%------------------------------------------------------------------------------
% 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_00088_002819__12073170_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2205 ( 680 unt; 906 typ;   0 def)
%            Number of atoms       : 3869 (1276 equ;  15 cnn)
%            Maximal formula atoms :   15 (   2 avg)
%            Number of connectives : 15469 ( 192   ~;  42   |; 443   &;13064   @)
%                                         (   0 <=>;1728  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :  166 ( 165 usr)
%            Number of type conns  : 3690 (3690   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  745 ( 741 usr;  29 con; 0-4 aty)
%            Number of variables   : 4842 ( 866   ^;3686   !; 290   ?;4842   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:27:57.782
%------------------------------------------------------------------------------
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    fChoic6978938873391328853at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_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__Ofm,type,
    fChoice_fm: ( fm > $o ) > fm ).

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

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

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

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

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

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__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__Product____Type__Oprod_It__Syntax__Orule_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    member8686995742288324264ist_fm: produc4630866025709511057ist_fm > set_Pr2241365836748572487ist_fm > $o ).

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

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

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_steps,type,
    steps: stream727092118206550309m_rule ).

% Relevant facts (1275)
thf(fact_0_assms,axiom,
    ( abstra8647779336736900908ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ steps ) ).

% assms
thf(fact_1_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_2_epath__sdrop,axiom,
    ! [Steps: stream727092118206550309m_rule,N: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) ) ).

% epath_sdrop
thf(fact_3_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra8647779336736900908ist_fm = abstra8647779336736900908ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_4_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_5_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_6_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_7_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_8_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_9_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_10_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_11_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_12_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_13_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_14_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 )
      = ( ! [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ X
              @ Steps ) ) ) ) ).

% Saturated_def
thf(fact_15_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_16_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_17_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_18_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_19_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_20_enabled__R,axiom,
    ! [S3: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ? [X3: rule] :
          ( ( member_rule @ X3 @ ( sset_rule @ rules ) )
          & ? [Sl2: fset_P661503646757059847ist_fm] :
              ( ( eff @ X3 @ S3 )
              = ( some_f2162114139285431020ist_fm @ Sl2 ) ) ) ) ).

% enabled_R
thf(fact_21_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_22_RuleSystem_Oper_Ocong,axiom,
    abstra4153889242452539894ist_fm = abstra4153889242452539894ist_fm ).

% RuleSystem.per.cong
thf(fact_23_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 )
       => ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P2467486768528901581at_nat] : ( Eff @ X3 @ 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 )
       => ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P5670320511379867111at_nat] : ( Eff @ X3 @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_25_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 )
       => ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_rule] : ( Eff @ X3 @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_26_RuleSystem_Oenabled__R,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_fm] : ( Eff @ X3 @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_27_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 )
       => ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P661503646757059847ist_fm] : ( Eff @ X3 @ S3 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
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_Oepath_Ocong,axiom,
    abstra5415202973855822231ist_fm = abstra5415202973855822231ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_30_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_31_RuleSystem__Defs_OmkTree_Ocong,axiom,
    abstra178918799166095863ist_fm = abstra178918799166095863ist_fm ).

% RuleSystem_Defs.mkTree.cong
thf(fact_32_RuleSystem__Defs_Oenabled_Ocong,axiom,
    abstra3055540779341473792ist_fm = abstra3055540779341473792ist_fm ).

% RuleSystem_Defs.enabled.cong
thf(fact_33_RuleSystem__Defs_Osaturated_Ocong,axiom,
    abstra7252176601000738572ist_fm = abstra7252176601000738572ist_fm ).

% RuleSystem_Defs.saturated.cong
thf(fact_34_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra8647779336736900908ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,Steps2: stream727092118206550309m_rule] :
        ! [X: rule] :
          ( ( member_rule @ X @ ( sset_rule @ Rules2 ) )
         => ( abstra7252176601000738572ist_fm @ Eff2 @ X @ Steps2 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_35_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_36_UNIV__I,axiom,
    ! [X5: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X5 @ top_to6833984726390702231at_nat ) ).

% UNIV_I
thf(fact_37_UNIV__I,axiom,
    ! [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ top_to4669805908274784177at_nat ) ).

% UNIV_I
thf(fact_38_UNIV__I,axiom,
    ! [X5: fm] : ( member_fm @ X5 @ top_top_set_fm ) ).

% UNIV_I
thf(fact_39_UNIV__I,axiom,
    ! [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ top_to728902761071950545ist_fm ) ).

% UNIV_I
thf(fact_40_UNIV__I,axiom,
    ! [X5: rule] : ( member_rule @ X5 @ top_top_set_rule ) ).

% UNIV_I
thf(fact_41_iso__tuple__UNIV__I,axiom,
    ! [X5: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X5 @ top_to6833984726390702231at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_42_iso__tuple__UNIV__I,axiom,
    ! [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ top_to4669805908274784177at_nat ) ).

% iso_tuple_UNIV_I
thf(fact_43_iso__tuple__UNIV__I,axiom,
    ! [X5: fm] : ( member_fm @ X5 @ top_top_set_fm ) ).

% iso_tuple_UNIV_I
thf(fact_44_iso__tuple__UNIV__I,axiom,
    ! [X5: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X5 @ top_to728902761071950545ist_fm ) ).

% iso_tuple_UNIV_I
thf(fact_45_iso__tuple__UNIV__I,axiom,
    ! [X5: rule] : ( member_rule @ X5 @ top_top_set_rule ) ).

% iso_tuple_UNIV_I
thf(fact_46_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_47_PersistentRuleSystem__def,axiom,
    ( abstra7887161495092839066ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S5: set_Pr7058068377845519745ist_fm] :
          ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules2 @ S5 )
          & ( abstra2778581409224112183ist_fm @ Eff2 @ Rules2 @ S5 ) ) ) ) ).

% PersistentRuleSystem_def
thf(fact_48_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_49_PersistentRuleSystem__axioms__def,axiom,
    ( abstra2778581409224112183ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S5: set_Pr7058068377845519745ist_fm] :
        ! [R: rule] :
          ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( abstra4153889242452539894ist_fm @ Eff2 @ Rules2 @ S5 @ R ) ) ) ) ).

% PersistentRuleSystem_axioms_def
thf(fact_50_eff__S,axiom,
    ! [S3: produc1996495991257130529ist_fm,R2: rule,Sl3: fset_P661503646757059847ist_fm,S6: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S3 @ top_to728902761071950545ist_fm )
     => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
       => ( ( ( eff @ R2 @ S3 )
            = ( some_f2162114139285431020ist_fm @ Sl3 ) )
         => ( ( fmembe3381613331217039976ist_fm @ S6 @ Sl3 )
           => ( member8102475879199740618ist_fm @ S6 @ top_to728902761071950545ist_fm ) ) ) ) ) ).

% eff_S
thf(fact_51_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_52_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_53_UNIV__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X: product_prod_nat_nat] : $true ) ) ).

% UNIV_def
thf(fact_54_UNIV__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm
      @ ^ [X: produc1996495991257130529ist_fm] : $true ) ) ).

% UNIV_def
thf(fact_55_UNIV__def,axiom,
    ( top_top_set_rule
    = ( collect_rule
      @ ^ [X: rule] : $true ) ) ).

% UNIV_def
thf(fact_56_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,S7: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S7 @ 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 )
                  @ S7 )
                @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_sdrop
thf(fact_57_RuleSystem__Defs_Ofair_Ocong,axiom,
    abstra1114877437032970289r_rule = abstra1114877437032970289r_rule ).

% RuleSystem_Defs.fair.cong
thf(fact_58_RuleSystem__Defs_OpickEff_Ocong,axiom,
    abstra458490714785879965ist_fm = abstra458490714785879965ist_fm ).

% RuleSystem_Defs.pickEff.cong
thf(fact_59_top__set__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat @ top_to3137496036531963500_nat_o ) ) ).

% top_set_def
thf(fact_60_top__set__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm @ top_to6798613745154499404t_fm_o ) ) ).

% top_set_def
thf(fact_61_top__set__def,axiom,
    ( top_top_set_rule
    = ( collect_rule @ top_top_rule_o ) ) ).

% top_set_def
thf(fact_62_sdrop__fair,axiom,
    ! [Rs: stream_rule,M: nat] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( sdrop_rule @ M @ Rs ) ) ) ).

% sdrop_fair
thf(fact_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_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_74_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_75_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_76_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_77_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_78_RuleSystem__def,axiom,
    ( abstra5167362717900046400m_rule
    = ( ^ [Eff2: fm > rule > fset_rule > $o,Rules2: stream_fm,S5: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ! [R: fm] :
                  ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S5 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ? [X: fm] :
                  ( ( member_fm @ X @ ( sset_fm @ Rules2 ) )
                  & ? [X4: fset_rule] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_79_RuleSystem__def,axiom,
    ( abstra886934874521163211_fm_fm
    = ( ^ [Eff2: fm > fm > fset_fm > $o,Rules2: stream_fm,S5: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S5 )
             => ! [R: fm] :
                  ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S5 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S5 )
             => ? [X: fm] :
                  ( ( member_fm @ X @ ( sset_fm @ Rules2 ) )
                  & ? [X4: fset_fm] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_80_RuleSystem__def,axiom,
    ( abstra2725590884229999157e_rule
    = ( ^ [Eff2: rule > rule > fset_rule > $o,Rules2: stream_rule,S5: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S5 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ? [X: rule] :
                  ( ( member_rule @ X @ ( sset_rule @ Rules2 ) )
                  & ? [X4: fset_rule] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_81_RuleSystem__def,axiom,
    ( abstra916514217863227968ule_fm
    = ( ^ [Eff2: rule > fm > fset_fm > $o,Rules2: stream_rule,S5: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S5 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S5 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S5 )
             => ? [X: rule] :
                  ( ( member_rule @ X @ ( sset_rule @ Rules2 ) )
                  & ? [X4: fset_fm] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_82_RuleSystem__def,axiom,
    ( abstra4023219294580590406at_nat
    = ( ^ [Eff2: fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules2: stream_fm,S5: set_Pr1261947904930325089at_nat] :
          ( ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S5 )
             => ! [R: fm] :
                  ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
                 => ! [Sl: fset_P5670320511379867111at_nat] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: product_prod_nat_nat] :
                          ( ( fmembe1449479052724974408at_nat @ S8 @ Sl )
                         => ( member8440522571783428010at_nat @ S8 @ S5 ) ) ) ) )
          & ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S5 )
             => ? [X: fm] :
                  ( ( member_fm @ X @ ( sset_fm @ Rules2 ) )
                  & ? [X4: fset_P5670320511379867111at_nat] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_83_RuleSystem__def,axiom,
    ( abstra1542008646636103517t_rule
    = ( ^ [Eff2: product_prod_nat_nat > rule > fset_rule > $o,Rules2: stream6724221391990029191at_nat,S5: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ! [R: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ R @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S5 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ? [X: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                  & ? [X4: fset_rule] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_84_RuleSystem__def,axiom,
    ( abstra8041801216803224424nat_fm
    = ( ^ [Eff2: product_prod_nat_nat > fm > fset_fm > $o,Rules2: stream6724221391990029191at_nat,S5: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S5 )
             => ! [R: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ R @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S5 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S5 )
             => ? [X: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                  & ? [X4: fset_fm] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_85_RuleSystem__def,axiom,
    ( abstra8657071042050908689at_nat
    = ( ^ [Eff2: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules2: stream_rule,S5: set_Pr1261947904930325089at_nat] :
          ( ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S5 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_P5670320511379867111at_nat] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: product_prod_nat_nat] :
                          ( ( fmembe1449479052724974408at_nat @ S8 @ Sl )
                         => ( member8440522571783428010at_nat @ S8 @ S5 ) ) ) ) )
          & ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S5 )
             => ? [X: rule] :
                  ( ( member_rule @ X @ ( sset_rule @ Rules2 ) )
                  & ? [X4: fset_P5670320511379867111at_nat] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_86_RuleSystem__def,axiom,
    ( abstra7719573905946618601at_nat
    = ( ^ [Eff2: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules2: stream6724221391990029191at_nat,S5: set_Pr1261947904930325089at_nat] :
          ( ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S5 )
             => ! [R: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ R @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                 => ! [Sl: fset_P5670320511379867111at_nat] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: product_prod_nat_nat] :
                          ( ( fmembe1449479052724974408at_nat @ S8 @ Sl )
                         => ( member8440522571783428010at_nat @ S8 @ S5 ) ) ) ) )
          & ! [S: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S @ S5 )
             => ? [X: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X @ ( sset_P8584168253256249621at_nat @ Rules2 ) )
                  & ? [X4: fset_P5670320511379867111at_nat] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_87_RuleSystem__def,axiom,
    ( abstra9106359784214577277m_rule
    = ( ^ [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S5: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ! [R: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff2 @ R @ S @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S5 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S5 )
             => ? [X: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                  & ? [X4: fset_rule] : ( Eff2 @ X @ S @ X4 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_88_mem__Collect__eq,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ A @ ( collec7231575776666803724ist_fm @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_89_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
    ! [A: rule,P: rule > $o] :
      ( ( member_rule @ A @ ( collect_rule @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
    ! [A: fm,P: fm > $o] :
      ( ( member_fm @ A @ ( collect_fm @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_92_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_93_Collect__mem__eq,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] :
      ( ( collec7231575776666803724ist_fm
        @ ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
    ! [A2: set_rule] :
      ( ( collect_rule
        @ ^ [X: rule] : ( member_rule @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
    ! [A2: set_fm] :
      ( ( collect_fm
        @ ^ [X: fm] : ( member_fm @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_98_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_99_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream_fm,Eff: fm > rule > fset_rule > $o] :
      ( ! [S9: rule] :
          ( ( member_rule @ S9 @ S4 )
         => ! [R3: fm] :
              ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: rule] :
            ( ( member_rule @ S9 @ S4 )
           => ? [X6: fm] :
                ( ( member_fm @ X6 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra5167362717900046400m_rule @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_100_RuleSystem_Ointro,axiom,
    ! [S4: set_fm,Rules: stream_fm,Eff: fm > fm > fset_fm > $o] :
      ( ! [S9: fm] :
          ( ( member_fm @ S9 @ S4 )
         => ! [R3: fm] :
              ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: fm] :
            ( ( member_fm @ S9 @ S4 )
           => ? [X6: fm] :
                ( ( member_fm @ X6 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra886934874521163211_fm_fm @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_101_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream_rule,Eff: rule > rule > fset_rule > $o] :
      ( ! [S9: rule] :
          ( ( member_rule @ S9 @ S4 )
         => ! [R3: rule] :
              ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: rule] :
            ( ( member_rule @ S9 @ S4 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_102_RuleSystem_Ointro,axiom,
    ! [S4: set_fm,Rules: stream_rule,Eff: rule > fm > fset_fm > $o] :
      ( ! [S9: fm] :
          ( ( member_fm @ S9 @ S4 )
         => ! [R3: rule] :
              ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: fm] :
            ( ( member_fm @ S9 @ S4 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_103_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr1261947904930325089at_nat,Rules: stream_fm,Eff: fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o] :
      ( ! [S9: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ S9 @ S4 )
         => ! [R3: fm] :
              ( ( member_fm @ R3 @ ( sset_fm @ Rules ) )
             => ! [Sl2: fset_P5670320511379867111at_nat] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: product_prod_nat_nat] :
                      ( ( fmembe1449479052724974408at_nat @ S7 @ Sl2 )
                     => ( member8440522571783428010at_nat @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ S9 @ S4 )
           => ? [X6: fm] :
                ( ( member_fm @ X6 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_P5670320511379867111at_nat] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra4023219294580590406at_nat @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_104_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream6724221391990029191at_nat,Eff: product_prod_nat_nat > rule > fset_rule > $o] :
      ( ! [S9: rule] :
          ( ( member_rule @ S9 @ S4 )
         => ! [R3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ R3 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: rule] :
            ( ( member_rule @ S9 @ S4 )
           => ? [X6: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X6 @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra1542008646636103517t_rule @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_105_RuleSystem_Ointro,axiom,
    ! [S4: set_fm,Rules: stream6724221391990029191at_nat,Eff: product_prod_nat_nat > fm > fset_fm > $o] :
      ( ! [S9: fm] :
          ( ( member_fm @ S9 @ S4 )
         => ! [R3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ R3 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: fm] :
            ( ( member_fm @ S9 @ S4 )
           => ? [X6: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X6 @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra8041801216803224424nat_fm @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_106_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr1261947904930325089at_nat,Rules: stream_rule,Eff: rule > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o] :
      ( ! [S9: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ S9 @ S4 )
         => ! [R3: rule] :
              ( ( member_rule @ R3 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_P5670320511379867111at_nat] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: product_prod_nat_nat] :
                      ( ( fmembe1449479052724974408at_nat @ S7 @ Sl2 )
                     => ( member8440522571783428010at_nat @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ S9 @ S4 )
           => ? [X6: rule] :
                ( ( member_rule @ X6 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_P5670320511379867111at_nat] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra8657071042050908689at_nat @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_107_RuleSystem_Ointro,axiom,
    ! [S4: set_Pr1261947904930325089at_nat,Rules: stream6724221391990029191at_nat,Eff: product_prod_nat_nat > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o] :
      ( ! [S9: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ S9 @ S4 )
         => ! [R3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ R3 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ! [Sl2: fset_P5670320511379867111at_nat] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: product_prod_nat_nat] :
                      ( ( fmembe1449479052724974408at_nat @ S7 @ Sl2 )
                     => ( member8440522571783428010at_nat @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ S9 @ S4 )
           => ? [X6: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X6 @ ( sset_P8584168253256249621at_nat @ Rules ) )
                & ? [X_12: fset_P5670320511379867111at_nat] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra7719573905946618601at_nat @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_108_RuleSystem_Ointro,axiom,
    ! [S4: set_rule,Rules: stream8299795917829157543ist_fm,Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o] :
      ( ! [S9: rule] :
          ( ( member_rule @ S9 @ S4 )
         => ! [R3: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R3 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff @ R3 @ S9 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule @ S7 @ S4 ) ) ) ) )
     => ( ! [S9: rule] :
            ( ( member_rule @ S9 @ S4 )
           => ? [X6: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X6 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff @ X6 @ S9 @ X_12 ) ) )
       => ( abstra9106359784214577277m_rule @ Eff @ Rules @ S4 ) ) ) ).

% RuleSystem.intro
thf(fact_109_RuleSystem_Oeff__S,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S4: set_rule,S3: rule,R2: fm,Sl3: fset_rule,S6: rule] :
      ( ( abstra5167362717900046400m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S6 @ Sl3 )
             => ( member_rule @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_110_RuleSystem_Oeff__S,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S4: set_fm,S3: fm,R2: fm,Sl3: fset_fm,S6: fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_fm @ S6 @ Sl3 )
             => ( member_fm @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_111_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > rule > fset_rule > $o,Rules: stream_rule,S4: set_rule,S3: rule,R2: rule,Sl3: fset_rule,S6: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S6 @ Sl3 )
             => ( member_rule @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_112_RuleSystem_Oeff__S,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,R2: rule,Sl3: fset_fm,S6: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_fm @ S6 @ Sl3 )
             => ( member_fm @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_113_RuleSystem_Oeff__S,axiom,
    ! [Eff: fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_fm,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,R2: fm,Sl3: fset_P5670320511379867111at_nat,S6: product_prod_nat_nat] :
      ( ( abstra4023219294580590406at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmembe1449479052724974408at_nat @ S6 @ Sl3 )
             => ( member8440522571783428010at_nat @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_114_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,S6: rule] :
      ( ( abstra1542008646636103517t_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member8440522571783428010at_nat @ R2 @ ( sset_P8584168253256249621at_nat @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S6 @ Sl3 )
             => ( member_rule @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_115_RuleSystem_Oeff__S,axiom,
    ! [Eff: product_prod_nat_nat > fm > fset_fm > $o,Rules: stream6724221391990029191at_nat,S4: set_fm,S3: fm,R2: product_prod_nat_nat,Sl3: fset_fm,S6: fm] :
      ( ( abstra8041801216803224424nat_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( member8440522571783428010at_nat @ R2 @ ( sset_P8584168253256249621at_nat @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_fm @ S6 @ Sl3 )
             => ( member_fm @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_116_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,S6: 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 @ S6 @ Sl3 )
             => ( member8440522571783428010at_nat @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_117_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,S6: 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 @ S6 @ Sl3 )
             => ( member8440522571783428010at_nat @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_118_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,S6: rule] :
      ( ( abstra9106359784214577277m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ( ( Eff @ R2 @ S3 @ Sl3 )
           => ( ( fmember_rule @ S6 @ Sl3 )
             => ( member_rule @ S6 @ S4 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_119_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_120_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_121_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_122_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ( abstra7168897066996117381ule_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_123_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_124_UNIV__rules,axiom,
    ( ( sset_rule @ rules )
    = top_top_set_rule ) ).

% UNIV_rules
thf(fact_125_UNIV__witness,axiom,
    ? [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ top_to6833984726390702231at_nat ) ).

% UNIV_witness
thf(fact_126_UNIV__witness,axiom,
    ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ top_to4669805908274784177at_nat ) ).

% UNIV_witness
thf(fact_127_UNIV__witness,axiom,
    ? [X3: fm] : ( member_fm @ X3 @ top_top_set_fm ) ).

% UNIV_witness
thf(fact_128_UNIV__witness,axiom,
    ? [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ top_to728902761071950545ist_fm ) ).

% UNIV_witness
thf(fact_129_UNIV__witness,axiom,
    ? [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ).

% UNIV_witness
thf(fact_130_UNIV__eq__I,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A2 )
     => ( top_to6833984726390702231at_nat = A2 ) ) ).

% UNIV_eq_I
thf(fact_131_UNIV__eq__I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 )
     => ( top_to4669805908274784177at_nat = A2 ) ) ).

% UNIV_eq_I
thf(fact_132_UNIV__eq__I,axiom,
    ! [A2: set_fm] :
      ( ! [X3: fm] : ( member_fm @ X3 @ A2 )
     => ( top_top_set_fm = A2 ) ) ).

% UNIV_eq_I
thf(fact_133_UNIV__eq__I,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ A2 )
     => ( top_to728902761071950545ist_fm = A2 ) ) ).

% UNIV_eq_I
thf(fact_134_UNIV__eq__I,axiom,
    ! [A2: set_rule] :
      ( ! [X3: rule] : ( member_rule @ X3 @ A2 )
     => ( top_top_set_rule = A2 ) ) ).

% UNIV_eq_I
thf(fact_135_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S4: set_rule,S3: rule,Rs: stream_fm,Steps: stream1800043145262542076ule_fm,R2: fm] :
      ( ( abstra4039608574777915031m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra8780395760585530021ule_fm @ ( abstra8242076542743448570m_rule @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra5359274141611169189m_rule @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_136_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S4: set_fm,S3: fm,Rs: stream_fm,Steps: stream1375025385734440967_fm_fm,R2: fm] :
      ( ( abstra1990789280842237858_fm_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra724614664783131952_fm_fm @ ( abstra5285091746773999237_fm_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra81079133870296496_fm_fm @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_137_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_138_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule,R2: rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra1108425641574350757ule_fm @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_139_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_fm,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_fm,Steps: stream2046136080190117392nat_fm,R2: fm] :
      ( ( abstra233951355244900655at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra3420385485006866193nat_fm @ ( abstra5015990347065291020at_nat @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra440462504249735841at_nat @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_140_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_141_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: product_prod_nat_nat > fm > fset_fm > $o,Rules: stream6724221391990029191at_nat,S4: set_fm,S3: fm,Rs: stream6724221391990029191at_nat,Steps: stream7763447396415229238at_nat,R2: product_prod_nat_nat] :
      ( ( abstra4252533277467534673nat_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1870481761539534101at_nat @ Rules @ Rs )
         => ( ( abstra244687536638030135at_nat @ ( abstra9034572269287925038nat_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member8440522571783428010at_nat @ R2 @ ( sset_P8584168253256249621at_nat @ Rules ) )
             => ( abstra4459044426472369859nat_fm @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_142_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_143_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_fm,Steps: stream5128679079952569136_fm_fm,R2: fm] :
      ( ( abstra6333699301575899727ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra2316067621365718065_fm_fm @ ( abstra982595312256735276ist_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra4899091444543819713ist_fm @ Eff @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_144_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_145_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule,M: nat] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_432990894411838815m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_146_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,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_2199339672054592340e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_147_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,S7: list_fm] :
                ( ( member_list_fm @ S7 @ S4 )
                & ( abstra5863983821914536912ist_fm @ ( abstra3846921640779263781ist_fm @ Eff @ ( sdrop_list_fm @ N2 @ Rs ) @ S7 ) @ ( sdrop_4442373711808556042ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_148_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,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( abstra4067143171202666092e_rule @ ( abstra4338455777355667597e_rule @ Eff @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_3185404920971892658e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_149_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc780299800278082923e_rule > fm > fset_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_fm,S3: fm,Rs: stream3588992565182678257e_rule,Steps: stream3505881084499982752e_rule,M: nat] :
      ( ( abstra6843014240698492626ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra8067268766995864225e_rule @ ( abstra5872912078930511000ule_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( abstra8067268766995864225e_rule @ ( abstra5872912078930511000ule_fm @ Eff @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_3146478060095727335e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_150_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,S7: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S7 @ S4 )
                & ( abstra3533331058055637382t_rule @ ( abstra3105372317759539415at_nat @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_4214913930985848524t_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_151_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,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) @ S7 ) @ ( sdrop_2468034597186633064ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_152_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S4: set_fm,S3: fm,Rs: stream8299795917829157543ist_fm,Steps: stream5068232101399153750ist_fm,M: nat] :
      ( ( abstra2919136520687232136_fm_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff @ Rs @ S3 ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) @ S7 ) @ ( sdrop_8120954597595522205ist_fm @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_153_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,S7: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S7 @ S4 )
                & ( abstra2308351064978115840e_rule @ ( abstra3792383586331528633at_nat @ Eff @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_7505857448455835194e_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_154_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,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S4 )
                & ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_155_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_156_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_157_fair__uu__def,axiom,
    abstra1913438439042302728u_rule = abstra1114877437032970289r_rule ).

% fair_uu_def
thf(fact_158_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_159_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_160_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_161_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_162_per__rules_H,axiom,
    ! [R2: rule,A2: list_fm,B: list_fm,A3: list_fm,B2: 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 @ B ) )
     => ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ ( produc381145313068854617ist_fm @ A3 @ B2 ) )
       => ( ( ( eff @ R4 @ ( produc381145313068854617ist_fm @ A2 @ B ) )
            = ( some_f2162114139285431020ist_fm @ Ss2 ) )
         => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B2 ) @ Ss2 )
           => ( R4 = R2 ) ) ) ) ) ).

% per_rules'
thf(fact_163_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_164_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_165_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_166_RuleSystem_Owf__mkTree,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra2794488811626936717ule_fm @ Eff @ Rules @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_167_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_168_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_169_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_170_RuleSystem_OminWait__ex,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7038331712872434097ule_fm @ Eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_171_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_172_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc780299800278082923e_rule > fm > fset_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_fm,S3: fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra6843014240698492626ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra1405456541278283553ule_fm @ Eff @ ( shd_Pr4573454438745646103e_rule @ ( sdrop_2199339672054592340e_rule @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_173_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_174_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_175_RuleSystem_OminWait__ex,axiom,
    ! [Eff: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S4: set_fm,S3: fm,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ? [N2: nat] : ( abstra8628512393757682903_fm_fm @ Eff @ ( shd_Pr772355297128350925ist_fm @ ( sdrop_4442373711808556042ist_fm @ N2 @ Rs ) ) @ S3 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_176_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_177_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_178_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_179_fair__fenum,axiom,
    abstra1114877437032970289r_rule @ rules @ ( abstra7284221463285775110m_rule @ rules ) ).

% fair_fenum
thf(fact_180_sset__fenum,axiom,
    ( ( sset_rule @ ( abstra7284221463285775110m_rule @ rules ) )
    = ( sset_rule @ rules ) ) ).

% sset_fenum
thf(fact_181_RuleSystem__Defs_Otrim_Ocong,axiom,
    abstra2363873759530810705ist_fm = abstra2363873759530810705ist_fm ).

% RuleSystem_Defs.trim.cong
thf(fact_182_RuleSystem__Defs_Owf_Ocong,axiom,
    abstra4476735159789068068ist_fm = abstra4476735159789068068ist_fm ).

% RuleSystem_Defs.wf.cong
thf(fact_183_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_184_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_185_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_186_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_187_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S2: stream8299795917829157543ist_fm] :
      ( ( shd_Pr2478396429205514213ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) )
      = ( produc7063755442767259331ist_fm @ ( shd_rule @ S1 ) @ ( shd_Pr772355297128350925ist_fm @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_188_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_189_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_190_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_191_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_192_szip_Osimps_I1_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S2: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) )
      = ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).

% szip.simps(1)
thf(fact_193_RuleSystem_OminWait_Ocong,axiom,
    abstra738319790293016300ist_fm = abstra738319790293016300ist_fm ).

% RuleSystem.minWait.cong
thf(fact_194_RuleSystem__Defs_Ofenum_Ocong,axiom,
    abstra7284221463285775110m_rule = abstra7284221463285775110m_rule ).

% RuleSystem_Defs.fenum.cong
thf(fact_195_shd__sset,axiom,
    ! [A: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) @ ( sset_P3431374123360452149ist_fm @ A ) ) ).

% shd_sset
thf(fact_196_shd__sset,axiom,
    ! [A: stream8032061917732916077at_nat] : ( member8757157785044589968at_nat @ ( shd_Pr4412363897699689747at_nat @ A ) @ ( sset_P1195531240007660155at_nat @ A ) ) ).

% shd_sset
thf(fact_197_shd__sset,axiom,
    ! [A: stream6724221391990029191at_nat] : ( member8440522571783428010at_nat @ ( shd_Pr4260400998323988397at_nat @ A ) @ ( sset_P8584168253256249621at_nat @ A ) ) ).

% shd_sset
thf(fact_198_shd__sset,axiom,
    ! [A: stream_fm] : ( member_fm @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).

% shd_sset
thf(fact_199_shd__sset,axiom,
    ! [A: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) @ ( sset_P4979403222752902913m_rule @ A ) ) ).

% shd_sset
thf(fact_200_shd__sset,axiom,
    ! [A: stream_rule] : ( member_rule @ ( shd_rule @ A ) @ ( sset_rule @ A ) ) ).

% shd_sset
thf(fact_201_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > rule > fset_rule > $o,Rules: stream_fm,S4: set_rule,S3: rule,Rs: stream_fm] :
      ( ( abstra5167362717900046400m_rule @ Eff @ Rules @ S4 )
     => ( ( member_rule @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra8496492133263135264m_rule @ Eff @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_202_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > fm > fset_fm > $o,Rules: stream_fm,S4: set_fm,S3: fm,Rs: stream_fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra5942290089224017835_fm_fm @ Eff @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_203_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_204_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_205_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > product_prod_nat_nat > fset_P5670320511379867111at_nat > $o,Rules: stream_fm,S4: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat,Rs: stream_fm] :
      ( ( abstra4023219294580590406at_nat @ Eff @ Rules @ S4 )
     => ( ( member8440522571783428010at_nat @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra1746128039397047654at_nat @ Eff @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_206_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_207_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: product_prod_nat_nat > fm > fset_fm > $o,Rules: stream6724221391990029191at_nat,S4: set_fm,S3: fm,Rs: stream6724221391990029191at_nat] :
      ( ( abstra8041801216803224424nat_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1870481761539534101at_nat @ Rules @ Rs )
         => ( member8440522571783428010at_nat @ ( shd_Pr4260400998323988397at_nat @ ( abstra5764709961619681672nat_fm @ Eff @ Rs @ S3 ) ) @ ( sset_P8584168253256249621at_nat @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_208_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_209_RuleSystem_Otrim__in__R,axiom,
    ! [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S4: set_Pr7058068377845519745ist_fm,S3: produc1996495991257130529ist_fm,Rs: stream_fm] :
      ( ( abstra7587511110199143014ist_fm @ Eff @ Rules @ S4 )
     => ( ( member8102475879199740618ist_fm @ S3 @ S4 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra948641862382362758ist_fm @ Eff @ Rs @ S3 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_210_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_211_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_212_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_213_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_214_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_215_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S4: set_fm,S3: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra7152516257876272129ule_fm @ Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_216_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_217_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_218_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_219_RuleSystem_Otrim__enabled,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra7038331712872434097ule_fm @ Eff @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_220_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_221_i_Osset__fenum,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
      = ( sset_rule @ Rules ) ) ).

% i.sset_fenum
thf(fact_222_RuleSystem__Defs_Osset__fenum,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ ( abstra7284221463285775110m_rule @ Rules ) )
      = ( sset_rule @ Rules ) ) ).

% RuleSystem_Defs.sset_fenum
thf(fact_223_RuleSystem__Defs_Ofair__fenum,axiom,
    ! [Rules: stream_rule] : ( abstra1114877437032970289r_rule @ Rules @ ( abstra7284221463285775110m_rule @ Rules ) ) ).

% RuleSystem_Defs.fair_fenum
thf(fact_224_i_Ofair__fenum,axiom,
    ! [Rules: stream_rule] : ( abstra1114877437032970289r_rule @ Rules @ ( abstra7284221463285775110m_rule @ Rules ) ) ).

% i.fair_fenum
thf(fact_225_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_226_RuleSystem_Otrim__alt,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 )
            = ( sdrop_rule @ ( abstra2294468009765627845ule_fm @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_227_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_228_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc780299800278082923e_rule > fm > fset_fm > $o,Rules: stream3588992565182678257e_rule,S4: set_fm,S3: fm,Rs: stream3588992565182678257e_rule] :
      ( ( abstra6843014240698492626ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4837122928373946751e_rule @ Rules @ Rs )
         => ( ( abstra7283367714565168370ule_fm @ Eff @ Rs @ S3 )
            = ( sdrop_2199339672054592340e_rule @ ( abstra1933929482952324109ule_fm @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_229_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_230_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_231_RuleSystem_Otrim__alt,axiom,
    ! [Eff: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S4: set_fm,S3: fm,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra5503639309725227688_fm_fm @ Eff @ Rs @ S3 )
            = ( sdrop_4442373711808556042ist_fm @ ( abstra8883821634369821123_fm_fm @ Eff @ Rs @ S3 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_232_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_233_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_234_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_235_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_236_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_237_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_238_RuleSystem_Otrim__fair,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_239_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_240_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_241_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_242_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_243_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_244_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_245_fenum__uu__def,axiom,
    abstra8121460317444261103u_rule = abstra7284221463285775110m_rule ).

% fenum_uu_def
thf(fact_246_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_247_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_248_top__empty__eq2,axiom,
    ( top_to5144090402559349552t_fm_o
    = ( ^ [X: rule,Y: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y ) @ top_to2837172092223966711ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_249_top__empty__eq2,axiom,
    ( top_to1326704617074743934_nat_o
    = ( ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ top_to5281548843312042519at_nat ) ) ) ).

% top_empty_eq2
thf(fact_250_top__empty__eq2,axiom,
    ( top_to8903224817164384382_nat_o
    = ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ top_to6833984726390702231at_nat ) ) ) ).

% top_empty_eq2
thf(fact_251_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_252_top__empty__eq2,axiom,
    ( top_to7326020031456794390t_fm_o
    = ( ^ [X: list_fm,Y: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_253_prod_Oinject,axiom,
    ! [X1: rule,X2: produc1996495991257130529ist_fm,Y1: rule,Y2: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ X1 @ X2 )
        = ( produc7063755442767259331ist_fm @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_254_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_255_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_256_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_257_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_258_old_Oprod_Oinject,axiom,
    ! [A: rule,B3: produc1996495991257130529ist_fm,A4: rule,B4: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B3 )
        = ( produc7063755442767259331ist_fm @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_259_old_Oprod_Oinject,axiom,
    ! [A: list_fm,B3: list_fm,A4: list_fm,B4: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B3 )
        = ( produc381145313068854617ist_fm @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_260_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B3 )
        = ( produc9060074326276436823at_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_261_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B3 )
        = ( produc2922128104949294807at_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_262_old_Oprod_Oinject,axiom,
    ! [A: nat,B3: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B3 )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_263_top__empty__eq,axiom,
    ( top_to1113057434437417734_nat_o
    = ( ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ top_to6833984726390702231at_nat ) ) ) ).

% top_empty_eq
thf(fact_264_top__empty__eq,axiom,
    ( top_to3137496036531963500_nat_o
    = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq
thf(fact_265_top__empty__eq,axiom,
    ( top_top_fm_o
    = ( ^ [X: fm] : ( member_fm @ X @ top_top_set_fm ) ) ) ).

% top_empty_eq
thf(fact_266_top__empty__eq,axiom,
    ( top_to6798613745154499404t_fm_o
    = ( ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq
thf(fact_267_top__empty__eq,axiom,
    ( top_top_rule_o
    = ( ^ [X: rule] : ( member_rule @ X @ top_top_set_rule ) ) ) ).

% top_empty_eq
thf(fact_268_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_269_order__refl,axiom,
    ! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).

% order_refl
thf(fact_270_order__refl,axiom,
    ! [X5: set_rule] : ( ord_less_eq_set_rule @ X5 @ X5 ) ).

% order_refl
thf(fact_271_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_272_dual__order_Orefl,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ A ) ).

% dual_order.refl
thf(fact_273_Least1I,axiom,
    ! [P: nat > $o] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y3: nat] :
              ( ( P @ Y3 )
             => ( ord_less_eq_nat @ X6 @ Y3 ) )
          & ! [Y3: nat] :
              ( ( ( P @ Y3 )
                & ! [Ya: nat] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_nat @ Y3 @ Ya ) ) )
             => ( Y3 = X6 ) ) )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% Least1I
thf(fact_274_Least1I,axiom,
    ! [P: set_rule > $o] :
      ( ? [X6: set_rule] :
          ( ( P @ X6 )
          & ! [Y3: set_rule] :
              ( ( P @ Y3 )
             => ( ord_less_eq_set_rule @ X6 @ Y3 ) )
          & ! [Y3: set_rule] :
              ( ( ( P @ Y3 )
                & ! [Ya: set_rule] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_set_rule @ Y3 @ Ya ) ) )
             => ( Y3 = X6 ) ) )
     => ( P @ ( ord_Least_set_rule @ P ) ) ) ).

% Least1I
thf(fact_275_nle__le,axiom,
    ! [A: nat,B3: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B3 ) )
      = ( ( ord_less_eq_nat @ B3 @ A )
        & ( B3 != A ) ) ) ).

% nle_le
thf(fact_276_Least1__le,axiom,
    ! [P: nat > $o,Z: nat] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y3: nat] :
              ( ( P @ Y3 )
             => ( ord_less_eq_nat @ X6 @ Y3 ) )
          & ! [Y3: nat] :
              ( ( ( P @ Y3 )
                & ! [Ya: nat] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_nat @ Y3 @ Ya ) ) )
             => ( Y3 = X6 ) ) )
     => ( ( P @ Z )
       => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ Z ) ) ) ).

% Least1_le
thf(fact_277_Least1__le,axiom,
    ! [P: set_rule > $o,Z: set_rule] :
      ( ? [X6: set_rule] :
          ( ( P @ X6 )
          & ! [Y3: set_rule] :
              ( ( P @ Y3 )
             => ( ord_less_eq_set_rule @ X6 @ Y3 ) )
          & ! [Y3: set_rule] :
              ( ( ( P @ Y3 )
                & ! [Ya: set_rule] :
                    ( ( P @ Ya )
                   => ( ord_less_eq_set_rule @ Y3 @ Ya ) ) )
             => ( Y3 = X6 ) ) )
     => ( ( P @ Z )
       => ( ord_less_eq_set_rule @ ( ord_Least_set_rule @ P ) @ Z ) ) ) ).

% Least1_le
thf(fact_278_LeastI2,axiom,
    ! [P: nat > $o,A: nat,Q: nat > $o] :
      ( ( P @ A )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2
thf(fact_279_le__cases3,axiom,
    ! [X5: nat,Y4: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X5 @ Y4 )
       => ~ ( ord_less_eq_nat @ Y4 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y4 @ X5 )
         => ~ ( ord_less_eq_nat @ X5 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X5 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y4 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y4 )
             => ~ ( ord_less_eq_nat @ Y4 @ X5 ) )
           => ( ( ( ord_less_eq_nat @ Y4 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X5 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X5 )
                 => ~ ( ord_less_eq_nat @ X5 @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_280_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_281_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_rule,Z2: set_rule] : ( Y5 = Z2 ) )
    = ( ^ [X: set_rule,Y: set_rule] :
          ( ( ord_less_eq_set_rule @ X @ Y )
          & ( ord_less_eq_set_rule @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_282_Least__le,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ K ) ) ).

% Least_le
thf(fact_283_ord__eq__le__trans,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( A = B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_284_ord__eq__le__trans,axiom,
    ! [A: set_rule,B3: set_rule,C: set_rule] :
      ( ( A = B3 )
     => ( ( ord_less_eq_set_rule @ B3 @ C )
       => ( ord_less_eq_set_rule @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_285_ord__le__eq__trans,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_286_ord__le__eq__trans,axiom,
    ! [A: set_rule,B3: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_set_rule @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_287_LeastI2__order,axiom,
    ! [P: nat > $o,X5: nat,Q: nat > $o] :
      ( ( P @ X5 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ X5 @ Y3 ) )
       => ( ! [X3: nat] :
              ( ( P @ X3 )
             => ( ! [Y6: nat] :
                    ( ( P @ Y6 )
                   => ( ord_less_eq_nat @ X3 @ Y6 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( ord_Least_nat @ P ) ) ) ) ) ).

% LeastI2_order
thf(fact_288_LeastI2__order,axiom,
    ! [P: set_rule > $o,X5: set_rule,Q: set_rule > $o] :
      ( ( P @ X5 )
     => ( ! [Y3: set_rule] :
            ( ( P @ Y3 )
           => ( ord_less_eq_set_rule @ X5 @ Y3 ) )
       => ( ! [X3: set_rule] :
              ( ( P @ X3 )
             => ( ! [Y6: set_rule] :
                    ( ( P @ Y6 )
                   => ( ord_less_eq_set_rule @ X3 @ Y6 ) )
               => ( Q @ X3 ) ) )
         => ( Q @ ( ord_Least_set_rule @ P ) ) ) ) ) ).

% LeastI2_order
thf(fact_289_order__antisym,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y4 )
     => ( ( ord_less_eq_nat @ Y4 @ X5 )
       => ( X5 = Y4 ) ) ) ).

% order_antisym
thf(fact_290_order__antisym,axiom,
    ! [X5: set_rule,Y4: set_rule] :
      ( ( ord_less_eq_set_rule @ X5 @ Y4 )
     => ( ( ord_less_eq_set_rule @ Y4 @ X5 )
       => ( X5 = Y4 ) ) ) ).

% order_antisym
thf(fact_291_LeastI__ex,axiom,
    ! [P: nat > $o] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI_ex
thf(fact_292_Least__equality,axiom,
    ! [P: nat > $o,X5: nat] :
      ( ( P @ X5 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ X5 @ Y3 ) )
       => ( ( ord_Least_nat @ P )
          = X5 ) ) ) ).

% Least_equality
thf(fact_293_Least__equality,axiom,
    ! [P: set_rule > $o,X5: set_rule] :
      ( ( P @ X5 )
     => ( ! [Y3: set_rule] :
            ( ( P @ Y3 )
           => ( ord_less_eq_set_rule @ X5 @ Y3 ) )
       => ( ( ord_Least_set_rule @ P )
          = X5 ) ) ) ).

% Least_equality
thf(fact_294_order_Otrans,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_295_order_Otrans,axiom,
    ! [A: set_rule,B3: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B3 )
     => ( ( ord_less_eq_set_rule @ B3 @ C )
       => ( ord_less_eq_set_rule @ A @ C ) ) ) ).

% order.trans
thf(fact_296_order__trans,axiom,
    ! [X5: nat,Y4: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y4 )
     => ( ( ord_less_eq_nat @ Y4 @ Z )
       => ( ord_less_eq_nat @ X5 @ Z ) ) ) ).

% order_trans
thf(fact_297_order__trans,axiom,
    ! [X5: set_rule,Y4: set_rule,Z: set_rule] :
      ( ( ord_less_eq_set_rule @ X5 @ Y4 )
     => ( ( ord_less_eq_set_rule @ Y4 @ Z )
       => ( ord_less_eq_set_rule @ X5 @ Z ) ) ) ).

% order_trans
thf(fact_298_LeastI2__ex,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_ex
thf(fact_299_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B3: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat,B5: nat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B3 ) ) ) ).

% linorder_wlog
thf(fact_300_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_301_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_rule,Z2: set_rule] : ( Y5 = Z2 ) )
    = ( ^ [A6: set_rule,B6: set_rule] :
          ( ( ord_less_eq_set_rule @ B6 @ A6 )
          & ( ord_less_eq_set_rule @ A6 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_302_dual__order_Oantisym,axiom,
    ! [B3: nat,A: nat] :
      ( ( ord_less_eq_nat @ B3 @ A )
     => ( ( ord_less_eq_nat @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_303_dual__order_Oantisym,axiom,
    ! [B3: set_rule,A: set_rule] :
      ( ( ord_less_eq_set_rule @ B3 @ A )
     => ( ( ord_less_eq_set_rule @ A @ B3 )
       => ( A = B3 ) ) ) ).

% dual_order.antisym
thf(fact_304_dual__order_Otrans,axiom,
    ! [B3: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A )
     => ( ( ord_less_eq_nat @ C @ B3 )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_305_dual__order_Otrans,axiom,
    ! [B3: set_rule,A: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ B3 @ A )
     => ( ( ord_less_eq_set_rule @ C @ B3 )
       => ( ord_less_eq_set_rule @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_306_LeastI2__wellorder,axiom,
    ! [P: nat > $o,A: nat,Q: nat > $o] :
      ( ( P @ A )
     => ( ! [A5: nat] :
            ( ( P @ A5 )
           => ( ! [B7: nat] :
                  ( ( P @ B7 )
                 => ( ord_less_eq_nat @ A5 @ B7 ) )
             => ( Q @ A5 ) ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_wellorder
thf(fact_307_LeastI2__wellorder__ex,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [A5: nat] :
            ( ( P @ A5 )
           => ( ! [B7: nat] :
                  ( ( P @ B7 )
                 => ( ord_less_eq_nat @ A5 @ B7 ) )
             => ( Q @ A5 ) ) )
       => ( Q @ ( ord_Least_nat @ P ) ) ) ) ).

% LeastI2_wellorder_ex
thf(fact_308_antisym,axiom,
    ! [A: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_309_antisym,axiom,
    ! [A: set_rule,B3: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B3 )
     => ( ( ord_less_eq_set_rule @ B3 @ A )
       => ( A = B3 ) ) ) ).

% antisym
thf(fact_310_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
          & ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_311_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_rule,Z2: set_rule] : ( Y5 = Z2 ) )
    = ( ^ [A6: set_rule,B6: set_rule] :
          ( ( ord_less_eq_set_rule @ A6 @ B6 )
          & ( ord_less_eq_set_rule @ B6 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_312_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_313_order__subst1,axiom,
    ! [A: nat,F: set_rule > nat,B3: set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_rule @ B3 @ C )
       => ( ! [X3: set_rule,Y3: set_rule] :
              ( ( ord_less_eq_set_rule @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_314_order__subst1,axiom,
    ! [A: set_rule,F: nat > set_rule,B3: nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_rule @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_315_order__subst1,axiom,
    ! [A: set_rule,F: set_rule > set_rule,B3: set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_rule @ B3 @ C )
       => ( ! [X3: set_rule,Y3: set_rule] :
              ( ( ord_less_eq_set_rule @ X3 @ Y3 )
             => ( ord_less_eq_set_rule @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_316_order__subst2,axiom,
    ! [A: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_317_order__subst2,axiom,
    ! [A: nat,B3: nat,F: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_eq_set_rule @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_rule @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_318_order__subst2,axiom,
    ! [A: set_rule,B3: set_rule,F: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: set_rule,Y3: set_rule] :
              ( ( ord_less_eq_set_rule @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_319_order__subst2,axiom,
    ! [A: set_rule,B3: set_rule,F: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B3 )
     => ( ( ord_less_eq_set_rule @ ( F @ B3 ) @ C )
       => ( ! [X3: set_rule,Y3: set_rule] :
              ( ( ord_less_eq_set_rule @ X3 @ Y3 )
             => ( ord_less_eq_set_rule @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_320_order__eq__refl,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( X5 = Y4 )
     => ( ord_less_eq_nat @ X5 @ Y4 ) ) ).

% order_eq_refl
thf(fact_321_order__eq__refl,axiom,
    ! [X5: set_rule,Y4: set_rule] :
      ( ( X5 = Y4 )
     => ( ord_less_eq_set_rule @ X5 @ Y4 ) ) ).

% order_eq_refl
thf(fact_322_linorder__linear,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( ord_less_eq_nat @ X5 @ Y4 )
      | ( ord_less_eq_nat @ Y4 @ X5 ) ) ).

% linorder_linear
thf(fact_323_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_324_ord__eq__le__subst,axiom,
    ! [A: set_rule,F: nat > set_rule,B3: nat,C: nat] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_rule @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_325_ord__eq__le__subst,axiom,
    ! [A: nat,F: set_rule > nat,B3: set_rule,C: set_rule] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_rule @ B3 @ C )
       => ( ! [X3: set_rule,Y3: set_rule] :
              ( ( ord_less_eq_set_rule @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_326_ord__eq__le__subst,axiom,
    ! [A: set_rule,F: set_rule > set_rule,B3: set_rule,C: set_rule] :
      ( ( A
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_rule @ B3 @ C )
       => ( ! [X3: set_rule,Y3: set_rule] :
              ( ( ord_less_eq_set_rule @ X3 @ Y3 )
             => ( ord_less_eq_set_rule @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_rule @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_327_ord__le__eq__subst,axiom,
    ! [A: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_328_ord__le__eq__subst,axiom,
    ! [A: nat,B3: nat,F: nat > set_rule,C: set_rule] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_set_rule @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_329_ord__le__eq__subst,axiom,
    ! [A: set_rule,B3: set_rule,F: set_rule > nat,C: nat] :
      ( ( ord_less_eq_set_rule @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: set_rule,Y3: set_rule] :
              ( ( ord_less_eq_set_rule @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_330_ord__le__eq__subst,axiom,
    ! [A: set_rule,B3: set_rule,F: set_rule > set_rule,C: set_rule] :
      ( ( ord_less_eq_set_rule @ A @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: set_rule,Y3: set_rule] :
              ( ( ord_less_eq_set_rule @ X3 @ Y3 )
             => ( ord_less_eq_set_rule @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_rule @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_331_linorder__le__cases,axiom,
    ! [X5: nat,Y4: nat] :
      ( ~ ( ord_less_eq_nat @ X5 @ Y4 )
     => ( ord_less_eq_nat @ Y4 @ X5 ) ) ).

% linorder_le_cases
thf(fact_332_order__antisym__conv,axiom,
    ! [Y4: nat,X5: nat] :
      ( ( ord_less_eq_nat @ Y4 @ X5 )
     => ( ( ord_less_eq_nat @ X5 @ Y4 )
        = ( X5 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_333_order__antisym__conv,axiom,
    ! [Y4: set_rule,X5: set_rule] :
      ( ( ord_less_eq_set_rule @ Y4 @ X5 )
     => ( ( ord_less_eq_set_rule @ X5 @ Y4 )
        = ( X5 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_334_LeastI,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI
thf(fact_335_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
     => ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_uniqueI
thf(fact_336_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_337_top_Oextremum__unique,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
      = ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_unique
thf(fact_338_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_339_top__greatest,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A @ top_to728902761071950545ist_fm ) ).

% top_greatest
thf(fact_340_top__greatest,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ top_top_set_rule ) ).

% top_greatest
thf(fact_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_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_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_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_369_prod__induct3,axiom,
    ! [P: produc4630866025709511057ist_fm > $o,X5: produc4630866025709511057ist_fm] :
      ( ! [A5: rule,B5: list_fm,C2: list_fm] : ( P @ ( produc7063755442767259331ist_fm @ A5 @ ( produc381145313068854617ist_fm @ B5 @ C2 ) ) )
     => ( P @ X5 ) ) ).

% prod_induct3
thf(fact_370_prod__cases3,axiom,
    ! [Y4: produc4630866025709511057ist_fm] :
      ~ ! [A5: rule,B5: list_fm,C2: list_fm] :
          ( Y4
         != ( produc7063755442767259331ist_fm @ A5 @ ( produc381145313068854617ist_fm @ B5 @ C2 ) ) ) ).

% prod_cases3
thf(fact_371_Pair__inject,axiom,
    ! [A: rule,B3: produc1996495991257130529ist_fm,A4: rule,B4: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B3 )
        = ( produc7063755442767259331ist_fm @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_372_Pair__inject,axiom,
    ! [A: list_fm,B3: list_fm,A4: list_fm,B4: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B3 )
        = ( produc381145313068854617ist_fm @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_373_Pair__inject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B3 )
        = ( produc9060074326276436823at_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_374_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B3 )
        = ( produc2922128104949294807at_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_375_Pair__inject,axiom,
    ! [A: nat,B3: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B3 )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_376_prod__cases,axiom,
    ! [P: produc4630866025709511057ist_fm > $o,P2: produc4630866025709511057ist_fm] :
      ( ! [A5: rule,B5: produc1996495991257130529ist_fm] : ( P @ ( produc7063755442767259331ist_fm @ A5 @ B5 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_377_prod__cases,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm] :
      ( ! [A5: list_fm,B5: list_fm] : ( P @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_378_prod__cases,axiom,
    ! [P: produc1319942482725812455at_nat > $o,P2: produc1319942482725812455at_nat] :
      ( ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : ( P @ ( produc9060074326276436823at_nat @ A5 @ B5 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_379_prod__cases,axiom,
    ! [P: produc3843707927480180839at_nat > $o,P2: produc3843707927480180839at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : ( P @ ( produc2922128104949294807at_nat @ A5 @ B5 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_380_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A5: nat,B5: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B5 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_381_surj__pair,axiom,
    ! [P2: produc4630866025709511057ist_fm] :
    ? [X3: rule,Y3: produc1996495991257130529ist_fm] :
      ( P2
      = ( produc7063755442767259331ist_fm @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_382_surj__pair,axiom,
    ! [P2: produc1996495991257130529ist_fm] :
    ? [X3: list_fm,Y3: list_fm] :
      ( P2
      = ( produc381145313068854617ist_fm @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_383_surj__pair,axiom,
    ! [P2: produc1319942482725812455at_nat] :
    ? [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( P2
      = ( produc9060074326276436823at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_384_surj__pair,axiom,
    ! [P2: produc3843707927480180839at_nat] :
    ? [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( P2
      = ( produc2922128104949294807at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_385_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_386_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc4630866025709511057ist_fm] :
      ~ ! [A5: rule,B5: produc1996495991257130529ist_fm] :
          ( Y4
         != ( produc7063755442767259331ist_fm @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_387_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc1996495991257130529ist_fm] :
      ~ ! [A5: list_fm,B5: list_fm] :
          ( Y4
         != ( produc381145313068854617ist_fm @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_388_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc1319942482725812455at_nat] :
      ~ ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
          ( Y4
         != ( produc9060074326276436823at_nat @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_389_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc3843707927480180839at_nat] :
      ~ ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( Y4
         != ( produc2922128104949294807at_nat @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_390_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_nat_nat] :
      ~ ! [A5: nat,B5: nat] :
          ( Y4
         != ( product_Pair_nat_nat @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_391_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_392_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_393_pred__equals__eq2,axiom,
    ! [R5: set_Pr2241365836748572487ist_fm,S4: set_Pr2241365836748572487ist_fm] :
      ( ( ( ^ [X: rule,Y: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y ) @ R5 ) )
        = ( ^ [X: rule,Y: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_394_pred__equals__eq2,axiom,
    ! [R5: set_Pr7058068377845519745ist_fm,S4: set_Pr7058068377845519745ist_fm] :
      ( ( ( ^ [X: list_fm,Y: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ R5 ) )
        = ( ^ [X: list_fm,Y: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_395_pred__equals__eq2,axiom,
    ! [R5: set_Pr7459493094073627847at_nat,S4: set_Pr7459493094073627847at_nat] :
      ( ( ( ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R5 ) )
        = ( ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_396_pred__equals__eq2,axiom,
    ! [R5: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R5 ) )
        = ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_397_pred__equals__eq2,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R5 ) )
        = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S4 ) ) )
      = ( R5 = S4 ) ) ).

% pred_equals_eq2
thf(fact_398_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_399_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_400_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_401_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_402_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,S7: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S7 @ 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 @ S7 @ 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 ) )
                  @ S7 ) ) ) ) ) ) ).

% in_cont_mkTree
thf(fact_403_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs: stream8299795917829157543ist_fm,S3: rule] :
      ( ( abstra7970568315424063040ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ Rs @ S3 ) )
      = ( produc7063755442767259331ist_fm @ S3 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs @ S3 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_404_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_405_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_406_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_407_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_408_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_409_Stream_Osmember__def,axiom,
    ( smembe2449974647026456671ist_fm
    = ( ^ [X: produc1996495991257130529ist_fm,S: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X @ ( sset_P3431374123360452149ist_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_410_Stream_Osmember__def,axiom,
    ( smembe7284261217439725989at_nat
    = ( ^ [X: produc3843707927480180839at_nat,S: stream8032061917732916077at_nat] : ( member8757157785044589968at_nat @ X @ ( sset_P1195531240007660155at_nat @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_411_Stream_Osmember__def,axiom,
    ( smembe7612037963498647359at_nat
    = ( ^ [X: product_prod_nat_nat,S: stream6724221391990029191at_nat] : ( member8440522571783428010at_nat @ X @ ( sset_P8584168253256249621at_nat @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_412_Stream_Osmember__def,axiom,
    ( smember_fm
    = ( ^ [X: fm,S: stream_fm] : ( member_fm @ X @ ( sset_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_413_Stream_Osmember__def,axiom,
    ( smember_rule
    = ( ^ [X: rule,S: stream_rule] : ( member_rule @ X @ ( sset_rule @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_414_relChain__def,axiom,
    ( bNF_Ca5108556014793540054fm_nat
    = ( ^ [R: set_Pr7058068377845519745ist_fm,As: list_fm > nat] :
        ! [I: list_fm,J: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_415_relChain__def,axiom,
    ( bNF_Ca8906692637563564659at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat,As: set_Pr4329608150637261639at_nat > nat] :
        ! [I: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_416_relChain__def,axiom,
    ( bNF_Ca4949091444211332621at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat,As: set_Pr1261947904930325089at_nat > nat] :
        ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_417_relChain__def,axiom,
    ( bNF_Ca968750328013420230at_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,As: nat > nat] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_418_relChain__def,axiom,
    ( bNF_Ca3669579366834765185t_rule
    = ( ^ [R: set_Pr7058068377845519745ist_fm,As: list_fm > set_rule] :
        ! [I: list_fm,J: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ I @ J ) @ R )
         => ( ord_less_eq_set_rule @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_419_relChain__def,axiom,
    ( bNF_Ca5756846265281756446t_rule
    = ( ^ [R: set_Pr7459493094073627847at_nat,As: set_Pr4329608150637261639at_nat > set_rule] :
        ! [I: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I @ J ) @ R )
         => ( ord_less_eq_set_rule @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_420_relChain__def,axiom,
    ( bNF_Ca2200319123417071288t_rule
    = ( ^ [R: set_Pr4329608150637261639at_nat,As: set_Pr1261947904930325089at_nat > set_rule] :
        ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R )
         => ( ord_less_eq_set_rule @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_421_relChain__def,axiom,
    ( bNF_Ca1736420049334796913t_rule
    = ( ^ [R: set_Pr1261947904930325089at_nat,As: nat > set_rule] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R )
         => ( ord_less_eq_set_rule @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_422_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_423_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_424_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_425_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_426_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_427_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_428_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_429_pred__subset__eq2,axiom,
    ! [R5: set_Pr2241365836748572487ist_fm,S4: set_Pr2241365836748572487ist_fm] :
      ( ( ord_le8079975967087607008t_fm_o
        @ ^ [X: rule,Y: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y ) @ R5 )
        @ ^ [X: rule,Y: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y ) @ S4 ) )
      = ( ord_le6285472588696295335ist_fm @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_430_pred__subset__eq2,axiom,
    ! [R5: set_Pr7058068377845519745ist_fm,S4: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le1774903256965159270t_fm_o
        @ ^ [X: list_fm,Y: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ R5 )
        @ ^ [X: list_fm,Y: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y ) @ S4 ) )
      = ( ord_le2055072402069232929ist_fm @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_431_pred__subset__eq2,axiom,
    ! [R5: set_Pr7459493094073627847at_nat,S4: set_Pr7459493094073627847at_nat] :
      ( ( ord_le3072208448688395470_nat_o
        @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R5 )
        @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ S4 ) )
      = ( ord_le5997549366648089703at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_432_pred__subset__eq2,axiom,
    ! [R5: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R5 )
        @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ S4 ) )
      = ( ord_le1268244103169919719at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_433_pred__subset__eq2,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R5 )
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_434_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_435_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_436_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_437_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_438_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_439_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_440_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_441_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_442_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_443_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_444_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_445_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_446_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_447_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_448_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_449_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_450_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_451_stl__sset,axiom,
    ! [X5: fm,A: stream_fm] :
      ( ( member_fm @ X5 @ ( sset_fm @ ( stl_fm @ A ) ) )
     => ( member_fm @ X5 @ ( sset_fm @ A ) ) ) ).

% stl_sset
thf(fact_452_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_453_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_454_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_455_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_456_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_457_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_458_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_459_sdrop__stl,axiom,
    ! [N: nat,S3: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S3 ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S3 ) ) ) ).

% sdrop_stl
thf(fact_460_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_461_subrelI,axiom,
    ! [R2: set_Pr2241365836748572487ist_fm,S3: set_Pr2241365836748572487ist_fm] :
      ( ! [X3: rule,Y3: produc1996495991257130529ist_fm] :
          ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) @ R2 )
         => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le6285472588696295335ist_fm @ R2 @ S3 ) ) ).

% subrelI
thf(fact_462_subrelI,axiom,
    ! [R2: set_Pr7058068377845519745ist_fm,S3: set_Pr7058068377845519745ist_fm] :
      ( ! [X3: list_fm,Y3: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ R2 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le2055072402069232929ist_fm @ R2 @ S3 ) ) ).

% subrelI
thf(fact_463_subrelI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S3: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le5997549366648089703at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_464_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_465_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_466_subset__UNIV,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A2 @ top_to728902761071950545ist_fm ) ).

% subset_UNIV
thf(fact_467_subset__UNIV,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ top_top_set_rule ) ).

% subset_UNIV
thf(fact_468_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_469_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_470_i_Ofair__stl,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( stl_Pr950425576149878629m_rule @ Rs ) ) ) ).

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

% i.fair_stl
thf(fact_472_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_473_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S3: rule] :
      ( ( abstra7970568315424063040ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) )
      = ( produc7063755442767259331ist_fm @ S3 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_474_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_475_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_476_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_477_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_478_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_479_sset__induct,axiom,
    ! [Y4: produc1996495991257130529ist_fm,S3: stream8299795917829157543ist_fm,P: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Y4 @ ( sset_P3431374123360452149ist_fm @ S3 ) )
     => ( ! [S9: stream8299795917829157543ist_fm] : ( P @ ( shd_Pr772355297128350925ist_fm @ S9 ) @ S9 )
       => ( ! [S9: stream8299795917829157543ist_fm,Y3: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Y3 @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ S9 ) ) )
             => ( ( P @ Y3 @ ( stl_Pr5027778045650968273ist_fm @ S9 ) )
               => ( P @ Y3 @ S9 ) ) )
         => ( P @ Y4 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_480_sset__induct,axiom,
    ! [Y4: produc3843707927480180839at_nat,S3: stream8032061917732916077at_nat,P: produc3843707927480180839at_nat > stream8032061917732916077at_nat > $o] :
      ( ( member8757157785044589968at_nat @ Y4 @ ( sset_P1195531240007660155at_nat @ S3 ) )
     => ( ! [S9: stream8032061917732916077at_nat] : ( P @ ( shd_Pr4412363897699689747at_nat @ S9 ) @ S9 )
       => ( ! [S9: stream8032061917732916077at_nat,Y3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ Y3 @ ( sset_P1195531240007660155at_nat @ ( stl_Pr6836137205557096727at_nat @ S9 ) ) )
             => ( ( P @ Y3 @ ( stl_Pr6836137205557096727at_nat @ S9 ) )
               => ( P @ Y3 @ S9 ) ) )
         => ( P @ Y4 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_481_sset__induct,axiom,
    ! [Y4: product_prod_nat_nat,S3: stream6724221391990029191at_nat,P: product_prod_nat_nat > stream6724221391990029191at_nat > $o] :
      ( ( member8440522571783428010at_nat @ Y4 @ ( sset_P8584168253256249621at_nat @ S3 ) )
     => ( ! [S9: stream6724221391990029191at_nat] : ( P @ ( shd_Pr4260400998323988397at_nat @ S9 ) @ S9 )
       => ( ! [S9: stream6724221391990029191at_nat,Y3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y3 @ ( sset_P8584168253256249621at_nat @ ( stl_Pr5027827701538482609at_nat @ S9 ) ) )
             => ( ( P @ Y3 @ ( stl_Pr5027827701538482609at_nat @ S9 ) )
               => ( P @ Y3 @ S9 ) ) )
         => ( P @ Y4 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_482_sset__induct,axiom,
    ! [Y4: fm,S3: stream_fm,P: fm > stream_fm > $o] :
      ( ( member_fm @ Y4 @ ( sset_fm @ S3 ) )
     => ( ! [S9: stream_fm] : ( P @ ( shd_fm @ S9 ) @ S9 )
       => ( ! [S9: stream_fm,Y3: fm] :
              ( ( member_fm @ Y3 @ ( sset_fm @ ( stl_fm @ S9 ) ) )
             => ( ( P @ Y3 @ ( stl_fm @ S9 ) )
               => ( P @ Y3 @ S9 ) ) )
         => ( P @ Y4 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_483_sset__induct,axiom,
    ! [Y4: produc164195504107695125m_rule,S3: stream727092118206550309m_rule,P: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ Y4 @ ( sset_P4979403222752902913m_rule @ S3 ) )
     => ( ! [S9: stream727092118206550309m_rule] : ( P @ ( shd_Pr7235097944458474089m_rule @ S9 ) @ S9 )
       => ( ! [S9: stream727092118206550309m_rule,Y3: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ S9 ) ) )
             => ( ( P @ Y3 @ ( stl_Pr950425576149878629m_rule @ S9 ) )
               => ( P @ Y3 @ S9 ) ) )
         => ( P @ Y4 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_484_sset__induct,axiom,
    ! [Y4: rule,S3: stream_rule,P: rule > stream_rule > $o] :
      ( ( member_rule @ Y4 @ ( sset_rule @ S3 ) )
     => ( ! [S9: stream_rule] : ( P @ ( shd_rule @ S9 ) @ S9 )
       => ( ! [S9: stream_rule,Y3: rule] :
              ( ( member_rule @ Y3 @ ( sset_rule @ ( stl_rule @ S9 ) ) )
             => ( ( P @ Y3 @ ( stl_rule @ S9 ) )
               => ( P @ Y3 @ S9 ) ) )
         => ( P @ Y4 @ S3 ) ) ) ) ).

% sset_induct
thf(fact_485_fair__stl,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).

% fair_stl
thf(fact_486_sdrop__while_Osimps,axiom,
    ( sdrop_5541336384294084785m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( if_str8948254419368749791m_rule @ ( P3 @ ( shd_Pr7235097944458474089m_rule @ S ) ) @ ( sdrop_5541336384294084785m_rule @ P3 @ ( stl_Pr950425576149878629m_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_487_sdrop__while_Osimps,axiom,
    ( sdrop_while_rule
    = ( ^ [P3: rule > $o,S: stream_rule] : ( if_stream_rule @ ( P3 @ ( shd_rule @ S ) ) @ ( sdrop_while_rule @ P3 @ ( stl_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_488_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_489_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_490_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_491_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_492_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_493_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_494_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_495_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_496_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,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmember_rule @ S7 @ Sl4 )
                & ( T3
                  = ( abstra7443870075974203631e_rule @ Eff @ ( stl_rule @ ( abstra7824469560916757525e_rule @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_497_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: rule > fm > fset_fm > $o,Rules: stream_rule,S4: set_fm,S3: fm,Rs: stream_rule,T3: abstra4524330545568143696m_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe2914889872492170903m_rule @ T3 @ ( abstra5134701107988495863m_rule @ ( abstra3991228042706630138ule_fm @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_fm,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmember_fm @ S7 @ Sl4 )
                & ( T3
                  = ( abstra3991228042706630138ule_fm @ Eff @ ( stl_rule @ ( abstra4245643633226316832ule_fm @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_498_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,S7: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S7 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra7711156300587845681at_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe1449479052724974408at_nat @ S7 @ Sl4 )
                & ( T3
                  = ( abstra3105372317759539415at_nat @ Eff @ ( stl_rule @ ( abstra7711156300587845681at_nat @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_499_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,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe3381613331217039976ist_fm @ S7 @ Sl4 )
                & ( T3
                  = ( abstra178918799166095863ist_fm @ Eff @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_500_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,S7: rule] :
                ( ( member_rule @ S7 @ S4 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmember_rule @ S7 @ Sl4 )
                & ( T3
                  = ( abstra1231468025953361855e_rule @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra5289413328258868709e_rule @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_501_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S4: set_fm,S3: fm,Rs: stream727092118206550309m_rule,T3: abstra4575282202256230560m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff @ Rules @ S4 )
     => ( ( member_fm @ S3 @ S4 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe8691679578407899495m_rule @ T3 @ ( abstra2433431901973683783m_rule @ ( abstra1207708450097661130ule_fm @ Eff @ Rs @ S3 ) ) )
           => ? [Sl4: fset_fm,S7: fm] :
                ( ( member_fm @ S7 @ S4 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmember_fm @ S7 @ Sl4 )
                & ( T3
                  = ( abstra1207708450097661130ule_fm @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra2315704231547878896ule_fm @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_502_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,S7: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S7 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra1865002795007470625m_rule @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe1635983029874454474m_rule @ S7 @ Sl4 )
                & ( T3
                  = ( abstra2861652164185304827m_rule @ Eff @ ( stl_rule @ ( abstra1865002795007470625m_rule @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_503_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,S7: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S7 @ S4 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra1939759854238243425at_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe1449479052724974408at_nat @ S7 @ Sl4 )
                & ( T3
                  = ( abstra5880731324333133319at_nat @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra1939759854238243425at_nat @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_504_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,S7: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ S7 @ S4 )
                & ( Eff @ ( shd_rule @ ( abstra1101146469844389783at_nat @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe9188685285568734254at_nat @ S7 @ Sl4 )
                & ( T3
                  = ( abstra2952254667104538429at_nat @ Eff @ ( stl_rule @ ( abstra1101146469844389783at_nat @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_505_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,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S4 )
                & ( Eff @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff @ Rs @ S3 ) ) @ S3 @ Sl4 )
                & ( fmembe3381613331217039976ist_fm @ S7 @ Sl4 )
                & ( T3
                  = ( abstra2161496378874750759ist_fm @ Eff @ ( stl_Pr950425576149878629m_rule @ ( abstra6201285654312316801ist_fm @ Eff @ Rs @ S3 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_506_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_507_i_OSaturated__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps )
      = ( ! [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
           => ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ X @ Steps ) ) ) ) ).

% i.Saturated_def
thf(fact_508_saturated__effG__uu__def,axiom,
    ( abstra7873564507807625473ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% saturated_effG_uu_def
thf(fact_509_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_510_mkTree__effG__uu__def,axiom,
    ( abstra3917079402916513406ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% mkTree_effG_uu_def
thf(fact_511_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_512_fsubsetI,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X3 @ A2 )
         => ( fmembe3381613331217039976ist_fm @ X3 @ B ) )
     => ( ord_le2064643713053750439ist_fm @ A2 @ B ) ) ).

% fsubsetI
thf(fact_513_fsubsetI,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule] :
      ( ! [X3: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X3 @ A2 )
         => ( fmembe1635983029874454474m_rule @ X3 @ B ) )
     => ( ord_le6951374404268211425m_rule @ A2 @ B ) ) ).

% fsubsetI
thf(fact_514_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_515_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_516_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_517_subsetI,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X3 @ A2 )
         => ( member8102475879199740618ist_fm @ X3 @ B ) )
     => ( ord_le2055072402069232929ist_fm @ A2 @ B ) ) ).

% subsetI
thf(fact_518_subsetI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ! [X3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A2 )
         => ( member8757157785044589968at_nat @ X3 @ B ) )
     => ( ord_le1268244103169919719at_nat @ A2 @ B ) ) ).

% subsetI
thf(fact_519_subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A2 )
         => ( member8440522571783428010at_nat @ X3 @ B ) )
     => ( ord_le3146513528884898305at_nat @ A2 @ B ) ) ).

% subsetI
thf(fact_520_subsetI,axiom,
    ! [A2: set_fm,B: set_fm] :
      ( ! [X3: fm] :
          ( ( member_fm @ X3 @ A2 )
         => ( member_fm @ X3 @ B ) )
     => ( ord_less_eq_set_fm @ A2 @ B ) ) ).

% subsetI
thf(fact_521_subsetI,axiom,
    ! [A2: set_rule,B: set_rule] :
      ( ! [X3: rule] :
          ( ( member_rule @ X3 @ A2 )
         => ( member_rule @ X3 @ B ) )
     => ( ord_less_eq_set_rule @ A2 @ B ) ) ).

% subsetI
thf(fact_522_subset__antisym,axiom,
    ! [A2: set_rule,B: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B )
     => ( ( ord_less_eq_set_rule @ B @ A2 )
       => ( A2 = B ) ) ) ).

% subset_antisym
thf(fact_523_fimage__eqI,axiom,
    ! [B3: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( B3
        = ( F @ X5 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
       => ( fmembe3381613331217039976ist_fm @ B3 @ ( fimage8674584286907012797ist_fm @ F @ A2 ) ) ) ) ).

% fimage_eqI
thf(fact_524_fimage__eqI,axiom,
    ! [B3: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( B3
        = ( F @ X5 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
       => ( fmembe1635983029874454474m_rule @ B3 @ ( fimage748043113761278389m_rule @ F @ A2 ) ) ) ) ).

% fimage_eqI
thf(fact_525_fimage__eqI,axiom,
    ! [B3: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( B3
        = ( F @ X5 ) )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
       => ( fmembe3381613331217039976ist_fm @ B3 @ ( fimage9164954772359936849ist_fm @ F @ A2 ) ) ) ) ).

% fimage_eqI
thf(fact_526_fimage__eqI,axiom,
    ! [B3: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( B3
        = ( F @ X5 ) )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
       => ( fmembe1635983029874454474m_rule @ B3 @ ( fimage3349360726776475169m_rule @ F @ A2 ) ) ) ) ).

% fimage_eqI
thf(fact_527_fimage__fsubsetI,axiom,
    ! [A2: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X3 @ A2 )
         => ( fmembe3381613331217039976ist_fm @ ( F @ X3 ) @ B ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage8674584286907012797ist_fm @ F @ A2 ) @ B ) ) ).

% fimage_fsubsetI
thf(fact_528_fimage__fsubsetI,axiom,
    ! [A2: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,B: fset_A5396818216090053761m_rule] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X3 @ A2 )
         => ( fmembe1635983029874454474m_rule @ ( F @ X3 ) @ B ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F @ A2 ) @ B ) ) ).

% fimage_fsubsetI
thf(fact_529_fimage__fsubsetI,axiom,
    ! [A2: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X3: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X3 @ A2 )
         => ( fmembe3381613331217039976ist_fm @ ( F @ X3 ) @ B ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F @ A2 ) @ B ) ) ).

% fimage_fsubsetI
thf(fact_530_fimage__fsubsetI,axiom,
    ! [A2: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,B: fset_A5396818216090053761m_rule] :
      ( ! [X3: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X3 @ A2 )
         => ( fmembe1635983029874454474m_rule @ ( F @ X3 ) @ B ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage3349360726776475169m_rule @ F @ A2 ) @ B ) ) ).

% fimage_fsubsetI
thf(fact_531_pred__subset__eq,axiom,
    ! [R5: set_Pr7058068377845519745ist_fm,S4: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le8518180823940269308t_fm_o
        @ ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ R5 )
        @ ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ S4 ) )
      = ( ord_le2055072402069232929ist_fm @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_532_pred__subset__eq,axiom,
    ! [R5: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1999416642142172342_nat_o
        @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ R5 )
        @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ S4 ) )
      = ( ord_le1268244103169919719at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_533_pred__subset__eq,axiom,
    ! [R5: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le704812498762024988_nat_o
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R5 )
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_534_pred__subset__eq,axiom,
    ! [R5: set_fm,S4: set_fm] :
      ( ( ord_less_eq_fm_o
        @ ^ [X: fm] : ( member_fm @ X @ R5 )
        @ ^ [X: fm] : ( member_fm @ X @ S4 ) )
      = ( ord_less_eq_set_fm @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_535_pred__subset__eq,axiom,
    ! [R5: set_rule,S4: set_rule] :
      ( ( ord_less_eq_rule_o
        @ ^ [X: rule] : ( member_rule @ X @ R5 )
        @ ^ [X: rule] : ( member_rule @ X @ S4 ) )
      = ( ord_less_eq_set_rule @ R5 @ S4 ) ) ).

% pred_subset_eq
thf(fact_536_subset__fimage__iff,axiom,
    ! [B: fset_A5396818216090053761m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A2: fset_P661503646757059847ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A2 ) )
      = ( ? [AA: fset_P661503646757059847ist_fm] :
            ( ( ord_le2064643713053750439ist_fm @ AA @ A2 )
            & ( B
              = ( fimage748043113761278389m_rule @ F @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_537_subset__fimage__iff,axiom,
    ! [B: fset_P661503646757059847ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A2: fset_A5396818216090053761m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A2 ) )
      = ( ? [AA: fset_A5396818216090053761m_rule] :
            ( ( ord_le6951374404268211425m_rule @ AA @ A2 )
            & ( B
              = ( fimage9164954772359936849ist_fm @ F @ AA ) ) ) ) ) ).

% subset_fimage_iff
thf(fact_538_fimage__mono,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( ord_le2064643713053750439ist_fm @ A2 @ B )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F @ A2 ) @ ( fimage748043113761278389m_rule @ F @ B ) ) ) ).

% fimage_mono
thf(fact_539_fimage__mono,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( ord_le6951374404268211425m_rule @ A2 @ B )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F @ A2 ) @ ( fimage9164954772359936849ist_fm @ F @ B ) ) ) ).

% fimage_mono
thf(fact_540_in__mono,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,X5: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A2 @ B )
     => ( ( member8102475879199740618ist_fm @ X5 @ A2 )
       => ( member8102475879199740618ist_fm @ X5 @ B ) ) ) ).

% in_mono
thf(fact_541_in__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X5: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ B )
     => ( ( member8757157785044589968at_nat @ X5 @ A2 )
       => ( member8757157785044589968at_nat @ X5 @ B ) ) ) ).

% in_mono
thf(fact_542_in__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B )
     => ( ( member8440522571783428010at_nat @ X5 @ A2 )
       => ( member8440522571783428010at_nat @ X5 @ B ) ) ) ).

% in_mono
thf(fact_543_in__mono,axiom,
    ! [A2: set_fm,B: set_fm,X5: fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B )
     => ( ( member_fm @ X5 @ A2 )
       => ( member_fm @ X5 @ B ) ) ) ).

% in_mono
thf(fact_544_in__mono,axiom,
    ! [A2: set_rule,B: set_rule,X5: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B )
     => ( ( member_rule @ X5 @ A2 )
       => ( member_rule @ X5 @ B ) ) ) ).

% in_mono
thf(fact_545_subsetD,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,B: set_Pr7058068377845519745ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ A2 @ B )
     => ( ( member8102475879199740618ist_fm @ C @ A2 )
       => ( member8102475879199740618ist_fm @ C @ B ) ) ) ).

% subsetD
thf(fact_546_subsetD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ B )
     => ( ( member8757157785044589968at_nat @ C @ A2 )
       => ( member8757157785044589968at_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_547_subsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B )
     => ( ( member8440522571783428010at_nat @ C @ A2 )
       => ( member8440522571783428010at_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_548_subsetD,axiom,
    ! [A2: set_fm,B: set_fm,C: fm] :
      ( ( ord_less_eq_set_fm @ A2 @ B )
     => ( ( member_fm @ C @ A2 )
       => ( member_fm @ C @ B ) ) ) ).

% subsetD
thf(fact_549_subsetD,axiom,
    ! [A2: set_rule,B: set_rule,C: rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B )
     => ( ( member_rule @ C @ A2 )
       => ( member_rule @ C @ B ) ) ) ).

% subsetD
thf(fact_550_equalityE,axiom,
    ! [A2: set_rule,B: set_rule] :
      ( ( A2 = B )
     => ~ ( ( ord_less_eq_set_rule @ A2 @ B )
         => ~ ( ord_less_eq_set_rule @ B @ A2 ) ) ) ).

% equalityE
thf(fact_551_subset__eq,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B8: set_Pr7058068377845519745ist_fm] :
        ! [X: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X @ A7 )
         => ( member8102475879199740618ist_fm @ X @ B8 ) ) ) ) ).

% subset_eq
thf(fact_552_subset__eq,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A7: set_Pr4329608150637261639at_nat,B8: set_Pr4329608150637261639at_nat] :
        ! [X: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X @ A7 )
         => ( member8757157785044589968at_nat @ X @ B8 ) ) ) ) ).

% subset_eq
thf(fact_553_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
        ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ A7 )
         => ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ).

% subset_eq
thf(fact_554_subset__eq,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B8: set_fm] :
        ! [X: fm] :
          ( ( member_fm @ X @ A7 )
         => ( member_fm @ X @ B8 ) ) ) ) ).

% subset_eq
thf(fact_555_subset__eq,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
        ! [X: rule] :
          ( ( member_rule @ X @ A7 )
         => ( member_rule @ X @ B8 ) ) ) ) ).

% subset_eq
thf(fact_556_equalityD1,axiom,
    ! [A2: set_rule,B: set_rule] :
      ( ( A2 = B )
     => ( ord_less_eq_set_rule @ A2 @ B ) ) ).

% equalityD1
thf(fact_557_equalityD2,axiom,
    ! [A2: set_rule,B: set_rule] :
      ( ( A2 = B )
     => ( ord_less_eq_set_rule @ B @ A2 ) ) ).

% equalityD2
thf(fact_558_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_559_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_560_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_561_subset__iff,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B8: set_fm] :
        ! [T2: fm] :
          ( ( member_fm @ T2 @ A7 )
         => ( member_fm @ T2 @ B8 ) ) ) ) ).

% subset_iff
thf(fact_562_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_563_subset__refl,axiom,
    ! [A2: set_rule] : ( ord_less_eq_set_rule @ A2 @ A2 ) ).

% subset_refl
thf(fact_564_Collect__mono,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_565_Collect__mono,axiom,
    ! [P: rule > $o,Q: rule > $o] :
      ( ! [X3: rule] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_rule @ ( collect_rule @ P ) @ ( collect_rule @ Q ) ) ) ).

% Collect_mono
thf(fact_566_subset__trans,axiom,
    ! [A2: set_rule,B: set_rule,C3: set_rule] :
      ( ( ord_less_eq_set_rule @ A2 @ B )
     => ( ( ord_less_eq_set_rule @ B @ C3 )
       => ( ord_less_eq_set_rule @ A2 @ C3 ) ) ) ).

% subset_trans
thf(fact_567_set__eq__subset,axiom,
    ( ( ^ [Y5: set_rule,Z2: set_rule] : ( Y5 = 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_568_less__eq__set__def,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B8: set_Pr7058068377845519745ist_fm] :
          ( ord_le8518180823940269308t_fm_o
          @ ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A7 )
          @ ^ [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_569_less__eq__set__def,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A7: set_Pr4329608150637261639at_nat,B8: set_Pr4329608150637261639at_nat] :
          ( ord_le1999416642142172342_nat_o
          @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ A7 )
          @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_570_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A7 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_571_less__eq__set__def,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B8: set_fm] :
          ( ord_less_eq_fm_o
          @ ^ [X: fm] : ( member_fm @ X @ A7 )
          @ ^ [X: fm] : ( member_fm @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_572_less__eq__set__def,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B8: set_rule] :
          ( ord_less_eq_rule_o
          @ ^ [X: rule] : ( member_rule @ X @ A7 )
          @ ^ [X: rule] : ( member_rule @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_573_Collect__mono__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) )
      = ( ! [X: product_prod_nat_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_574_Collect__mono__iff,axiom,
    ! [P: rule > $o,Q: rule > $o] :
      ( ( ord_less_eq_set_rule @ ( collect_rule @ P ) @ ( collect_rule @ Q ) )
      = ( ! [X: rule] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_575_fimageE,axiom,
    ! [B3: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ B3 @ ( fimage8674584286907012797ist_fm @ F @ A2 ) )
     => ~ ! [X3: produc1996495991257130529ist_fm] :
            ( ( B3
              = ( F @ X3 ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X3 @ A2 ) ) ) ).

% fimageE
thf(fact_576_fimageE,axiom,
    ! [B3: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A2: fset_A5396818216090053761m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ B3 @ ( fimage9164954772359936849ist_fm @ F @ A2 ) )
     => ~ ! [X3: abstra4572470701423150929m_rule] :
            ( ( B3
              = ( F @ X3 ) )
           => ~ ( fmembe1635983029874454474m_rule @ X3 @ A2 ) ) ) ).

% fimageE
thf(fact_577_fimageE,axiom,
    ! [B3: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A2: fset_P661503646757059847ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ B3 @ ( fimage748043113761278389m_rule @ F @ A2 ) )
     => ~ ! [X3: produc1996495991257130529ist_fm] :
            ( ( B3
              = ( F @ X3 ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X3 @ A2 ) ) ) ).

% fimageE
thf(fact_578_fimageE,axiom,
    ! [B3: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ B3 @ ( fimage3349360726776475169m_rule @ F @ A2 ) )
     => ~ ! [X3: abstra4572470701423150929m_rule] :
            ( ( B3
              = ( F @ X3 ) )
           => ~ ( fmembe1635983029874454474m_rule @ X3 @ A2 ) ) ) ).

% fimageE
thf(fact_579_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_580_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_581_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_582_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_583_fimage__cong,axiom,
    ! [M2: fset_P661503646757059847ist_fm,N4: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( M2 = N4 )
     => ( ! [X3: produc1996495991257130529ist_fm] :
            ( ( fmembe3381613331217039976ist_fm @ X3 @ N4 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( fimage748043113761278389m_rule @ F @ M2 )
          = ( fimage748043113761278389m_rule @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_584_fimage__cong,axiom,
    ! [M2: fset_A5396818216090053761m_rule,N4: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( M2 = N4 )
     => ( ! [X3: abstra4572470701423150929m_rule] :
            ( ( fmembe1635983029874454474m_rule @ X3 @ N4 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( fimage9164954772359936849ist_fm @ F @ M2 )
          = ( fimage9164954772359936849ist_fm @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_585_rev__fimage__eqI,axiom,
    ! [X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm,B3: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
     => ( ( B3
          = ( F @ X5 ) )
       => ( fmembe3381613331217039976ist_fm @ B3 @ ( fimage8674584286907012797ist_fm @ F @ A2 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_586_rev__fimage__eqI,axiom,
    ! [X5: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm,B3: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
     => ( ( B3
          = ( F @ X5 ) )
       => ( fmembe1635983029874454474m_rule @ B3 @ ( fimage748043113761278389m_rule @ F @ A2 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_587_rev__fimage__eqI,axiom,
    ! [X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule,B3: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
     => ( ( B3
          = ( F @ X5 ) )
       => ( fmembe3381613331217039976ist_fm @ B3 @ ( fimage9164954772359936849ist_fm @ F @ A2 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_588_rev__fimage__eqI,axiom,
    ! [X5: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule,B3: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
     => ( ( B3
          = ( F @ X5 ) )
       => ( fmembe1635983029874454474m_rule @ B3 @ ( fimage3349360726776475169m_rule @ F @ A2 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_589_Collect__subset,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_590_Collect__subset,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ord_le1268244103169919719at_nat
      @ ( collec6321179662152712658at_nat
        @ ^ [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_591_Collect__subset,axiom,
    ! [A2: set_fm,P: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X: fm] :
            ( ( member_fm @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_592_Collect__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_593_Collect__subset,axiom,
    ! [A2: set_rule,P: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X: rule] :
            ( ( member_rule @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_594_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
        @ ^ [X: produc1996495991257130529ist_fm] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_595_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
        @ ^ [X: abstra4572470701423150929m_rule] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_596_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
        @ ^ [X: produc1996495991257130529ist_fm] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_597_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
        @ ^ [X: abstra4572470701423150929m_rule] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_598_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
        @ ^ [X: abstra4572470701423150929m_rule] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_599_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
        @ ^ [X: produc1996495991257130529ist_fm] : ( F @ ( G @ X ) )
        @ A2 ) ) ).

% fimage_fimage
thf(fact_600_fst__conv,axiom,
    ! [X1: rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_601_fst__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
      ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_602_fst__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_603_fst__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_604_fst__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_605_fst__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_606_fst__eqD,axiom,
    ! [X5: rule,Y4: produc1996495991257130529ist_fm,A: rule] :
      ( ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y4 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_607_fst__eqD,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_608_fst__eqD,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_609_fst__eqD,axiom,
    ! [X5: nat,Y4: nat,A: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_610_fst__eqD,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y4: rule,A: produc1996495991257130529ist_fm] :
      ( ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X5 @ Y4 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_611_fst__eqD,axiom,
    ! [X5: list_fm,Y4: list_fm,A: list_fm] :
      ( ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y4 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_612_ftree__no__ipath,axiom,
    ! [T: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra3507476289341504315m_rule @ T )
     => ~ ( abstra2285467793620430502m_rule @ T @ Steps ) ) ).

% ftree_no_ipath
thf(fact_613_eqfelem__imp__iff,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y4: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( X5 = Y4 )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
        = ( fmembe3381613331217039976ist_fm @ Y4 @ A2 ) ) ) ).

% eqfelem_imp_iff
thf(fact_614_eqfelem__imp__iff,axiom,
    ! [X5: abstra4572470701423150929m_rule,Y4: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( X5 = Y4 )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
        = ( fmembe1635983029874454474m_rule @ Y4 @ A2 ) ) ) ).

% eqfelem_imp_iff
thf(fact_615_if__split__fmem2,axiom,
    ! [A: produc1996495991257130529ist_fm,Q: $o,X5: fset_P661503646757059847ist_fm,Y4: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A @ ( if_fse3714977293194272717ist_fm @ Q @ X5 @ Y4 ) )
      = ( ( Q
         => ( fmembe3381613331217039976ist_fm @ A @ X5 ) )
        & ( ~ Q
         => ( fmembe3381613331217039976ist_fm @ A @ Y4 ) ) ) ) ).

% if_split_fmem2
thf(fact_616_if__split__fmem2,axiom,
    ! [A: abstra4572470701423150929m_rule,Q: $o,X5: fset_A5396818216090053761m_rule,Y4: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ A @ ( if_fse3370020817956878651m_rule @ Q @ X5 @ Y4 ) )
      = ( ( Q
         => ( fmembe1635983029874454474m_rule @ A @ X5 ) )
        & ( ~ Q
         => ( fmembe1635983029874454474m_rule @ A @ Y4 ) ) ) ) ).

% if_split_fmem2
thf(fact_617_if__split__fmem1,axiom,
    ! [Q: $o,X5: produc1996495991257130529ist_fm,Y4: produc1996495991257130529ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ ( if_Pro3930376587665744871ist_fm @ Q @ X5 @ Y4 ) @ B3 )
      = ( ( Q
         => ( fmembe3381613331217039976ist_fm @ X5 @ B3 ) )
        & ( ~ Q
         => ( fmembe3381613331217039976ist_fm @ Y4 @ B3 ) ) ) ) ).

% if_split_fmem1
thf(fact_618_if__split__fmem1,axiom,
    ! [Q: $o,X5: abstra4572470701423150929m_rule,Y4: abstra4572470701423150929m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ ( if_Abs4566652138351786251m_rule @ Q @ X5 @ Y4 ) @ B3 )
      = ( ( Q
         => ( fmembe1635983029874454474m_rule @ X5 @ B3 ) )
        & ( ~ Q
         => ( fmembe1635983029874454474m_rule @ Y4 @ B3 ) ) ) ) ).

% if_split_fmem1
thf(fact_619_eqfset__imp__iff,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,X5: produc1996495991257130529ist_fm] :
      ( ( A2 = B )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
        = ( fmembe3381613331217039976ist_fm @ X5 @ B ) ) ) ).

% eqfset_imp_iff
thf(fact_620_eqfset__imp__iff,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,X5: abstra4572470701423150929m_rule] :
      ( ( A2 = B )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
        = ( fmembe1635983029874454474m_rule @ X5 @ B ) ) ) ).

% eqfset_imp_iff
thf(fact_621_eq__fmem__trans,axiom,
    ! [A: produc1996495991257130529ist_fm,B3: produc1996495991257130529ist_fm,A2: fset_P661503646757059847ist_fm] :
      ( ( A = B3 )
     => ( ( fmembe3381613331217039976ist_fm @ B3 @ A2 )
       => ( fmembe3381613331217039976ist_fm @ A @ A2 ) ) ) ).

% eq_fmem_trans
thf(fact_622_eq__fmem__trans,axiom,
    ! [A: abstra4572470701423150929m_rule,B3: abstra4572470701423150929m_rule,A2: fset_A5396818216090053761m_rule] :
      ( ( A = B3 )
     => ( ( fmembe1635983029874454474m_rule @ B3 @ A2 )
       => ( fmembe1635983029874454474m_rule @ A @ A2 ) ) ) ).

% eq_fmem_trans
thf(fact_623_fequalityCE,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( A2 = B )
     => ( ( ( fmembe3381613331217039976ist_fm @ C @ A2 )
         => ~ ( fmembe3381613331217039976ist_fm @ C @ B ) )
       => ~ ( ~ ( fmembe3381613331217039976ist_fm @ C @ A2 )
           => ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ) ).

% fequalityCE
thf(fact_624_fequalityCE,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( A2 = B )
     => ( ( ( fmembe1635983029874454474m_rule @ C @ A2 )
         => ~ ( fmembe1635983029874454474m_rule @ C @ B ) )
       => ~ ( ~ ( fmembe1635983029874454474m_rule @ C @ A2 )
           => ( fmembe1635983029874454474m_rule @ C @ B ) ) ) ) ).

% fequalityCE
thf(fact_625_fset__eqI,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ! [X3: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X3 @ A2 )
          = ( fmembe3381613331217039976ist_fm @ X3 @ B ) )
     => ( A2 = B ) ) ).

% fset_eqI
thf(fact_626_fset__eqI,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule] :
      ( ! [X3: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X3 @ A2 )
          = ( fmembe1635983029874454474m_rule @ X3 @ B ) )
     => ( A2 = B ) ) ).

% fset_eqI
thf(fact_627_fsubsetD,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A2 @ B )
     => ( ( fmembe3381613331217039976ist_fm @ C @ A2 )
       => ( fmembe3381613331217039976ist_fm @ C @ B ) ) ) ).

% fsubsetD
thf(fact_628_fsubsetD,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A2 @ B )
     => ( ( fmembe1635983029874454474m_rule @ C @ A2 )
       => ( fmembe1635983029874454474m_rule @ C @ B ) ) ) ).

% fsubsetD
thf(fact_629_fin__mono,axiom,
    ! [A2: fset_P661503646757059847ist_fm,B: fset_P661503646757059847ist_fm,X5: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A2 @ B )
     => ( ( fmembe3381613331217039976ist_fm @ X5 @ A2 )
       => ( fmembe3381613331217039976ist_fm @ X5 @ B ) ) ) ).

% fin_mono
thf(fact_630_fin__mono,axiom,
    ! [A2: fset_A5396818216090053761m_rule,B: fset_A5396818216090053761m_rule,X5: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A2 @ B )
     => ( ( fmembe1635983029874454474m_rule @ X5 @ A2 )
       => ( fmembe1635983029874454474m_rule @ X5 @ B ) ) ) ).

% fin_mono
thf(fact_631_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_632_tfinite_Osimps,axiom,
    ( abstra3507476289341504315m_rule
    = ( ^ [A6: abstra4572470701423150929m_rule] :
        ? [T2: abstra4572470701423150929m_rule] :
          ( ( A6 = T2 )
          & ! [X: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra3507476289341504315m_rule @ X ) ) ) ) ) ).

% tfinite.simps
thf(fact_633_tfinite,axiom,
    ! [T: abstra4572470701423150929m_rule] :
      ( ! [T4: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T4 @ ( abstra4962253831105676628m_rule @ T ) )
         => ( abstra3507476289341504315m_rule @ T4 ) )
     => ( abstra3507476289341504315m_rule @ T ) ) ).

% tfinite
thf(fact_634_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,S4: set_list_fm,S3: list_fm] :
      ( ( abstra1616580364776158018ist_fm @ Eff @ Rules @ S4 )
     => ( ( member_list_fm @ S3 @ S4 )
       => ( ? [T8: abstra5042247676209264507ist_fm] :
              ( ( ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T8 ) )
                = S3 )
              & ( abstra8964572462466433336ist_fm @ Eff @ Rules @ T8 )
              & ( abstra5066959810000083067ist_fm @ T8 ) )
          | ? [Steps4: stream8299795917829157543ist_fm] :
              ( ( ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps4 ) )
                = S3 )
              & ( abstra6579460235205289861ist_fm @ Eff @ Rules @ Steps4 )
              & ( abstra8946172763262761776ist_fm @ Eff @ Rules @ Steps4 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_635_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_636_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_637_i_OmkTree_Ocode,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S3: rule] :
      ( ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 )
      = ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S3 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) ) @ ( abstra3109574944306299113m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) @ S3 ) ) ) ) ).

% i.mkTree.code
thf(fact_638_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_639_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_640_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_641_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_642_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_643_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra2830003028686515011m_rule
    = ( ^ [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs2: stream8299795917829157543ist_fm,S: rule] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs2 @ S ) ) ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs2 @ S ) ) ) @ ( abstra3109574944306299113m_rule @ Eff2 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_644_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_645_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_646_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_647_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_648_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_649_epath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X5: stream727092118206550309m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream727092118206550309m_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream727092118206550309m_rule,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ 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_650_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_651_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_652_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_653_epath__eff,axiom,
    ! [Steps: stream727092118206550309m_rule,Ss3: fset_P661503646757059847ist_fm] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
          = ( some_f2162114139285431020ist_fm @ Ss3 ) )
       => ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Ss3 ) ) ) ).

% epath_eff
thf(fact_654_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_655_prod_Ocollapse,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_656_prod_Ocollapse,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_657_prod_Ocollapse,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_658_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_659_prod_Ocollapse,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_660_prod_Ocollapse,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_661_tree_Ocollapse,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) )
      = Tree ) ).

% tree.collapse
thf(fact_662_tree_Oexhaust,axiom,
    ! [Y4: abstra4572470701423150929m_rule] :
      ~ ! [X12: produc164195504107695125m_rule,X22: fset_A5396818216090053761m_rule] :
          ( Y4
         != ( abstra7467780926338743076m_rule @ X12 @ X22 ) ) ).

% tree.exhaust
thf(fact_663_snd__eqD,axiom,
    ! [X5: rule,Y4: produc1996495991257130529ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_664_snd__eqD,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_665_snd__eqD,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_666_snd__eqD,axiom,
    ! [X5: nat,Y4: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_667_snd__eqD,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y4: rule,A: rule] :
      ( ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X5 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_668_snd__eqD,axiom,
    ! [X5: list_fm,Y4: list_fm,A: list_fm] :
      ( ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_669_snd__conv,axiom,
    ! [X1: rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_670_snd__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
      ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_671_snd__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_672_snd__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_673_snd__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_674_snd__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_675_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc164195504107695125m_rule,Z2: produc164195504107695125m_rule] : ( Y5 = Z2 ) )
    = ( ^ [S: produc164195504107695125m_rule,T2: produc164195504107695125m_rule] :
          ( ( ( produc6879501374131015971m_rule @ S )
            = ( produc6879501374131015971m_rule @ T2 ) )
          & ( ( produc7165828336582415457m_rule @ S )
            = ( produc7165828336582415457m_rule @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_676_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc1996495991257130529ist_fm,Z2: produc1996495991257130529ist_fm] : ( Y5 = Z2 ) )
    = ( ^ [S: produc1996495991257130529ist_fm,T2: produc1996495991257130529ist_fm] :
          ( ( ( produc1501393135466168645ist_fm @ S )
            = ( produc1501393135466168645ist_fm @ T2 ) )
          & ( ( produc4588648349897876871ist_fm @ S )
            = ( produc4588648349897876871ist_fm @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_677_prod__eqI,axiom,
    ! [P2: produc164195504107695125m_rule,Q2: produc164195504107695125m_rule] :
      ( ( ( produc6879501374131015971m_rule @ P2 )
        = ( produc6879501374131015971m_rule @ Q2 ) )
     => ( ( ( produc7165828336582415457m_rule @ P2 )
          = ( produc7165828336582415457m_rule @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_678_prod__eqI,axiom,
    ! [P2: produc1996495991257130529ist_fm,Q2: produc1996495991257130529ist_fm] :
      ( ( ( produc1501393135466168645ist_fm @ P2 )
        = ( produc1501393135466168645ist_fm @ Q2 ) )
     => ( ( ( produc4588648349897876871ist_fm @ P2 )
          = ( produc4588648349897876871ist_fm @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_679_prod_Oexpand,axiom,
    ! [Prod: produc164195504107695125m_rule,Prod2: produc164195504107695125m_rule] :
      ( ( ( ( produc6879501374131015971m_rule @ Prod )
          = ( produc6879501374131015971m_rule @ Prod2 ) )
        & ( ( produc7165828336582415457m_rule @ Prod )
          = ( produc7165828336582415457m_rule @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_680_prod_Oexpand,axiom,
    ! [Prod: produc1996495991257130529ist_fm,Prod2: produc1996495991257130529ist_fm] :
      ( ( ( ( produc1501393135466168645ist_fm @ Prod )
          = ( produc1501393135466168645ist_fm @ Prod2 ) )
        & ( ( produc4588648349897876871ist_fm @ Prod )
          = ( produc4588648349897876871ist_fm @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_681_tree_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra3503897793822247108m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X2 ) )
      = X1 ) ).

% tree.sel(1)
thf(fact_682_tree_Osel_I2_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra4962253831105676628m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X2 ) )
      = X2 ) ).

% tree.sel(2)
thf(fact_683_surjective__pairing,axiom,
    ! [T: produc4630866025709511057ist_fm] :
      ( T
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ T ) @ ( produc4514744107061996309ist_fm @ T ) ) ) ).

% surjective_pairing
thf(fact_684_surjective__pairing,axiom,
    ! [T: produc1319942482725812455at_nat] :
      ( T
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ T ) @ ( produc826222791318859909at_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_685_surjective__pairing,axiom,
    ! [T: produc3843707927480180839at_nat] :
      ( T
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ T ) @ ( produc2937545259423399429at_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_686_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_687_surjective__pairing,axiom,
    ! [T: produc164195504107695125m_rule] :
      ( T
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ T ) @ ( produc7165828336582415457m_rule @ T ) ) ) ).

% surjective_pairing
thf(fact_688_surjective__pairing,axiom,
    ! [T: produc1996495991257130529ist_fm] :
      ( T
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ T ) @ ( produc4588648349897876871ist_fm @ T ) ) ) ).

% surjective_pairing
thf(fact_689_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( Prod
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_690_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( Prod
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_691_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( Prod
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_692_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_693_prod_Oexhaust__sel,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( Prod
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_694_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( Prod
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_695_tree_Oexhaust__sel,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( Tree
      = ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) ) ) ).

% tree.exhaust_sel
thf(fact_696_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream8299795917829157543ist_fm > $o,X5: stream8299795917829157543ist_fm,Rules: stream_list_fm,Eff: list_fm > list_fm > fset_list_fm > $o] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream8299795917829157543ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream8299795917829157543ist_fm,Sl5: fset_list_fm] :
                ( ( X3 = Steps3 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules ) )
                & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) )
                  | ( abstra6579460235205289861ist_fm @ Eff @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) ) )
       => ( abstra6579460235205289861ist_fm @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_697_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream5128679079952569136_fm_fm > $o,X5: stream5128679079952569136_fm_fm,Rules: stream_fm,Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream5128679079952569136_fm_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream5128679079952569136_fm_fm,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) )
                  | ( abstra9155489748554225612ist_fm @ Eff @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_698_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 )
     => ( ! [X3: stream727092118206550309m_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream727092118206550309m_rule,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ 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_699_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 )
     => ( ! [X3: stream5916762827556431437at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream5916762827556431437at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = 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_700_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X7: stream812778442145099016ule_fm > $o,X5: stream812778442145099016ule_fm,Rules: stream_fm,Eff: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream812778442145099016ule_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream812778442145099016ule_fm,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl5 )
                & ( Eff @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) )
                  | ( abstra1364558218721404774m_rule @ Eff @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_701_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 )
     => ( ! [X3: stream7755889244956737277e_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7755889244956737277e_rule,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X3 = 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_702_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 )
     => ( ! [X3: stream3409308193418444653ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream3409308193418444653ist_fm,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = 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_703_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 )
     => ( ! [X3: stream6060259936882193013at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream6060259936882193013at_nat,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X3 = 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_704_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 )
     => ( ! [X3: stream7989562319868115861ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7989562319868115861ist_fm,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X3 = 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_705_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 )
     => ( ! [X3: stream7788502890425800371at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7788502890425800371at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = 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_706_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream8299795917829157543ist_fm,Rules: stream_list_fm,Sl3: fset_list_fm,Eff: list_fm > list_fm > fset_list_fm > $o] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra6579460235205289861ist_fm @ Eff @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps ) )
           => ( abstra6579460235205289861ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_707_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream5128679079952569136_fm_fm,Rules: stream_fm,Sl3: fset_P661503646757059847ist_fm,Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra9155489748554225612ist_fm @ Eff @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) )
           => ( abstra9155489748554225612ist_fm @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_708_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_709_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_710_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream812778442145099016ule_fm,Rules: stream_fm,Sl3: fset_A5396818216090053761m_rule,Eff: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra1364558218721404774m_rule @ Eff @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps ) )
           => ( abstra1364558218721404774m_rule @ Eff @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_711_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_712_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_713_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_714_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_715_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_716_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6579460235205289861ist_fm
    = ( ^ [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,A6: stream8299795917829157543ist_fm] :
        ? [Steps2: stream8299795917829157543ist_fm,Sl: fset_list_fm] :
          ( ( A6 = Steps2 )
          & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules2 ) )
          & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ Sl )
          & ( abstra6579460235205289861ist_fm @ Eff2 @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_717_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra9155489748554225612ist_fm
    = ( ^ [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,A6: stream5128679079952569136_fm_fm] :
        ? [Steps2: stream5128679079952569136_fm_fm,Sl: fset_P661503646757059847ist_fm] :
          ( ( A6 = Steps2 )
          & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( sset_fm @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ Sl )
          & ( abstra9155489748554225612ist_fm @ Eff2 @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_718_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5415202973855822231ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A6: stream727092118206550309m_rule] :
        ? [Steps2: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
          ( ( A6 = 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_719_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra4041229200784676899ist_fm
    = ( ^ [Eff2: product_prod_nat_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream6724221391990029191at_nat,A6: stream5916762827556431437at_nat] :
        ? [Steps2: stream5916762827556431437at_nat,Sl: fset_P661503646757059847ist_fm] :
          ( ( A6 = 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_720_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1364558218721404774m_rule
    = ( ^ [Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_fm,A6: stream812778442145099016ule_fm] :
        ? [Steps2: stream812778442145099016ule_fm,Sl: fset_A5396818216090053761m_rule] :
          ( ( A6 = Steps2 )
          & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ Sl )
          & ( abstra1364558218721404774m_rule @ Eff2 @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_721_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5823804033175701339m_rule
    = ( ^ [Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_rule,A6: stream7755889244956737277e_rule] :
        ? [Steps2: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
          ( ( A6 = 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_722_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5061764961602482947ist_fm
    = ( ^ [Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8299795917829157543ist_fm,A6: stream3409308193418444653ist_fm] :
        ? [Steps2: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
          ( ( A6 = 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_723_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra9014086451575537615m_rule
    = ( ^ [Eff2: product_prod_nat_nat > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream6724221391990029191at_nat,A6: stream6060259936882193013at_nat] :
        ? [Steps2: stream6060259936882193013at_nat,Sl: fset_A5396818216090053761m_rule] :
          ( ( A6 = 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_724_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra8066765647771630831m_rule
    = ( ^ [Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8299795917829157543ist_fm,A6: stream7989562319868115861ist_fm] :
        ? [Steps2: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
          ( ( A6 = 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_725_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6262887151753371453ist_fm
    = ( ^ [Eff2: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8032061917732916077at_nat,A6: stream7788502890425800371at_nat] :
        ? [Steps2: stream7788502890425800371at_nat,Sl: fset_P661503646757059847ist_fm] :
          ( ( A6 = 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_726_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ Eff @ Rules @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( sset_list_fm @ Rules ) )
         => ! [Sl2: fset_list_fm] :
              ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra6579460235205289861ist_fm @ Eff @ Rules @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_727_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ Eff @ Rules @ A )
     => ~ ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ Sl2 )
               => ~ ( abstra9155489748554225612ist_fm @ Eff @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_728_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_729_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_730_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ Eff @ Rules @ A )
     => ~ ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) @ Sl2 )
             => ( ( Eff @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ Sl2 )
               => ~ ( abstra1364558218721404774m_rule @ Eff @ Rules @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_731_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_732_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_733_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_734_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_735_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_736_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream8299795917829157543ist_fm > $o,X5: stream8299795917829157543ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream8299795917829157543ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream8299795917829157543ist_fm,Sl5: fset_list_fm] :
                ( ( X3 = Steps3 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( sset_list_fm @ Rules ) )
                & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) @ Sl5 )
                & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) )
                  | ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) ) )
       => ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_737_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream5128679079952569136_fm_fm > $o,X5: stream5128679079952569136_fm_fm,Rules: stream_fm,Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream5128679079952569136_fm_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream5128679079952569136_fm_fm,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) @ Sl5 )
                & ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) )
                  | ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_738_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream727092118206550309m_rule > $o,X5: stream727092118206550309m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream727092118206550309m_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream727092118206550309m_rule,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ 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_739_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 )
     => ( ! [X3: stream5916762827556431437at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream5916762827556431437at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = 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_740_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream812778442145099016ule_fm > $o,X5: stream812778442145099016ule_fm,Rules: stream_fm,Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream812778442145099016ule_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream812778442145099016ule_fm,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X3 = Steps3 )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl5 )
                & ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl5 )
                & ( ( X7 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) )
                  | ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.epath.coinduct
thf(fact_741_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7755889244956737277e_rule > $o,X5: stream7755889244956737277e_rule,Rules: stream_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream7755889244956737277e_rule] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7755889244956737277e_rule,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X3 = 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_742_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream3409308193418444653ist_fm > $o,X5: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream3409308193418444653ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream3409308193418444653ist_fm,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = 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_743_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 )
     => ( ! [X3: stream6060259936882193013at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream6060259936882193013at_nat,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X3 = 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_744_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7989562319868115861ist_fm > $o,X5: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream7989562319868115861ist_fm] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7989562319868115861ist_fm,Sl5: fset_A5396818216090053761m_rule] :
                ( ( X3 = 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_745_i_Oepath_Ocoinduct,axiom,
    ! [X7: stream7788502890425800371at_nat > $o,X5: stream7788502890425800371at_nat,Rules: stream8032061917732916077at_nat,Eff3: produc3843707927480180839at_nat > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X3: stream7788502890425800371at_nat] :
            ( ( X7 @ X3 )
           => ? [Steps3: stream7788502890425800371at_nat,Sl5: fset_P661503646757059847ist_fm] :
                ( ( X3 = 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_746_i_Oepath_Ointros,axiom,
    ! [Steps: stream8299795917829157543ist_fm,Rules: stream_list_fm,Sl3: fset_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps ) )
           => ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_747_i_Oepath_Ointros,axiom,
    ! [Steps: stream5128679079952569136_fm_fm,Rules: stream_fm,Sl3: fset_P661503646757059847ist_fm,Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) )
           => ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_748_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_749_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_750_i_Oepath_Ointros,axiom,
    ! [Steps: stream812778442145099016ule_fm,Rules: stream_fm,Sl3: fset_A5396818216090053761m_rule,Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps ) ) ) @ Sl3 )
       => ( ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps ) )
           => ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_751_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_752_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_753_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_754_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_755_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_756_i_Oepath_Osimps,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream8299795917829157543ist_fm,Sl: fset_list_fm] :
            ( ( A = Steps2 )
            & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( sset_list_fm @ Rules ) )
            & ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ Sl )
            & ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_757_i_Oepath_Osimps,axiom,
    ! [Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream5128679079952569136_fm_fm,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ Sl )
            & ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_758_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_759_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_760_i_Oepath_Osimps,axiom,
    ! [Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream812778442145099016ule_fm,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) @ Sl )
            & ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ Sl )
            & ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_761_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_762_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_763_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_764_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_765_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_766_i_Oepath_Ocases,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( sset_list_fm @ Rules ) )
         => ! [Sl2: fset_list_fm] :
              ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) @ Sl2 )
             => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_767_i_Oepath_Ocases,axiom,
    ! [Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) @ Sl2 )
             => ( ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ Sl2 )
               => ~ ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_768_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_769_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_770_i_Oepath_Ocases,axiom,
    ! [Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( sset_fm @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) @ Sl2 )
             => ( ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ Sl2 )
               => ~ ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_771_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_772_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_773_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_774_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_775_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_776_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,S8: 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 ) )
                @ S8 )
              @ ( 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_777_i_OmkTree__unfold,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S3: rule] :
      ( ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 )
      = ( case_s482378023405642050ist_fm
        @ ^ [R: produc1996495991257130529ist_fm,S8: stream8299795917829157543ist_fm] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S3 @ R ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ S8 ) @ ( abstra3109574944306299113m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_778_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,S8: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S3 @ R ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ S8 ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_779_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,S8: stream3627166509194609229at_nat] : ( abstra6446703985809605400at_nat @ ( produc9060074326276436823at_nat @ S3 @ R ) @ ( fimage7170464556407549111at_nat @ ( abstra8108511269830108963at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ S8 ) @ ( abstra5031851239426275837at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra5441914787988489033at_nat @ ( abstra7959051331525504128at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_780_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,S8: stream7255200869668761831at_nat] : ( abstra4133525815659287960at_nat @ ( produc2922128104949294807at_nat @ S3 @ R ) @ ( fimage9092832869046556317at_nat @ ( abstra7709754627474349731at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ S8 ) @ ( abstra6518908551485222781at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra6541677642950016201at_nat @ ( abstra2633104702069076736at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_781_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,S8: stream_nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S3 @ R ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ S8 ) @ ( abstra969719415724737503at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra7573781713853518635at_nat @ ( abstra4671362528648695778at_nat @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_782_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,S8: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S3 @ R ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ S8 ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S3 ) ) )
        @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S3 ) ) ) ).

% i.mkTree_unfold
thf(fact_783_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra2830003028686515011m_rule
    = ( ^ [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs2: stream8299795917829157543ist_fm,S: rule] :
          ( case_s482378023405642050ist_fm
          @ ^ [R: produc1996495991257130529ist_fm,S8: stream8299795917829157543ist_fm] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S @ R ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ S8 ) @ ( abstra3109574944306299113m_rule @ Eff2 @ R @ S ) ) )
          @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_784_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,S8: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S @ R ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ S8 ) @ ( abstra1097029126060538623ist_fm @ Eff2 @ R @ S ) ) )
          @ ( abstra2983386638146376267ist_fm @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_785_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,S8: stream3627166509194609229at_nat] : ( abstra6446703985809605400at_nat @ ( produc9060074326276436823at_nat @ S @ R ) @ ( fimage7170464556407549111at_nat @ ( abstra8108511269830108963at_nat @ Eff2 @ S8 ) @ ( abstra5031851239426275837at_nat @ Eff2 @ R @ S ) ) )
          @ ( abstra5441914787988489033at_nat @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_786_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,S8: stream7255200869668761831at_nat] : ( abstra4133525815659287960at_nat @ ( produc2922128104949294807at_nat @ S @ R ) @ ( fimage9092832869046556317at_nat @ ( abstra7709754627474349731at_nat @ Eff2 @ S8 ) @ ( abstra6518908551485222781at_nat @ Eff2 @ R @ S ) ) )
          @ ( abstra6541677642950016201at_nat @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_787_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra1192804020863382533at_nat
    = ( ^ [Eff2: nat > nat > fset_nat > $o,Rs2: stream_nat,S: nat] :
          ( case_s2280698752336340829at_nat
          @ ^ [R: nat,S8: stream_nat] : ( abstra4999521449322675122at_nat @ ( product_Pair_nat_nat @ S @ R ) @ ( fimage8970742173175350476at_nat @ ( abstra1192804020863382533at_nat @ Eff2 @ S8 ) @ ( abstra969719415724737503at_nat @ Eff2 @ R @ S ) ) )
          @ ( abstra7573781713853518635at_nat @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_788_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,S8: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ R ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ S8 ) @ ( abstra458490714785879965ist_fm @ Eff2 @ R @ S ) ) )
          @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_789_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_790_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 ) ) ) )
            & ! [X: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X @ ( abstra4962253831105676628m_rule @ T2 ) )
               => ( abstra4476735159789068068ist_fm
                  @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ rules
                  @ X ) ) ) ) ) ).

% wf.simps
thf(fact_791_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_792_wf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X5: abstra4572470701423150929m_rule] :
      ( ( X7 @ X5 )
     => ( ! [X3: abstra4572470701423150929m_rule] :
            ( ( X7 @ X3 )
           => ? [T6: abstra4572470701423150929m_rule] :
                ( ( X3 = 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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_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_803_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_804_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_805_K__record__comp,axiom,
    ! [C: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P4378417847848123130m_rule
        @ ^ [X: produc164195504107695125m_rule] : C
        @ F )
      = ( ^ [X: abstra4572470701423150929m_rule] : C ) ) ).

% K_record_comp
thf(fact_806_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_807_shd__def,axiom,
    ( shd_rule
    = ( case_s3980788006751379256e_rule
      @ ^ [X13: rule,X23: stream_rule] : X13 ) ) ).

% shd_def
thf(fact_808_shd__def,axiom,
    ( shd_Pr7235097944458474089m_rule
    = ( case_s4599053733867249880m_rule
      @ ^ [X13: produc164195504107695125m_rule,X23: stream727092118206550309m_rule] : X13 ) ) ).

% shd_def
thf(fact_809_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_810_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ Eff @ Rules @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( sset_list_fm @ Rules ) )
         => ( ( Eff @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ A ) ) )
           => ~ ! [T7: abstra5042247676209264507ist_fm] :
                  ( ( fmembe913231786053615938ist_fm @ T7 @ ( abstra1905332785217431714ist_fm @ A ) )
                 => ( abstra8964572462466433336ist_fm @ Eff @ Rules @ T7 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_811_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_812_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra8964572462466433336ist_fm
    = ( ^ [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,A6: abstra5042247676209264507ist_fm] :
        ? [T2: abstra5042247676209264507ist_fm] :
          ( ( A6 = T2 )
          & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( sset_list_fm @ Rules2 ) )
          & ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T2 ) ) )
          & ! [X: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ X @ ( abstra1905332785217431714ist_fm @ T2 ) )
             => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules2 @ X ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_813_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra4476735159789068068ist_fm
    = ( ^ [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A6: abstra4572470701423150929m_rule] :
        ? [T2: abstra4572470701423150929m_rule] :
          ( ( A6 = 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 ) ) )
          & ! [X: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules2 @ X ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_814_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff: list_fm > list_fm > fset_list_fm > $o] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( Eff @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T ) ) )
       => ( ! [T4: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ T4 @ ( abstra1905332785217431714ist_fm @ T ) )
             => ( abstra8964572462466433336ist_fm @ Eff @ Rules @ T4 ) )
         => ( abstra8964572462466433336ist_fm @ Eff @ Rules @ T ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_815_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_816_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X7: abstra5042247676209264507ist_fm > $o,X5: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff: list_fm > list_fm > fset_list_fm > $o] :
      ( ( X7 @ X5 )
     => ( ! [X3: abstra5042247676209264507ist_fm] :
            ( ( X7 @ X3 )
           => ? [T6: abstra5042247676209264507ist_fm] :
                ( ( X3 = T6 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( sset_list_fm @ Rules ) )
                & ( Eff @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T6 ) ) )
                & ! [Xa: abstra5042247676209264507ist_fm] :
                    ( ( fmembe913231786053615938ist_fm @ Xa @ ( abstra1905332785217431714ist_fm @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra8964572462466433336ist_fm @ Eff @ Rules @ Xa ) ) ) ) )
       => ( abstra8964572462466433336ist_fm @ Eff @ Rules @ X5 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_817_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 )
     => ( ! [X3: abstra4572470701423150929m_rule] :
            ( ( X7 @ X3 )
           => ? [T6: abstra4572470701423150929m_rule] :
                ( ( X3 = 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_818_i_Owf_Ocases,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( sset_list_fm @ Rules ) )
         => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ A ) ) )
           => ~ ! [T7: abstra5042247676209264507ist_fm] :
                  ( ( fmembe913231786053615938ist_fm @ T7 @ ( abstra1905332785217431714ist_fm @ A ) )
                 => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ T7 ) ) ) ) ) ).

% i.wf.cases
thf(fact_819_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_820_i_Owf_Osimps,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rules: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [T2: abstra5042247676209264507ist_fm] :
            ( ( A = T2 )
            & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( sset_list_fm @ Rules ) )
            & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T2 ) ) )
            & ! [X: abstra5042247676209264507ist_fm] :
                ( ( fmembe913231786053615938ist_fm @ X @ ( abstra1905332785217431714ist_fm @ T2 ) )
               => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ X ) ) ) ) ) ).

% i.wf.simps
thf(fact_821_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 ) ) )
            & ! [X: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X @ ( abstra4962253831105676628m_rule @ T2 ) )
               => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X ) ) ) ) ) ).

% i.wf.simps
thf(fact_822_i_Owf_Ointros,axiom,
    ! [T: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T ) ) )
       => ( ! [T4: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ T4 @ ( abstra1905332785217431714ist_fm @ T ) )
             => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ T4 ) )
         => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ T ) ) ) ) ).

% i.wf.intros
thf(fact_823_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_824_i_Owf_Ocoinduct,axiom,
    ! [X7: abstra5042247676209264507ist_fm > $o,X5: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( X7 @ X5 )
     => ( ! [X3: abstra5042247676209264507ist_fm] :
            ( ( X7 @ X3 )
           => ? [T6: abstra5042247676209264507ist_fm] :
                ( ( X3 = T6 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( sset_list_fm @ Rules ) )
                & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T6 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T6 ) ) )
                & ! [Xa: abstra5042247676209264507ist_fm] :
                    ( ( fmembe913231786053615938ist_fm @ Xa @ ( abstra1905332785217431714ist_fm @ T6 ) )
                   => ( ( X7 @ Xa )
                      | ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ Xa ) ) ) ) )
       => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ X5 ) ) ) ).

% i.wf.coinduct
thf(fact_825_i_Owf_Ocoinduct,axiom,
    ! [X7: abstra4572470701423150929m_rule > $o,X5: abstra4572470701423150929m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X7 @ X5 )
     => ( ! [X3: abstra4572470701423150929m_rule] :
            ( ( X7 @ X3 )
           => ? [T6: abstra4572470701423150929m_rule] :
                ( ( X3 = 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_826_exI__realizer,axiom,
    ! [P: produc1996495991257130529ist_fm > rule > $o,Y4: produc1996495991257130529ist_fm,X5: rule] :
      ( ( P @ Y4 @ X5 )
     => ( P @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y4 ) ) @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_827_exI__realizer,axiom,
    ! [P: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,Y4: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat] :
      ( ( P @ Y4 @ X5 )
     => ( P @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) ) @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_828_exI__realizer,axiom,
    ! [P: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Y4: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( P @ Y4 @ X5 )
     => ( P @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) ) @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_829_exI__realizer,axiom,
    ! [P: nat > nat > $o,Y4: nat,X5: nat] :
      ( ( P @ Y4 @ X5 )
     => ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_830_exI__realizer,axiom,
    ! [P: rule > produc1996495991257130529ist_fm > $o,Y4: rule,X5: produc1996495991257130529ist_fm] :
      ( ( P @ Y4 @ X5 )
     => ( P @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X5 @ Y4 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X5 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_831_exI__realizer,axiom,
    ! [P: list_fm > list_fm > $o,Y4: list_fm,X5: list_fm] :
      ( ( P @ Y4 @ X5 )
     => ( P @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y4 ) ) @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_832_conjI__realizer,axiom,
    ! [P: rule > $o,P2: rule,Q: produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ P2 @ Q2 ) ) )
          & ( Q @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_833_conjI__realizer,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,P2: set_Pr4329608150637261639at_nat,Q: set_Pr4329608150637261639at_nat > $o,Q2: set_Pr4329608150637261639at_nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_834_conjI__realizer,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,P2: set_Pr1261947904930325089at_nat,Q: set_Pr1261947904930325089at_nat > $o,Q2: set_Pr1261947904930325089at_nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_835_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q: nat > $o,Q2: nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_836_conjI__realizer,axiom,
    ! [P: produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm,Q: rule > $o,Q2: rule] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_837_conjI__realizer,axiom,
    ! [P: list_fm > $o,P2: list_fm,Q: list_fm > $o,Q2: list_fm] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ P2 @ Q2 ) ) )
          & ( Q @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_838_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: rule > produc1996495991257130529ist_fm > $o,X5: rule,Y4: produc1996495991257130529ist_fm,A: produc4630866025709511057ist_fm] :
      ( ( P @ X5 @ Y4 )
     => ( ( A
          = ( produc7063755442767259331ist_fm @ X5 @ Y4 ) )
       => ( P @ ( produc4228417144610596823ist_fm @ A ) @ ( produc4514744107061996309ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_839_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat,A: produc1319942482725812455at_nat] :
      ( ( P @ X5 @ Y4 )
     => ( ( A
          = ( produc9060074326276436823at_nat @ X5 @ Y4 ) )
       => ( P @ ( produc4486455499028916547at_nat @ A ) @ ( produc826222791318859909at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_840_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat] :
      ( ( P @ X5 @ Y4 )
     => ( ( A
          = ( produc2922128104949294807at_nat @ X5 @ Y4 ) )
       => ( P @ ( produc6011863565407846595at_nat @ A ) @ ( produc2937545259423399429at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_841_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: nat > nat > $o,X5: nat,Y4: nat,A: product_prod_nat_nat] :
      ( ( P @ X5 @ Y4 )
     => ( ( A
          = ( product_Pair_nat_nat @ X5 @ Y4 ) )
       => ( P @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_842_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: produc1996495991257130529ist_fm > rule > $o,X5: produc1996495991257130529ist_fm,Y4: rule,A: produc164195504107695125m_rule] :
      ( ( P @ X5 @ Y4 )
     => ( ( A
          = ( produc491467635432902671m_rule @ X5 @ Y4 ) )
       => ( P @ ( produc6879501374131015971m_rule @ A ) @ ( produc7165828336582415457m_rule @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_843_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: list_fm > list_fm > $o,X5: list_fm,Y4: list_fm,A: produc1996495991257130529ist_fm] :
      ( ( P @ X5 @ Y4 )
     => ( ( A
          = ( produc381145313068854617ist_fm @ X5 @ Y4 ) )
       => ( P @ ( produc1501393135466168645ist_fm @ A ) @ ( produc4588648349897876871ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_844_eq__snd__iff,axiom,
    ! [B3: produc1996495991257130529ist_fm,P2: produc4630866025709511057ist_fm] :
      ( ( B3
        = ( produc4514744107061996309ist_fm @ P2 ) )
      = ( ? [A6: rule] :
            ( P2
            = ( produc7063755442767259331ist_fm @ A6 @ B3 ) ) ) ) ).

% eq_snd_iff
thf(fact_845_eq__snd__iff,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,P2: produc1319942482725812455at_nat] :
      ( ( B3
        = ( produc826222791318859909at_nat @ P2 ) )
      = ( ? [A6: set_Pr4329608150637261639at_nat] :
            ( P2
            = ( produc9060074326276436823at_nat @ A6 @ B3 ) ) ) ) ).

% eq_snd_iff
thf(fact_846_eq__snd__iff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,P2: produc3843707927480180839at_nat] :
      ( ( B3
        = ( produc2937545259423399429at_nat @ P2 ) )
      = ( ? [A6: set_Pr1261947904930325089at_nat] :
            ( P2
            = ( produc2922128104949294807at_nat @ A6 @ B3 ) ) ) ) ).

% eq_snd_iff
thf(fact_847_eq__snd__iff,axiom,
    ! [B3: nat,P2: product_prod_nat_nat] :
      ( ( B3
        = ( product_snd_nat_nat @ P2 ) )
      = ( ? [A6: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A6 @ B3 ) ) ) ) ).

% eq_snd_iff
thf(fact_848_eq__snd__iff,axiom,
    ! [B3: rule,P2: produc164195504107695125m_rule] :
      ( ( B3
        = ( produc7165828336582415457m_rule @ P2 ) )
      = ( ? [A6: produc1996495991257130529ist_fm] :
            ( P2
            = ( produc491467635432902671m_rule @ A6 @ B3 ) ) ) ) ).

% eq_snd_iff
thf(fact_849_eq__snd__iff,axiom,
    ! [B3: list_fm,P2: produc1996495991257130529ist_fm] :
      ( ( B3
        = ( produc4588648349897876871ist_fm @ P2 ) )
      = ( ? [A6: list_fm] :
            ( P2
            = ( produc381145313068854617ist_fm @ A6 @ B3 ) ) ) ) ).

% eq_snd_iff
thf(fact_850_sndI,axiom,
    ! [X5: produc4630866025709511057ist_fm,Y4: rule,Z: produc1996495991257130529ist_fm] :
      ( ( X5
        = ( produc7063755442767259331ist_fm @ Y4 @ Z ) )
     => ( ( produc4514744107061996309ist_fm @ X5 )
        = Z ) ) ).

% sndI
thf(fact_851_sndI,axiom,
    ! [X5: produc1319942482725812455at_nat,Y4: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( X5
        = ( produc9060074326276436823at_nat @ Y4 @ Z ) )
     => ( ( produc826222791318859909at_nat @ X5 )
        = Z ) ) ).

% sndI
thf(fact_852_sndI,axiom,
    ! [X5: produc3843707927480180839at_nat,Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( X5
        = ( produc2922128104949294807at_nat @ Y4 @ Z ) )
     => ( ( produc2937545259423399429at_nat @ X5 )
        = Z ) ) ).

% sndI
thf(fact_853_sndI,axiom,
    ! [X5: product_prod_nat_nat,Y4: nat,Z: nat] :
      ( ( X5
        = ( product_Pair_nat_nat @ Y4 @ Z ) )
     => ( ( product_snd_nat_nat @ X5 )
        = Z ) ) ).

% sndI
thf(fact_854_sndI,axiom,
    ! [X5: produc164195504107695125m_rule,Y4: produc1996495991257130529ist_fm,Z: rule] :
      ( ( X5
        = ( produc491467635432902671m_rule @ Y4 @ Z ) )
     => ( ( produc7165828336582415457m_rule @ X5 )
        = Z ) ) ).

% sndI
thf(fact_855_sndI,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y4: list_fm,Z: list_fm] :
      ( ( X5
        = ( produc381145313068854617ist_fm @ Y4 @ Z ) )
     => ( ( produc4588648349897876871ist_fm @ X5 )
        = Z ) ) ).

% sndI
thf(fact_856_eq__fst__iff,axiom,
    ! [A: rule,P2: produc4630866025709511057ist_fm] :
      ( ( A
        = ( produc4228417144610596823ist_fm @ P2 ) )
      = ( ? [B6: produc1996495991257130529ist_fm] :
            ( P2
            = ( produc7063755442767259331ist_fm @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_857_eq__fst__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,P2: produc1319942482725812455at_nat] :
      ( ( A
        = ( produc4486455499028916547at_nat @ P2 ) )
      = ( ? [B6: set_Pr4329608150637261639at_nat] :
            ( P2
            = ( produc9060074326276436823at_nat @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_858_eq__fst__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,P2: produc3843707927480180839at_nat] :
      ( ( A
        = ( produc6011863565407846595at_nat @ P2 ) )
      = ( ? [B6: set_Pr1261947904930325089at_nat] :
            ( P2
            = ( produc2922128104949294807at_nat @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_859_eq__fst__iff,axiom,
    ! [A: nat,P2: product_prod_nat_nat] :
      ( ( A
        = ( product_fst_nat_nat @ P2 ) )
      = ( ? [B6: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_860_eq__fst__iff,axiom,
    ! [A: produc1996495991257130529ist_fm,P2: produc164195504107695125m_rule] :
      ( ( A
        = ( produc6879501374131015971m_rule @ P2 ) )
      = ( ? [B6: rule] :
            ( P2
            = ( produc491467635432902671m_rule @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_861_eq__fst__iff,axiom,
    ! [A: list_fm,P2: produc1996495991257130529ist_fm] :
      ( ( A
        = ( produc1501393135466168645ist_fm @ P2 ) )
      = ( ? [B6: list_fm] :
            ( P2
            = ( produc381145313068854617ist_fm @ A @ B6 ) ) ) ) ).

% eq_fst_iff
thf(fact_862_snd__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc6725306161561501521e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X: rule] : ( produc1769597910952761123e_rule @ X @ X )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% snd_diag_snd
thf(fact_863_snd__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X: list_fm] : ( produc381145313068854617ist_fm @ X @ X )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% snd_diag_snd
thf(fact_864_fst__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc5894184038225292483ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X @ X )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% fst_diag_fst
thf(fact_865_fst__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X: list_fm] : ( produc381145313068854617ist_fm @ X @ X )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% fst_diag_fst
thf(fact_866_fst__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc7264528503596239631e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X: rule] : ( produc1769597910952761123e_rule @ X @ X )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% fst_diag_snd
thf(fact_867_fst__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X: list_fm] : ( produc381145313068854617ist_fm @ X @ X )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% fst_diag_snd
thf(fact_868_snd__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc1639249265332130309ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X @ X )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% snd_diag_fst
thf(fact_869_snd__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X: list_fm] : ( produc381145313068854617ist_fm @ X @ X )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% snd_diag_fst
thf(fact_870_stl__def,axiom,
    ( stl_rule
    = ( case_s5295311805034529096m_rule
      @ ^ [X13: rule,X23: stream_rule] : X23 ) ) ).

% stl_def
thf(fact_871_stl__def,axiom,
    ( stl_Pr950425576149878629m_rule
    = ( case_s474603282864825704m_rule
      @ ^ [X13: produc164195504107695125m_rule,X23: stream727092118206550309m_rule] : X23 ) ) ).

% stl_def
thf(fact_872_subset__CollectI,axiom,
    ! [B: set_Pr7058068377845519745ist_fm,A2: set_Pr7058068377845519745ist_fm,Q: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B @ A2 )
     => ( ! [X3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le2055072402069232929ist_fm
          @ ( collec7231575776666803724ist_fm
            @ ^ [X: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X @ B )
                & ( Q @ X ) ) )
          @ ( collec7231575776666803724ist_fm
            @ ^ [X: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_873_subset__CollectI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,Q: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ B @ A2 )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le1268244103169919719at_nat
          @ ( collec6321179662152712658at_nat
            @ ^ [X: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X @ B )
                & ( Q @ X ) ) )
          @ ( collec6321179662152712658at_nat
            @ ^ [X: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_874_subset__CollectI,axiom,
    ! [B: set_fm,A2: set_fm,Q: fm > $o,P: fm > $o] :
      ( ( ord_less_eq_set_fm @ B @ A2 )
     => ( ! [X3: fm] :
            ( ( member_fm @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_fm
          @ ( collect_fm
            @ ^ [X: fm] :
                ( ( member_fm @ X @ B )
                & ( Q @ X ) ) )
          @ ( collect_fm
            @ ^ [X: fm] :
                ( ( member_fm @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_875_subset__CollectI,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B @ A2 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ B )
                & ( Q @ X ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_876_subset__CollectI,axiom,
    ! [B: set_rule,A2: set_rule,Q: rule > $o,P: rule > $o] :
      ( ( ord_less_eq_set_rule @ B @ A2 )
     => ( ! [X3: rule] :
            ( ( member_rule @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_rule
          @ ( collect_rule
            @ ^ [X: rule] :
                ( ( member_rule @ X @ B )
                & ( Q @ X ) ) )
          @ ( collect_rule
            @ ^ [X: rule] :
                ( ( member_rule @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_877_subset__Collect__iff,axiom,
    ! [B: set_Pr7058068377845519745ist_fm,A2: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B @ A2 )
     => ( ( ord_le2055072402069232929ist_fm @ B
          @ ( collec7231575776666803724ist_fm
            @ ^ [X: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( ! [X: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_878_subset__Collect__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ B @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ B
          @ ( collec6321179662152712658at_nat
            @ ^ [X: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( ! [X: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_879_subset__Collect__iff,axiom,
    ! [B: set_fm,A2: set_fm,P: fm > $o] :
      ( ( ord_less_eq_set_fm @ B @ A2 )
     => ( ( ord_less_eq_set_fm @ B
          @ ( collect_fm
            @ ^ [X: fm] :
                ( ( member_fm @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( ! [X: fm] :
              ( ( member_fm @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_880_subset__Collect__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ B
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_881_subset__Collect__iff,axiom,
    ! [B: set_rule,A2: set_rule,P: rule > $o] :
      ( ( ord_less_eq_set_rule @ B @ A2 )
     => ( ( ord_less_eq_set_rule @ B
          @ ( collect_rule
            @ ^ [X: rule] :
                ( ( member_rule @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( ! [X: rule] :
              ( ( member_rule @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_882_fstI,axiom,
    ! [X5: produc4630866025709511057ist_fm,Y4: rule,Z: produc1996495991257130529ist_fm] :
      ( ( X5
        = ( produc7063755442767259331ist_fm @ Y4 @ Z ) )
     => ( ( produc4228417144610596823ist_fm @ X5 )
        = Y4 ) ) ).

% fstI
thf(fact_883_fstI,axiom,
    ! [X5: produc1319942482725812455at_nat,Y4: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( X5
        = ( produc9060074326276436823at_nat @ Y4 @ Z ) )
     => ( ( produc4486455499028916547at_nat @ X5 )
        = Y4 ) ) ).

% fstI
thf(fact_884_fstI,axiom,
    ! [X5: produc3843707927480180839at_nat,Y4: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( X5
        = ( produc2922128104949294807at_nat @ Y4 @ Z ) )
     => ( ( produc6011863565407846595at_nat @ X5 )
        = Y4 ) ) ).

% fstI
thf(fact_885_fstI,axiom,
    ! [X5: product_prod_nat_nat,Y4: nat,Z: nat] :
      ( ( X5
        = ( product_Pair_nat_nat @ Y4 @ Z ) )
     => ( ( product_fst_nat_nat @ X5 )
        = Y4 ) ) ).

% fstI
thf(fact_886_fstI,axiom,
    ! [X5: produc164195504107695125m_rule,Y4: produc1996495991257130529ist_fm,Z: rule] :
      ( ( X5
        = ( produc491467635432902671m_rule @ Y4 @ Z ) )
     => ( ( produc6879501374131015971m_rule @ X5 )
        = Y4 ) ) ).

% fstI
thf(fact_887_fstI,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y4: list_fm,Z: list_fm] :
      ( ( X5
        = ( produc381145313068854617ist_fm @ Y4 @ Z ) )
     => ( ( produc1501393135466168645ist_fm @ X5 )
        = Y4 ) ) ).

% fstI
thf(fact_888_Konig,axiom,
    ! [T: abstra4572470701423150929m_rule] :
      ( ~ ( abstra3507476289341504315m_rule @ T )
     => ( abstra2285467793620430502m_rule @ T @ ( abstra3079604473823497664m_rule @ T ) ) ) ).

% Konig
thf(fact_889_konig_Osimps_I1_J,axiom,
    ! [T: abstract_tree_rule] :
      ( ( shd_rule @ ( abstract_konig_rule @ T ) )
      = ( abstract_root_rule @ T ) ) ).

% konig.simps(1)
thf(fact_890_konig_Osimps_I1_J,axiom,
    ! [T: abstra4572470701423150929m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( abstra3079604473823497664m_rule @ T ) )
      = ( abstra3503897793822247108m_rule @ T ) ) ).

% konig.simps(1)
thf(fact_891_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_892_cont__def,axiom,
    ( abstra4962253831105676628m_rule
    = ( abstra1216198984563783442m_rule
      @ ^ [X13: produc164195504107695125m_rule,X23: fset_A5396818216090053761m_rule] : X23 ) ) ).

% cont_def
thf(fact_893_tree_Oroot__def,axiom,
    ( abstra3503897793822247108m_rule
    = ( abstra6581154509029876902m_rule
      @ ^ [X13: produc164195504107695125m_rule,X23: fset_A5396818216090053761m_rule] : X13 ) ) ).

% tree.root_def
thf(fact_894_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_895_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_896_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_897_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_898_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_899_snd__comp__apsnd,axiom,
    ! [F: rule > rule] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F ) )
      = ( comp_r641804198332660996m_rule @ F @ produc7165828336582415457m_rule ) ) ).

% snd_comp_apsnd
thf(fact_900_snd__comp__apsnd,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc9047611296996456864ist_fm @ F ) )
      = ( comp_l1868218281415653180ist_fm @ F @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_apsnd
thf(fact_901_fst__comp__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F ) )
      = ( comp_P8245093624798776376m_rule @ F @ produc6879501374131015971m_rule ) ) ).

% fst_comp_apfst
thf(fact_902_fst__comp__apfst,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc1130293176790411102ist_fm @ F ) )
      = ( comp_l1868218281415653180ist_fm @ F @ produc1501393135466168645ist_fm ) ) ).

% fst_comp_apfst
thf(fact_903_Collect__restrict,axiom,
    ! [X7: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_904_Collect__restrict,axiom,
    ! [X7: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ord_le1268244103169919719at_nat
      @ ( collec6321179662152712658at_nat
        @ ^ [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_905_Collect__restrict,axiom,
    ! [X7: set_fm,P: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X: fm] :
            ( ( member_fm @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_906_Collect__restrict,axiom,
    ! [X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_907_Collect__restrict,axiom,
    ! [X7: set_rule,P: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X: rule] :
            ( ( member_rule @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_908_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
            @ ^ [X: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_909_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
            @ ^ [X: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_910_prop__restrict,axiom,
    ! [X5: fm,Z3: set_fm,X7: set_fm,P: fm > $o] :
      ( ( member_fm @ X5 @ Z3 )
     => ( ( ord_less_eq_set_fm @ Z3
          @ ( collect_fm
            @ ^ [X: fm] :
                ( ( member_fm @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_911_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
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_912_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
            @ ^ [X: rule] :
                ( ( member_rule @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_913_apfst__conv,axiom,
    ! [F: rule > rule,X5: rule,Y4: produc1996495991257130529ist_fm] :
      ( ( produc5534506254562563289ist_fm @ F @ ( produc7063755442767259331ist_fm @ X5 @ Y4 ) )
      = ( produc7063755442767259331ist_fm @ ( F @ X5 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_914_apfst__conv,axiom,
    ! [F: list_fm > list_fm,X5: list_fm,Y4: list_fm] :
      ( ( produc1130293176790411102ist_fm @ F @ ( produc381145313068854617ist_fm @ X5 @ Y4 ) )
      = ( produc381145313068854617ist_fm @ ( F @ X5 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_915_apfst__conv,axiom,
    ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] :
      ( ( produc6529295065807390923at_nat @ F @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) )
      = ( produc9060074326276436823at_nat @ ( F @ X5 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_916_apfst__conv,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
      ( ( produc5380969780118646373at_nat @ F @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) )
      = ( produc2922128104949294807at_nat @ ( F @ X5 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_917_apfst__conv,axiom,
    ! [F: nat > nat,X5: nat,Y4: nat] :
      ( ( produc986720760941809198at_nat @ F @ ( product_Pair_nat_nat @ X5 @ Y4 ) )
      = ( product_Pair_nat_nat @ ( F @ X5 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_918_apsnd__conv,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X5: rule,Y4: produc1996495991257130529ist_fm] :
      ( ( produc4332487942798490499m_rule @ F @ ( produc7063755442767259331ist_fm @ X5 @ Y4 ) )
      = ( produc7063755442767259331ist_fm @ X5 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_919_apsnd__conv,axiom,
    ! [F: list_fm > list_fm,X5: list_fm,Y4: list_fm] :
      ( ( produc9047611296996456864ist_fm @ F @ ( produc381145313068854617ist_fm @ X5 @ Y4 ) )
      = ( produc381145313068854617ist_fm @ X5 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_920_apsnd__conv,axiom,
    ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] :
      ( ( produc8057050257526108681at_nat @ F @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) )
      = ( produc9060074326276436823at_nat @ X5 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_921_apsnd__conv,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
      ( ( produc6468969725218530979at_nat @ F @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) )
      = ( produc2922128104949294807at_nat @ X5 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_922_apsnd__conv,axiom,
    ! [F: nat > nat,X5: nat,Y4: nat] :
      ( ( produc3094765310956027504at_nat @ F @ ( product_Pair_nat_nat @ X5 @ Y4 ) )
      = ( product_Pair_nat_nat @ X5 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_923_fst__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X5: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F @ X5 ) )
      = ( F @ ( produc6879501374131015971m_rule @ X5 ) ) ) ).

% fst_apfst
thf(fact_924_fst__apfst,axiom,
    ! [F: list_fm > list_fm,X5: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc1130293176790411102ist_fm @ F @ X5 ) )
      = ( F @ ( produc1501393135466168645ist_fm @ X5 ) ) ) ).

% fst_apfst
thf(fact_925_snd__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X5: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F @ X5 ) )
      = ( produc7165828336582415457m_rule @ X5 ) ) ).

% snd_apfst
thf(fact_926_snd__apfst,axiom,
    ! [F: list_fm > list_fm,X5: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F @ X5 ) )
      = ( produc4588648349897876871ist_fm @ X5 ) ) ).

% snd_apfst
thf(fact_927_fst__apsnd,axiom,
    ! [F: rule > rule,X5: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F @ X5 ) )
      = ( produc6879501374131015971m_rule @ X5 ) ) ).

% fst_apsnd
thf(fact_928_fst__apsnd,axiom,
    ! [F: list_fm > list_fm,X5: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F @ X5 ) )
      = ( produc1501393135466168645ist_fm @ X5 ) ) ).

% fst_apsnd
thf(fact_929_snd__apsnd,axiom,
    ! [F: rule > rule,X5: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F @ X5 ) )
      = ( F @ ( produc7165828336582415457m_rule @ X5 ) ) ) ).

% snd_apsnd
thf(fact_930_snd__apsnd,axiom,
    ! [F: list_fm > list_fm,X5: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc9047611296996456864ist_fm @ F @ X5 ) )
      = ( F @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ).

% snd_apsnd
thf(fact_931_snd__comp__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F ) )
      = produc7165828336582415457m_rule ) ).

% snd_comp_apfst
thf(fact_932_snd__comp__apfst,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F ) )
      = produc4588648349897876871ist_fm ) ).

% snd_comp_apfst
thf(fact_933_fst__comp__apsnd,axiom,
    ! [F: rule > rule] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F ) )
      = produc6879501374131015971m_rule ) ).

% fst_comp_apsnd
thf(fact_934_fst__comp__apsnd,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F ) )
      = produc1501393135466168645ist_fm ) ).

% fst_comp_apsnd
thf(fact_935_apsnd__apfst,axiom,
    ! [F: rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > rule,X5: produc164195504107695125m_rule] :
      ( ( produc5714219303521602403m_rule @ F @ ( produc1151914582733876081e_rule @ G @ X5 ) )
      = ( produc7063755442767259331ist_fm @ ( G @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_936_apsnd__apfst,axiom,
    ! [F: rule > list_fm,G: produc1996495991257130529ist_fm > list_fm,X5: produc164195504107695125m_rule] :
      ( ( produc7464737497110386437ist_fm @ F @ ( produc7276395001879883020m_rule @ G @ X5 ) )
      = ( produc381145313068854617ist_fm @ ( G @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_937_apsnd__apfst,axiom,
    ! [F: rule > set_Pr4329608150637261639at_nat,G: produc1996495991257130529ist_fm > set_Pr4329608150637261639at_nat,X5: produc164195504107695125m_rule] :
      ( ( produc5145205034683326211at_nat @ F @ ( produc8664965124788370535t_rule @ G @ X5 ) )
      = ( produc9060074326276436823at_nat @ ( G @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_938_apsnd__apfst,axiom,
    ! [F: rule > set_Pr1261947904930325089at_nat,G: produc1996495991257130529ist_fm > set_Pr1261947904930325089at_nat,X5: produc164195504107695125m_rule] :
      ( ( produc5534148443436299907at_nat @ F @ ( produc547843410706184321t_rule @ G @ X5 ) )
      = ( produc2922128104949294807at_nat @ ( G @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_939_apsnd__apfst,axiom,
    ! [F: rule > nat,G: produc1996495991257130529ist_fm > nat,X5: produc164195504107695125m_rule] :
      ( ( produc7007998740204313573at_nat @ F @ ( produc6002204213176801020t_rule @ G @ X5 ) )
      = ( product_Pair_nat_nat @ ( G @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_940_apsnd__apfst,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X5: produc1996495991257130529ist_fm] :
      ( ( produc7454059569673786888m_rule @ F @ ( produc8579845201498058179ist_fm @ G @ X5 ) )
      = ( produc7063755442767259331ist_fm @ ( G @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_941_apsnd__apfst,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm,X5: produc1996495991257130529ist_fm] :
      ( ( produc9047611296996456864ist_fm @ F @ ( produc1130293176790411102ist_fm @ G @ X5 ) )
      = ( produc381145313068854617ist_fm @ ( G @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_942_apsnd__apfst,axiom,
    ! [F: list_fm > set_Pr4329608150637261639at_nat,G: list_fm > set_Pr4329608150637261639at_nat,X5: produc1996495991257130529ist_fm] :
      ( ( produc6043827135038106142at_nat @ F @ ( produc1224987502424935947ist_fm @ G @ X5 ) )
      = ( produc9060074326276436823at_nat @ ( G @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_943_apsnd__apfst,axiom,
    ! [F: list_fm > set_Pr1261947904930325089at_nat,G: list_fm > set_Pr1261947904930325089at_nat,X5: produc1996495991257130529ist_fm] :
      ( ( produc8437310097721350814at_nat @ F @ ( produc6442639821993327525ist_fm @ G @ X5 ) )
      = ( produc2922128104949294807at_nat @ ( G @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_944_apsnd__apfst,axiom,
    ! [F: list_fm > nat,G: list_fm > nat,X5: produc1996495991257130529ist_fm] :
      ( ( produc8916614678339685504at_nat @ F @ ( produc4198509902575289422ist_fm @ G @ X5 ) )
      = ( product_Pair_nat_nat @ ( G @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apsnd_apfst
thf(fact_945_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > rule,G: rule > produc1996495991257130529ist_fm,X5: produc164195504107695125m_rule] :
      ( ( produc1080319409639782133ist_fm @ F @ ( produc4729144337346781827ist_fm @ G @ X5 ) )
      = ( produc7063755442767259331ist_fm @ ( F @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_946_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,X5: produc164195504107695125m_rule] :
      ( ( produc6153989603652719015ist_fm @ F @ ( produc2413407552729308604ist_fm @ G @ X5 ) )
      = ( produc381145313068854617ist_fm @ ( F @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_947_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > set_Pr4329608150637261639at_nat,G: rule > set_Pr4329608150637261639at_nat,X5: produc164195504107695125m_rule] :
      ( ( produc1927988930686815397at_nat @ F @ ( produc7253952550029104093ist_fm @ G @ X5 ) )
      = ( produc9060074326276436823at_nat @ ( F @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_948_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > set_Pr1261947904930325089at_nat,G: rule > set_Pr1261947904930325089at_nat,X5: produc164195504107695125m_rule] :
      ( ( produc2555876096144603685at_nat @ F @ ( produc2579366291402042947ist_fm @ G @ X5 ) )
      = ( produc2922128104949294807at_nat @ ( F @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_949_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > nat,G: rule > nat,X5: produc164195504107695125m_rule] :
      ( ( produc8027278558657866375at_nat @ F @ ( produc5400006205771932108ist_fm @ G @ X5 ) )
      = ( product_Pair_nat_nat @ ( F @ ( produc6879501374131015971m_rule @ X5 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_950_apfst__apsnd,axiom,
    ! [F: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,X5: produc1996495991257130529ist_fm] :
      ( ( produc5155535142916265854ist_fm @ F @ ( produc2501461190424677027ist_fm @ G @ X5 ) )
      = ( produc7063755442767259331ist_fm @ ( F @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_951_apfst__apsnd,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm,X5: produc1996495991257130529ist_fm] :
      ( ( produc1130293176790411102ist_fm @ F @ ( produc9047611296996456864ist_fm @ G @ X5 ) )
      = ( produc381145313068854617ist_fm @ ( F @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_952_apfst__apsnd,axiom,
    ! [F: list_fm > set_Pr4329608150637261639at_nat,G: list_fm > set_Pr4329608150637261639at_nat,X5: produc1996495991257130529ist_fm] :
      ( ( produc4764139051848938716at_nat @ F @ ( produc237184781397498441ist_fm @ G @ X5 ) )
      = ( produc9060074326276436823at_nat @ ( F @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_953_apfst__apsnd,axiom,
    ! [F: list_fm > set_Pr1261947904930325089at_nat,G: list_fm > set_Pr1261947904930325089at_nat,X5: produc1996495991257130529ist_fm] :
      ( ( produc1058546750110232412at_nat @ F @ ( produc1127123323018611939ist_fm @ G @ X5 ) )
      = ( produc2922128104949294807at_nat @ ( F @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_954_apfst__apsnd,axiom,
    ! [F: list_fm > nat,G: list_fm > nat,X5: produc1996495991257130529ist_fm] :
      ( ( produc4433738330514286142at_nat @ F @ ( produc5009473360894124176ist_fm @ G @ X5 ) )
      = ( product_Pair_nat_nat @ ( F @ ( produc1501393135466168645ist_fm @ X5 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X5 ) ) ) ) ).

% apfst_apsnd
thf(fact_955_ssubst__Pair__rhs,axiom,
    ! [R2: rule,S3: produc1996495991257130529ist_fm,R5: set_Pr2241365836748572487ist_fm,S6: produc1996495991257130529ist_fm] :
      ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S3 ) @ R5 )
     => ( ( S6 = S3 )
       => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S6 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_956_ssubst__Pair__rhs,axiom,
    ! [R2: list_fm,S3: list_fm,R5: set_Pr7058068377845519745ist_fm,S6: list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S3 ) @ R5 )
     => ( ( S6 = S3 )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S6 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_957_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat,R5: set_Pr7459493094073627847at_nat,S6: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S3 ) @ R5 )
     => ( ( S6 = S3 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S6 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_958_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,R5: set_Pr4329608150637261639at_nat,S6: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S3 ) @ R5 )
     => ( ( S6 = S3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S6 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_959_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S3: nat,R5: set_Pr1261947904930325089at_nat,S6: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R5 )
     => ( ( S6 = S3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S6 ) @ R5 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_960_some__sym__eq__trivial,axiom,
    ! [X5: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ( ^ [Y5: fset_P661503646757059847ist_fm,Z2: fset_P661503646757059847ist_fm] : ( Y5 = Z2 )
          @ X5 ) )
      = X5 ) ).

% some_sym_eq_trivial
thf(fact_961_some__eq__trivial,axiom,
    ! [X5: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ^ [Y: fset_P661503646757059847ist_fm] : ( Y = X5 ) )
      = X5 ) ).

% some_eq_trivial
thf(fact_962_some__equality,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ( P @ A )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ( fChoic3892542562059901019ist_fm @ P )
          = A ) ) ) ).

% some_equality
thf(fact_963_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_964_apfst__convE,axiom,
    ! [Q2: produc4630866025709511057ist_fm,F: rule > rule,P2: produc4630866025709511057ist_fm] :
      ( ( Q2
        = ( produc5534506254562563289ist_fm @ F @ P2 ) )
     => ~ ! [X3: rule,Y3: produc1996495991257130529ist_fm] :
            ( ( P2
              = ( produc7063755442767259331ist_fm @ X3 @ Y3 ) )
           => ( Q2
             != ( produc7063755442767259331ist_fm @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_965_apfst__convE,axiom,
    ! [Q2: produc1996495991257130529ist_fm,F: list_fm > list_fm,P2: produc1996495991257130529ist_fm] :
      ( ( Q2
        = ( produc1130293176790411102ist_fm @ F @ P2 ) )
     => ~ ! [X3: list_fm,Y3: list_fm] :
            ( ( P2
              = ( produc381145313068854617ist_fm @ X3 @ Y3 ) )
           => ( Q2
             != ( produc381145313068854617ist_fm @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_966_apfst__convE,axiom,
    ! [Q2: produc1319942482725812455at_nat,F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,P2: produc1319942482725812455at_nat] :
      ( ( Q2
        = ( produc6529295065807390923at_nat @ F @ P2 ) )
     => ~ ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
            ( ( P2
              = ( produc9060074326276436823at_nat @ X3 @ Y3 ) )
           => ( Q2
             != ( produc9060074326276436823at_nat @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_967_apfst__convE,axiom,
    ! [Q2: produc3843707927480180839at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P2: produc3843707927480180839at_nat] :
      ( ( Q2
        = ( produc5380969780118646373at_nat @ F @ P2 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P2
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ( Q2
             != ( produc2922128104949294807at_nat @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_968_apfst__convE,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat,P2: product_prod_nat_nat] :
      ( ( Q2
        = ( produc986720760941809198at_nat @ F @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ( Q2
             != ( product_Pair_nat_nat @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_969_stream_Ocollapse,axiom,
    ! [Stream: stream_rule] :
      ( ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_970_stream_Ocollapse,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_971_szip__unfold,axiom,
    ! [A: produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,B3: rule,S2: stream_rule] :
      ( ( szip_P1977448745965526924m_rule @ ( sCons_307821682964077496ist_fm @ A @ S1 ) @ ( sCons_rule @ B3 @ S2 ) )
      = ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ A @ B3 ) @ ( szip_P1977448745965526924m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_972_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B3: rule,S2: stream_rule] :
      ( ( szip_rule_rule @ ( sCons_rule @ A @ S1 ) @ ( sCons_rule @ B3 @ S2 ) )
      = ( sCons_7940479310257918722e_rule @ ( produc1769597910952761123e_rule @ A @ B3 ) @ ( szip_rule_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_973_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B3: produc164195504107695125m_rule,S2: stream727092118206550309m_rule] :
      ( ( szip_r2254114822203693110m_rule @ ( sCons_rule @ A @ S1 ) @ ( sCons_5731567480799343806m_rule @ B3 @ S2 ) )
      = ( sCons_2207919253591517394m_rule @ ( produc6084188410630241011m_rule @ A @ B3 ) @ ( szip_r2254114822203693110m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_974_szip__unfold,axiom,
    ! [A: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B3: rule,S2: stream_rule] :
      ( ( szip_P553291425819358518e_rule @ ( sCons_5731567480799343806m_rule @ A @ S1 ) @ ( sCons_rule @ B3 @ S2 ) )
      = ( sCons_6835861040388153298e_rule @ ( produc4383365014245906419e_rule @ A @ B3 ) @ ( szip_P553291425819358518e_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_975_szip__unfold,axiom,
    ! [A: produc164195504107695125m_rule,S1: stream727092118206550309m_rule,B3: produc164195504107695125m_rule,S2: stream727092118206550309m_rule] :
      ( ( szip_P2499414959592755846m_rule @ ( sCons_5731567480799343806m_rule @ A @ S1 ) @ ( sCons_5731567480799343806m_rule @ B3 @ S2 ) )
      = ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ A @ B3 ) @ ( szip_P2499414959592755846m_rule @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_976_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B3: produc1996495991257130529ist_fm,S2: stream8299795917829157543ist_fm] :
      ( ( szip_r8549736553299883584ist_fm @ ( sCons_rule @ A @ S1 ) @ ( sCons_307821682964077496ist_fm @ B3 @ S2 ) )
      = ( sCons_974865965546383930ist_fm @ ( produc7063755442767259331ist_fm @ A @ B3 ) @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_977_szip__unfold,axiom,
    ! [A: list_fm,S1: stream_list_fm,B3: list_fm,S2: stream_list_fm] :
      ( ( szip_list_fm_list_fm @ ( sCons_list_fm @ A @ S1 ) @ ( sCons_list_fm @ B3 @ S2 ) )
      = ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ A @ B3 ) @ ( szip_list_fm_list_fm @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_978_szip__unfold,axiom,
    ! [A: set_Pr4329608150637261639at_nat,S1: stream3627166509194609229at_nat,B3: set_Pr4329608150637261639at_nat,S2: stream3627166509194609229at_nat] :
      ( ( szip_s1528810966652963738at_nat @ ( sCons_4876977544976522334at_nat @ A @ S1 ) @ ( sCons_4876977544976522334at_nat @ B3 @ S2 ) )
      = ( sCons_796194910919348478at_nat @ ( produc9060074326276436823at_nat @ A @ B3 ) @ ( szip_s1528810966652963738at_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_979_szip__unfold,axiom,
    ! [A: set_Pr1261947904930325089at_nat,S1: stream7255200869668761831at_nat,B3: set_Pr1261947904930325089at_nat,S2: stream7255200869668761831at_nat] :
      ( ( szip_s6495482949952704538at_nat @ ( sCons_2777945275570954744at_nat @ A @ S1 ) @ ( sCons_2777945275570954744at_nat @ B3 @ S2 ) )
      = ( sCons_3175934388574233982at_nat @ ( produc2922128104949294807at_nat @ A @ B3 ) @ ( szip_s6495482949952704538at_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_980_szip__unfold,axiom,
    ! [A: nat,S1: stream_nat,B3: nat,S2: stream_nat] :
      ( ( szip_nat_nat @ ( sCons_nat @ A @ S1 ) @ ( sCons_nat @ B3 @ S2 ) )
      = ( sCons_2147322719763279000at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ ( szip_nat_nat @ S1 @ S2 ) ) ) ).

% szip_unfold
thf(fact_981_stream_Osel_I1_J,axiom,
    ! [X1: rule,X2: stream_rule] :
      ( ( shd_rule @ ( sCons_rule @ X1 @ X2 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_982_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_983_stream_Osel_I2_J,axiom,
    ! [X1: rule,X2: stream_rule] :
      ( ( stl_rule @ ( sCons_rule @ X1 @ X2 ) )
      = X2 ) ).

% stream.sel(2)
thf(fact_984_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_985_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_986_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_987_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_988_stream_Oset__induct,axiom,
    ! [X5: fm,A: stream_fm,P: fm > stream_fm > $o] :
      ( ( member_fm @ X5 @ ( sset_fm @ A ) )
     => ( ! [Z1: fm,Z22: stream_fm] : ( P @ Z1 @ ( sCons_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: fm,Z22: stream_fm,Xa: fm] :
              ( ( member_fm @ Xa @ ( sset_fm @ Z22 ) )
             => ( ( P @ Xa @ Z22 )
               => ( P @ Xa @ ( sCons_fm @ Z1 @ Z22 ) ) ) )
         => ( P @ X5 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_989_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_990_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_991_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_992_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_993_stream_Oset__cases,axiom,
    ! [E: fm,A: stream_fm] :
      ( ( member_fm @ E @ ( sset_fm @ A ) )
     => ( ! [Z22: stream_fm] :
            ( A
           != ( sCons_fm @ E @ Z22 ) )
       => ~ ! [Z1: fm,Z22: stream_fm] :
              ( ( A
                = ( sCons_fm @ Z1 @ Z22 ) )
             => ~ ( member_fm @ E @ ( sset_fm @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_994_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_995_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_996_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_997_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_998_stream_Oset__intros_I1_J,axiom,
    ! [X1: fm,X2: stream_fm] : ( member_fm @ X1 @ ( sset_fm @ ( sCons_fm @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_999_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_1000_stream_Oset__intros_I2_J,axiom,
    ! [Y4: produc1996495991257130529ist_fm,X2: stream8299795917829157543ist_fm,X1: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ Y4 @ ( sset_P3431374123360452149ist_fm @ X2 ) )
     => ( member8102475879199740618ist_fm @ Y4 @ ( sset_P3431374123360452149ist_fm @ ( sCons_307821682964077496ist_fm @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_1001_stream_Oset__intros_I2_J,axiom,
    ! [Y4: produc3843707927480180839at_nat,X2: stream8032061917732916077at_nat,X1: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ Y4 @ ( sset_P1195531240007660155at_nat @ X2 ) )
     => ( member8757157785044589968at_nat @ Y4 @ ( sset_P1195531240007660155at_nat @ ( sCons_3175934388574233982at_nat @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_1002_stream_Oset__intros_I2_J,axiom,
    ! [Y4: product_prod_nat_nat,X2: stream6724221391990029191at_nat,X1: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Y4 @ ( sset_P8584168253256249621at_nat @ X2 ) )
     => ( member8440522571783428010at_nat @ Y4 @ ( sset_P8584168253256249621at_nat @ ( sCons_2147322719763279000at_nat @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_1003_stream_Oset__intros_I2_J,axiom,
    ! [Y4: fm,X2: stream_fm,X1: fm] :
      ( ( member_fm @ Y4 @ ( sset_fm @ X2 ) )
     => ( member_fm @ Y4 @ ( sset_fm @ ( sCons_fm @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_1004_stream_Oset__intros_I2_J,axiom,
    ! [Y4: rule,X2: stream_rule,X1: rule] :
      ( ( member_rule @ Y4 @ ( sset_rule @ X2 ) )
     => ( member_rule @ Y4 @ ( sset_rule @ ( sCons_rule @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_1005_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_1006_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_1007_stream_Oexhaust__sel,axiom,
    ! [Stream: stream_rule] :
      ( Stream
      = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_1008_stream_Oexhaust__sel,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( Stream
      = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_szip_Ocode,axiom,
    ( szip_r8549736553299883584ist_fm
    = ( ^ [S12: stream_rule,S22: stream8299795917829157543ist_fm] : ( sCons_974865965546383930ist_fm @ ( produc7063755442767259331ist_fm @ ( shd_rule @ S12 ) @ ( shd_Pr772355297128350925ist_fm @ S22 ) ) @ ( szip_r8549736553299883584ist_fm @ ( stl_rule @ S12 ) @ ( stl_Pr5027778045650968273ist_fm @ S22 ) ) ) ) ) ).

% szip.code
thf(fact_1017_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_1018_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_1019_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_1020_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_1021_someI2,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm,Q: fset_P661503646757059847ist_fm > $o] :
      ( ( P @ A )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
           => ( Q @ X3 ) )
       => ( Q @ ( fChoic3892542562059901019ist_fm @ P ) ) ) ) ).

% someI2
thf(fact_1022_someI__ex,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_12: fset_P661503646757059847ist_fm] : ( P @ X_12 )
     => ( P @ ( fChoic3892542562059901019ist_fm @ P ) ) ) ).

% someI_ex
thf(fact_1023_someI2__ex,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,Q: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_12: fset_P661503646757059847ist_fm] : ( P @ X_12 )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
           => ( Q @ X3 ) )
       => ( Q @ ( fChoic3892542562059901019ist_fm @ P ) ) ) ) ).

% someI2_ex
thf(fact_1024_someI2__bex,axiom,
    ! [A2: set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm > $o] :
      ( ? [X6: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X3: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ X3 @ A2 )
              & ( P @ X3 ) )
           => ( Q @ X3 ) )
       => ( Q
          @ ( fChoic1980167584695909237ist_fm
            @ ^ [X: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1025_someI2__bex,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ? [X6: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( ( member8757157785044589968at_nat @ X3 @ A2 )
              & ( P @ X3 ) )
           => ( Q @ X3 ) )
       => ( Q
          @ ( fChoic2372384411376717755at_nat
            @ ^ [X: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1026_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 ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( ( member8440522571783428010at_nat @ X3 @ A2 )
              & ( P @ X3 ) )
           => ( Q @ X3 ) )
       => ( Q
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1027_someI2__bex,axiom,
    ! [A2: set_rule,P: rule > $o,Q: rule > $o] :
      ( ? [X6: rule] :
          ( ( member_rule @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X3: rule] :
            ( ( ( member_rule @ X3 @ A2 )
              & ( P @ X3 ) )
           => ( Q @ X3 ) )
       => ( Q
          @ ( fChoice_rule
            @ ^ [X: rule] :
                ( ( member_rule @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1028_someI2__bex,axiom,
    ! [A2: set_fm,P: fm > $o,Q: fm > $o] :
      ( ? [X6: fm] :
          ( ( member_fm @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X3: fm] :
            ( ( ( member_fm @ X3 @ A2 )
              & ( P @ X3 ) )
           => ( Q @ X3 ) )
       => ( Q
          @ ( fChoice_fm
            @ ^ [X: fm] :
                ( ( member_fm @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1029_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 ) )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( ( member3493611713653719344ist_fm @ X3 @ A2 )
              & ( P @ X3 ) )
           => ( Q @ X3 ) )
       => ( Q
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X: fset_P661503646757059847ist_fm] :
                ( ( member3493611713653719344ist_fm @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_1030_some__eq__ex,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o] :
      ( ( P @ ( fChoic3892542562059901019ist_fm @ P ) )
      = ( ? [X4: fset_P661503646757059847ist_fm] : ( P @ X4 ) ) ) ).

% some_eq_ex
thf(fact_1031_some1__equality,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ? [X6: fset_P661503646757059847ist_fm] :
          ( ( P @ X6 )
          & ! [Y3: fset_P661503646757059847ist_fm] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ A )
       => ( ( fChoic3892542562059901019ist_fm @ P )
          = A ) ) ) ).

% some1_equality
thf(fact_1032_sfilter_Ocode,axiom,
    ( sfilte7579662470208054517m_rule
    = ( ^ [P3: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P3 ) @ S ) ) @ ( sfilte7579662470208054517m_rule @ P3 @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P3 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_1033_sfilter_Ocode,axiom,
    ( sfilter_rule
    = ( ^ [P3: rule > $o,S: stream_rule] : ( sCons_rule @ ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P3 ) @ S ) ) @ ( sfilter_rule @ P3 @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P3 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_1034_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_1035_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_1036_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_1037_verit__sko__forall,axiom,
    ( ( ^ [P4: fset_P661503646757059847ist_fm > $o] :
        ! [X8: fset_P661503646757059847ist_fm] : ( P4 @ X8 ) )
    = ( ^ [P3: fset_P661503646757059847ist_fm > $o] :
          ( P3
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X: fset_P661503646757059847ist_fm] :
                ~ ( P3 @ X ) ) ) ) ) ).

% verit_sko_forall
thf(fact_1038_verit__sko__forall_H,axiom,
    ! [P: fset_P661503646757059847ist_fm > $o,A2: $o] :
      ( ( ( P
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X: fset_P661503646757059847ist_fm] :
                ~ ( P @ X ) ) )
        = A2 )
     => ( ( ! [X4: fset_P661503646757059847ist_fm] : ( P @ X4 ) )
        = A2 ) ) ).

% verit_sko_forall'
thf(fact_1039_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_1040_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_1041_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_1042_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_1043_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1044_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1045_verit__la__disequality,axiom,
    ! [A: nat,B3: nat] :
      ( ( A = B3 )
      | ~ ( ord_less_eq_nat @ A @ B3 )
      | ~ ( ord_less_eq_nat @ B3 @ A ) ) ).

% verit_la_disequality
thf(fact_1046_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_1047_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_1048_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_1049_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_1050_verit__sko__forall__indirect2,axiom,
    ! [X5: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o,P5: fset_P661503646757059847ist_fm > $o] :
      ( ( X5
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X: fset_P661503646757059847ist_fm] :
              ~ ( P @ X ) ) )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
            = ( P5 @ X3 ) )
       => ( ( ! [X4: fset_P661503646757059847ist_fm] : ( P5 @ X4 ) )
          = ( P @ X5 ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_1051_verit__sko__forall__indirect,axiom,
    ! [X5: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o] :
      ( ( X5
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X: fset_P661503646757059847ist_fm] :
              ~ ( P @ X ) ) )
     => ( ( ! [X4: fset_P661503646757059847ist_fm] : ( P @ X4 ) )
        = ( P @ X5 ) ) ) ).

% verit_sko_forall_indirect
thf(fact_1052_verit__sko__ex__indirect2,axiom,
    ! [X5: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o,P5: fset_P661503646757059847ist_fm > $o] :
      ( ( X5
        = ( fChoic3892542562059901019ist_fm @ P ) )
     => ( ! [X3: fset_P661503646757059847ist_fm] :
            ( ( P @ X3 )
            = ( P5 @ X3 ) )
       => ( ( ? [X4: fset_P661503646757059847ist_fm] : ( P5 @ X4 ) )
          = ( P @ X5 ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_1053_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_1054_verit__sko__forall_H_H,axiom,
    ! [B: fset_P661503646757059847ist_fm,A2: fset_P661503646757059847ist_fm,P: fset_P661503646757059847ist_fm > $o] :
      ( ( B = A2 )
     => ( ( ( fChoic3892542562059901019ist_fm @ P )
          = A2 )
        = ( ( fChoic3892542562059901019ist_fm @ P )
          = B ) ) ) ).

% verit_sko_forall''
thf(fact_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_snth__szip,axiom,
    ! [S1: stream_rule,S2: stream8299795917829157543ist_fm,N: nat] :
      ( ( snth_P6179250367719980013ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S2 ) @ N )
      = ( produc7063755442767259331ist_fm @ ( snth_rule @ S1 @ N ) @ ( snth_P2624749845746815685ist_fm @ S2 @ N ) ) ) ).

% snth_szip
thf(fact_1079_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_1080_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_1081_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_1082_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_1083_snth__sset,axiom,
    ! [S3: stream8299795917829157543ist_fm,N: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S3 @ N ) @ ( sset_P3431374123360452149ist_fm @ S3 ) ) ).

% snth_sset
thf(fact_1084_snth__sset,axiom,
    ! [S3: stream8032061917732916077at_nat,N: nat] : ( member8757157785044589968at_nat @ ( snth_P8412462321310240523at_nat @ S3 @ N ) @ ( sset_P1195531240007660155at_nat @ S3 ) ) ).

% snth_sset
thf(fact_1085_snth__sset,axiom,
    ! [S3: stream6724221391990029191at_nat,N: nat] : ( member8440522571783428010at_nat @ ( snth_P9042041890044241317at_nat @ S3 @ N ) @ ( sset_P8584168253256249621at_nat @ S3 ) ) ).

% snth_sset
thf(fact_1086_snth__sset,axiom,
    ! [S3: stream_fm,N: nat] : ( member_fm @ ( snth_fm @ S3 @ N ) @ ( sset_fm @ S3 ) ) ).

% snth_sset
thf(fact_1087_snth__sset,axiom,
    ! [S3: stream_rule,N: nat] : ( member_rule @ ( snth_rule @ S3 @ N ) @ ( sset_rule @ S3 ) ) ).

% snth_sset
thf(fact_1088_conj__subset__def,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A2
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le3146513528884898305at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
        & ( ord_le3146513528884898305at_nat @ A2 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1089_conj__subset__def,axiom,
    ! [A2: set_rule,P: rule > $o,Q: rule > $o] :
      ( ( ord_less_eq_set_rule @ A2
        @ ( collect_rule
          @ ^ [X: rule] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_rule @ A2 @ ( collect_rule @ P ) )
        & ( ord_less_eq_set_rule @ A2 @ ( collect_rule @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1090_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_1091_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_1092_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_1093_snth__sset__smerge,axiom,
    ! [Ss3: stream_stream_fm,N: nat,M: nat] : ( member_fm @ ( snth_fm @ ( snth_stream_fm @ Ss3 @ N ) @ M ) @ ( sset_fm @ ( smerge_fm @ Ss3 ) ) ) ).

% snth_sset_smerge
thf(fact_1094_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_1095_Fair__Stream_Ofair__def,axiom,
    ( fair_fair_rule
    = ( ^ [S: stream_rule] :
        ! [X: rule] :
          ( ( member_rule @ X @ ( sset_rule @ S ) )
         => ! [M3: nat] :
            ? [N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
              & ( ( snth_rule @ S @ N3 )
                = X ) ) ) ) ) ).

% Fair_Stream.fair_def
thf(fact_1096_epath__Pre__sdrop_I2_J,axiom,
    ! [Steps: stream727092118206550309m_rule,P: nat,Ts: list_tm,M: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) ) ) )
       => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ) ).

% epath_Pre_sdrop(2)
thf(fact_1097_epath__Pre__sdrop_I1_J,axiom,
    ! [Steps: stream727092118206550309m_rule,P: nat,Ts: list_tm,M: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) ) ) )
       => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M @ Steps ) ) ) ) ) ) ) ) ).

% epath_Pre_sdrop(1)
thf(fact_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_eff__preserves__Pre_I1_J,axiom,
    ! [A2: list_fm,B: list_fm,R2: rule,Ss3: fset_P661503646757059847ist_fm,A3: list_fm,B2: list_fm,P: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A2 @ B ) @ R2 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A2 @ B ) @ R2 ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss3 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B2 ) @ Ss3 )
       => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A2 ) )
         => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A3 ) ) ) ) ) ).

% eff_preserves_Pre(1)
thf(fact_1125_eff__preserves__Pre_I2_J,axiom,
    ! [A2: list_fm,B: list_fm,R2: rule,Ss3: fset_P661503646757059847ist_fm,A3: list_fm,B2: list_fm,P: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A2 @ B ) @ R2 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A2 @ B ) @ R2 ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss3 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B2 ) @ Ss3 )
       => ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B ) )
         => ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B2 ) ) ) ) ) ).

% eff_preserves_Pre(2)
thf(fact_1126_subset__code_I1_J,axiom,
    ! [Xs: list_P5616295576739893671ist_fm,B: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ ( set_Pr8767716839810916150ist_fm @ Xs ) @ B )
      = ( ! [X: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X @ ( set_Pr8767716839810916150ist_fm @ Xs ) )
           => ( member8102475879199740618ist_fm @ X @ B ) ) ) ) ).

% subset_code(1)
thf(fact_1127_subset__code_I1_J,axiom,
    ! [Xs: list_P5464809261938338413at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ B )
      = ( ! [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ ( set_Pr3765526544606949372at_nat @ Xs ) )
           => ( member8757157785044589968at_nat @ X @ B ) ) ) ) ).

% subset_code(1)
thf(fact_1128_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ X @ B ) ) ) ) ).

% subset_code(1)
thf(fact_1129_subset__code_I1_J,axiom,
    ! [Xs: list_fm,B: set_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B )
      = ( ! [X: fm] :
            ( ( member_fm @ X @ ( set_fm2 @ Xs ) )
           => ( member_fm @ X @ B ) ) ) ) ).

% subset_code(1)
thf(fact_1130_subset__code_I1_J,axiom,
    ! [Xs: list_rule,B: set_rule] :
      ( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ B )
      = ( ! [X: rule] :
            ( ( member_rule @ X @ ( set_rule2 @ Xs ) )
           => ( member_rule @ X @ B ) ) ) ) ).

% subset_code(1)
thf(fact_1131_fset__of__list__subset,axiom,
    ! [Xs: list_fm,Ys: list_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) )
     => ( ord_less_eq_fset_fm @ ( fset_of_list_fm @ Xs ) @ ( fset_of_list_fm @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_1132_fset__of__list__subset,axiom,
    ! [Xs: list_rule,Ys: list_rule] :
      ( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ ( set_rule2 @ Ys ) )
     => ( ord_le7171078382243635157t_rule @ ( fset_of_list_rule @ Xs ) @ ( fset_of_list_rule @ Ys ) ) ) ).

% fset_of_list_subset
thf(fact_1133_prod_Oswap__def,axiom,
    ( produc5396952528824381687at_nat
    = ( ^ [P6: produc1319942482725812455at_nat] : ( produc9060074326276436823at_nat @ ( produc826222791318859909at_nat @ P6 ) @ ( produc4486455499028916547at_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1134_prod_Oswap__def,axiom,
    ( produc3946833401584108663at_nat
    = ( ^ [P6: produc3843707927480180839at_nat] : ( produc2922128104949294807at_nat @ ( produc2937545259423399429at_nat @ P6 ) @ ( produc6011863565407846595at_nat @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1135_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_1136_prod_Oswap__def,axiom,
    ( produc2641608828089375343m_rule
    = ( ^ [P6: produc164195504107695125m_rule] : ( produc7063755442767259331ist_fm @ ( produc7165828336582415457m_rule @ P6 ) @ ( produc6879501374131015971m_rule @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1137_prod_Oswap__def,axiom,
    ( produc7936585717479832313ist_fm
    = ( ^ [P6: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc4588648349897876871ist_fm @ P6 ) @ ( produc1501393135466168645ist_fm @ P6 ) ) ) ) ).

% prod.swap_def
thf(fact_1138_fst__snd__flip,axiom,
    ( produc4486455499028916547at_nat
    = ( comp_P3728553842126764068at_nat @ produc826222791318859909at_nat
      @ ( produc5456455677105495163at_nat
        @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1139_fst__snd__flip,axiom,
    ( produc6011863565407846595at_nat
    = ( comp_P4974406650942564362at_nat @ produc2937545259423399429at_nat
      @ ( produc7548845685434208379at_nat
        @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1140_fst__snd__flip,axiom,
    ( product_fst_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1141_fst__snd__flip,axiom,
    ( produc6879501374131015971m_rule
    = ( comp_P6646430376872094402m_rule @ produc4514744107061996309ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X: produc1996495991257130529ist_fm,Y: rule] : ( produc7063755442767259331ist_fm @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1142_fst__snd__flip,axiom,
    ( produc4228417144610596823ist_fm
    = ( comp_P4888803072877744336ist_fm @ produc7165828336582415457m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X: rule,Y: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1143_fst__snd__flip,axiom,
    ( produc1501393135466168645ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X: list_fm,Y: list_fm] : ( produc381145313068854617ist_fm @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1144_snd__fst__flip,axiom,
    ( produc826222791318859909at_nat
    = ( comp_P3728553842126764068at_nat @ produc4486455499028916547at_nat
      @ ( produc5456455677105495163at_nat
        @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_1145_snd__fst__flip,axiom,
    ( produc2937545259423399429at_nat
    = ( comp_P4974406650942564362at_nat @ produc6011863565407846595at_nat
      @ ( produc7548845685434208379at_nat
        @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_1146_snd__fst__flip,axiom,
    ( product_snd_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_1147_snd__fst__flip,axiom,
    ( produc4514744107061996309ist_fm
    = ( comp_P2707267495825058490ist_fm @ produc6879501374131015971m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X: rule,Y: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_1148_snd__fst__flip,axiom,
    ( produc7165828336582415457m_rule
    = ( comp_P3413117018928051920m_rule @ produc4228417144610596823ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X: produc1996495991257130529ist_fm,Y: rule] : ( produc7063755442767259331ist_fm @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_1149_snd__fst__flip,axiom,
    ( produc4588648349897876871ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X: list_fm,Y: list_fm] : ( produc381145313068854617ist_fm @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_1150_swap__simp,axiom,
    ! [X5: produc1996495991257130529ist_fm,Y4: rule] :
      ( ( produc2641608828089375343m_rule @ ( produc491467635432902671m_rule @ X5 @ Y4 ) )
      = ( produc7063755442767259331ist_fm @ Y4 @ X5 ) ) ).

% swap_simp
thf(fact_1151_swap__simp,axiom,
    ! [X5: rule,Y4: produc1996495991257130529ist_fm] :
      ( ( produc9213896635423732003ist_fm @ ( produc7063755442767259331ist_fm @ X5 @ Y4 ) )
      = ( produc491467635432902671m_rule @ Y4 @ X5 ) ) ).

% swap_simp
thf(fact_1152_swap__simp,axiom,
    ! [X5: list_fm,Y4: list_fm] :
      ( ( produc7936585717479832313ist_fm @ ( produc381145313068854617ist_fm @ X5 @ Y4 ) )
      = ( produc381145313068854617ist_fm @ Y4 @ X5 ) ) ).

% swap_simp
thf(fact_1153_swap__simp,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] :
      ( ( produc5396952528824381687at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) )
      = ( produc9060074326276436823at_nat @ Y4 @ X5 ) ) ).

% swap_simp
thf(fact_1154_swap__simp,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
      ( ( produc3946833401584108663at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) )
      = ( produc2922128104949294807at_nat @ Y4 @ X5 ) ) ).

% swap_simp
thf(fact_1155_swap__simp,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) )
      = ( product_Pair_nat_nat @ Y4 @ X5 ) ) ).

% swap_simp
thf(fact_1156_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A: nat,B3: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B3 ) )
      = ( F @ A @ B3 ) ) ).

% case_prod_conv
thf(fact_1157_case__swap,axiom,
    ! [F: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [Y: nat,X: nat] : ( F @ X @ Y )
        @ ( product_swap_nat_nat @ P2 ) )
      = ( produc6081775807080527818_nat_o @ F @ P2 ) ) ).

% case_swap
thf(fact_1158_snd__swap,axiom,
    ! [X5: produc164195504107695125m_rule] :
      ( ( produc4514744107061996309ist_fm @ ( produc2641608828089375343m_rule @ X5 ) )
      = ( produc6879501374131015971m_rule @ X5 ) ) ).

% snd_swap
thf(fact_1159_snd__swap,axiom,
    ! [X5: produc4630866025709511057ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc9213896635423732003ist_fm @ X5 ) )
      = ( produc4228417144610596823ist_fm @ X5 ) ) ).

% snd_swap
thf(fact_1160_snd__swap,axiom,
    ! [X5: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc7936585717479832313ist_fm @ X5 ) )
      = ( produc1501393135466168645ist_fm @ X5 ) ) ).

% snd_swap
thf(fact_1161_fst__swap,axiom,
    ! [X5: produc4630866025709511057ist_fm] :
      ( ( produc6879501374131015971m_rule @ ( produc9213896635423732003ist_fm @ X5 ) )
      = ( produc4514744107061996309ist_fm @ X5 ) ) ).

% fst_swap
thf(fact_1162_fst__swap,axiom,
    ! [X5: produc164195504107695125m_rule] :
      ( ( produc4228417144610596823ist_fm @ ( produc2641608828089375343m_rule @ X5 ) )
      = ( produc7165828336582415457m_rule @ X5 ) ) ).

% fst_swap
thf(fact_1163_fst__swap,axiom,
    ! [X5: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc7936585717479832313ist_fm @ X5 ) )
      = ( produc4588648349897876871ist_fm @ X5 ) ) ).

% fst_swap
thf(fact_1164_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X2: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_1165_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X13: nat,X23: nat] : ( H @ ( F @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_1166_case__prod__Pair__iden,axiom,
    ! [P2: produc4630866025709511057ist_fm] :
      ( ( produc6651706081449720449ist_fm @ produc7063755442767259331ist_fm @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1167_case__prod__Pair__iden,axiom,
    ! [P2: produc1996495991257130529ist_fm] :
      ( ( produc4544959197559615795ist_fm @ produc381145313068854617ist_fm @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1168_case__prod__Pair__iden,axiom,
    ! [P2: produc1319942482725812455at_nat] :
      ( ( produc5456455677105495163at_nat @ produc9060074326276436823at_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1169_case__prod__Pair__iden,axiom,
    ! [P2: produc3843707927480180839at_nat] :
      ( ( produc7548845685434208379at_nat @ produc2922128104949294807at_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1170_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1171_cond__case__prod__eta,axiom,
    ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_1172_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_1173_case__prodE2,axiom,
    ! [Q: $o > $o,P: nat > nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_1174_fst__def,axiom,
    ( produc6879501374131015971m_rule
    = ( produc8807799049144449253ist_fm
      @ ^ [X13: produc1996495991257130529ist_fm,X23: rule] : X13 ) ) ).

% fst_def
thf(fact_1175_fst__def,axiom,
    ( produc1501393135466168645ist_fm
    = ( produc4154610405322092878ist_fm
      @ ^ [X13: list_fm,X23: list_fm] : X13 ) ) ).

% fst_def
thf(fact_1176_snd__def,axiom,
    ( produc7165828336582415457m_rule
    = ( produc1213517383330063745e_rule
      @ ^ [X13: produc1996495991257130529ist_fm,X23: rule] : X23 ) ) ).

% snd_def
thf(fact_1177_snd__def,axiom,
    ( produc4588648349897876871ist_fm
    = ( produc4154610405322092878ist_fm
      @ ^ [X13: list_fm,X23: list_fm] : X23 ) ) ).

% snd_def
thf(fact_1178_case__prod__beta,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F2: nat > nat > $o,P6: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P6 ) @ ( product_snd_nat_nat @ P6 ) ) ) ) ).

% case_prod_beta
thf(fact_1179_split__beta,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F2: nat > nat > $o,Prod3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ Prod3 ) @ ( product_snd_nat_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_1180_internal__case__prod__def,axiom,
    produc9139807813499881076_nat_o = produc6081775807080527818_nat_o ).

% internal_case_prod_def
thf(fact_1181_case__prod__unfold,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [C4: nat > nat > $o,P6: product_prod_nat_nat] : ( C4 @ ( product_fst_nat_nat @ P6 ) @ ( product_snd_nat_nat @ P6 ) ) ) ) ).

% case_prod_unfold
thf(fact_1182_case__prod__beta_H,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F2: nat > nat > $o,X: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_1183_exE__realizer,axiom,
    ! [P: nat > nat > $o,P2: product_prod_nat_nat,Q: $o > $o,F: nat > nat > $o] :
      ( ( P @ ( product_snd_nat_nat @ P2 ) @ ( product_fst_nat_nat @ P2 ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc6081775807080527818_nat_o @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_1184_fset__of__list__elem,axiom,
    ! [X5: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( fmembe9188685285568734254at_nat @ X5 @ ( fset_o6030404570504654125at_nat @ Xs ) )
      = ( member8757157785044589968at_nat @ X5 @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1185_fset__of__list__elem,axiom,
    ! [X5: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( fmembe1449479052724974408at_nat @ X5 @ ( fset_o7974655632839724743at_nat @ Xs ) )
      = ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1186_fset__of__list__elem,axiom,
    ! [X5: rule,Xs: list_rule] :
      ( ( fmember_rule @ X5 @ ( fset_of_list_rule @ Xs ) )
      = ( member_rule @ X5 @ ( set_rule2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1187_fset__of__list__elem,axiom,
    ! [X5: produc1996495991257130529ist_fm,Xs: list_P5616295576739893671ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X5 @ ( fset_o3706400737857578983ist_fm @ Xs ) )
      = ( member8102475879199740618ist_fm @ X5 @ ( set_Pr8767716839810916150ist_fm @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1188_fset__of__list__elem,axiom,
    ! [X5: abstra4572470701423150929m_rule,Xs: list_A1849579841884561377m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X5 @ ( fset_o8294491781805274379m_rule @ Xs ) )
      = ( member1590433669991072232m_rule @ X5 @ ( set_Ab5312821370972875900m_rule @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1189_fset__of__list__elem,axiom,
    ! [X5: fm,Xs: list_fm] :
      ( ( fmember_fm @ X5 @ ( fset_of_list_fm @ Xs ) )
      = ( member_fm @ X5 @ ( set_fm2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_1190_prod_Osplit__sel,axiom,
    ! [P: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_1191_prod_Osplit__sel__asm,axiom,
    ! [P: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_1192_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat > $o,G: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc6081775807080527818_nat_o @ F @ P2 )
          = ( produc6081775807080527818_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_1193_abort__Bleast__def,axiom,
    ( abort_Bleast_nat
    = ( ^ [S5: set_nat,P3: nat > $o] :
          ( ord_Least_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ S5 )
              & ( P3 @ X ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_1194_Bleast__def,axiom,
    ( bleast_nat
    = ( ^ [S5: set_nat,P3: nat > $o] :
          ( ord_Least_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ S5 )
              & ( P3 @ X ) ) ) ) ) ).

% Bleast_def
thf(fact_1195_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_1196_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
          @ ^ [Xs2: 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 ) )
                @ Xs2 )
             => ( ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] :
                      ( ( produc7165828336582415457m_rule @ Step )
                      = R2 )
                  @ Xs2 )
                | ( 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 ) ) )
                  @ Xs2 ) ) )
          @ Steps ) ) ) ).

% per_alw
thf(fact_1197_case__prodI,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( F @ A @ B3 )
     => ( produc410239310623530412_nat_o @ F @ ( produc2922128104949294807at_nat @ A @ B3 ) ) ) ).

% case_prodI
thf(fact_1198_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B3: nat] :
      ( ( F @ A @ B3 )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B3 ) ) ) ).

% case_prodI
thf(fact_1199_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 )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K2 @ I2 )
                 => ( P @ I2 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_1200_diff__diff__cancel,axiom,
    ! [I3: nat,N: nat] :
      ( ( ord_less_eq_nat @ I3 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
        = I3 ) ) ).

% diff_diff_cancel
thf(fact_1201_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_1202_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_1203_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1204_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_1205_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_1206_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_1207_le__trans,axiom,
    ! [I3: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K )
       => ( ord_less_eq_nat @ I3 @ K ) ) ) ).

% le_trans
thf(fact_1208_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_1209_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1210_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_1211_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1212_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_1213_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_1214_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1215_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_1216_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_1217_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_1218_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R5: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R5 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z4: nat] :
              ( ( R5 @ X3 @ Y3 )
             => ( ( R5 @ Y3 @ Z4 )
               => ( R5 @ X3 @ Z4 ) ) )
         => ( ! [N2: nat] : ( R5 @ N2 @ ( suc @ N2 ) )
           => ( R5 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1219_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
          & ( M3 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_1220_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_1221_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_1222_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_1223_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_1224_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I3: nat,J2: nat] :
      ( ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I3 @ J2 )
       => ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1225_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_1226_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_1227_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_1228_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_1229_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1230_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B3 ) )
          = ( ord_less_eq_nat @ B3 @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1231_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_1232_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
        @ ^ [Xs2: 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 ) )
              @ Xs2 )
           => ( linear2722748333883190224m_rule
              @ ( linear7082939723987557601m_rule
                @ ^ [Step: produc164195504107695125m_rule] :
                    ( ( produc7165828336582415457m_rule @ Step )
                    = R2 ) )
              @ Xs2 ) ) ) ) ).

% saturated_def
thf(fact_1233_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_1234_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1235_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_1236_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_1237_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_1238_inc__induct,axiom,
    ! [I3: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I3 @ J2 )
     => ( ( P @ J2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I3 ) ) ) ) ).

% inc_induct
thf(fact_1239_dec__induct,axiom,
    ! [I3: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I3 @ J2 )
     => ( ( P @ I3 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J2 ) ) ) ) ).

% dec_induct
thf(fact_1240_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_1241_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1242_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_1243_diff__less__mono,axiom,
    ! [A: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A @ B3 )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1244_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_1245_Prover_Ofair__def,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
      = ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs ) @ ( sset_rule @ rules ) )
        & ! [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ rules ) )
           => ( linear7625967296368300489w_rule
              @ ( linear6831045468625226112v_rule
                @ ( linear4670012741531116049s_rule
                  @ ( ^ [Y5: rule,Z2: rule] : ( Y5 = Z2 )
                    @ X ) ) )
              @ Rs ) ) ) ) ).

% Prover.fair_def
thf(fact_1246_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_1247_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K3 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_1248_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_1249_prod__decode__aux_Oelims,axiom,
    ! [X5: nat,Xa2: nat,Y4: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X5 @ Xa2 )
        = Y4 )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X5 )
         => ( Y4
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X5 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X5 )
         => ( Y4
            = ( nat_prod_decode_aux @ ( suc @ X5 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X5 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_1250_pair__lessI2,axiom,
    ! [A: nat,B3: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1251_pair__less__iff1,axiom,
    ! [X5: nat,Y4: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ ( product_Pair_nat_nat @ X5 @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y4 @ Z ) ) ).

% pair_less_iff1
thf(fact_1252_pair__lessI1,axiom,
    ! [A: nat,B3: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_1253_pair__leqI2,axiom,
    ! [A: nat,B3: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B3 )
     => ( ( ord_less_eq_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1254_pair__leqI1,axiom,
    ! [A: nat,B3: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_1255_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M3: nat,N3: nat] :
            ( N3
            = ( suc @ M3 ) ) ) ) ) ).

% pred_nat_def
thf(fact_1256_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,S8: 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 @ S8 @ Sl6 ) )
           => ( abstra3055540779341473792ist_fm
              @ ^ [R: rule,T2: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ T2 )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R2
              @ S8 ) ) ) ) ).

% per_def
thf(fact_1257_NE__R,axiom,
    ( ( sset_rule @ rules )
   != bot_bot_set_rule ) ).

% NE_R
thf(fact_1258_wmax__insertI,axiom,
    ! [Y4: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y4 @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y4 ) @ 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_1259_wmin__insertI,axiom,
    ! [X5: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y4: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X5 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y4 ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y4 @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_1260_wmin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_1261_smax__insertI,axiom,
    ! [Y4: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y4 @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y4 ) @ 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_1262_smin__insertI,axiom,
    ! [X5: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y4: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X5 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y4 ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y4 @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_1263_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_1264_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_1265_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_1266_eff_Osimps_I2_J,axiom,
    ! [P: nat,Ts: list_tm,A2: list_fm,B: list_fm] :
      ( ( ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A2 ) )
          & ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B ) ) )
       => ( ( eff @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A2 @ B ) )
          = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
      & ( ~ ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A2 ) )
            & ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B ) ) )
       => ( ( eff @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A2 @ B ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(2)
thf(fact_1267_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
thf(fact_1268_eff_Osimps_I1_J,axiom,
    ! [A2: list_fm,B: list_fm] :
      ( ( eff @ idle @ ( produc381145313068854617ist_fm @ A2 @ B ) )
      = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A2 @ B ) @ bot_bo2367426573206113139ist_fm ) ) ) ).

% eff.simps(1)
thf(fact_1269_less__than__iff,axiom,
    ! [X5: nat,Y4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ less_than )
      = ( ord_less_nat @ X5 @ Y4 ) ) ).

% less_than_iff
thf(fact_1270_eff_Opsimps_I2_J,axiom,
    ! [P: nat,Ts: list_tm,A2: list_fm,B: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A2 @ B ) ) )
     => ( ( ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A2 ) )
            & ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B ) ) )
         => ( ( eff @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A2 @ B ) )
            = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
        & ( ~ ( ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ A2 ) )
              & ( member_fm @ ( pre @ P @ Ts ) @ ( set_fm2 @ B ) ) )
         => ( ( eff @ ( axiom @ P @ Ts ) @ ( produc381145313068854617ist_fm @ A2 @ B ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(2)
thf(fact_1271_eff_Opsimps_I1_J,axiom,
    ! [A2: list_fm,B: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ idle @ ( produc381145313068854617ist_fm @ A2 @ B ) ) )
     => ( ( eff @ idle @ ( produc381145313068854617ist_fm @ A2 @ B ) )
        = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A2 @ B ) @ bot_bo2367426573206113139ist_fm ) ) ) ) ).

% eff.psimps(1)
thf(fact_1272_eff_Opsimps_I3_J,axiom,
    ! [A2: list_fm,B: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ flsL @ ( produc381145313068854617ist_fm @ A2 @ B ) ) )
     => ( ( ( member_fm @ falsity @ ( set_fm2 @ A2 ) )
         => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A2 @ B ) )
            = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
        & ( ~ ( member_fm @ falsity @ ( set_fm2 @ A2 ) )
         => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A2 @ B ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(3)
thf(fact_1273_eff_Osimps_I3_J,axiom,
    ! [A2: list_fm,B: list_fm] :
      ( ( ( member_fm @ falsity @ ( set_fm2 @ A2 ) )
       => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A2 @ B ) )
          = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
      & ( ~ ( member_fm @ falsity @ ( set_fm2 @ A2 ) )
       => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A2 @ B ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(3)
thf(fact_1274_prod__decode__aux_Opelims,axiom,
    ! [X5: nat,Xa2: nat,Y4: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X5 @ Xa2 )
        = Y4 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X5 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X5 )
               => ( Y4
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X5 @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X5 )
               => ( Y4
                  = ( nat_prod_decode_aux @ ( suc @ X5 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X5 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X5 @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims

% Helper facts (23)
thf(help_fChoice_1_1_fChoice_001t__Syntax__Ofm_T,axiom,
    ! [P: fm > $o] :
      ( ( P @ ( fChoice_fm @ P ) )
      = ( ? [X4: fm] : ( P @ X4 ) ) ) ).

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,Y4: stream_rule] :
      ( ( if_stream_rule @ $false @ X5 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X5: stream_rule,Y4: stream_rule] :
      ( ( if_stream_rule @ $true @ X5 @ Y4 )
      = 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,Y4: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X5 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X5 @ Y4 )
      = 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,Y4: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $false @ X5 @ Y4 )
      = Y4 ) ).

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,Y4: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $true @ X5 @ Y4 )
      = 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,Y4: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $false @ X5 @ Y4 )
      = Y4 ) ).

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,Y4: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $true @ X5 @ Y4 )
      = 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_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,Y4: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $false @ X5 @ Y4 )
      = Y4 ) ).

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,Y4: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $true @ X5 @ Y4 )
      = 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,Y4: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $false @ X5 @ Y4 )
      = Y4 ) ).

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,Y4: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $true @ X5 @ Y4 )
      = 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,Y4: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $false @ X5 @ Y4 )
      = Y4 ) ).

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,Y4: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $true @ X5 @ Y4 )
      = X5 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( abstra8647779336736900908ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ ( sdrop_7224736112439592940m_rule @ n @ steps ) ) ).

%------------------------------------------------------------------------------