & ~ ( abstra6797871900832723819e_rule @ T5 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_864_konig_Osimps_I2_J,axiom,
    ! [T: abstra4572470701423150929m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( abstra3079604473823497664m_rule @ T ) )
      = ( abstra3079604473823497664m_rule
        @ ( fChoic7543385803022166781m_rule
          @ ^ [T5: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T ) )
              & ~ ( abstra3507476289341504315m_rule @ T5 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_865_fst__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc5894184038225292483ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X3 @ X3 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% fst_diag_fst
thf(fact_866_snd__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc6725306161561501521e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X3: rule] : ( produc1769597910952761123e_rule @ X3 @ X3 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% snd_diag_snd
thf(fact_867_fst__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc7264528503596239631e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X3: rule] : ( produc1769597910952761123e_rule @ X3 @ X3 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% fst_diag_snd
thf(fact_868_snd__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc1639249265332130309ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X3 @ X3 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% snd_diag_fst
thf(fact_869_cont__def,axiom,
    ( abstra4962253831105676628m_rule
    = ( abstra1216198984563783442m_rule
      @ ^ [X13: produc164195504107695125m_rule,X23: fset_A5396818216090053761m_rule] : X23 ) ) ).

% cont_def
thf(fact_870_tree_Oroot__def,axiom,
    ( abstra3503897793822247108m_rule
    = ( abstra6581154509029876902m_rule
      @ ^ [X13: produc164195504107695125m_rule,X23: fset_A5396818216090053761m_rule] : X13 ) ) ).

% tree.root_def
thf(fact_871_some__sym__eq__trivial,axiom,
    ! [X5: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ( ^ [Y4: fset_P661503646757059847ist_fm,Z2: fset_P661503646757059847ist_fm] : ( Y4 = Z2 )
          @ X5 ) )
      = X5 ) ).

% some_sym_eq_trivial
thf(fact_872_some__eq__trivial,axiom,
    ! [X5: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ^ [Y5: fset_P661503646757059847ist_fm] : ( Y5 = X5 ) )
      = X5 ) ).

% some_eq_trivial
thf(fact_873_some__equality,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ( P @ A )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P @ X )
           => ( X = A ) )
       => ( ( fChoic3892542562059901019ist_fm @ P )
          = A ) ) ) ).

% some_equality
thf(fact_874_konig_Ocode,axiom,
    ( abstra3079604473823497664m_rule
    = ( ^ [T2: abstra4572470701423150929m_rule] :
          ( sCons_5731567480799343806m_rule @ ( abstra3503897793822247108m_rule @ T2 )
          @ ( abstra3079604473823497664m_rule
            @ ( fChoic7543385803022166781m_rule
              @ ^ [T5: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T2 ) )
                  & ~ ( abstra3507476289341504315m_rule @ T5 ) ) ) ) ) ) ) ).

% konig.code
thf(fact_875_verit__sko__forall__indirect2,axiom,
    ! [X5: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o,P4: fset_P661503646757059847ist_fm > $o] :
      ( ( X5
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X3: fset_P661503646757059847ist_fm] :
              ~ ( P @ X3 ) ) )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P @ X )
            = ( P4 @ X ) )
       => ( ( ! [X4: fset_P661503646757059847ist_fm] : ( P4 @ X4 ) )
          = ( P @ X5 ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_876_verit__sko__forall__indirect,axiom,
    ! [X5: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o] :
      ( ( X5
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X3: fset_P661503646757059847ist_fm] :
              ~ ( P @ X3 ) ) )
     => ( ( ! [X4: fset_P661503646757059847ist_fm] : ( P @ X4 ) )
        = ( P @ X5 ) ) ) ).

% verit_sko_forall_indirect
thf(fact_877_stream_Ocollapse,axiom,
    ! [Stream: stream_rule] :
      ( ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_878_stream_Ocollapse,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_879_szip__unfold,axiom,
    ! [A: produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,B: rule,S2: stream_rule] :
      ( ( szip_P1977448745965526924m_rule @ ( sCons_307821682964077496ist_fm @ A @ S1 ) @ ( sCons_rule @ B @ S2 ) )
      = ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ A @ B ) @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_880_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B: rule,S2: stream_rule] :
      ( ( szip_rule_rule @ ( sCons_rule @ A @ S1 ) @ ( sCons_rule @ B @ S2 ) )
      = ( sCons_7940479310257918722e_rule @ ( produc1769597910952761123e_rule @ A @ B ) @ ( szip_rule_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_881_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B: produc164195504107695125m_rule,S2: stream727092118206550309m_rule] :
      ( ( szip_r2254114822203693110m_rule @ ( sCons_rule @ A @ S1 ) @ ( sCons_5731567480799343806m_rule @ B @ S2 ) )
      = ( sCons_2207919253591517394m_rule @ ( produc6084188410630241011m_rule @ A @ B ) @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_882_szip__unfold,axiom,
    ! [A: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B: rule,S2: stream_rule] :
      ( ( szip_P553291425819358518e_rule @ ( sCons_5731567480799343806m_rule @ A @ S1 ) @ ( sCons_rule @ B @ S2 ) )
      = ( sCons_6835861040388153298e_rule @ ( produc4383365014245906419e_rule @ A @ B ) @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_883_szip__unfold,axiom,
    ! [A: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B: produc164195504107695125m_rule,S2: stream727092118206550309m_rule] :
      ( ( szip_P2499414959592755846m_rule @ ( sCons_5731567480799343806m_rule @ A @ S1 ) @ ( sCons_5731567480799343806m_rule @ B @ S2 ) )
      = ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ A @ B ) @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_884_szip__unfold,axiom,
    ! [A: product_prod_nat_nat,S1: stream6724221391990029191at_nat,B: product_prod_nat_nat,S2: stream6724221391990029191at_nat] :
      ( ( szip_P4314639285670189082at_nat @ ( sCons_2147322719763279000at_nat @ A @ S1 ) @ ( sCons_2147322719763279000at_nat @ B @ S2 ) )
      = ( sCons_5297226365640217982at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_885_szip__unfold,axiom,
    ! [A: list_fm,S1: stream_list_fm,B: list_fm,S2: stream_list_fm] :
      ( ( szip_list_fm_list_fm @ ( sCons_list_fm @ A @ S1 ) @ ( sCons_list_fm @ B @ S2 ) )
      = ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ A @ B ) @ ( szip_list_fm_list_fm @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_886_szip__unfold,axiom,
    ! [A: set_Pr4329608150637261639at_nat,S1: stream3627166509194609229at_nat,B: set_Pr4329608150637261639at_nat,S2: stream3627166509194609229at_nat] :
      ( ( szip_s1528810966652963738at_nat @ ( sCons_4876977544976522334at_nat @ A @ S1 ) @ ( sCons_4876977544976522334at_nat @ B @ S2 ) )
      = ( sCons_796194910919348478at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( szip_s1528810966652963738at_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_887_szip__unfold,axiom,
    ! [A: set_Pr1261947904930325089at_nat,S1: stream7255200869668761831at_nat,B: set_Pr1261947904930325089at_nat,S2: stream7255200869668761831at_nat] :
      ( ( szip_s6495482949952704538at_nat @ ( sCons_2777945275570954744at_nat @ A @ S1 ) @ ( sCons_2777945275570954744at_nat @ B @ S2 ) )
      = ( sCons_3175934388574233982at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( szip_s6495482949952704538at_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_888_szip__unfold,axiom,
    ! [A: nat,S1: stream_nat,B: nat,S2: stream_nat] :
      ( ( szip_nat_nat @ ( sCons_nat @ A @ S1 ) @ ( sCons_nat @ B @ S2 ) )
      = ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( szip_nat_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_889_stream_Osel_I2_J,axiom,
    ! [X1: rule,X2: stream_rule] :
      ( ( stl_rule @ ( sCons_rule @ X1 @ X2 ) )
      = X2 ) ).

% stream.sel(2)
thf(fact_890_stream_Osel_I2_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) )
      = X2 ) ).

% stream.sel(2)
thf(fact_891_stream_Osel_I1_J,axiom,
    ! [X1: rule,X2: stream_rule] :
      ( ( shd_rule @ ( sCons_rule @ X1 @ X2 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_892_stream_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_893_stream_Oset__intros_I2_J,axiom,
    ! [Y3: produc1996495991257130529ist_fm,X2: stream8299795917829157543ist_fm,X1: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ Y3 @ ( sset_P3431374123360452149ist_fm @ X2 ) )
     => ( member8102475879199740618ist_fm @ Y3 @ ( sset_P3431374123360452149ist_fm @ ( sCons_307821682964077496ist_fm @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_894_stream_Oset__intros_I2_J,axiom,
    ! [Y3: produc859450856879609959at_nat,X2: stream8372878641218411373at_nat,X1: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Y3 @ ( sset_P9164763103296375419at_nat @ X2 ) )
     => ( member8206827879206165904at_nat @ Y3 @ ( sset_P9164763103296375419at_nat @ ( sCons_5297226365640217982at_nat @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_895_stream_Oset__intros_I2_J,axiom,
    ! [Y3: product_prod_nat_nat,X2: stream6724221391990029191at_nat,X1: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Y3 @ ( sset_P8584168253256249621at_nat @ X2 ) )
     => ( member8440522571783428010at_nat @ Y3 @ ( sset_P8584168253256249621at_nat @ ( sCons_2147322719763279000at_nat @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_896_stream_Oset__intros_I2_J,axiom,
    ! [Y3: produc3843707927480180839at_nat,X2: stream8032061917732916077at_nat,X1: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ Y3 @ ( sset_P1195531240007660155at_nat @ X2 ) )
     => ( member8757157785044589968at_nat @ Y3 @ ( sset_P1195531240007660155at_nat @ ( sCons_3175934388574233982at_nat @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_897_stream_Oset__intros_I2_J,axiom,
    ! [Y3: rule,X2: stream_rule,X1: rule] :
      ( ( member_rule @ Y3 @ ( sset_rule @ X2 ) )
     => ( member_rule @ Y3 @ ( sset_rule @ ( sCons_rule @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_898_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X1 @ ( sset_P3431374123360452149ist_fm @ ( sCons_307821682964077496ist_fm @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_899_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc859450856879609959at_nat,X2: stream8372878641218411373at_nat] : ( member8206827879206165904at_nat @ X1 @ ( sset_P9164763103296375419at_nat @ ( sCons_5297226365640217982at_nat @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_900_stream_Oset__intros_I1_J,axiom,
    ! [X1: product_prod_nat_nat,X2: stream6724221391990029191at_nat] : ( member8440522571783428010at_nat @ X1 @ ( sset_P8584168253256249621at_nat @ ( sCons_2147322719763279000at_nat @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_901_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc3843707927480180839at_nat,X2: stream8032061917732916077at_nat] : ( member8757157785044589968at_nat @ X1 @ ( sset_P1195531240007660155at_nat @ ( sCons_3175934388574233982at_nat @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_902_stream_Oset__intros_I1_J,axiom,
    ! [X1: rule,X2: stream_rule] : ( member_rule @ X1 @ ( sset_rule @ ( sCons_rule @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_903_stream_Oset__cases,axiom,
    ! [E: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm] :
      ( ( member8102475879199740618ist_fm @ E @ ( sset_P3431374123360452149ist_fm @ A ) )
     => ( ! [Z22: stream8299795917829157543ist_fm] :
            ( A
           != ( sCons_307821682964077496ist_fm @ E @ Z22 ) )
       => ~ ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm] :
              ( ( A
                = ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) )
             => ~ ( member8102475879199740618ist_fm @ E @ ( sset_P3431374123360452149ist_fm @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_904_stream_Oset__cases,axiom,
    ! [E: produc859450856879609959at_nat,A: stream8372878641218411373at_nat] :
      ( ( member8206827879206165904at_nat @ E @ ( sset_P9164763103296375419at_nat @ A ) )
     => ( ! [Z22: stream8372878641218411373at_nat] :
            ( A
           != ( sCons_5297226365640217982at_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc859450856879609959at_nat,Z22: stream8372878641218411373at_nat] :
              ( ( A
                = ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) )
             => ~ ( member8206827879206165904at_nat @ E @ ( sset_P9164763103296375419at_nat @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_905_stream_Oset__cases,axiom,
    ! [E: product_prod_nat_nat,A: stream6724221391990029191at_nat] :
      ( ( member8440522571783428010at_nat @ E @ ( sset_P8584168253256249621at_nat @ A ) )
     => ( ! [Z22: stream6724221391990029191at_nat] :
            ( A
           != ( sCons_2147322719763279000at_nat @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_nat_nat,Z22: stream6724221391990029191at_nat] :
              ( ( A
                = ( sCons_2147322719763279000at_nat @ Z1 @ Z22 ) )
             => ~ ( member8440522571783428010at_nat @ E @ ( sset_P8584168253256249621at_nat @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_906_stream_Oset__cases,axiom,
    ! [E: produc3843707927480180839at_nat,A: stream8032061917732916077at_nat] :
      ( ( member8757157785044589968at_nat @ E @ ( sset_P1195531240007660155at_nat @ A ) )
     => ( ! [Z22: stream8032061917732916077at_nat] :
            ( A
           != ( sCons_3175934388574233982at_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc3843707927480180839at_nat,Z22: stream8032061917732916077at_nat] :
              ( ( A
                = ( sCons_3175934388574233982at_nat @ Z1 @ Z22 ) )
             => ~ ( member8757157785044589968at_nat @ E @ ( sset_P1195531240007660155at_nat @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_907_stream_Oset__cases,axiom,
    ! [E: rule,A: stream_rule] :
      ( ( member_rule @ E @ ( sset_rule @ A ) )
     => ( ! [Z22: stream_rule] :
            ( A
           != ( sCons_rule @ E @ Z22 ) )
       => ~ ! [Z1: rule,Z22: stream_rule] :
              ( ( A
                = ( sCons_rule @ Z1 @ Z22 ) )
             => ~ ( member_rule @ E @ ( sset_rule @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_908_stream_Oset__induct,axiom,
    ! [X5: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm,P: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X5 @ ( sset_P3431374123360452149ist_fm @ A ) )
     => ( ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm] : ( P @ Z1 @ ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm,Xa: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Xa @ ( sset_P3431374123360452149ist_fm @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) ) ) )
         => ( P @ X5 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_909_stream_Oset__induct,axiom,
    ! [X5: produc859450856879609959at_nat,A: stream8372878641218411373at_nat,P: produc859450856879609959at_nat > stream8372878641218411373at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X5 @ ( sset_P9164763103296375419at_nat @ A ) )
     => ( ! [Z1: produc859450856879609959at_nat,Z22: stream8372878641218411373at_nat] : ( P @ Z1 @ ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) )
       => ( ! [Z1: produc859450856879609959at_nat,Z22: stream8372878641218411373at_nat,Xa: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ Xa @ ( sset_P9164763103296375419at_nat @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_5297226365640217982at_nat @ Z1 @ Z22 ) ) ) )
         => ( P @ X5 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_910_stream_Oset__induct,axiom,
    ! [X5: product_prod_nat_nat,A: stream6724221391990029191at_nat,P: product_prod_nat_nat > stream6724221391990029191at_nat > $o] :
      ( ( member8440522571783428010at_nat @ X5 @ ( sset_P8584168253256249621at_nat @ A ) )
     => ( ! [Z1: product_prod_nat_nat,Z22: stream6724221391990029191at_nat] : ( P @ Z1 @ ( sCons_2147322719763279000at_nat @ Z1 @ Z22 ) )
       => ( ! [Z1: product_prod_nat_nat,Z22: stream6724221391990029191at_nat,Xa: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Xa @ ( sset_P8584168253256249621at_nat @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_2147322719763279000at_nat @ Z1 @ Z22 ) ) ) )
         => ( P @ X5 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_911_stream_Oset__induct,axiom,
    ! [X5: produc3843707927480180839at_nat,A: stream8032061917732916077at_nat,P: produc3843707927480180839at_nat > stream8032061917732916077at_nat > $o] :
      ( ( member8757157785044589968at_nat @ X5 @ ( sset_P1195531240007660155at_nat @ A ) )
     => ( ! [Z1: produc3843707927480180839at_nat,Z22: stream8032061917732916077at_nat] : ( P @ Z1 @ ( sCons_3175934388574233982at_nat @ Z1 @ Z22 ) )
       => ( ! [Z1: produc3843707927480180839at_nat,Z22: stream8032061917732916077at_nat,Xa: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Xa @ ( sset_P1195531240007660155at_nat @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_3175934388574233982at_nat @ Z1 @ Z22 ) ) ) )
         => ( P @ X5 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_912_stream_Oset__induct,axiom,
    ! [X5: rule,A: stream_rule,P: rule > stream_rule > $o] :
      ( ( member_rule @ X5 @ ( sset_rule @ A ) )
     => ( ! [Z1: rule,Z22: stream_rule] : ( P @ Z1 @ ( sCons_rule @ Z1 @ Z22 ) )
       => ( ! [Z1: rule,Z22: stream_rule,Xa: rule] :
              ( ( member_rule @ Xa @ ( sset_rule @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_rule @ Z1 @ Z22 ) ) ) )
         => ( P @ X5 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_913_stream_Ocase,axiom,
    ! [F: rule > stream_rule > abstra4572470701423150929m_rule,X1: rule,X2: stream_rule] :
      ( ( case_s2494923709252486212m_rule @ F @ ( sCons_rule @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% stream.case
thf(fact_914_sdrop__while__SCons,axiom,
    ! [P: rule > $o,A: rule,S3: stream_rule] :
      ( ( ( P @ A )
       => ( ( sdrop_while_rule @ P @ ( sCons_rule @ A @ S3 ) )
          = ( sdrop_while_rule @ P @ S3 ) ) )
      & ( ~ ( P @ A )
       => ( ( sdrop_while_rule @ P @ ( sCons_rule @ A @ S3 ) )
          = ( sCons_rule @ A @ S3 ) ) ) ) ).

% sdrop_while_SCons
thf(fact_915_stream_Oexhaust__sel,axiom,
    ! [Stream: stream_rule] :
      ( Stream
      = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_916_stream_Oexhaust__sel,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( Stream
      = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_917_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_918_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_919_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_920_stream_Osplit__sel,axiom,
    ! [P: abstra4572470701423150929m_rule > $o,F: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( P @ ( case_s2494923709252486212m_rule @ F @ Stream ) )
      = ( ( Stream
          = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) )
       => ( P @ ( F @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ) ) ).

% stream.split_sel
thf(fact_921_stream_Osplit__sel__asm,axiom,
    ! [P: abstra4572470701423150929m_rule > $o,F: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( P @ ( case_s2494923709252486212m_rule @ F @ Stream ) )
      = ( ~ ( ( Stream
              = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) )
            & ~ ( P @ ( F @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ) ) ) ).

% stream.split_sel_asm
thf(fact_922_szip_Ocode,axiom,
    ( szip_P4314639285670189082at_nat
    = ( ^ [S12: stream6724221391990029191at_nat,S22: stream6724221391990029191at_nat] : ( sCons_5297226365640217982at_nat @ ( produc6161850002892822231at_nat @ ( shd_Pr4260400998323988397at_nat @ S12 ) @ ( shd_Pr4260400998323988397at_nat @ S22 ) ) @ ( szip_P4314639285670189082at_nat @ ( stl_Pr5027827701538482609at_nat @ S12 ) @ ( stl_Pr5027827701538482609at_nat @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_923_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_924_szip_Ocode,axiom,
    ( szip_s1528810966652963738at_nat
    = ( ^ [S12: stream3627166509194609229at_nat,S22: stream3627166509194609229at_nat] : ( sCons_796194910919348478at_nat @ ( produc9060074326276436823at_nat @ ( shd_se2672623163517224947at_nat @ S12 ) @ ( shd_se2672623163517224947at_nat @ S22 ) ) @ ( szip_s1528810966652963738at_nat @ ( stl_se8095006981682938871at_nat @ S12 ) @ ( stl_se8095006981682938871at_nat @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_925_szip_Ocode,axiom,
    ( szip_s6495482949952704538at_nat
    = ( ^ [S12: stream7255200869668761831at_nat,S22: stream7255200869668761831at_nat] : ( sCons_3175934388574233982at_nat @ ( produc2922128104949294807at_nat @ ( shd_se6818656645295851789at_nat @ S12 ) @ ( shd_se6818656645295851789at_nat @ S22 ) ) @ ( szip_s6495482949952704538at_nat @ ( stl_se6007078998265101073at_nat @ S12 ) @ ( stl_se6007078998265101073at_nat @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_926_szip_Ocode,axiom,
    ( szip_nat_nat
    = ( ^ [S12: stream_nat,S22: stream_nat] : ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ ( shd_nat @ S12 ) @ ( shd_nat @ S22 ) ) @ ( szip_nat_nat @ ( stl_nat @ S12 ) @ ( stl_nat @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_927_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_928_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_929_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_930_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_931_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_932_verit__sko__ex_H,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A2: $o] :
      ( ( ( P @ ( fChoic3892542562059901019ist_fm @ P ) )
        = A2 )
     => ( ( ? [X4: fset_P661503646757059847ist_fm] : ( P @ X4 ) )
        = A2 ) ) ).

% verit_sko_ex'
thf(fact_933_verit__sko__forall,axiom,
    ( ( ^ [P5: fset_P661503646757059847ist_fm > $o] :
        ! [X8: fset_P661503646757059847ist_fm] : ( P5 @ X8 ) )
    = ( ^ [P2: fset_P661503646757059847ist_fm > $o] :
          ( P2
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X3: fset_P661503646757059847ist_fm] :
                ~ ( P2 @ X3 ) ) ) ) ) ).

% verit_sko_forall
thf(fact_934_someI2,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm,Q: fset_P661503646757059847ist_fm > $o] :
      ( ( P @ A )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P @ X )
           => ( Q @ X ) )
       => ( Q @ ( fChoic3892542562059901019ist_fm @ P ) ) ) ) ).

% someI2
thf(fact_935_verit__sko__forall_H,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A2: $o] :
      ( ( ( P
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X3: fset_P661503646757059847ist_fm] :
                ~ ( P @ X3 ) ) )
        = A2 )
     => ( ( ! [X4: fset_P661503646757059847ist_fm] : ( P @ X4 ) )
        = A2 ) ) ).

% verit_sko_forall'
thf(fact_936_verit__sko__forall_H_H,axiom,
    ! [B5: fset_P661503646757059847ist_fm,A2: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o] :
      ( ( B5 = A2 )
     => ( ( ( fChoic3892542562059901019ist_fm @ P )
          = A2 )
        = ( ( fChoic3892542562059901019ist_fm @ P )
          = B5 ) ) ) ).

% verit_sko_forall''
thf(fact_937_someI__ex,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_12: fset_P661503646757059847ist_fm] : ( P @ X_12 )
     => ( P @ ( fChoic3892542562059901019ist_fm @ P ) ) ) ).

% someI_ex
thf(fact_938_someI2__ex,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,Q: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_12: fset_P661503646757059847ist_fm] : ( P @ X_12 )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P @ X )
           => ( Q @ X ) )
       => ( Q @ ( fChoic3892542562059901019ist_fm @ P ) ) ) ) ).

% someI2_ex
thf(fact_939_someI2__bex,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm > $o] :
      ( ? [X6: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ X @ A2 )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoic1980167584695909237ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_940_someI2__bex,axiom,
    ! [A2: set_rule,P: rule > $o,Q: rule > $o] :
      ( ? [X6: rule] :
          ( ( member_rule @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X: rule] :
            ( ( ( member_rule @ X @ A2 )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoice_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_941_someI2__bex,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
      ( ? [X6: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X: produc859450856879609959at_nat] :
            ( ( ( member8206827879206165904at_nat @ X @ A2 )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoic6673916560228149179at_nat
            @ ^ [X3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_942_someI2__bex,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X: product_prod_nat_nat] :
            ( ( ( member8440522571783428010at_nat @ X @ A2 )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_943_someI2__bex,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ? [X6: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X: produc3843707927480180839at_nat] :
            ( ( ( member8757157785044589968at_nat @ X @ A2 )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoic2372384411376717755at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_944_someI2__bex,axiom,
    ! [A2: set_fs7534060674568567527ist_fm,P: fset_P661503646757059847ist_fm > $o,Q: fset_P661503646757059847ist_fm > $o] :
      ( ? [X6: fset_P661503646757059847ist_fm] :
          ( ( member3493611713653719344ist_fm @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( ( member3493611713653719344ist_fm @ X @ A2 )
              & ( P @ X ) )
           => ( Q @ X ) )
       => ( Q
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X3: fset_P661503646757059847ist_fm] :
                ( ( member3493611713653719344ist_fm @ X3 @ A2 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_945_some__eq__ex,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o] :
      ( ( P @ ( fChoic3892542562059901019ist_fm @ P ) )
      = ( ? [X4: fset_P661503646757059847ist_fm] : ( P @ X4 ) ) ) ).

% some_eq_ex
thf(fact_946_verit__sko__ex__indirect,axiom,
    ! [X5: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o] :
      ( ( X5
        = ( fChoic3892542562059901019ist_fm @ P ) )
     => ( ( ? [X4: fset_P661503646757059847ist_fm] : ( P @ X4 ) )
        = ( P @ X5 ) ) ) ).

% verit_sko_ex_indirect
thf(fact_947_verit__sko__ex__indirect2,axiom,
    ! [X5: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o,P4: fset_P661503646757059847ist_fm > $o] :
      ( ( X5
        = ( fChoic3892542562059901019ist_fm @ P ) )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P @ X )
            = ( P4 @ X ) )
       => ( ( ? [X4: fset_P661503646757059847ist_fm] : ( P4 @ X4 ) )
          = ( P @ X5 ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_948_some1__equality,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ? [X6: fset_P661503646757059847ist_fm] :
          ( ( P @ X6 )
          & ! [Y: fset_P661503646757059847ist_fm] :
              ( ( P @ Y )
             => ( Y = X6 ) ) )
     => ( ( P @ A )
       => ( ( fChoic3892542562059901019ist_fm @ P )
          = A ) ) ) ).

% some1_equality
thf(fact_949_sfilter_Ocode,axiom,
    ( sfilte7579662470208054517m_rule
    = ( ^ [P2: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P2 ) @ S ) ) @ ( sfilte7579662470208054517m_rule @ P2 @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P2 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_950_sfilter_Ocode,axiom,
    ( sfilter_rule
    = ( ^ [P2: rule > $o,S: stream_rule] : ( sCons_rule @ ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P2 ) @ S ) ) @ ( sfilter_rule @ P2 @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P2 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_951_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_952_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_953_convol__expand__snd,axiom,
    ! [F: abstra4572470701423150929m_rule > produc164195504107695125m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ F )
        = G )
     => ( ( bNF_co2617238127527879852m_rule @ G @ ( comp_P1643750715206316048m_rule @ produc7165828336582415457m_rule @ F ) )
        = F ) ) ).

% convol_expand_snd
thf(fact_954_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
      = ( smap2_rule_rule_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_955_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc780299800278082923e_rule > rule,S1: stream_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_rule @ N @ ( smap2_4638655784531377133e_rule @ F @ S1 @ S2 ) )
      = ( smap2_4638655784531377133e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_956_sdrop__smap2,axiom,
    ! [N: nat,F: produc780299800278082923e_rule > rule > rule,S1: stream3588992565182678257e_rule,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_4273976090187245369e_rule @ F @ S1 @ S2 ) )
      = ( smap2_4273976090187245369e_rule @ F @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_957_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc780299800278082923e_rule,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( smap2_2301042431756194977e_rule @ F @ S1 @ S2 ) )
      = ( smap2_2301042431756194977e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_958_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc1996495991257130529ist_fm > rule,S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( sdrop_rule @ N @ ( smap2_3943848181979139747m_rule @ F @ S1 @ S2 ) )
      = ( smap2_3943848181979139747m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_4442373711808556042ist_fm @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_959_sdrop__smap2,axiom,
    ! [N: nat,F: produc1996495991257130529ist_fm > rule > rule,S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_6133544317484809199e_rule @ F @ S1 @ S2 ) )
      = ( smap2_6133544317484809199e_rule @ F @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_960_sdrop__smap2,axiom,
    ! [N: nat,F: produc780299800278082923e_rule > produc780299800278082923e_rule > rule,S1: stream3588992565182678257e_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_rule @ N @ ( smap2_6314367768709263299e_rule @ F @ S1 @ S2 ) )
      = ( smap2_6314367768709263299e_rule @ F @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_961_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc1996495991257130529ist_fm,S1: stream_rule,S2: stream_rule] :
      ( ( sdrop_4442373711808556042ist_fm @ N @ ( smap2_1292763952458720599ist_fm @ F @ S1 @ S2 ) )
      = ( smap2_1292763952458720599ist_fm @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_962_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc780299800278082923e_rule > produc780299800278082923e_rule,S1: stream_rule,S2: stream3588992565182678257e_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( smap2_644143445760767683e_rule @ F @ S1 @ S2 ) )
      = ( smap2_644143445760767683e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_2199339672054592340e_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_963_sdrop__smap2,axiom,
    ! [N: nat,F: produc780299800278082923e_rule > rule > produc780299800278082923e_rule,S1: stream3588992565182678257e_rule,S2: stream_rule] :
      ( ( sdrop_2199339672054592340e_rule @ N @ ( smap2_3976754415934081143e_rule @ F @ S1 @ S2 ) )
      = ( smap2_3976754415934081143e_rule @ F @ ( sdrop_2199339672054592340e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_smap2
thf(fact_964_smap2_Osimps_I2_J,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( stl_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
      = ( smap2_rule_rule_rule @ F @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_965_smap2_Osimps_I2_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_2737362159257330783e_rule @ F @ S1 @ S2 ) )
      = ( smap2_2737362159257330783e_rule @ F @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_966_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( stl_rule @ ( smap2_190827247425732959e_rule @ F @ S1 @ S2 ) )
      = ( smap2_190827247425732959e_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_967_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_1391242002117124143e_rule @ F @ S1 @ S2 ) )
      = ( smap2_1391242002117124143e_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_968_smap2_Osimps_I2_J,axiom,
    ! [F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_4438185555641665375m_rule @ F @ S1 @ S2 ) )
      = ( smap2_4438185555641665375m_rule @ F @ ( stl_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_969_smap2_Osimps_I2_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_7722561687528711727m_rule @ F @ S1 @ S2 ) )
      = ( smap2_7722561687528711727m_rule @ F @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_970_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3092065398501458735m_rule @ F @ S1 @ S2 ) )
      = ( smap2_3092065398501458735m_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_971_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S2 ) )
      = ( smap2_3865418324013074943m_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% smap2.simps(2)
thf(fact_972_smap2_Osimps_I1_J,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_973_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_2737362159257330783e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_974_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( shd_rule @ ( smap2_190827247425732959e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_975_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_1391242002117124143e_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_976_smap2_Osimps_I1_J,axiom,
    ! [F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_4438185555641665375m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_977_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_7722561687528711727m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_978_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3092065398501458735m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_979_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S2 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S2 ) ) ) ).

% smap2.simps(1)
thf(fact_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_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_988_convol__o,axiom,
    ! [F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P4378935762333416640m_rule @ ( bNF_co8656752567580110654ist_fm @ F @ G ) @ H )
      = ( bNF_co5722988055202470650ist_fm @ ( comp_P4378417847848123130m_rule @ F @ H ) @ ( comp_P4378417847848123130m_rule @ G @ H ) ) ) ).

% convol_o
thf(fact_989_convol__o,axiom,
    ! [F: produc164195504107695125m_rule > list_fm,G: produc164195504107695125m_rule > list_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P4378417847848123130m_rule @ ( bNF_co6210242947678836288ist_fm @ F @ G ) @ H )
      = ( bNF_co6484617322090750972ist_fm @ ( comp_P3077181278802195115m_rule @ F @ H ) @ ( comp_P3077181278802195115m_rule @ G @ H ) ) ) ).

% convol_o
thf(fact_990_convol__expand__snd_H,axiom,
    ! [F: abstra4572470701423150929m_rule > produc164195504107695125m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > rule] :
      ( ( ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ F )
        = G )
     => ( ( H
          = ( comp_P1643750715206316048m_rule @ produc7165828336582415457m_rule @ F ) )
        = ( ( bNF_co2617238127527879852m_rule @ G @ H )
          = F ) ) ) ).

% convol_expand_snd'
thf(fact_991_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_992_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_993_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_994_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_995_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_996_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,S3: stream8299795917829157543ist_fm] :
      ( ? [N5: nat] : ( P @ ( snth_P2624749845746815685ist_fm @ S3 @ N5 ) )
     => ( ( sdrop_7013889347380152453ist_fm @ ( comp_o6353595986216377982ist_fm @ (~) @ P ) @ S3 )
        = ( sdrop_4442373711808556042ist_fm
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P2624749845746815685ist_fm @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_997_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc780299800278082923e_rule > $o,S3: stream3588992565182678257e_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P8487267234472522255e_rule @ S3 @ N5 ) )
     => ( ( sdrop_4395986648051443663e_rule @ ( comp_o7949119091651446216e_rule @ (~) @ P ) @ S3 )
        = ( sdrop_2199339672054592340e_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P8487267234472522255e_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_998_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc1660549859252256699m_rule > $o,S3: stream1878305877988316353m_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P7967347330282200671m_rule @ S3 @ N5 ) )
     => ( ( sdrop_6423752018218906655m_rule @ ( comp_o6304513743058443928m_rule @ (~) @ P ) @ S3 )
        = ( sdrop_1764812713384251940m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P7967347330282200671m_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_999_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc6288491646048892603e_rule > $o,S3: stream2312013117288958913e_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P3371917080224060767e_rule @ S3 @ N5 ) )
     => ( ( sdrop_1828321768160766751e_rule @ ( comp_o1709083493000304024e_rule @ (~) @ P ) @ S3 )
        = ( sdrop_6392754500180887844e_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P3371917080224060767e_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_1000_sdrop__while__sdrop__LEAST,axiom,
    ! [P: produc3793571666044074251m_rule > $o,S3: stream6494289010434245521m_rule] :
      ( ? [N5: nat] : ( P @ ( snth_P8950227884416379311m_rule @ S3 @ N5 ) )
     => ( ( sdrop_7771878303739699055m_rule @ ( comp_o8292784708554675816m_rule @ (~) @ P ) @ S3 )
        = ( sdrop_7373388980841566196m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P @ ( snth_P8950227884416379311m_rule @ S3 @ N3 ) ) )
          @ S3 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_1001_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_1002_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream8299795917829157543ist_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S3 ) )
      = ( snth_P2624749845746815685ist_fm @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_1003_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream3588992565182678257e_rule] :
      ( ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S3 ) )
      = ( snth_P8487267234472522255e_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_1004_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream1878305877988316353m_rule] :
      ( ( shd_Pr3564872132220007m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S3 ) )
      = ( snth_P7967347330282200671m_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_1005_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream2312013117288958913e_rule] :
      ( ( shd_Pr4631506658928855911e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S3 ) )
      = ( snth_P3371917080224060767e_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_1006_sdrop__simps_I1_J,axiom,
    ! [N: nat,S3: stream6494289010434245521m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S3 ) )
      = ( snth_P8950227884416379311m_rule @ S3 @ N ) ) ).

% sdrop_simps(1)
thf(fact_1007_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_1008_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_1009_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_1010_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream_rule,N: nat] :
      ( ( snth_P8487267234472522255e_rule @ ( szip_rule_rule @ S1 @ S2 ) @ N )
      = ( produc1769597910952761123e_rule @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1011_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P7967347330282200671m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) @ N )
      = ( produc6084188410630241011m_rule @ ( snth_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1012_snth__szip,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream_rule,N: nat] :
      ( ( snth_P3371917080224060767e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) @ N )
      = ( produc4383365014245906419e_rule @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1013_snth__szip,axiom,
    ! [S1: stream727092118206550309m_rule,S2: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P8950227884416379311m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) @ N )
      = ( produc5927390650430071747m_rule @ ( snth_P1712579846118164081m_rule @ S1 @ N ) @ ( snth_P1712579846118164081m_rule @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1014_snth__szip,axiom,
    ! [S1: stream6724221391990029191at_nat,S2: stream6724221391990029191at_nat,N: nat] :
      ( ( snth_P9098630209796499211at_nat @ ( szip_P4314639285670189082at_nat @ S1 @ S2 ) @ N )
      = ( produc6161850002892822231at_nat @ ( snth_P9042041890044241317at_nat @ S1 @ N ) @ ( snth_P9042041890044241317at_nat @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1015_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_1016_snth__szip,axiom,
    ! [S1: stream3627166509194609229at_nat,S2: stream3627166509194609229at_nat,N: nat] :
      ( ( snth_P3703150499108313995at_nat @ ( szip_s1528810966652963738at_nat @ S1 @ S2 ) @ N )
      = ( produc9060074326276436823at_nat @ ( snth_s1413033584812429803at_nat @ S1 @ N ) @ ( snth_s1413033584812429803at_nat @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1017_snth__szip,axiom,
    ! [S1: stream7255200869668761831at_nat,S2: stream7255200869668761831at_nat,N: nat] :
      ( ( snth_P8412462321310240523at_nat @ ( szip_s6495482949952704538at_nat @ S1 @ S2 ) @ N )
      = ( produc2922128104949294807at_nat @ ( snth_s5805196001064724229at_nat @ S1 @ N ) @ ( snth_s5805196001064724229at_nat @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1018_snth__szip,axiom,
    ! [S1: stream_nat,S2: stream_nat,N: nat] :
      ( ( snth_P9042041890044241317at_nat @ ( szip_nat_nat @ S1 @ S2 ) @ N )
      = ( product_Pair_nat_nat @ ( snth_nat @ S1 @ N ) @ ( snth_nat @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1019_snth__sset,axiom,
    ! [S3: stream8299795917829157543ist_fm,N: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S3 @ N ) @ ( sset_P3431374123360452149ist_fm @ S3 ) ) ).

% snth_sset
thf(fact_1020_snth__sset,axiom,
    ! [S3: stream8372878641218411373at_nat,N: nat] : ( member8206827879206165904at_nat @ ( snth_P9098630209796499211at_nat @ S3 @ N ) @ ( sset_P9164763103296375419at_nat @ S3 ) ) ).

% snth_sset
thf(fact_1021_snth__sset,axiom,
    ! [S3: stream6724221391990029191at_nat,N: nat] : ( member8440522571783428010at_nat @ ( snth_P9042041890044241317at_nat @ S3 @ N ) @ ( sset_P8584168253256249621at_nat @ S3 ) ) ).

% snth_sset
thf(fact_1022_snth__sset,axiom,
    ! [S3: stream8032061917732916077at_nat,N: nat] : ( member8757157785044589968at_nat @ ( snth_P8412462321310240523at_nat @ S3 @ N ) @ ( sset_P1195531240007660155at_nat @ S3 ) ) ).

% snth_sset
thf(fact_1023_snth__sset,axiom,
    ! [S3: stream_rule,N: nat] : ( member_rule @ ( snth_rule @ S3 @ N ) @ ( sset_rule @ S3 ) ) ).

% snth_sset
thf(fact_1024_conj__subset__def,axiom,
    ! [A2: set_rule,P: rule > $o,Q: rule > $o] :
      ( ( ord_less_eq_set_rule @ A2
        @ ( collect_rule
          @ ^ [X3: rule] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) )
      = ( ( ord_less_eq_set_rule @ A2 @ ( collect_rule @ P ) )
        & ( ord_less_eq_set_rule @ A2 @ ( collect_rule @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1025_snth__sset__smerge,axiom,
    ! [Ss3: stream8183306005803988909ist_fm,N: nat,M: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ ( snth_s4276058539312847179ist_fm @ Ss3 @ N ) @ M ) @ ( sset_P3431374123360452149ist_fm @ ( smerge8067294617398470191ist_fm @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_1026_snth__sset__smerge,axiom,
    ! [Ss3: stream7445237232777090803at_nat,N: nat,M: nat] : ( member8206827879206165904at_nat @ ( snth_P9098630209796499211at_nat @ ( snth_s7872458624752591889at_nat @ Ss3 @ N ) @ M ) @ ( sset_P9164763103296375419at_nat @ ( smerge8841642037904234357at_nat @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_1027_snth__sset__smerge,axiom,
    ! [Ss3: stream6397635269783887501at_nat,N: nat,M: nat] : ( member8440522571783428010at_nat @ ( snth_P9042041890044241317at_nat @ ( snth_s1632096110156728875at_nat @ Ss3 @ N ) @ M ) @ ( sset_P8584168253256249621at_nat @ ( smerge3155118954755218703at_nat @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_1028_snth__sset__smerge,axiom,
    ! [Ss3: stream9081085636389990131at_nat,N: nat,M: nat] : ( member8757157785044589968at_nat @ ( snth_P8412462321310240523at_nat @ ( snth_s1140275531675071505at_nat @ Ss3 @ N ) @ M ) @ ( sset_P1195531240007660155at_nat @ ( smerge2707736165599096693at_nat @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_1029_snth__sset__smerge,axiom,
    ! [Ss3: stream_stream_rule,N: nat,M: nat] : ( member_rule @ ( snth_rule @ ( snth_stream_rule @ Ss3 @ N ) @ M ) @ ( sset_rule @ ( smerge_rule @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_1030_Fair__Stream_Ofair__def,axiom,
    ( fair_fair_rule
    = ( ^ [S: stream_rule] :
        ! [X3: rule] :
          ( ( member_rule @ X3 @ ( sset_rule @ S ) )
         => ! [M3: nat] :
            ? [N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
              & ( ( snth_rule @ S @ N3 )
                = X3 ) ) ) ) ) ).

% Fair_Stream.fair_def
thf(fact_1031_prop__restrict,axiom,
    ! [X5: produc1996495991257130529ist_fm,Z3: set_Pr7058068377845519745ist_fm,X7: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X5 @ Z3 )
     => ( ( ord_le2055072402069232929ist_fm @ Z3
          @ ( collec7231575776666803724ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ X7 )
                & ( P @ X3 ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_1032_prop__restrict,axiom,
    ! [X5: produc859450856879609959at_nat,Z3: set_Pr8693737435421807431at_nat,X7: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X5 @ Z3 )
     => ( ( ord_le3000389064537975527at_nat @ Z3
          @ ( collec7088162979684241874at_nat
            @ ^ [X3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X3 @ X7 )
                & ( P @ X3 ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_1033_prop__restrict,axiom,
    ! [X5: product_prod_nat_nat,Z3: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X5 @ Z3 )
     => ( ( ord_le3146513528884898305at_nat @ Z3
          @ ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ X7 )
                & ( P @ X3 ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_1034_prop__restrict,axiom,
    ! [X5: produc3843707927480180839at_nat,Z3: set_Pr4329608150637261639at_nat,X7: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ X5 @ Z3 )
     => ( ( ord_le1268244103169919719at_nat @ Z3
          @ ( collec6321179662152712658at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X3 @ X7 )
                & ( P @ X3 ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_1035_prop__restrict,axiom,
    ! [X5: rule,Z3: set_rule,X7: set_rule,P: rule > $o] :
      ( ( member_rule @ X5 @ Z3 )
     => ( ( ord_less_eq_set_rule @ Z3
          @ ( collect_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ X7 )
                & ( P @ X3 ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_1036_Collect__restrict,axiom,
    ! [X7: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X3 @ X7 )
            & ( P @ X3 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1037_Collect__restrict,axiom,
    ! [X7: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ X7 )
            & ( P @ X3 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1038_Collect__restrict,axiom,
    ! [X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ X7 )
            & ( P @ X3 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1039_Collect__restrict,axiom,
    ! [X7: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ord_le1268244103169919719at_nat
      @ ( collec6321179662152712658at_nat
        @ ^ [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ X7 )
            & ( P @ X3 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1040_Collect__restrict,axiom,
    ! [X7: set_rule,P: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X3: rule] :
            ( ( member_rule @ X3 @ X7 )
            & ( P @ X3 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1041_prod_Oswap__def,axiom,
    ( produc7225117575323628663at_nat
    = ( ^ [P6: produc859450856879609959at_nat] : ( produc6161850002892822231at_nat @ ( produc6408287024330202629at_nat @ P6 ) @ ( produc3213797794245857475at_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1042_prod_Oswap__def,axiom,
    ( produc7936585717479832313ist_fm
    = ( ^ [P6: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc4588648349897876871ist_fm @ P6 ) @ ( produc1501393135466168645ist_fm @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1043_prod_Oswap__def,axiom,
    ( produc5396952528824381687at_nat
    = ( ^ [P6: produc1319942482725812455at_nat] : ( produc9060074326276436823at_nat @ ( produc826222791318859909at_nat @ P6 ) @ ( produc4486455499028916547at_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1044_prod_Oswap__def,axiom,
    ( produc3946833401584108663at_nat
    = ( ^ [P6: produc3843707927480180839at_nat] : ( produc2922128104949294807at_nat @ ( produc2937545259423399429at_nat @ P6 ) @ ( produc6011863565407846595at_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1045_prod_Oswap__def,axiom,
    ( product_swap_nat_nat
    = ( ^ [P6: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P6 ) @ ( product_fst_nat_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1046_prod_Oswap__def,axiom,
    ( produc2641608828089375343m_rule
    = ( ^ [P6: produc164195504107695125m_rule] : ( produc7063755442767259331ist_fm @ ( produc7165828336582415457m_rule @ P6 ) @ ( produc6879501374131015971m_rule @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1047_swap__simp,axiom,
    ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( produc7225117575323628663at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) )
      = ( produc6161850002892822231at_nat @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_1048_swap__simp,axiom,
    ! [X5: list_fm,Y3: list_fm] :
      ( ( produc7936585717479832313ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y3 ) )
      = ( produc381145313068854617ist_fm @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_1049_swap__simp,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( ( produc5396952528824381687at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y3 ) )
      = ( produc9060074326276436823at_nat @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_1050_swap__simp,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( produc3946833401584108663at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y3 ) )
      = ( produc2922128104949294807at_nat @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_1051_swap__simp,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) )
      = ( product_Pair_nat_nat @ Y3 @ X5 ) ) ).

% swap_simp
thf(fact_1052_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S3: product_prod_nat_nat,R5: set_Pr8693737435421807431at_nat,S7: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S3 ) @ R5 )
     => ( ( S7 = S3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S7 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1053_ssubst__Pair__rhs,axiom,
    ! [R2: list_fm,S3: list_fm,R5: set_Pr7058068377845519745ist_fm,S7: list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S3 ) @ R5 )
     => ( ( S7 = S3 )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S7 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1054_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat,R5: set_Pr7459493094073627847at_nat,S7: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S3 ) @ R5 )
     => ( ( S7 = S3 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S7 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1055_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,R5: set_Pr4329608150637261639at_nat,S7: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S3 ) @ R5 )
     => ( ( S7 = S3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S7 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1056_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S3: nat,R5: set_Pr1261947904930325089at_nat,S7: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R5 )
     => ( ( S7 = S3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S7 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1057_fst__snd__flip,axiom,
    ( produc3213797794245857475at_nat
    = ( comp_P4814319827393324394at_nat @ produc6408287024330202629at_nat
      @ ( produc6744312701629110395at_nat
        @ ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_1058_fst__snd__flip,axiom,
    ( produc1501393135466168645ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X3: list_fm,Y5: list_fm] : ( produc381145313068854617ist_fm @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_1059_fst__snd__flip,axiom,
    ( produc4486455499028916547at_nat
    = ( comp_P3728553842126764068at_nat @ produc826222791318859909at_nat
      @ ( produc5456455677105495163at_nat
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_1060_fst__snd__flip,axiom,
    ( produc6011863565407846595at_nat
    = ( comp_P4974406650942564362at_nat @ produc2937545259423399429at_nat
      @ ( produc7548845685434208379at_nat
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_1061_fst__snd__flip,axiom,
    ( product_fst_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X3: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_1062_fst__snd__flip,axiom,
    ( produc6879501374131015971m_rule
    = ( comp_P6646430376872094402m_rule @ produc4514744107061996309ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm,Y5: rule] : ( produc7063755442767259331ist_fm @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_1063_fst__snd__flip,axiom,
    ( produc4228417144610596823ist_fm
    = ( comp_P4888803072877744336ist_fm @ produc7165828336582415457m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X3: rule,Y5: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_1064_snd__fst__flip,axiom,
    ( produc6408287024330202629at_nat
    = ( comp_P4814319827393324394at_nat @ produc3213797794245857475at_nat
      @ ( produc6744312701629110395at_nat
        @ ^ [X3: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_1065_snd__fst__flip,axiom,
    ( produc4588648349897876871ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X3: list_fm,Y5: list_fm] : ( produc381145313068854617ist_fm @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_1066_snd__fst__flip,axiom,
    ( produc826222791318859909at_nat
    = ( comp_P3728553842126764068at_nat @ produc4486455499028916547at_nat
      @ ( produc5456455677105495163at_nat
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_1067_snd__fst__flip,axiom,
    ( produc2937545259423399429at_nat
    = ( comp_P4974406650942564362at_nat @ produc6011863565407846595at_nat
      @ ( produc7548845685434208379at_nat
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_1068_snd__fst__flip,axiom,
    ( product_snd_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X3: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_1069_snd__fst__flip,axiom,
    ( produc4514744107061996309ist_fm
    = ( comp_P2707267495825058490ist_fm @ produc6879501374131015971m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X3: rule,Y5: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_1070_snd__fst__flip,axiom,
    ( produc7165828336582415457m_rule
    = ( comp_P3413117018928051920m_rule @ produc4228417144610596823ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm,Y5: rule] : ( produc7063755442767259331ist_fm @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_1071_stake__pos__minWait,axiom,
    ! [Rs: stream_rule,S3: produc1996495991257130529ist_fm,R2: rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( ( ord_less_nat
          @ ( abstra738319790293016300ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S3 )
          @ ( abstract_pos_rule @ Rs @ R2 ) )
       => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
         => ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
           => ( ( abstract_pos_rule
                @ ( stl_rule
                  @ ( abstra2363873759530810705ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ Rs
                    @ S3 ) )
                @ R2 )
              = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R2 )
                @ ( suc
                  @ ( abstra738319790293016300ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ Rs
                    @ S3 ) ) ) ) ) ) ) ) ).

% stake_pos_minWait
thf(fact_1072_per__alw,axiom,
    ! [R2: rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4153889242452539894ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R2 )
     => ( ( ( abstra5415202973855822231ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ Steps )
          & ( member8102475879199740618ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ top_to728902761071950545ist_fm ) )
       => ( linear5684299538428340377m_rule
          @ ^ [Xs: stream727092118206550309m_rule] :
              ( ( linear7082939723987557601m_rule
                @ ^ [Step: produc164195504107695125m_rule] :
                    ( abstra3055540779341473792ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ R2
                    @ ( produc6879501374131015971m_rule @ Step ) )
                @ Xs )
             => ( ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] :
                      ( ( produc7165828336582415457m_rule @ Step )
                      = R2 )
                  @ Xs )
                | ( linear7357459635800698592rule_o
                  @ ( linear7082939723987557601m_rule
                    @ ^ [Step: produc164195504107695125m_rule] :
                        ( abstra3055540779341473792ist_fm
                        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                            ( ( eff @ R @ S )
                            = ( some_f2162114139285431020ist_fm @ Ss ) )
                        @ R2
                        @ ( produc6879501374131015971m_rule @ Step ) ) )
                  @ Xs ) ) )
          @ Steps ) ) ) ).

% per_alw
thf(fact_1073_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream8299795917829157543ist_fm] :
      ( ( stl_Pr5027778045650968273ist_fm @ ( sdrop_4442373711808556042ist_fm @ N @ S3 ) )
      = ( sdrop_4442373711808556042ist_fm @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1074_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream3588992565182678257e_rule] :
      ( ( stl_Pr4904654673476115995e_rule @ ( sdrop_2199339672054592340e_rule @ N @ S3 ) )
      = ( sdrop_2199339672054592340e_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1075_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream1878305877988316353m_rule] :
      ( ( stl_Pr5425948690297933931m_rule @ ( sdrop_1764812713384251940m_rule @ N @ S3 ) )
      = ( sdrop_1764812713384251940m_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1076_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream2312013117288958913e_rule] :
      ( ( stl_Pr830518440239794027e_rule @ ( sdrop_6392754500180887844e_rule @ N @ S3 ) )
      = ( sdrop_6392754500180887844e_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1077_sdrop__simps_I2_J,axiom,
    ! [N: nat,S3: stream6494289010434245521m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( sdrop_7373388980841566196m_rule @ N @ S3 ) )
      = ( sdrop_7373388980841566196m_rule @ ( suc @ N ) @ S3 ) ) ).

% sdrop_simps(2)
thf(fact_1078_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_1079_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_1080_order__less__imp__not__less,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X5 ) ) ).

% order_less_imp_not_less
thf(fact_1081_order__less__imp__not__eq2,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ( Y3 != X5 ) ) ).

% order_less_imp_not_eq2
thf(fact_1082_order__less__imp__not__eq,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ( X5 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_1083_linorder__less__linear,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
      | ( X5 = Y3 )
      | ( ord_less_nat @ Y3 @ X5 ) ) ).

% linorder_less_linear
thf(fact_1084_order__less__imp__triv,axiom,
    ! [X5: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X5 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1085_order__less__not__sym,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X5 ) ) ).

% order_less_not_sym
thf(fact_1086_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1087_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1088_order__less__irrefl,axiom,
    ! [X5: nat] :
      ~ ( ord_less_nat @ X5 @ X5 ) ).

% order_less_irrefl
thf(fact_1089_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1090_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1091_order__less__trans,axiom,
    ! [X5: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z )
       => ( ord_less_nat @ X5 @ Z ) ) ) ).

% order_less_trans
thf(fact_1092_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1093_linorder__neq__iff,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( X5 != Y3 )
      = ( ( ord_less_nat @ X5 @ Y3 )
        | ( ord_less_nat @ Y3 @ X5 ) ) ) ).

% linorder_neq_iff
thf(fact_1094_order__less__asym,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X5 ) ) ).

% order_less_asym
thf(fact_1095_linorder__neqE,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( X5 != Y3 )
     => ( ~ ( ord_less_nat @ X5 @ Y3 )
       => ( ord_less_nat @ Y3 @ X5 ) ) ) ).

% linorder_neqE
thf(fact_1096_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1097_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1098_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_1099_not__less__iff__gr__or__eq,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X5 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X5 )
        | ( X5 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1100_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_1101_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( ord_less_nat @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B2: nat] :
              ( ( P @ B2 @ A3 )
             => ( P @ A3 @ B2 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1102_exists__least__iff,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X8: nat] : ( P5 @ X8 ) )
    = ( ^ [P2: nat > $o] :
        ? [N3: nat] :
          ( ( P2 @ N3 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ~ ( P2 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1103_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1104_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1105_linorder__cases,axiom,
    ! [X5: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X5 @ Y3 )
     => ( ( X5 != Y3 )
       => ( ord_less_nat @ Y3 @ X5 ) ) ) ).

% linorder_cases
thf(fact_1106_antisym__conv3,axiom,
    ! [Y3: nat,X5: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X5 )
     => ( ( ~ ( ord_less_nat @ X5 @ Y3 ) )
        = ( X5 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_1107_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X )
             => ( P @ Y6 ) )
         => ( P @ X ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_1108_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1109_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1110_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_1111_less__imp__neq,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ( X5 != Y3 ) ) ).

% less_imp_neq
thf(fact_1112_gt__ex,axiom,
    ! [X5: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X5 @ X_1 ) ).

% gt_ex
thf(fact_1113_top_Onot__eq__extremum,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( A != top_to728902761071950545ist_fm )
      = ( ord_le5784518313675440685ist_fm @ A @ top_to728902761071950545ist_fm ) ) ).

% top.not_eq_extremum
thf(fact_1114_top_Onot__eq__extremum,axiom,
    ! [A: set_rule] :
      ( ( A != top_top_set_rule )
      = ( ord_less_set_rule @ A @ top_top_set_rule ) ) ).

% top.not_eq_extremum
thf(fact_1115_top_Oextremum__strict,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ~ ( ord_le5784518313675440685ist_fm @ top_to728902761071950545ist_fm @ A ) ).

% top.extremum_strict
thf(fact_1116_top_Oextremum__strict,axiom,
    ! [A: set_rule] :
      ~ ( ord_less_set_rule @ top_top_set_rule @ A ) ).

% top.extremum_strict
thf(fact_1117_leD,axiom,
    ! [Y3: nat,X5: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X5 )
     => ~ ( ord_less_nat @ X5 @ Y3 ) ) ).

% leD
thf(fact_1118_leD,axiom,
    ! [Y3: set_rule,X5: set_rule] :
      ( ( ord_less_eq_set_rule @ Y3 @ X5 )
     => ~ ( ord_less_set_rule @ X5 @ Y3 ) ) ).

% leD
thf(fact_1119_leI,axiom,
    ! [X5: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X5 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X5 ) ) ).

% leI
thf(fact_1120_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1121_nless__le,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( ~ ( ord_less_set_rule @ A @ B ) )
      = ( ~ ( ord_less_eq_set_rule @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1122_antisym__conv1,axiom,
    ! [X5: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X5 @ Y3 )
     => ( ( ord_less_eq_nat @ X5 @ Y3 )
        = ( X5 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1123_antisym__conv1,axiom,
    ! [X5: set_rule,Y3: set_rule] :
      ( ~ ( ord_less_set_rule @ X5 @ Y3 )
     => ( ( ord_less_eq_set_rule @ X5 @ Y3 )
        = ( X5 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1124_antisym__conv2,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X5 @ Y3 ) )
        = ( X5 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1125_antisym__conv2,axiom,
    ! [X5: set_rule,Y3: set_rule] :
      ( ( ord_less_eq_set_rule @ X5 @ Y3 )
     => ( ( ~ ( ord_less_set_rule @ X5 @ Y3 ) )
        = ( X5 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1126_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_1127_less__le__not__le,axiom,
    ( ord_less_set_rule
    = ( ^ [X3: set_rule,Y5: set_rule] :
          ( ( ord_less_eq_set_rule @ X3 @ Y5 )
          & ~ ( ord_less_eq_set_rule @ Y5 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_1128_not__le__imp__less,axiom,
    ! [Y3: nat,X5: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X5 )
     => ( ord_less_nat @ X5 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1129_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_nat @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1130_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( ( ord_less_set_rule @ A4 @ B3 )
          | ( A4 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1131_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1132_order_Ostrict__iff__order,axiom,
    ( ord_less_set_rule
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( ( ord_less_eq_set_rule @ A4 @ B3 )
          & ( A4 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1133_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1134_order_Ostrict__trans1,axiom,
    ! [A: set_rule,B: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_set_rule @ B @ C )
       => ( ord_less_set_rule @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1135_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1136_order_Ostrict__trans2,axiom,
    ! [A: set_rule,B: set_rule,C: set_rule] :
      ( ( ord_less_set_rule @ A @ B )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ord_less_set_rule @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1137_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1138_order_Ostrict__iff__not,axiom,
    ( ord_less_set_rule
    = ( ^ [A4: set_rule,B3: set_rule] :
          ( ( ord_less_eq_set_rule @ A4 @ B3 )
          & ~ ( ord_less_eq_set_rule @ B3 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1139_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_nat @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1140_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [B3: set_rule,A4: set_rule] :
          ( ( ord_less_set_rule @ B3 @ A4 )
          | ( A4 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1141_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1142_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_rule
    = ( ^ [B3: set_rule,A4: set_rule] :
          ( ( ord_less_eq_set_rule @ B3 @ A4 )
          & ( A4 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1143_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1144_dual__order_Ostrict__trans1,axiom,
    ! [B: set_rule,A: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ B @ A )
     => ( ( ord_less_set_rule @ C @ B )
       => ( ord_less_set_rule @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1145_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1146_dual__order_Ostrict__trans2,axiom,
    ! [B: set_rule,A: set_rule,C: set_rule] :
      ( ( ord_less_set_rule @ B @ A )
     => ( ( ord_less_eq_set_rule @ C @ B )
       => ( ord_less_set_rule @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1147_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B3 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1148_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_rule
    = ( ^ [B3: set_rule,A4: set_rule] :
          ( ( ord_less_eq_set_rule @ B3 @ A4 )
          & ~ ( ord_less_eq_set_rule @ A4 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1149_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1150_order_Ostrict__implies__order,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( ord_less_set_rule @ A @ B )
     => ( ord_less_eq_set_rule @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1151_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1152_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_rule,A: set_rule] :
      ( ( ord_less_set_rule @ B @ A )
     => ( ord_less_eq_set_rule @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1153_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y5: nat] :
          ( ( ord_less_nat @ X3 @ Y5 )
          | ( X3 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1154_order__le__less,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [X3: set_rule,Y5: set_rule] :
          ( ( ord_less_set_rule @ X3 @ Y5 )
          | ( X3 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1155_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y5 )
          & ( X3 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1156_order__less__le,axiom,
    ( ord_less_set_rule
    = ( ^ [X3: set_rule,Y5: set_rule] :
          ( ( ord_less_eq_set_rule @ X3 @ Y5 )
          & ( X3 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1157_linorder__not__le,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X5 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X5 ) ) ).

% linorder_not_le
thf(fact_1158_linorder__not__less,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X5 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X5 ) ) ).

% linorder_not_less
thf(fact_1159_order__less__imp__le,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ( ord_less_eq_nat @ X5 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1160_order__less__imp__le,axiom,
    ! [X5: set_rule,Y3: set_rule] :
      ( ( ord_less_set_rule @ X5 @ Y3 )
     => ( ord_less_eq_set_rule @ X5 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1161_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1162_order__le__neq__trans,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_rule @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1163_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1164_order__neq__le__trans,axiom,
    ! [A: set_rule,B: set_rule] :
      ( ( A != B )
     => ( ( ord_less_eq_set_rule @ A @ B )
       => ( ord_less_set_rule @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1165_order__le__less__trans,axiom,
    ! [X5: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z )
       => ( ord_less_nat @ X5 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1166_order__le__less__trans,axiom,
    ! [X5: set_rule,Y3: set_rule,Z: set_rule] :
      ( ( ord_less_eq_set_rule @ X5 @ Y3 )
     => ( ( ord_less_set_rule @ Y3 @ Z )
       => ( ord_less_set_rule @ X5 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1167_order__less__le__trans,axiom,
    ! [X5: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ X5 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_nat @ X5 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1168_order__less__le__trans,axiom,
    ! [X5: set_rule,Y3: set_rule,Z: set_rule] :
      ( ( ord_less_set_rule @ X5 @ Y3 )
     => ( ( ord_less_eq_set_rule @ Y3 @ Z )
       => ( ord_less_set_rule @ X5 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1169_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1170_order__le__less__subst1,axiom,
    ! [A: set_rule,F: nat > set_rule,B: nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1171_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1172_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_set_rule @ ( F @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1173_order__le__less__subst2,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1174_order__le__less__subst2,axiom,
    ! [A: set_rule,B: set_rule,F: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B )
     => ( ( ord_less_set_rule @ ( F @ B ) @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1175_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1176_order__less__le__subst1,axiom,
    ! [A: set_rule,F: nat > set_rule,B: nat,C: nat] :
      ( ( ord_less_set_rule @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1177_order__less__le__subst1,axiom,
    ! [A: nat,F: set_rule > nat,B: set_rule,C: set_rule] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1178_order__less__le__subst1,axiom,
    ! [A: set_rule,F: set_rule > set_rule,B: set_rule,C: set_rule] :
      ( ( ord_less_set_rule @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_rule @ B @ C )
       => ( ! [X: set_rule,Y: set_rule] :
              ( ( ord_less_eq_set_rule @ X @ Y )
             => ( ord_less_eq_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1179_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1180_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > set_rule,C: set_rule] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_set_rule @ ( F @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ Y )
             => ( ord_less_set_rule @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1181_linorder__le__less__linear,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
      | ( ord_less_nat @ Y3 @ X5 ) ) ).

% linorder_le_less_linear
thf(fact_1182_order__le__imp__less__or__eq,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y3 )
     => ( ( ord_less_nat @ X5 @ Y3 )
        | ( X5 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1183_order__le__imp__less__or__eq,axiom,
    ! [X5: set_rule,Y3: set_rule] :
      ( ( ord_less_eq_set_rule @ X5 @ Y3 )
     => ( ( ord_less_set_rule @ X5 @ Y3 )
        | ( X5 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1184_verit__comp__simplify1_I3_J,axiom,
    ! [B7: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
      = ( ord_less_nat @ A6 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1185_alw__iff__sdrop,axiom,
    ( linear1269638131612670621ist_fm
    = ( ^ [P2: stream8299795917829157543ist_fm > $o,Omega: stream8299795917829157543ist_fm] :
        ! [M3: nat] : ( P2 @ ( sdrop_4442373711808556042ist_fm @ M3 @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_1186_alw__iff__sdrop,axiom,
    ( linear6469244230239975911e_rule
    = ( ^ [P2: stream3588992565182678257e_rule > $o,Omega: stream3588992565182678257e_rule] :
        ! [M3: nat] : ( P2 @ ( sdrop_2199339672054592340e_rule @ M3 @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_1187_alw__iff__sdrop,axiom,
    ( linear1765130979247723063m_rule
    = ( ^ [P2: stream1878305877988316353m_rule > $o,Omega: stream1878305877988316353m_rule] :
        ! [M3: nat] : ( P2 @ ( sdrop_1764812713384251940m_rule @ M3 @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_1188_alw__iff__sdrop,axiom,
    ( linear6393072766044358967e_rule
    = ( ^ [P2: stream2312013117288958913e_rule > $o,Omega: stream2312013117288958913e_rule] :
        ! [M3: nat] : ( P2 @ ( sdrop_6392754500180887844e_rule @ M3 @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_1189_alw__iff__sdrop,axiom,
    ( linear4946900228879878535m_rule
    = ( ^ [P2: stream6494289010434245521m_rule > $o,Omega: stream6494289010434245521m_rule] :
        ! [M3: nat] : ( P2 @ ( sdrop_7373388980841566196m_rule @ M3 @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_1190_alw__iff__sdrop,axiom,
    ( linear5684299538428340377m_rule
    = ( ^ [P2: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
        ! [M3: nat] : ( P2 @ ( sdrop_7224736112439592940m_rule @ M3 @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_1191_alw__iff__sdrop,axiom,
    ( linear7625967296368300489w_rule
    = ( ^ [P2: stream_rule > $o,Omega: stream_rule] :
        ! [M3: nat] : ( P2 @ ( sdrop_rule @ M3 @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_1192_alw__sdrop,axiom,
    ! [Phi: stream8299795917829157543ist_fm > $o,Xs2: stream8299795917829157543ist_fm,N: nat] :
      ( ( linear1269638131612670621ist_fm @ Phi @ Xs2 )
     => ( linear1269638131612670621ist_fm @ Phi @ ( sdrop_4442373711808556042ist_fm @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_1193_alw__sdrop,axiom,
    ! [Phi: stream3588992565182678257e_rule > $o,Xs2: stream3588992565182678257e_rule,N: nat] :
      ( ( linear6469244230239975911e_rule @ Phi @ Xs2 )
     => ( linear6469244230239975911e_rule @ Phi @ ( sdrop_2199339672054592340e_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_1194_alw__sdrop,axiom,
    ! [Phi: stream1878305877988316353m_rule > $o,Xs2: stream1878305877988316353m_rule,N: nat] :
      ( ( linear1765130979247723063m_rule @ Phi @ Xs2 )
     => ( linear1765130979247723063m_rule @ Phi @ ( sdrop_1764812713384251940m_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_1195_alw__sdrop,axiom,
    ! [Phi: stream2312013117288958913e_rule > $o,Xs2: stream2312013117288958913e_rule,N: nat] :
      ( ( linear6393072766044358967e_rule @ Phi @ Xs2 )
     => ( linear6393072766044358967e_rule @ Phi @ ( sdrop_6392754500180887844e_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_1196_alw__sdrop,axiom,
    ! [Phi: stream6494289010434245521m_rule > $o,Xs2: stream6494289010434245521m_rule,N: nat] :
      ( ( linear4946900228879878535m_rule @ Phi @ Xs2 )
     => ( linear4946900228879878535m_rule @ Phi @ ( sdrop_7373388980841566196m_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_1197_alw__sdrop,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule,N: nat] :
      ( ( linear5684299538428340377m_rule @ Phi @ Xs2 )
     => ( linear5684299538428340377m_rule @ Phi @ ( sdrop_7224736112439592940m_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_1198_alw__sdrop,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule,N: nat] :
      ( ( linear7625967296368300489w_rule @ Phi @ Xs2 )
     => ( linear7625967296368300489w_rule @ Phi @ ( sdrop_rule @ N @ Xs2 ) ) ) ).

% alw_sdrop
thf(fact_1199_alw_Ocases,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,A: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ Phi @ A )
     => ~ ( ( Phi @ A )
         => ~ ( linear5684299538428340377m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ).

% alw.cases
thf(fact_1200_alw_Ocases,axiom,
    ! [Phi: stream_rule > $o,A: stream_rule] :
      ( ( linear7625967296368300489w_rule @ Phi @ A )
     => ~ ( ( Phi @ A )
         => ~ ( linear7625967296368300489w_rule @ Phi @ ( stl_rule @ A ) ) ) ) ).

% alw.cases
thf(fact_1201_alw_Osimps,axiom,
    ( linear5684299538428340377m_rule
    = ( ^ [Phi2: stream727092118206550309m_rule > $o,A4: stream727092118206550309m_rule] :
        ? [Xs: stream727092118206550309m_rule] :
          ( ( A4 = Xs )
          & ( Phi2 @ Xs )
          & ( linear5684299538428340377m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ Xs ) ) ) ) ) ).

% alw.simps
thf(fact_1202_alw_Osimps,axiom,
    ( linear7625967296368300489w_rule
    = ( ^ [Phi2: stream_rule > $o,A4: stream_rule] :
        ? [Xs: stream_rule] :
          ( ( A4 = Xs )
          & ( Phi2 @ Xs )
          & ( linear7625967296368300489w_rule @ Phi2 @ ( stl_rule @ Xs ) ) ) ) ) ).

% alw.simps
thf(fact_1203_alw_Ointros,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] :
      ( ( Phi @ Xs2 )
     => ( ( linear5684299538428340377m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ Xs2 ) )
       => ( linear5684299538428340377m_rule @ Phi @ Xs2 ) ) ) ).

% alw.intros
thf(fact_1204_alw_Ointros,axiom,
    ! [Phi: stream_rule > $o,Xs2: stream_rule] :
      ( ( Phi @ Xs2 )
     => ( ( linear7625967296368300489w_rule @ Phi @ ( stl_rule @ Xs2 ) )
       => ( linear7625967296368300489w_rule @ Phi @ Xs2 ) ) ) ).

% alw.intros
thf(fact_1205_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1206_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1207_ipath__mkTree__ev,axiom,
    ! [S3: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( abstra1114877437032970289r_rule @ rules @ Rs )
       => ( ( abstra2285467793620430502m_rule
            @ ( abstra178918799166095863ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S3 )
            @ Steps )
         => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
           => ( ( linear5684299538428340377m_rule
                @ ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] :
                      ( abstra3055540779341473792ist_fm
                      @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                          ( ( eff @ R @ S )
                          = ( some_f2162114139285431020ist_fm @ Ss ) )
                      @ R2
                      @ ( produc6879501374131015971m_rule @ Step ) ) )
                @ Steps )
             => ( linear2722748333883190224m_rule
                @ ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] :
                      ( ( produc7165828336582415457m_rule @ Step )
                      = R2 ) )
                @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_ev
thf(fact_1208_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1209_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y: nat] :
            ( ( P @ Y )
           => ( ord_less_eq_nat @ Y @ B ) )
       => ? [X: nat] :
            ( ( P @ X )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1210_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_1211_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_1212_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_1213_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_1214_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_1215_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y: nat] :
            ( ( P @ Y )
           => ( ord_less_eq_nat @ Y @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_1216_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y: nat] :
            ( ( P @ Y )
           => ( ord_less_eq_nat @ Y @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_1217_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y: nat] :
            ( ( P @ Y )
           => ( ord_less_eq_nat @ Y @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_1218_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1219_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1220_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1221_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
     => ? [M5: nat] :
          ( M4
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_1222_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1223_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1224_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1225_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M6: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M6 ) @ N2 )
             => ( P @ M6 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1226_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1227_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R5: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X: nat] : ( R5 @ X @ X )
       => ( ! [X: nat,Y: nat,Z4: nat] :
              ( ( R5 @ X @ Y )
             => ( ( R5 @ Y @ Z4 )
               => ( R5 @ X @ Z4 ) ) )
         => ( ! [N2: nat] : ( R5 @ N2 @ ( suc @ N2 ) )
           => ( R5 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1228_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
          & ( M3 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_1229_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_1230_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
          | ( M3 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1231_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_1232_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_1233_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1234_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1235_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1236_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1237_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1238_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1239_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1240_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_1241_saturated__def,axiom,
    ! [R2: rule] :
      ( ( abstra7252176601000738572ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2 )
      = ( linear5684299538428340377m_rule
        @ ^ [Xs: stream727092118206550309m_rule] :
            ( ( linear7082939723987557601m_rule
              @ ^ [Step: produc164195504107695125m_rule] :
                  ( abstra3055540779341473792ist_fm
                  @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ R2
                  @ ( produc6879501374131015971m_rule @ Step ) )
              @ Xs )
           => ( linear2722748333883190224m_rule
              @ ( linear7082939723987557601m_rule
                @ ^ [Step: produc164195504107695125m_rule] :
                    ( ( produc7165828336582415457m_rule @ Step )
                    = R2 ) )
              @ Xs ) ) ) ) ).

% saturated_def
thf(fact_1242_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1243_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1244_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1245_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1246_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1247_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1248_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1249_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1250_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1251_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1252_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1253_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K2 @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_1254_Prover_Ofair__def,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
      = ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs ) @ ( sset_rule @ rules ) )
        & ! [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ rules ) )
           => ( linear7625967296368300489w_rule
              @ ( linear6831045468625226112v_rule
                @ ( linear4670012741531116049s_rule
                  @ ( ^ [Y4: rule,Z2: rule] : ( Y4 = Z2 )
                    @ X3 ) ) )
              @ Rs ) ) ) ) ).

% Prover.fair_def
thf(fact_1255_prod__decode__aux_Oelims,axiom,
    ! [X5: nat,Xa2: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X5 @ Xa2 )
        = Y3 )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X5 )
         => ( Y3
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X5 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X5 )
         => ( Y3
            = ( nat_prod_decode_aux @ ( suc @ X5 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X5 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_1256_prod__decode__aux_Ocases,axiom,
    ! [X5: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] :
          ( X5
         != ( product_Pair_nat_nat @ K2 @ M5 ) ) ).

% prod_decode_aux.cases
thf(fact_1257_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K3 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_1258_pair__lessI2,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1259_pair__less__iff1,axiom,
    ! [X5: nat,Y3: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ ( product_Pair_nat_nat @ X5 @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y3 @ Z ) ) ).

% pair_less_iff1
thf(fact_1260_pair__lessI1,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_1261_pair__leqI2,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1262_pair__leqI1,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_1263_per__def,axiom,
    ! [R2: rule] :
      ( ( abstra4153889242452539894ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ top_to728902761071950545ist_fm
        @ R2 )
      = ( ! [S: produc1996495991257130529ist_fm,R12: rule,Sl6: fset_P661503646757059847ist_fm,S9: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
              & ( abstra3055540779341473792ist_fm
                @ ^ [R: rule,T2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ T2 )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R2
                @ S )
              & ( member_rule @ R12 @ ( minus_minus_set_rule @ ( sset_rule @ rules ) @ ( insert_rule @ R2 @ bot_bot_set_rule ) ) )
              & ( ( eff @ R12 @ S )
                = ( some_f2162114139285431020ist_fm @ Sl6 ) )
              & ( fmembe3381613331217039976ist_fm @ S9 @ Sl6 ) )
           => ( abstra3055540779341473792ist_fm
              @ ^ [R: rule,T2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ T2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R2
              @ S9 ) ) ) ) ).

% per_def
thf(fact_1264_less__than__iff,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ less_than )
      = ( ord_less_nat @ X5 @ Y3 ) ) ).

% less_than_iff
thf(fact_1265_NE__R,axiom,
    ( ( sset_rule @ rules )
   != bot_bot_set_rule ) ).

% NE_R
thf(fact_1266_wmax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y3 @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X5 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_1267_wmin__insertI,axiom,
    ! [X5: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X5 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_1268_smax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Y7 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X5 @ X7 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_1269_wmin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_1270_smin__insertI,axiom,
    ! [X5: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X5 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_1271_smin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] :
      ( ( X7 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_1272_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_1273_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_1274_pickEff__def,axiom,
    ! [R2: rule,S3: produc1996495991257130529ist_fm] :
      ( ( ( abstra3055540779341473792ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ R2
          @ S3 )
       => ( ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 )
          = ( fChoic3892542562059901019ist_fm
            @ ^ [Sl: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S3 )
                = ( some_f2162114139285431020ist_fm @ Sl ) ) ) ) )
      & ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 )
       => ( ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S3 )
          = ( the_fs400605564276472219ist_fm @ none_f2701453242279119720ist_fm ) ) ) ) ).

% pickEff_def

% Helper facts (23)
thf(help_fChoice_1_1_fChoice_001t__Syntax__Orule_T,axiom,
    ! [P: rule > $o] :
      ( ( P @ ( fChoice_rule @ P ) )
      = ( ? [X4: rule] : ( P @ X4 ) ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X5: stream_rule,Y3: stream_rule] :
      ( ( if_stream_rule @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X5: stream_rule,Y3: stream_rule] :
      ( ( if_stream_rule @ $true @ X5 @ Y3 )
      = X5 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X5 @ Y3 )
      = X5 ) ).

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 ) )
      = ( ? [X4: abstract_tree_rule] : ( P @ X4 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( P @ ( fChoic6978938873391328853at_nat @ P ) )
      = ( ? [X4: product_prod_nat_nat] : ( P @ X4 ) ) ) ).

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,
    ! [X5: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_T,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y3: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $true @ X5 @ Y3 )
      = X5 ) ).

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 ) )
      = ( ? [X4: produc1996495991257130529ist_fm] : ( P @ X4 ) ) ) ).

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,
    ! [X5: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [X5: fset_P661503646757059847ist_fm,Y3: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $true @ X5 @ Y3 )
      = X5 ) ).

thf(help_fChoice_1_1_fChoice_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J_T,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o] :
      ( ( P @ ( fChoic3892542562059901019ist_fm @ P ) )
      = ( ? [X4: fset_P661503646757059847ist_fm] : ( P @ X4 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [P: produc859450856879609959at_nat > $o] :
      ( ( P @ ( fChoic6673916560228149179at_nat @ P ) )
      = ( ? [X4: produc859450856879609959at_nat] : ( P @ X4 ) ) ) ).

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,
    ! [X5: stream727092118206550309m_rule,Y3: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J_T,axiom,
    ! [X5: stream727092118206550309m_rule,Y3: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $true @ X5 @ Y3 )
      = X5 ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( P @ ( fChoic2372384411376717755at_nat @ P ) )
      = ( ? [X4: produc3843707927480180839at_nat] : ( P @ X4 ) ) ) ).

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,
    ! [X5: abstra4572470701423150929m_rule,Y3: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $false @ X5 @ Y3 )
      = Y3 ) ).

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,
    ! [X5: abstra4572470701423150929m_rule,Y3: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $true @ X5 @ Y3 )
      = X5 ) ).

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 ) )
      = ( ? [X4: abstra4572470701423150929m_rule] : ( P @ X4 ) ) ) ).

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,
    ! [X5: fset_A5396818216090053761m_rule,Y3: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $false @ X5 @ Y3 )
      = Y3 ) ).

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,
    ! [X5: fset_A5396818216090053761m_rule,Y3: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $true @ X5 @ Y3 )
      = X5 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ( abstra5415202973855822231ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ steps ) ).

thf(conj_1,conjecture,
    ( abstra5415202973855822231ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ ( sdrop_7224736112439592940m_rule @ n @ steps ) ) ).

%------------------------------------------------------------------------------