TPTP Problem File: SLH0493^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FOL_Seq_Calc3/0011_Completeness/prob_00136_004866__12187548_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2007 ( 662 unt; 710 typ;   0 def)
%            Number of atoms       : 4007 (1258 equ;  10 cnn)
%            Maximal formula atoms :   15 (   3 avg)
%            Number of connectives : 15551 ( 270   ~;  54   |; 452   &;13127   @)
%                                         (   0 <=>;1648  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :  107 ( 106 usr)
%            Number of type conns  : 3247 (3247   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  608 ( 604 usr;  30 con; 0-4 aty)
%            Number of variables   : 4786 ( 929   ^;3583   !; 274   ?;4786   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:28:19.405
%------------------------------------------------------------------------------
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thf(sy_c_fChoice_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J,type,
    fChoic1980167584695909237ist_fm: ( produc1996495991257130529ist_fm > $o ) > produc1996495991257130529ist_fm ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J,type,
    fChoic9196029409387999681m_rule: ( produc164195504107695125m_rule > $o ) > produc164195504107695125m_rule ).

thf(sy_c_fChoice_001t__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,
    fChoic4439327894135039805ist_fm: ( produc4630866025709511057ist_fm > $o ) > produc4630866025709511057ist_fm ).

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__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__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__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_J,type,
    member5006235884692811216ist_fm: stream8299795917829157543ist_fm > set_st6731251264560614279ist_fm > $o ).

thf(sy_c_member_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_J,type,
    member6117863459850561468m_rule: stream727092118206550309m_rule > set_st6921181922238103643m_rule > $o ).

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

thf(sy_c_member_001t__Stream__Ostream_It__Syntax__Orule_J,type,
    member_stream_rule: stream_rule > set_stream_rule > $o ).

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

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

thf(sy_v_P____,type,
    p: nat ).

thf(sy_v_k____,type,
    k: nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_steps,type,
    steps: stream727092118206550309m_rule ).

thf(sy_v_thesis____,type,
    thesis: $o ).

thf(sy_v_ts____,type,
    ts: list_tm ).

% Relevant facts (1274)
thf(fact_0__092_060open_062_092_060_094bold_062_092_060ddagger_062_AP_Ats_A_092_060in_062_AtreeA_Asteps_092_060close_062,axiom,
    member_fm @ ( pre @ p @ ts ) @ ( treeA @ steps ) ).

% \<open>\<^bold>\<ddagger> P ts \<in> treeA steps\<close>
thf(fact_1__092_060open_062_092_060_094bold_062_092_060ddagger_062_AP_Ats_A_092_060in_062_AtreeB_Asteps_092_060close_062,axiom,
    member_fm @ ( pre @ p @ ts ) @ ( treeB @ steps ) ).

% \<open>\<^bold>\<ddagger> P ts \<in> treeB steps\<close>
thf(fact_2_rule_Oinject_I1_J,axiom,
    ! [X21: nat,X22: list_tm,Y21: nat,Y22: list_tm] :
      ( ( ( axiom @ X21 @ X22 )
        = ( axiom @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% rule.inject(1)
thf(fact_3__092_060open_062enabledAtStep_A_IAxiom_AP_Ats_J_A_Ishd_A_Isdrop_A_Im_A_L_Ak_J_Asteps_J_J_092_060close_062,axiom,
    ( abstra3055540779341473792ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ ( axiom @ p @ ts )
    @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ ( plus_plus_nat @ m @ k ) @ steps ) ) ) ) ).

% \<open>enabledAtStep (Axiom P ts) (shd (sdrop (m + k) steps))\<close>
thf(fact_4_m,axiom,
    member_fm @ ( pre @ p @ ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ m @ steps ) ) ) ) ) ).

% m
thf(fact_5_k,axiom,
    member_fm @ ( pre @ p @ ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ k @ steps ) ) ) ) ) ).

% k
thf(fact_6__092_060open_062_092_060And_062r_An_O_AenabledAtStep_Ar_A_Ishd_A_Isdrop_An_Asteps_J_J_A_092_060Longrightarrow_062_A_092_060exists_062k_O_Asnd_A_Ishd_A_Isdrop_Ak_A_Isdrop_An_Asteps_J_J_J_A_061_Ar_092_060close_062,axiom,
    ! [R2: rule,N: nat] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ steps ) ) ) )
     => ? [K: nat] :
          ( ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ K @ ( sdrop_7224736112439592940m_rule @ N @ steps ) ) ) )
          = R2 ) ) ).

% \<open>\<And>r n. enabledAtStep r (shd (sdrop n steps)) \<Longrightarrow> \<exists>k. snd (shd (sdrop k (sdrop n steps))) = r\<close>
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A_092_060_094bold_062_092_060ddagger_062_AP_Ats_A_091_092_060in_062_093_Alhsd_A_Isdrop_Am_Asteps_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [M: nat] :
        ~ ( member_fm @ ( pre @ p @ ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M @ steps ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>m. \<^bold>\<ddagger> P ts [\<in>] lhsd (sdrop m steps) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062k_O_A_092_060_094bold_062_092_060ddagger_062_AP_Ats_A_091_092_060in_062_093_Arhsd_A_Isdrop_Ak_Asteps_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [K: nat] :
        ~ ( member_fm @ ( pre @ p @ ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ K @ steps ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>k. \<^bold>\<ddagger> P ts [\<in>] rhsd (sdrop k steps) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_sdrop__add,axiom,
    ! [N: nat,M2: nat,S2: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( sdrop_7224736112439592940m_rule @ M2 @ S2 ) )
      = ( sdrop_7224736112439592940m_rule @ ( plus_plus_nat @ M2 @ N ) @ S2 ) ) ).

% sdrop_add
thf(fact_10_sdrop__add,axiom,
    ! [N: nat,M2: nat,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( sdrop_rule @ M2 @ S2 ) )
      = ( sdrop_rule @ ( plus_plus_nat @ M2 @ N ) @ S2 ) ) ).

% sdrop_add
thf(fact_11_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( sdrop_7373388980841566196m_rule @ N @ ( szip_P2499414959592755846m_rule @ S1 @ S22 ) )
      = ( szip_P2499414959592755846m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_12_sdrop__szip,axiom,
    ! [N: nat,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( sdrop_6392754500180887844e_rule @ N @ ( szip_P553291425819358518e_rule @ S1 @ S22 ) )
      = ( szip_P553291425819358518e_rule @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_13_sdrop__szip,axiom,
    ! [N: nat,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( sdrop_1764812713384251940m_rule @ N @ ( szip_r2254114822203693110m_rule @ S1 @ S22 ) )
      = ( szip_r2254114822203693110m_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S22 ) ) ) ).

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

% sdrop_szip
thf(fact_15_sdrop__szip,axiom,
    ! [N: nat,S1: stream8299795917829157543ist_fm,S22: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) )
      = ( szip_P1977448745965526924m_rule @ ( sdrop_4442373711808556042ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_szip
thf(fact_16__092_060open_062_092_060_094bold_062_092_060ddagger_062_AP_Ats_A_091_092_060in_062_093_Alhsd_A_Isdrop_A_Im_A_L_Ak_J_Asteps_J_092_060close_062,axiom,
    member_fm @ ( pre @ p @ ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ ( plus_plus_nat @ m @ k ) @ steps ) ) ) ) ) ).

% \<open>\<^bold>\<ddagger> P ts [\<in>] lhsd (sdrop (m + k) steps)\<close>
thf(fact_17_fm_Oinject_I1_J,axiom,
    ! [X21: nat,X22: list_tm,Y21: nat,Y22: list_tm] :
      ( ( ( pre @ X21 @ X22 )
        = ( pre @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% fm.inject(1)
thf(fact_18__092_060open_062_092_060_094bold_062_092_060ddagger_062_AP_Ats_A_091_092_060in_062_093_Arhsd_A_Isdrop_A_Im_A_L_Ak_J_Asteps_J_092_060close_062,axiom,
    member_fm @ ( pre @ p @ ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ ( plus_plus_nat @ m @ k ) @ steps ) ) ) ) ) ).

% \<open>\<^bold>\<ddagger> P ts [\<in>] rhsd (sdrop (m + k) steps)\<close>
thf(fact_19_treeB__snth,axiom,
    ! [P: fm,Steps: stream727092118206550309m_rule] :
      ( ( member_fm @ P @ ( treeB @ Steps ) )
     => ? [N2: nat] : ( member_fm @ P @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) ) ) ) ) ) ).

% treeB_snth
thf(fact_20_treeA__snth,axiom,
    ! [P: fm,Steps: stream727092118206550309m_rule] :
      ( ( member_fm @ P @ ( treeA @ Steps ) )
     => ? [N2: nat] : ( member_fm @ P @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Steps ) ) ) ) ) ) ) ).

% treeA_snth
thf(fact_21_enabled__def,axiom,
    ! [R2: rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ S2 )
      = ( ? [Sl: fset_P661503646757059847ist_fm] :
            ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Sl ) ) ) ) ).

% enabled_def
thf(fact_22_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_23_assms_I2_J,axiom,
    ( abstra8647779336736900908ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ steps ) ).

% assms(2)
thf(fact_24_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_25_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_26_prod__eq__iff,axiom,
    ( ( ^ [Y: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y = Z ) )
    = ( ^ [S: produc164195504107695125m_rule,T: produc164195504107695125m_rule] :
          ( ( ( produc6879501374131015971m_rule @ S )
            = ( produc6879501374131015971m_rule @ T ) )
          & ( ( produc7165828336582415457m_rule @ S )
            = ( produc7165828336582415457m_rule @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_27_prod__eq__iff,axiom,
    ( ( ^ [Y: produc1996495991257130529ist_fm,Z: produc1996495991257130529ist_fm] : ( Y = Z ) )
    = ( ^ [S: produc1996495991257130529ist_fm,T: produc1996495991257130529ist_fm] :
          ( ( ( produc1501393135466168645ist_fm @ S )
            = ( produc1501393135466168645ist_fm @ T ) )
          & ( ( produc4588648349897876871ist_fm @ S )
            = ( produc4588648349897876871ist_fm @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_28_exE__realizer_H,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,P: produc164195504107695125m_rule] :
      ( ( P2 @ ( produc7165828336582415457m_rule @ P ) @ ( produc6879501374131015971m_rule @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y3: rule] :
            ~ ( P2 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_29_exE__realizer_H,axiom,
    ! [P2: list_fm > list_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ( P2 @ ( produc4588648349897876871ist_fm @ P ) @ ( produc1501393135466168645ist_fm @ P ) )
     => ~ ! [X: list_fm,Y3: list_fm] :
            ~ ( P2 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_30_prod__eqI,axiom,
    ! [P: produc164195504107695125m_rule,Q: produc164195504107695125m_rule] :
      ( ( ( produc6879501374131015971m_rule @ P )
        = ( produc6879501374131015971m_rule @ Q ) )
     => ( ( ( produc7165828336582415457m_rule @ P )
          = ( produc7165828336582415457m_rule @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_31_prod__eqI,axiom,
    ! [P: produc1996495991257130529ist_fm,Q: produc1996495991257130529ist_fm] :
      ( ( ( produc1501393135466168645ist_fm @ P )
        = ( produc1501393135466168645ist_fm @ Q ) )
     => ( ( ( produc4588648349897876871ist_fm @ P )
          = ( produc4588648349897876871ist_fm @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_32_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_33_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_34_assms_I1_J,axiom,
    ( abstra5415202973855822231ist_fm
    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
        ( ( eff @ R @ S )
        = ( some_f2162114139285431020ist_fm @ Ss ) )
    @ rules
    @ steps ) ).

% assms(1)
thf(fact_35_epath__Pre__sdrop_I2_J,axiom,
    ! [Steps: stream727092118206550309m_rule,P2: nat,Ts: list_tm,M2: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) ) ) )
       => ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M2 @ Steps ) ) ) ) ) ) ) ) ).

% epath_Pre_sdrop(2)
thf(fact_36_epath__Pre__sdrop_I1_J,axiom,
    ! [Steps: stream727092118206550309m_rule,P2: nat,Ts: list_tm,M2: nat] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) ) ) )
       => ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ M2 @ Steps ) ) ) ) ) ) ) ) ).

% epath_Pre_sdrop(1)
thf(fact_37_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_38_Saturated__sdrop,axiom,
    ! [Steps: stream727092118206550309m_rule,N: nat] :
      ( ( abstra8647779336736900908ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( abstra8647779336736900908ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) ) ).

% Saturated_sdrop
thf(fact_39_enabled__ex__taken,axiom,
    ! [Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( abstra5415202973855822231ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
     => ( ( abstra8647779336736900908ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ rules
          @ Steps )
       => ( ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) )
         => ? [K: nat] :
              ( ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ K @ Steps ) ) )
              = R2 ) ) ) ) ).

% enabled_ex_taken
thf(fact_40_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_41_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_42_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_43_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_44_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_45_group__cancel_Oadd2,axiom,
    ! [B3: nat,K2: nat,B: nat,A: nat] :
      ( ( B3
        = ( plus_plus_nat @ K2 @ B ) )
     => ( ( plus_plus_nat @ A @ B3 )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_46_group__cancel_Oadd1,axiom,
    ! [A3: nat,K2: nat,A: nat,B: nat] :
      ( ( A3
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( plus_plus_nat @ A3 @ B )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_47_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( ( plus_plus_nat @ I @ K2 )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_48_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_49_pickEff,axiom,
    ! [R2: rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ S2 )
     => ( ( eff @ R2 @ S2 )
        = ( some_f2162114139285431020ist_fm
          @ ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S2 ) ) ) ) ).

% pickEff
thf(fact_50_RuleSystem__Defs_OSaturated_Ocong,axiom,
    abstra8647779336736900908ist_fm = abstra8647779336736900908ist_fm ).

% RuleSystem_Defs.Saturated.cong
thf(fact_51_mem__Collect__eq,axiom,
    ! [A: fm,P2: fm > $o] :
      ( ( member_fm @ A @ ( collect_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
    ! [A: rule,P2: rule > $o] :
      ( ( member_rule @ A @ ( collect_rule @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
    ! [A: produc1996495991257130529ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ A @ ( collec7231575776666803724ist_fm @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
    ! [A3: set_fm] :
      ( ( collect_fm
        @ ^ [X3: fm] : ( member_fm @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
    ! [A3: set_rule] :
      ( ( collect_rule
        @ ^ [X3: rule] : ( member_rule @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
    ! [A3: set_Pr7058068377845519745ist_fm] :
      ( ( collec7231575776666803724ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_57_fst__swap,axiom,
    ! [X4: produc4630866025709511057ist_fm] :
      ( ( produc6879501374131015971m_rule @ ( produc9213896635423732003ist_fm @ X4 ) )
      = ( produc4514744107061996309ist_fm @ X4 ) ) ).

% fst_swap
thf(fact_58_fst__swap,axiom,
    ! [X4: produc164195504107695125m_rule] :
      ( ( produc4228417144610596823ist_fm @ ( produc2641608828089375343m_rule @ X4 ) )
      = ( produc7165828336582415457m_rule @ X4 ) ) ).

% fst_swap
thf(fact_59_fst__swap,axiom,
    ! [X4: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc7936585717479832313ist_fm @ X4 ) )
      = ( produc4588648349897876871ist_fm @ X4 ) ) ).

% fst_swap
thf(fact_60_snd__swap,axiom,
    ! [X4: produc164195504107695125m_rule] :
      ( ( produc4514744107061996309ist_fm @ ( produc2641608828089375343m_rule @ X4 ) )
      = ( produc6879501374131015971m_rule @ X4 ) ) ).

% snd_swap
thf(fact_61_snd__swap,axiom,
    ! [X4: produc4630866025709511057ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc9213896635423732003ist_fm @ X4 ) )
      = ( produc4228417144610596823ist_fm @ X4 ) ) ).

% snd_swap
thf(fact_62_snd__swap,axiom,
    ! [X4: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc7936585717479832313ist_fm @ X4 ) )
      = ( produc1501393135466168645ist_fm @ X4 ) ) ).

% snd_swap
thf(fact_63_RuleSystem__Defs_Oepath_Ocong,axiom,
    abstra5415202973855822231ist_fm = abstra5415202973855822231ist_fm ).

% RuleSystem_Defs.epath.cong
thf(fact_64_RuleSystem__Defs_Oenabled_Ocong,axiom,
    abstra3055540779341473792ist_fm = abstra3055540779341473792ist_fm ).

% RuleSystem_Defs.enabled.cong
thf(fact_65_RuleSystem__Defs_Oenabled__def,axiom,
    ( abstra3055540779341473792ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R: rule,S: produc1996495991257130529ist_fm] :
        ? [X5: fset_P661503646757059847ist_fm] : ( Eff @ R @ S @ X5 ) ) ) ).

% RuleSystem_Defs.enabled_def
thf(fact_66_epath__eff,axiom,
    ! [Steps: stream727092118206550309m_rule,Ss2: 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 @ Ss2 ) )
       => ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps ) ) ) @ Ss2 ) ) ) ).

% epath_eff
thf(fact_67_szip_Osimps_I1_J,axiom,
    ! [S1: stream_list_fm,S22: stream_list_fm] :
      ( ( shd_Pr772355297128350925ist_fm @ ( szip_list_fm_list_fm @ S1 @ S22 ) )
      = ( produc381145313068854617ist_fm @ ( shd_list_fm @ S1 ) @ ( shd_list_fm @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_68_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr2340755956080993207m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S22 ) )
      = ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_69_szip_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( shd_Pr4631506658928855911e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S22 ) )
      = ( produc4383365014245906419e_rule @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_70_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream8299795917829157543ist_fm] :
      ( ( shd_Pr2478396429205514213ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S22 ) )
      = ( produc7063755442767259331ist_fm @ ( shd_rule @ S1 ) @ ( shd_Pr772355297128350925ist_fm @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_71_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr3564872132220007m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S22 ) )
      = ( produc6084188410630241011m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_72_szip_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( shd_Pr4573454438745646103e_rule @ ( szip_rule_rule @ S1 @ S22 ) )
      = ( produc1769597910952761123e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_73_szip_Osimps_I1_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S22: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) )
      = ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% szip.simps(1)
thf(fact_74_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_75_old_Oprod_Oinject,axiom,
    ! [A: list_fm,B: list_fm,A4: list_fm,B4: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_76_old_Oprod_Oinject,axiom,
    ! [A: produc1996495991257130529ist_fm,B: rule,A4: produc1996495991257130529ist_fm,B4: rule] :
      ( ( ( produc491467635432902671m_rule @ A @ B )
        = ( produc491467635432902671m_rule @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_77_old_Oprod_Oinject,axiom,
    ! [A: rule,B: produc1996495991257130529ist_fm,A4: rule,B4: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B )
        = ( produc7063755442767259331ist_fm @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_78_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_79_prod_Oinject,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule,Y1: produc1996495991257130529ist_fm,Y2: rule] :
      ( ( ( produc491467635432902671m_rule @ X1 @ X2 )
        = ( produc491467635432902671m_rule @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_80_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_81_swap__simp,axiom,
    ! [X4: list_fm,Y4: list_fm] :
      ( ( produc7936585717479832313ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y4 ) )
      = ( produc381145313068854617ist_fm @ Y4 @ X4 ) ) ).

% swap_simp
thf(fact_82_swap__simp,axiom,
    ! [X4: produc1996495991257130529ist_fm,Y4: rule] :
      ( ( produc2641608828089375343m_rule @ ( produc491467635432902671m_rule @ X4 @ Y4 ) )
      = ( produc7063755442767259331ist_fm @ Y4 @ X4 ) ) ).

% swap_simp
thf(fact_83_swap__simp,axiom,
    ! [X4: rule,Y4: produc1996495991257130529ist_fm] :
      ( ( produc9213896635423732003ist_fm @ ( produc7063755442767259331ist_fm @ X4 @ Y4 ) )
      = ( produc491467635432902671m_rule @ Y4 @ X4 ) ) ).

% swap_simp
thf(fact_84_prod_Ocollapse,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_85_prod_Ocollapse,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_86_prod_Ocollapse,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_87_prod__induct3,axiom,
    ! [P2: produc4630866025709511057ist_fm > $o,X4: produc4630866025709511057ist_fm] :
      ( ! [A5: rule,B5: list_fm,C2: list_fm] : ( P2 @ ( produc7063755442767259331ist_fm @ A5 @ ( produc381145313068854617ist_fm @ B5 @ C2 ) ) )
     => ( P2 @ X4 ) ) ).

% prod_induct3
thf(fact_88_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_89_Pair__inject,axiom,
    ! [A: list_fm,B: list_fm,A4: list_fm,B4: list_fm] :
      ( ( ( produc381145313068854617ist_fm @ A @ B )
        = ( produc381145313068854617ist_fm @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_90_Pair__inject,axiom,
    ! [A: produc1996495991257130529ist_fm,B: rule,A4: produc1996495991257130529ist_fm,B4: rule] :
      ( ( ( produc491467635432902671m_rule @ A @ B )
        = ( produc491467635432902671m_rule @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_91_Pair__inject,axiom,
    ! [A: rule,B: produc1996495991257130529ist_fm,A4: rule,B4: produc1996495991257130529ist_fm] :
      ( ( ( produc7063755442767259331ist_fm @ A @ B )
        = ( produc7063755442767259331ist_fm @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_92_prod__cases,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ! [A5: list_fm,B5: list_fm] : ( P2 @ ( produc381145313068854617ist_fm @ A5 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_93_prod__cases,axiom,
    ! [P2: produc164195504107695125m_rule > $o,P: produc164195504107695125m_rule] :
      ( ! [A5: produc1996495991257130529ist_fm,B5: rule] : ( P2 @ ( produc491467635432902671m_rule @ A5 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_94_prod__cases,axiom,
    ! [P2: produc4630866025709511057ist_fm > $o,P: produc4630866025709511057ist_fm] :
      ( ! [A5: rule,B5: produc1996495991257130529ist_fm] : ( P2 @ ( produc7063755442767259331ist_fm @ A5 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_95_surj__pair,axiom,
    ! [P: produc1996495991257130529ist_fm] :
    ? [X: list_fm,Y3: list_fm] :
      ( P
      = ( produc381145313068854617ist_fm @ X @ Y3 ) ) ).

% surj_pair
thf(fact_96_surj__pair,axiom,
    ! [P: produc164195504107695125m_rule] :
    ? [X: produc1996495991257130529ist_fm,Y3: rule] :
      ( P
      = ( produc491467635432902671m_rule @ X @ Y3 ) ) ).

% surj_pair
thf(fact_97_surj__pair,axiom,
    ! [P: produc4630866025709511057ist_fm] :
    ? [X: rule,Y3: produc1996495991257130529ist_fm] :
      ( P
      = ( produc7063755442767259331ist_fm @ X @ Y3 ) ) ).

% surj_pair
thf(fact_98_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc1996495991257130529ist_fm] :
      ~ ! [A5: list_fm,B5: list_fm] :
          ( Y4
         != ( produc381145313068854617ist_fm @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_99_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc164195504107695125m_rule] :
      ~ ! [A5: produc1996495991257130529ist_fm,B5: rule] :
          ( Y4
         != ( produc491467635432902671m_rule @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_100_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc4630866025709511057ist_fm] :
      ~ ! [A5: rule,B5: produc1996495991257130529ist_fm] :
          ( Y4
         != ( produc7063755442767259331ist_fm @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_101_RuleSystem__Defs_OmkTree_Ocong,axiom,
    abstra178918799166095863ist_fm = abstra178918799166095863ist_fm ).

% RuleSystem_Defs.mkTree.cong
thf(fact_102_RuleSystem__Defs_OpickEff_Ocong,axiom,
    abstra458490714785879965ist_fm = abstra458490714785879965ist_fm ).

% RuleSystem_Defs.pickEff.cong
thf(fact_103_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr2200982372225425851m_rule @ ( szip_P2499414959592755846m_rule @ S1 @ S22 ) )
      = ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_104_szip_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( stl_Pr830518440239794027e_rule @ ( szip_P553291425819358518e_rule @ S1 @ S22 ) )
      = ( szip_P553291425819358518e_rule @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_105_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr5425948690297933931m_rule @ ( szip_r2254114822203693110m_rule @ S1 @ S22 ) )
      = ( szip_r2254114822203693110m_rule @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_106_szip_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( stl_Pr4904654673476115995e_rule @ ( szip_rule_rule @ S1 @ S22 ) )
      = ( szip_rule_rule @ ( stl_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_107_szip_Osimps_I2_J,axiom,
    ! [S1: stream8299795917829157543ist_fm,S22: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) )
      = ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% szip.simps(2)
thf(fact_108_RuleSystem__Defs_OpickEff,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R2: rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ Eff2 @ R2 @ S2 )
     => ( Eff2 @ R2 @ S2 @ ( abstra458490714785879965ist_fm @ Eff2 @ R2 @ S2 ) ) ) ).

% RuleSystem_Defs.pickEff
thf(fact_109_stream_Ocoinduct__strong,axiom,
    ! [R3: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( ( R3 @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) )
                | ( ( stl_Pr950425576149878629m_rule @ Stream3 )
                  = ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_110_stream_Ocoinduct__strong,axiom,
    ! [R3: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( ( R3 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) )
                | ( ( stl_rule @ Stream3 )
                  = ( stl_rule @ Stream4 ) ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct_strong
thf(fact_111_stream_Ocoinduct,axiom,
    ! [R3: stream727092118206550309m_rule > stream727092118206550309m_rule > $o,Stream: stream727092118206550309m_rule,Stream2: stream727092118206550309m_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream727092118206550309m_rule,Stream4: stream727092118206550309m_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_Pr7235097944458474089m_rule @ Stream3 )
                = ( shd_Pr7235097944458474089m_rule @ Stream4 ) )
              & ( R3 @ ( stl_Pr950425576149878629m_rule @ Stream3 ) @ ( stl_Pr950425576149878629m_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_112_stream_Ocoinduct,axiom,
    ! [R3: stream_rule > stream_rule > $o,Stream: stream_rule,Stream2: stream_rule] :
      ( ( R3 @ Stream @ Stream2 )
     => ( ! [Stream3: stream_rule,Stream4: stream_rule] :
            ( ( R3 @ Stream3 @ Stream4 )
           => ( ( ( shd_rule @ Stream3 )
                = ( shd_rule @ Stream4 ) )
              & ( R3 @ ( stl_rule @ Stream3 ) @ ( stl_rule @ Stream4 ) ) ) )
       => ( Stream = Stream2 ) ) ) ).

% stream.coinduct
thf(fact_113_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_114_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_115_sdrop__stl,axiom,
    ! [N: nat,S2: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( stl_Pr950425576149878629m_rule @ S2 ) )
      = ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) ) ).

% sdrop_stl
thf(fact_116_sdrop__stl,axiom,
    ! [N: nat,S2: stream_rule] :
      ( ( sdrop_rule @ N @ ( stl_rule @ S2 ) )
      = ( stl_rule @ ( sdrop_rule @ N @ S2 ) ) ) ).

% sdrop_stl
thf(fact_117_fst__eqD,axiom,
    ! [X4: rule,Y4: produc1996495991257130529ist_fm,A: rule] :
      ( ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X4 @ Y4 ) )
        = A )
     => ( X4 = A ) ) ).

% fst_eqD
thf(fact_118_fst__eqD,axiom,
    ! [X4: produc1996495991257130529ist_fm,Y4: rule,A: produc1996495991257130529ist_fm] :
      ( ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X4 @ Y4 ) )
        = A )
     => ( X4 = A ) ) ).

% fst_eqD
thf(fact_119_fst__eqD,axiom,
    ! [X4: list_fm,Y4: list_fm,A: list_fm] :
      ( ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y4 ) )
        = A )
     => ( X4 = A ) ) ).

% fst_eqD
thf(fact_120_fst__conv,axiom,
    ! [X1: rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_121_fst__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_122_fst__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_123_snd__conv,axiom,
    ! [X1: rule,X2: produc1996495991257130529ist_fm] :
      ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_124_snd__conv,axiom,
    ! [X1: produc1996495991257130529ist_fm,X2: rule] :
      ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_125_snd__conv,axiom,
    ! [X1: list_fm,X2: list_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_126_snd__eqD,axiom,
    ! [X4: rule,Y4: produc1996495991257130529ist_fm,A: produc1996495991257130529ist_fm] :
      ( ( ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X4 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_127_snd__eqD,axiom,
    ! [X4: produc1996495991257130529ist_fm,Y4: rule,A: rule] :
      ( ( ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X4 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_128_snd__eqD,axiom,
    ! [X4: list_fm,Y4: list_fm,A: list_fm] :
      ( ( ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_129_prod_Oswap__def,axiom,
    ( produc9213896635423732003ist_fm
    = ( ^ [P3: produc4630866025709511057ist_fm] : ( produc491467635432902671m_rule @ ( produc4514744107061996309ist_fm @ P3 ) @ ( produc4228417144610596823ist_fm @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_130_prod_Oswap__def,axiom,
    ( produc2641608828089375343m_rule
    = ( ^ [P3: produc164195504107695125m_rule] : ( produc7063755442767259331ist_fm @ ( produc7165828336582415457m_rule @ P3 ) @ ( produc6879501374131015971m_rule @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_131_prod_Oswap__def,axiom,
    ( produc7936585717479832313ist_fm
    = ( ^ [P3: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc4588648349897876871ist_fm @ P3 ) @ ( produc1501393135466168645ist_fm @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_132_per__rules_H,axiom,
    ! [R2: rule,A3: list_fm,B3: list_fm,A6: list_fm,B6: list_fm,R4: rule,Ss3: fset_P661503646757059847ist_fm] :
      ( ( abstra3055540779341473792ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ R2
        @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
     => ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ ( produc381145313068854617ist_fm @ A6 @ B6 ) )
       => ( ( ( eff @ R4 @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
            = ( some_f2162114139285431020ist_fm @ Ss3 ) )
         => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B6 ) @ Ss3 )
           => ( R4 = R2 ) ) ) ) ) ).

% per_rules'
thf(fact_133_eff__preserves__Pre_I2_J,axiom,
    ! [A3: list_fm,B3: list_fm,R2: rule,Ss2: fset_P661503646757059847ist_fm,A6: list_fm,B6: list_fm,P2: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ R2 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ R2 ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B6 ) @ Ss2 )
       => ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B3 ) )
         => ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B6 ) ) ) ) ) ).

% eff_preserves_Pre(2)
thf(fact_134_eff__preserves__Pre_I1_J,axiom,
    ! [A3: list_fm,B3: list_fm,R2: rule,Ss2: fset_P661503646757059847ist_fm,A6: list_fm,B6: list_fm,P2: nat,Ts: list_tm] :
      ( ( ( eff @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ R2 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ R2 ) ) )
        = ( some_f2162114139285431020ist_fm @ Ss2 ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc381145313068854617ist_fm @ A6 @ B6 ) @ Ss2 )
       => ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A3 ) )
         => ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A6 ) ) ) ) ) ).

% eff_preserves_Pre(1)
thf(fact_135_surjective__pairing,axiom,
    ! [T2: produc4630866025709511057ist_fm] :
      ( T2
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ T2 ) @ ( produc4514744107061996309ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_136_surjective__pairing,axiom,
    ! [T2: produc164195504107695125m_rule] :
      ( T2
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ T2 ) @ ( produc7165828336582415457m_rule @ T2 ) ) ) ).

% surjective_pairing
thf(fact_137_surjective__pairing,axiom,
    ! [T2: produc1996495991257130529ist_fm] :
      ( T2
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ T2 ) @ ( produc4588648349897876871ist_fm @ T2 ) ) ) ).

% surjective_pairing
thf(fact_138_conjI__realizer,axiom,
    ! [P2: rule > $o,P: rule,Q2: produc1996495991257130529ist_fm > $o,Q: produc1996495991257130529ist_fm] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ P @ Q ) ) )
          & ( Q2 @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_139_conjI__realizer,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,P: produc1996495991257130529ist_fm,Q2: rule > $o,Q: rule] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) )
          & ( Q2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_140_conjI__realizer,axiom,
    ! [P2: list_fm > $o,P: list_fm,Q2: list_fm > $o,Q: list_fm] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) )
          & ( Q2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_141_exI__realizer,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,Y4: produc1996495991257130529ist_fm,X4: rule] :
      ( ( P2 @ Y4 @ X4 )
     => ( P2 @ ( produc4514744107061996309ist_fm @ ( produc7063755442767259331ist_fm @ X4 @ Y4 ) ) @ ( produc4228417144610596823ist_fm @ ( produc7063755442767259331ist_fm @ X4 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_142_exI__realizer,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,Y4: rule,X4: produc1996495991257130529ist_fm] :
      ( ( P2 @ Y4 @ X4 )
     => ( P2 @ ( produc7165828336582415457m_rule @ ( produc491467635432902671m_rule @ X4 @ Y4 ) ) @ ( produc6879501374131015971m_rule @ ( produc491467635432902671m_rule @ X4 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_143_exI__realizer,axiom,
    ! [P2: list_fm > list_fm > $o,Y4: list_fm,X4: list_fm] :
      ( ( P2 @ Y4 @ X4 )
     => ( P2 @ ( produc4588648349897876871ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y4 ) ) @ ( produc1501393135466168645ist_fm @ ( produc381145313068854617ist_fm @ X4 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_144_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4630866025709511057ist_fm] :
      ( Prod
      = ( produc7063755442767259331ist_fm @ ( produc4228417144610596823ist_fm @ Prod ) @ ( produc4514744107061996309ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_145_prod_Oexhaust__sel,axiom,
    ! [Prod: produc164195504107695125m_rule] :
      ( Prod
      = ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Prod ) @ ( produc7165828336582415457m_rule @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_146_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1996495991257130529ist_fm] :
      ( Prod
      = ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Prod ) @ ( produc4588648349897876871ist_fm @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_147_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: rule > produc1996495991257130529ist_fm > $o,X4: rule,Y4: produc1996495991257130529ist_fm,A: produc4630866025709511057ist_fm] :
      ( ( P2 @ X4 @ Y4 )
     => ( ( A
          = ( produc7063755442767259331ist_fm @ X4 @ Y4 ) )
       => ( P2 @ ( produc4228417144610596823ist_fm @ A ) @ ( produc4514744107061996309ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_148_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,X4: produc1996495991257130529ist_fm,Y4: rule,A: produc164195504107695125m_rule] :
      ( ( P2 @ X4 @ Y4 )
     => ( ( A
          = ( produc491467635432902671m_rule @ X4 @ Y4 ) )
       => ( P2 @ ( produc6879501374131015971m_rule @ A ) @ ( produc7165828336582415457m_rule @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_149_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_fm > list_fm > $o,X4: list_fm,Y4: list_fm,A: produc1996495991257130529ist_fm] :
      ( ( P2 @ X4 @ Y4 )
     => ( ( A
          = ( produc381145313068854617ist_fm @ X4 @ Y4 ) )
       => ( P2 @ ( produc1501393135466168645ist_fm @ A ) @ ( produc4588648349897876871ist_fm @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_150_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_151_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_152_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_153_epath_Ocoinduct,axiom,
    ! [X6: stream727092118206550309m_rule > $o,X4: stream727092118206550309m_rule] :
      ( ( X6 @ X4 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X6 @ X )
           => ? [Steps3: stream727092118206550309m_rule,Sl4: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl4 )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) )
                  = ( some_f2162114139285431020ist_fm @ Sl4 ) )
                & ( ( X6 @ ( 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
          @ X4 ) ) ) ).

% epath.coinduct
thf(fact_154_eq__snd__iff,axiom,
    ! [B: produc1996495991257130529ist_fm,P: produc4630866025709511057ist_fm] :
      ( ( B
        = ( produc4514744107061996309ist_fm @ P ) )
      = ( ? [A2: rule] :
            ( P
            = ( produc7063755442767259331ist_fm @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_155_eq__snd__iff,axiom,
    ! [B: rule,P: produc164195504107695125m_rule] :
      ( ( B
        = ( produc7165828336582415457m_rule @ P ) )
      = ( ? [A2: produc1996495991257130529ist_fm] :
            ( P
            = ( produc491467635432902671m_rule @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_156_eq__snd__iff,axiom,
    ! [B: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( B
        = ( produc4588648349897876871ist_fm @ P ) )
      = ( ? [A2: list_fm] :
            ( P
            = ( produc381145313068854617ist_fm @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_157_sndI,axiom,
    ! [X4: produc4630866025709511057ist_fm,Y4: rule,Z2: produc1996495991257130529ist_fm] :
      ( ( X4
        = ( produc7063755442767259331ist_fm @ Y4 @ Z2 ) )
     => ( ( produc4514744107061996309ist_fm @ X4 )
        = Z2 ) ) ).

% sndI
thf(fact_158_sndI,axiom,
    ! [X4: produc164195504107695125m_rule,Y4: produc1996495991257130529ist_fm,Z2: rule] :
      ( ( X4
        = ( produc491467635432902671m_rule @ Y4 @ Z2 ) )
     => ( ( produc7165828336582415457m_rule @ X4 )
        = Z2 ) ) ).

% sndI
thf(fact_159_sndI,axiom,
    ! [X4: produc1996495991257130529ist_fm,Y4: list_fm,Z2: list_fm] :
      ( ( X4
        = ( produc381145313068854617ist_fm @ Y4 @ Z2 ) )
     => ( ( produc4588648349897876871ist_fm @ X4 )
        = Z2 ) ) ).

% sndI
thf(fact_160_fstI,axiom,
    ! [X4: produc4630866025709511057ist_fm,Y4: rule,Z2: produc1996495991257130529ist_fm] :
      ( ( X4
        = ( produc7063755442767259331ist_fm @ Y4 @ Z2 ) )
     => ( ( produc4228417144610596823ist_fm @ X4 )
        = Y4 ) ) ).

% fstI
thf(fact_161_fstI,axiom,
    ! [X4: produc164195504107695125m_rule,Y4: produc1996495991257130529ist_fm,Z2: rule] :
      ( ( X4
        = ( produc491467635432902671m_rule @ Y4 @ Z2 ) )
     => ( ( produc6879501374131015971m_rule @ X4 )
        = Y4 ) ) ).

% fstI
thf(fact_162_fstI,axiom,
    ! [X4: produc1996495991257130529ist_fm,Y4: list_fm,Z2: list_fm] :
      ( ( X4
        = ( produc381145313068854617ist_fm @ Y4 @ Z2 ) )
     => ( ( produc1501393135466168645ist_fm @ X4 )
        = Y4 ) ) ).

% fstI
thf(fact_163_eq__fst__iff,axiom,
    ! [A: rule,P: produc4630866025709511057ist_fm] :
      ( ( A
        = ( produc4228417144610596823ist_fm @ P ) )
      = ( ? [B2: produc1996495991257130529ist_fm] :
            ( P
            = ( produc7063755442767259331ist_fm @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_164_eq__fst__iff,axiom,
    ! [A: produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
      ( ( A
        = ( produc6879501374131015971m_rule @ P ) )
      = ( ? [B2: rule] :
            ( P
            = ( produc491467635432902671m_rule @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_165_eq__fst__iff,axiom,
    ! [A: list_fm,P: produc1996495991257130529ist_fm] :
      ( ( A
        = ( produc1501393135466168645ist_fm @ P ) )
      = ( ? [B2: list_fm] :
            ( P
            = ( produc381145313068854617ist_fm @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_166_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,A: stream5128679079952569136_fm_fm] :
      ( ( abstra9155489748554225612ist_fm @ Eff2 @ 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 )
             => ( ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ A ) ) @ Sl2 )
               => ~ ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_167_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream8299795917829157543ist_fm,A: stream3409308193418444653ist_fm] :
      ( ( abstra5061764961602482947ist_fm @ Eff2 @ 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 )
             => ( ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ ( stl_Pr8700568296466786583ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_168_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_fm,A: stream812778442145099016ule_fm] :
      ( ( abstra1364558218721404774m_rule @ Eff2 @ 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 )
             => ( ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ A ) ) @ Sl2 )
               => ~ ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ ( stl_Pr2064664776838053682ule_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_169_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream8299795917829157543ist_fm,A: stream7989562319868115861ist_fm] :
      ( ( abstra8066765647771630831m_rule @ Eff2 @ 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 )
             => ( ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ ( stl_Pr6727363551237990101ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_170_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_rule,A: stream7755889244956737277e_rule] :
      ( ( abstra5823804033175701339m_rule @ Eff2 @ 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 )
             => ( ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ A ) ) @ Sl2 )
               => ~ ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ ( stl_Pr8695508334200742183e_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_171_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,A: stream642806101564963573m_rule] :
      ( ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ A ) ) ) @ Sl2 )
             => ( ( Eff2 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ ( stl_Pr8578586445745536693m_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_172_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: produc164195504107695125m_rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream727092118206550309m_rule,A: stream5838387310478709965m_rule] :
      ( ( abstra3709527095716548139m_rule @ Eff2 @ Rules @ A )
     => ~ ( ( member4220325220686508332m_rule @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ A ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ ( stl_Pr7399785186797918583m_rule @ A ) ) ) @ Sl2 )
             => ( ( Eff2 @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ A ) ) @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra3709527095716548139m_rule @ Eff2 @ Rules @ ( stl_Pr7399785186797918583m_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_173_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,A: stream8299795917829157543ist_fm] :
      ( ( abstra6579460235205289861ist_fm @ Eff2 @ 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 )
             => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) ) @ Sl2 )
               => ~ ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ ( stl_Pr5027778045650968273ist_fm @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_174_RuleSystem__Defs_Oepath_Ocases,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A: stream727092118206550309m_rule] :
      ( ( abstra5415202973855822231ist_fm @ Eff2 @ 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 )
             => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ) ) ).

% RuleSystem_Defs.epath.cases
thf(fact_175_shd__sset,axiom,
    ! [A: stream_fm] : ( member_fm @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).

% shd_sset
thf(fact_176_shd__sset,axiom,
    ! [A: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ A ) @ ( sset_P3431374123360452149ist_fm @ A ) ) ).

% shd_sset
thf(fact_177_shd__sset,axiom,
    ! [A: stream_rule] : ( member_rule @ ( shd_rule @ A ) @ ( sset_rule @ A ) ) ).

% shd_sset
thf(fact_178_shd__sset,axiom,
    ! [A: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ A ) @ ( sset_P4979403222752902913m_rule @ A ) ) ).

% shd_sset
thf(fact_179_stl__sset,axiom,
    ! [X4: fm,A: stream_fm] :
      ( ( member_fm @ X4 @ ( sset_fm @ ( stl_fm @ A ) ) )
     => ( member_fm @ X4 @ ( sset_fm @ A ) ) ) ).

% stl_sset
thf(fact_180_stl__sset,axiom,
    ! [X4: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm] :
      ( ( member8102475879199740618ist_fm @ X4 @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ A ) ) )
     => ( member8102475879199740618ist_fm @ X4 @ ( sset_P3431374123360452149ist_fm @ A ) ) ) ).

% stl_sset
thf(fact_181_stl__sset,axiom,
    ! [X4: rule,A: stream_rule] :
      ( ( member_rule @ X4 @ ( sset_rule @ ( stl_rule @ A ) ) )
     => ( member_rule @ X4 @ ( sset_rule @ A ) ) ) ).

% stl_sset
thf(fact_182_stl__sset,axiom,
    ! [X4: produc164195504107695125m_rule,A: stream727092118206550309m_rule] :
      ( ( member4220325220686508332m_rule @ X4 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ A ) ) )
     => ( member4220325220686508332m_rule @ X4 @ ( sset_P4979403222752902913m_rule @ A ) ) ) ).

% stl_sset
thf(fact_183_sset__induct,axiom,
    ! [Y4: fm,S2: stream_fm,P2: fm > stream_fm > $o] :
      ( ( member_fm @ Y4 @ ( sset_fm @ S2 ) )
     => ( ! [S3: stream_fm] : ( P2 @ ( shd_fm @ S3 ) @ S3 )
       => ( ! [S3: stream_fm,Y3: fm] :
              ( ( member_fm @ Y3 @ ( sset_fm @ ( stl_fm @ S3 ) ) )
             => ( ( P2 @ Y3 @ ( stl_fm @ S3 ) )
               => ( P2 @ Y3 @ S3 ) ) )
         => ( P2 @ Y4 @ S2 ) ) ) ) ).

% sset_induct
thf(fact_184_sset__induct,axiom,
    ! [Y4: produc1996495991257130529ist_fm,S2: stream8299795917829157543ist_fm,P2: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ Y4 @ ( sset_P3431374123360452149ist_fm @ S2 ) )
     => ( ! [S3: stream8299795917829157543ist_fm] : ( P2 @ ( shd_Pr772355297128350925ist_fm @ S3 ) @ S3 )
       => ( ! [S3: stream8299795917829157543ist_fm,Y3: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Y3 @ ( sset_P3431374123360452149ist_fm @ ( stl_Pr5027778045650968273ist_fm @ S3 ) ) )
             => ( ( P2 @ Y3 @ ( stl_Pr5027778045650968273ist_fm @ S3 ) )
               => ( P2 @ Y3 @ S3 ) ) )
         => ( P2 @ Y4 @ S2 ) ) ) ) ).

% sset_induct
thf(fact_185_sset__induct,axiom,
    ! [Y4: rule,S2: stream_rule,P2: rule > stream_rule > $o] :
      ( ( member_rule @ Y4 @ ( sset_rule @ S2 ) )
     => ( ! [S3: stream_rule] : ( P2 @ ( shd_rule @ S3 ) @ S3 )
       => ( ! [S3: stream_rule,Y3: rule] :
              ( ( member_rule @ Y3 @ ( sset_rule @ ( stl_rule @ S3 ) ) )
             => ( ( P2 @ Y3 @ ( stl_rule @ S3 ) )
               => ( P2 @ Y3 @ S3 ) ) )
         => ( P2 @ Y4 @ S2 ) ) ) ) ).

% sset_induct
thf(fact_186_sset__induct,axiom,
    ! [Y4: produc164195504107695125m_rule,S2: stream727092118206550309m_rule,P2: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ Y4 @ ( sset_P4979403222752902913m_rule @ S2 ) )
     => ( ! [S3: stream727092118206550309m_rule] : ( P2 @ ( shd_Pr7235097944458474089m_rule @ S3 ) @ S3 )
       => ( ! [S3: stream727092118206550309m_rule,Y3: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Y3 @ ( sset_P4979403222752902913m_rule @ ( stl_Pr950425576149878629m_rule @ S3 ) ) )
             => ( ( P2 @ Y3 @ ( stl_Pr950425576149878629m_rule @ S3 ) )
               => ( P2 @ Y3 @ S3 ) ) )
         => ( P2 @ Y4 @ S2 ) ) ) ) ).

% sset_induct
thf(fact_187_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X6: stream5128679079952569136_fm_fm > $o,X4: stream5128679079952569136_fm_fm,Rules: stream_fm,Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream5128679079952569136_fm_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream5128679079952569136_fm_fm,Sl4: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) @ Sl4 )
                & ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) )
                  | ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_188_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X6: stream3409308193418444653ist_fm > $o,X4: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream3409308193418444653ist_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream3409308193418444653ist_fm,Sl4: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl4 )
                & ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) )
                  | ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_189_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X6: stream812778442145099016ule_fm > $o,X4: stream812778442145099016ule_fm,Rules: stream_fm,Eff2: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream812778442145099016ule_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream812778442145099016ule_fm,Sl4: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl4 )
                & ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) )
                  | ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_190_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X6: stream7989562319868115861ist_fm > $o,X4: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream7989562319868115861ist_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream7989562319868115861ist_fm,Sl4: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl4 )
                & ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) )
                  | ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_191_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X6: stream7755889244956737277e_rule > $o,X4: stream7755889244956737277e_rule,Rules: stream_rule,Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream7755889244956737277e_rule] :
            ( ( X6 @ X )
           => ? [Steps3: stream7755889244956737277e_rule,Sl4: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl4 )
                & ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) )
                  | ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_192_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X6: stream642806101564963573m_rule > $o,X4: stream642806101564963573m_rule,Rules: stream727092118206550309m_rule,Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream642806101564963573m_rule] :
            ( ( X6 @ X )
           => ? [Steps3: stream642806101564963573m_rule,Sl4: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) ) ) @ Sl4 )
                & ( Eff2 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) )
                  | ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) ) ) ) )
       => ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_193_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X6: stream5838387310478709965m_rule > $o,X4: stream5838387310478709965m_rule,Rules: stream727092118206550309m_rule,Eff2: produc164195504107695125m_rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream5838387310478709965m_rule] :
            ( ( X6 @ X )
           => ? [Steps3: stream5838387310478709965m_rule,Sl4: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member4220325220686508332m_rule @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps3 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ ( stl_Pr7399785186797918583m_rule @ Steps3 ) ) ) @ Sl4 )
                & ( Eff2 @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps3 ) ) @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr7399785186797918583m_rule @ Steps3 ) )
                  | ( abstra3709527095716548139m_rule @ Eff2 @ Rules @ ( stl_Pr7399785186797918583m_rule @ Steps3 ) ) ) ) )
       => ( abstra3709527095716548139m_rule @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_194_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X6: stream8299795917829157543ist_fm > $o,X4: stream8299795917829157543ist_fm,Rules: stream_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream8299795917829157543ist_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream8299795917829157543ist_fm,Sl4: fset_list_fm] :
                ( ( X = 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 ) ) ) @ Sl4 )
                & ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) )
                  | ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) ) )
       => ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_195_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
    ! [X6: stream727092118206550309m_rule > $o,X4: stream727092118206550309m_rule,Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X6 @ X )
           => ? [Steps3: stream727092118206550309m_rule,Sl4: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl4 )
                & ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.epath.coinduct
thf(fact_196_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream5128679079952569136_fm_fm,Rules: stream_fm,Sl3: fset_P661503646757059847ist_fm,Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( sset_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps ) )
           => ( abstra9155489748554225612ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_197_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_P661503646757059847ist_fm,Eff2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps ) )
           => ( abstra5061764961602482947ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_198_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream812778442145099016ule_fm,Rules: stream_fm,Sl3: fset_A5396818216090053761m_rule,Eff2: 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 )
       => ( ( Eff2 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps ) )
           => ( abstra1364558218721404774m_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_199_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Sl3: fset_A5396818216090053761m_rule,Eff2: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps ) )
           => ( abstra8066765647771630831m_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_200_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream7755889244956737277e_rule,Rules: stream_rule,Sl3: fset_A5396818216090053761m_rule,Eff2: 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 )
       => ( ( Eff2 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps ) )
           => ( abstra5823804033175701339m_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_201_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream642806101564963573m_rule,Rules: stream727092118206550309m_rule,Sl3: fset_P661503646757059847ist_fm,Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps ) )
           => ( abstra6058711572139910855ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_202_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream5838387310478709965m_rule,Rules: stream727092118206550309m_rule,Sl3: fset_A5396818216090053761m_rule,Eff2: produc164195504107695125m_rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o] :
      ( ( member4220325220686508332m_rule @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ ( stl_Pr7399785186797918583m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps ) ) @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra3709527095716548139m_rule @ Eff2 @ Rules @ ( stl_Pr7399785186797918583m_rule @ Steps ) )
           => ( abstra3709527095716548139m_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_203_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream8299795917829157543ist_fm,Rules: stream_list_fm,Sl3: fset_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( fmember_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( stl_Pr5027778045650968273ist_fm @ Steps ) ) ) @ Sl3 )
       => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ Sl3 )
         => ( ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps ) )
           => ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_204_RuleSystem__Defs_Oepath_Ointros,axiom,
    ! [Steps: stream727092118206550309m_rule,Rules: stream_rule,Sl3: fset_P661503646757059847ist_fm,Eff2: 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 )
       => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps ) )
           => ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% RuleSystem_Defs.epath.intros
thf(fact_205_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra9155489748554225612ist_fm
    = ( ^ [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,A2: stream5128679079952569136_fm_fm] :
        ? [Steps2: stream5128679079952569136_fm_fm,Sl: fset_P661503646757059847ist_fm] :
          ( ( A2 = 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 )
          & ( Eff @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps2 ) ) @ Sl )
          & ( abstra9155489748554225612ist_fm @ Eff @ Rules2 @ ( stl_Pr2715999368752424048_fm_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_206_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5061764961602482947ist_fm
    = ( ^ [Eff: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream8299795917829157543ist_fm,A2: stream3409308193418444653ist_fm] :
        ? [Steps2: stream3409308193418444653ist_fm,Sl: fset_P661503646757059847ist_fm] :
          ( ( A2 = 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 )
          & ( Eff @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps2 ) ) @ Sl )
          & ( abstra5061764961602482947ist_fm @ Eff @ Rules2 @ ( stl_Pr8700568296466786583ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_207_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra1364558218721404774m_rule
    = ( ^ [Eff: fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_fm,A2: stream812778442145099016ule_fm] :
        ? [Steps2: stream812778442145099016ule_fm,Sl: fset_A5396818216090053761m_rule] :
          ( ( A2 = Steps2 )
          & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( sset_fm @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps2 ) ) @ Sl )
          & ( abstra1364558218721404774m_rule @ Eff @ Rules2 @ ( stl_Pr2064664776838053682ule_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_208_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra8066765647771630831m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream8299795917829157543ist_fm,A2: stream7989562319868115861ist_fm] :
        ? [Steps2: stream7989562319868115861ist_fm,Sl: fset_A5396818216090053761m_rule] :
          ( ( A2 = 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 )
          & ( Eff @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps2 ) ) @ Sl )
          & ( abstra8066765647771630831m_rule @ Eff @ Rules2 @ ( stl_Pr6727363551237990101ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_209_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5823804033175701339m_rule
    = ( ^ [Eff: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream_rule,A2: stream7755889244956737277e_rule] :
        ? [Steps2: stream7755889244956737277e_rule,Sl: fset_A5396818216090053761m_rule] :
          ( ( A2 = Steps2 )
          & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps2 ) ) @ Sl )
          & ( abstra5823804033175701339m_rule @ Eff @ Rules2 @ ( stl_Pr8695508334200742183e_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_210_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6058711572139910855ist_fm
    = ( ^ [Eff: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream727092118206550309m_rule,A2: stream642806101564963573m_rule] :
        ? [Steps2: stream642806101564963573m_rule,Sl: fset_P661503646757059847ist_fm] :
          ( ( A2 = Steps2 )
          & ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ Sl )
          & ( abstra6058711572139910855ist_fm @ Eff @ Rules2 @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_211_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra3709527095716548139m_rule
    = ( ^ [Eff: produc164195504107695125m_rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules2: stream727092118206550309m_rule,A2: stream5838387310478709965m_rule] :
        ? [Steps2: stream5838387310478709965m_rule,Sl: fset_A5396818216090053761m_rule] :
          ( ( A2 = Steps2 )
          & ( member4220325220686508332m_rule @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps2 ) ) @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
          & ( fmembe1635983029874454474m_rule @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ ( stl_Pr7399785186797918583m_rule @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps2 ) ) @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps2 ) ) @ Sl )
          & ( abstra3709527095716548139m_rule @ Eff @ Rules2 @ ( stl_Pr7399785186797918583m_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_212_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra6579460235205289861ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,A2: stream8299795917829157543ist_fm] :
        ? [Steps2: stream8299795917829157543ist_fm,Sl: fset_list_fm] :
          ( ( A2 = 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 )
          & ( Eff @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps2 ) ) @ Sl )
          & ( abstra6579460235205289861ist_fm @ Eff @ Rules2 @ ( stl_Pr5027778045650968273ist_fm @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_213_RuleSystem__Defs_Oepath_Osimps,axiom,
    ( abstra5415202973855822231ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A2: stream727092118206550309m_rule] :
        ? [Steps2: stream727092118206550309m_rule,Sl: fset_P661503646757059847ist_fm] :
          ( ( A2 = Steps2 )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( sset_rule @ Rules2 ) )
          & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) @ Sl )
          & ( Eff @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps2 ) ) @ Sl )
          & ( abstra5415202973855822231ist_fm @ Eff @ Rules2 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% RuleSystem_Defs.epath.simps
thf(fact_214_Saturated__def,axiom,
    ! [Steps: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ Steps )
      = ( ! [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ rules ) )
           => ( abstra7252176601000738572ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ X3
              @ Steps ) ) ) ) ).

% Saturated_def
thf(fact_215_i_Oepath_Ocoinduct,axiom,
    ! [X6: stream5128679079952569136_fm_fm > $o,X4: stream5128679079952569136_fm_fm,Rules: stream_fm,Eff3: fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X6 @ X4 )
     => ( ! [X: stream5128679079952569136_fm_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream5128679079952569136_fm_fm,Sl4: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member_fm @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) @ Sl4 )
                & ( abstra5048403109753943183ist_fm @ Eff3 @ ( produc6738438136904575596_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ ( produc1217498474891896878_fm_fm @ ( shd_Pr7265661315679113588_fm_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) )
                  | ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ ( stl_Pr2715999368752424048_fm_fm @ Steps3 ) ) ) ) )
       => ( abstra9155489748554225612ist_fm @ ( abstra5048403109753943183ist_fm @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.epath.coinduct
thf(fact_216_i_Oepath_Ocoinduct,axiom,
    ! [X6: stream3409308193418444653ist_fm > $o,X4: stream3409308193418444653ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X6 @ X4 )
     => ( ! [X: stream3409308193418444653ist_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream3409308193418444653ist_fm,Sl4: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) @ Sl4 )
                & ( abstra4222672346834617088ist_fm @ Eff3 @ ( produc1639249265332130309ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ ( produc5894184038225292483ist_fm @ ( shd_Pr4587271278095355155ist_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) )
                  | ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8700568296466786583ist_fm @ Steps3 ) ) ) ) )
       => ( abstra5061764961602482947ist_fm @ ( abstra4222672346834617088ist_fm @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.epath.coinduct
thf(fact_217_i_Oepath_Ocoinduct,axiom,
    ! [X6: stream812778442145099016ule_fm > $o,X4: stream812778442145099016ule_fm,Rules: stream_fm,Eff3: fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X6 @ X4 )
     => ( ! [X: stream812778442145099016ule_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream812778442145099016ule_fm,Sl4: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member_fm @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( sset_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) @ Sl4 )
                & ( abstra7119683891065068131m_rule @ Eff3 @ ( produc8096733172076065128ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ ( produc8194852094735076902ule_fm @ ( shd_Pr849711437766469422ule_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) )
                  | ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ ( stl_Pr2064664776838053682ule_fm @ Steps3 ) ) ) ) )
       => ( abstra1364558218721404774m_rule @ ( abstra7119683891065068131m_rule @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.epath.coinduct
thf(fact_218_i_Oepath_Ocoinduct,axiom,
    ! [X6: stream7989562319868115861ist_fm > $o,X4: stream7989562319868115861ist_fm,Rules: stream8299795917829157543ist_fm,Eff3: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X6 @ X4 )
     => ( ! [X: stream7989562319868115861ist_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream7989562319868115861ist_fm,Sl4: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member8102475879199740618ist_fm @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) @ Sl4 )
                & ( abstra1645840942258654514m_rule @ Eff3 @ ( produc3674566753905654281ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ ( produc2646333476271629515ist_fm @ ( shd_Pr563299216415026649ist_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) )
                  | ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ ( stl_Pr6727363551237990101ist_fm @ Steps3 ) ) ) ) )
       => ( abstra8066765647771630831m_rule @ ( abstra1645840942258654514m_rule @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.epath.coinduct
thf(fact_219_i_Oepath_Ocoinduct,axiom,
    ! [X6: stream7755889244956737277e_rule > $o,X4: stream7755889244956737277e_rule,Rules: stream_rule,Eff3: rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X6 @ X4 )
     => ( ! [X: stream7755889244956737277e_rule] :
            ( ( X6 @ X )
           => ? [Steps3: stream7755889244956737277e_rule,Sl4: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) @ Sl4 )
                & ( abstra3964157444165176280m_rule @ Eff3 @ ( produc4652457655725677917e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ ( produc7822628117027650331e_rule @ ( shd_Pr1235285222789368611e_rule @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) )
                  | ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ ( stl_Pr8695508334200742183e_rule @ Steps3 ) ) ) ) )
       => ( abstra5823804033175701339m_rule @ ( abstra3964157444165176280m_rule @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.epath.coinduct
thf(fact_220_i_Oepath_Ocoinduct,axiom,
    ! [X6: stream642806101564963573m_rule > $o,X4: stream642806101564963573m_rule,Rules: stream727092118206550309m_rule,Eff3: produc164195504107695125m_rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X6 @ X4 )
     => ( ! [X: stream642806101564963573m_rule] :
            ( ( X6 @ X )
           => ? [Steps3: stream642806101564963573m_rule,Sl4: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) ) ) @ Sl4 )
                & ( abstra2934389420647555978ist_fm @ Eff3 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) )
                  | ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps3 ) ) ) ) )
       => ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.epath.coinduct
thf(fact_221_i_Oepath_Ocoinduct,axiom,
    ! [X6: stream5838387310478709965m_rule > $o,X4: stream5838387310478709965m_rule,Rules: stream727092118206550309m_rule,Eff3: produc164195504107695125m_rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( X6 @ X4 )
     => ( ! [X: stream5838387310478709965m_rule] :
            ( ( X6 @ X )
           => ? [Steps3: stream5838387310478709965m_rule,Sl4: fset_A5396818216090053761m_rule] :
                ( ( X = Steps3 )
                & ( member4220325220686508332m_rule @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps3 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
                & ( fmembe1635983029874454474m_rule @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ ( stl_Pr7399785186797918583m_rule @ Steps3 ) ) ) @ Sl4 )
                & ( abstra1666401625646948648m_rule @ Eff3 @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps3 ) ) @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr7399785186797918583m_rule @ Steps3 ) )
                  | ( abstra3709527095716548139m_rule @ ( abstra1666401625646948648m_rule @ Eff3 ) @ Rules @ ( stl_Pr7399785186797918583m_rule @ Steps3 ) ) ) ) )
       => ( abstra3709527095716548139m_rule @ ( abstra1666401625646948648m_rule @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.epath.coinduct
thf(fact_222_i_Oepath_Ocoinduct,axiom,
    ! [X6: stream8299795917829157543ist_fm > $o,X4: stream8299795917829157543ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( X6 @ X4 )
     => ( ! [X: stream8299795917829157543ist_fm] :
            ( ( X6 @ X )
           => ? [Steps3: stream8299795917829157543ist_fm,Sl4: fset_list_fm] :
                ( ( X = 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 ) ) ) @ Sl4 )
                & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) )
                  | ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ ( stl_Pr5027778045650968273ist_fm @ Steps3 ) ) ) ) )
       => ( abstra6579460235205289861ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.epath.coinduct
thf(fact_223_i_Oepath_Ocoinduct,axiom,
    ! [X6: stream727092118206550309m_rule > $o,X4: stream727092118206550309m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X6 @ X4 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X6 @ X )
           => ? [Steps3: stream727092118206550309m_rule,Sl4: fset_P661503646757059847ist_fm] :
                ( ( X = Steps3 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( sset_rule @ Rules ) )
                & ( fmembe3381613331217039976ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) @ Sl4 )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps3 ) ) @ Sl4 )
                & ( ( X6 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.epath.coinduct
thf(fact_224_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_225_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_226_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_227_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_228_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_229_i_Oepath_Ointros,axiom,
    ! [Steps: stream642806101564963573m_rule,Rules: stream727092118206550309m_rule,Sl3: fset_P661503646757059847ist_fm,Eff3: produc164195504107695125m_rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
     => ( ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( abstra2934389420647555978ist_fm @ Eff3 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps ) )
           => ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_230_i_Oepath_Ointros,axiom,
    ! [Steps: stream5838387310478709965m_rule,Rules: stream727092118206550309m_rule,Sl3: fset_A5396818216090053761m_rule,Eff3: produc164195504107695125m_rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule] :
      ( ( member4220325220686508332m_rule @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
     => ( ( fmembe1635983029874454474m_rule @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ ( stl_Pr7399785186797918583m_rule @ Steps ) ) ) @ Sl3 )
       => ( ( abstra1666401625646948648m_rule @ Eff3 @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps ) ) @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps ) ) @ Sl3 )
         => ( ( abstra3709527095716548139m_rule @ ( abstra1666401625646948648m_rule @ Eff3 ) @ Rules @ ( stl_Pr7399785186797918583m_rule @ Steps ) )
           => ( abstra3709527095716548139m_rule @ ( abstra1666401625646948648m_rule @ Eff3 ) @ Rules @ Steps ) ) ) ) ) ).

% i.epath.intros
thf(fact_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_i_Oepath_Osimps,axiom,
    ! [Eff3: produc164195504107695125m_rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream727092118206550309m_rule,A: stream642806101564963573m_rule] :
      ( ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream642806101564963573m_rule,Sl: fset_P661503646757059847ist_fm] :
            ( ( A = Steps2 )
            & ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) ) ) @ Sl )
            & ( abstra2934389420647555978ist_fm @ Eff3 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ Steps2 ) ) @ Sl )
            & ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8578586445745536693m_rule @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_239_i_Oepath_Osimps,axiom,
    ! [Eff3: produc164195504107695125m_rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream727092118206550309m_rule,A: stream5838387310478709965m_rule] :
      ( ( abstra3709527095716548139m_rule @ ( abstra1666401625646948648m_rule @ Eff3 ) @ Rules @ A )
      = ( ? [Steps2: stream5838387310478709965m_rule,Sl: fset_A5396818216090053761m_rule] :
            ( ( A = Steps2 )
            & ( member4220325220686508332m_rule @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps2 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ( fmembe1635983029874454474m_rule @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ ( stl_Pr7399785186797918583m_rule @ Steps2 ) ) ) @ Sl )
            & ( abstra1666401625646948648m_rule @ Eff3 @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps2 ) ) @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ Steps2 ) ) @ Sl )
            & ( abstra3709527095716548139m_rule @ ( abstra1666401625646948648m_rule @ Eff3 ) @ Rules @ ( stl_Pr7399785186797918583m_rule @ Steps2 ) ) ) ) ) ).

% i.epath.simps
thf(fact_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_i_Oepath_Ocases,axiom,
    ! [Eff3: produc164195504107695125m_rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream727092118206550309m_rule,A: stream642806101564963573m_rule] :
      ( ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ A )
     => ~ ( ( member4220325220686508332m_rule @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
         => ! [Sl2: fset_P661503646757059847ist_fm] :
              ( ( fmembe3381613331217039976ist_fm @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ ( stl_Pr8578586445745536693m_rule @ A ) ) ) @ Sl2 )
             => ( ( abstra2934389420647555978ist_fm @ Eff3 @ ( produc2016132375285593905m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ ( produc8602611229794383091m_rule @ ( shd_Pr1407159933805374393m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra6058711572139910855ist_fm @ ( abstra2934389420647555978ist_fm @ Eff3 ) @ Rules @ ( stl_Pr8578586445745536693m_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_248_i_Oepath_Ocases,axiom,
    ! [Eff3: produc164195504107695125m_rule > abstra4572470701423150929m_rule > option7479661550464326865m_rule,Rules: stream727092118206550309m_rule,A: stream5838387310478709965m_rule] :
      ( ( abstra3709527095716548139m_rule @ ( abstra1666401625646948648m_rule @ Eff3 ) @ Rules @ A )
     => ~ ( ( member4220325220686508332m_rule @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ A ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) )
         => ! [Sl2: fset_A5396818216090053761m_rule] :
              ( ( fmembe1635983029874454474m_rule @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ ( stl_Pr7399785186797918583m_rule @ A ) ) ) @ Sl2 )
             => ( ( abstra1666401625646948648m_rule @ Eff3 @ ( produc4721672646055640621m_rule @ ( shd_Pr1655981415546798451m_rule @ A ) ) @ ( produc3513566032450624235m_rule @ ( shd_Pr1655981415546798451m_rule @ A ) ) @ Sl2 )
               => ~ ( abstra3709527095716548139m_rule @ ( abstra1666401625646948648m_rule @ Eff3 ) @ Rules @ ( stl_Pr7399785186797918583m_rule @ A ) ) ) ) ) ) ).

% i.epath.cases
thf(fact_249_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_250_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_251_eff__S,axiom,
    ! [S2: produc1996495991257130529ist_fm,R2: rule,Sl3: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S2 @ top_to728902761071950545ist_fm )
     => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
       => ( ( ( eff @ R2 @ S2 )
            = ( some_f2162114139285431020ist_fm @ Sl3 ) )
         => ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl3 )
           => ( member8102475879199740618ist_fm @ S4 @ top_to728902761071950545ist_fm ) ) ) ) ) ).

% eff_S
thf(fact_252_enabled__R,axiom,
    ! [S2: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S2 @ top_to728902761071950545ist_fm )
     => ? [X: rule] :
          ( ( member_rule @ X @ ( sset_rule @ rules ) )
          & ? [Sl2: fset_P661503646757059847ist_fm] :
              ( ( eff @ X @ S2 )
              = ( some_f2162114139285431020ist_fm @ Sl2 ) ) ) ) ).

% enabled_R
thf(fact_253_RuleSystem__Defs_OSaturated__def,axiom,
    ( abstra8647779336736900908ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,Steps2: stream727092118206550309m_rule] :
        ! [X3: rule] :
          ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
         => ( abstra7252176601000738572ist_fm @ Eff @ X3 @ Steps2 ) ) ) ) ).

% RuleSystem_Defs.Saturated_def
thf(fact_254_pred__equals__eq2,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S5: set_Pr7058068377845519745ist_fm] :
      ( ( ( ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ R3 ) )
        = ( ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ S5 ) ) )
      = ( R3 = S5 ) ) ).

% pred_equals_eq2
thf(fact_255_pred__equals__eq2,axiom,
    ! [R3: set_Pr1008144964186165195m_rule,S5: set_Pr1008144964186165195m_rule] :
      ( ( ( ^ [X3: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X3 @ Y5 ) @ R3 ) )
        = ( ^ [X3: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X3 @ Y5 ) @ S5 ) ) )
      = ( R3 = S5 ) ) ).

% pred_equals_eq2
thf(fact_256_pred__equals__eq2,axiom,
    ! [R3: set_Pr2241365836748572487ist_fm,S5: set_Pr2241365836748572487ist_fm] :
      ( ( ( ^ [X3: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y5 ) @ R3 ) )
        = ( ^ [X3: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y5 ) @ S5 ) ) )
      = ( R3 = S5 ) ) ).

% pred_equals_eq2
thf(fact_257_RuleSystem__Defs_Osaturated_Ocong,axiom,
    abstra7252176601000738572ist_fm = abstra7252176601000738572ist_fm ).

% RuleSystem_Defs.saturated.cong
thf(fact_258_i_Oenabled__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R2: rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S2 )
      = ( ? [X5: fset_P661503646757059847ist_fm] : ( abstra5459341389059251418ist_fm @ Eff3 @ R2 @ S2 @ X5 ) ) ) ).

% i.enabled_def
thf(fact_259_i_OSaturated__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra8647779336736900908ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps )
      = ( ! [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Rules ) )
           => ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ X3 @ Steps ) ) ) ) ).

% i.Saturated_def
thf(fact_260_i_OpickEff,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R2: rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S2 )
     => ( abstra5459341389059251418ist_fm @ Eff3 @ R2 @ S2 @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ S2 ) ) ) ).

% i.pickEff
thf(fact_261_saturated__effG__uu__def,axiom,
    ( abstra7873564507807625473ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% saturated_effG_uu_def
thf(fact_262_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_263_mkTree__effG__uu__def,axiom,
    ( abstra3917079402916513406ist_fm
    = ( ^ [Eff4: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] : ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff4 ) ) ) ) ).

% mkTree_effG_uu_def
thf(fact_264_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_265_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_266_UNIV__I,axiom,
    ! [X4: fm] : ( member_fm @ X4 @ top_top_set_fm ) ).

% UNIV_I
thf(fact_267_UNIV__I,axiom,
    ! [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ top_to728902761071950545ist_fm ) ).

% UNIV_I
thf(fact_268_UNIV__I,axiom,
    ! [X4: rule] : ( member_rule @ X4 @ top_top_set_rule ) ).

% UNIV_I
thf(fact_269_iso__tuple__UNIV__I,axiom,
    ! [X4: fm] : ( member_fm @ X4 @ top_top_set_fm ) ).

% iso_tuple_UNIV_I
thf(fact_270_iso__tuple__UNIV__I,axiom,
    ! [X4: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X4 @ top_to728902761071950545ist_fm ) ).

% iso_tuple_UNIV_I
thf(fact_271_iso__tuple__UNIV__I,axiom,
    ! [X4: rule] : ( member_rule @ X4 @ top_top_set_rule ) ).

% iso_tuple_UNIV_I
thf(fact_272_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_273_top__set__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm @ top_to6798613745154499404t_fm_o ) ) ).

% top_set_def
thf(fact_274_top__set__def,axiom,
    ( top_top_set_rule
    = ( collect_rule @ top_top_rule_o ) ) ).

% top_set_def
thf(fact_275_RuleSystem_Oper_Ocong,axiom,
    abstra4153889242452539894ist_fm = abstra4153889242452539894ist_fm ).

% RuleSystem.per.cong
thf(fact_276_top__empty__eq,axiom,
    ( top_top_fm_o
    = ( ^ [X3: fm] : ( member_fm @ X3 @ top_top_set_fm ) ) ) ).

% top_empty_eq
thf(fact_277_top__empty__eq,axiom,
    ( top_to6798613745154499404t_fm_o
    = ( ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq
thf(fact_278_top__empty__eq,axiom,
    ( top_top_rule_o
    = ( ^ [X3: rule] : ( member_rule @ X3 @ top_top_set_rule ) ) ) ).

% top_empty_eq
thf(fact_279_PersistentRuleSystem_Oper,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,R2: rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
       => ( abstra4153889242452539894ist_fm @ Eff2 @ Rules @ S5 @ R2 ) ) ) ).

% PersistentRuleSystem.per
thf(fact_280_top__empty__eq2,axiom,
    ( top_to1955928418919630078rule_o
    = ( ^ [X3: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X3 @ Y5 ) @ top_to1603951219661559419m_rule ) ) ) ).

% top_empty_eq2
thf(fact_281_top__empty__eq2,axiom,
    ( top_to5144090402559349552t_fm_o
    = ( ^ [X3: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y5 ) @ top_to2837172092223966711ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_282_top__empty__eq2,axiom,
    ( top_to7326020031456794390t_fm_o
    = ( ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ top_to728902761071950545ist_fm ) ) ) ).

% top_empty_eq2
thf(fact_283_UNIV__rules,axiom,
    ( ( sset_rule @ rules )
    = top_top_set_rule ) ).

% UNIV_rules
thf(fact_284_UNIV__witness,axiom,
    ? [X: fm] : ( member_fm @ X @ top_top_set_fm ) ).

% UNIV_witness
thf(fact_285_UNIV__witness,axiom,
    ? [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ top_to728902761071950545ist_fm ) ).

% UNIV_witness
thf(fact_286_UNIV__witness,axiom,
    ? [X: rule] : ( member_rule @ X @ top_top_set_rule ) ).

% UNIV_witness
thf(fact_287_UNIV__eq__I,axiom,
    ! [A3: set_fm] :
      ( ! [X: fm] : ( member_fm @ X @ A3 )
     => ( top_top_set_fm = A3 ) ) ).

% UNIV_eq_I
thf(fact_288_UNIV__eq__I,axiom,
    ! [A3: set_Pr7058068377845519745ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X @ A3 )
     => ( top_to728902761071950545ist_fm = A3 ) ) ).

% UNIV_eq_I
thf(fact_289_UNIV__eq__I,axiom,
    ! [A3: set_rule] :
      ( ! [X: rule] : ( member_rule @ X @ A3 )
     => ( top_top_set_rule = A3 ) ) ).

% UNIV_eq_I
thf(fact_290_UNIV__def,axiom,
    ( top_to728902761071950545ist_fm
    = ( collec7231575776666803724ist_fm
      @ ^ [X3: produc1996495991257130529ist_fm] : $true ) ) ).

% UNIV_def
thf(fact_291_UNIV__def,axiom,
    ( top_top_set_rule
    = ( collect_rule
      @ ^ [X3: rule] : $true ) ) ).

% UNIV_def
thf(fact_292_minWait__ex,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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 ) )
            @ S2 ) ) ) ).

% minWait_ex
thf(fact_293_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_294_PersistentRuleSystem__axioms_Ointro,axiom,
    ! [Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,S5: set_Pr7058068377845519745ist_fm] :
      ( ! [R5: rule] :
          ( ( member_rule @ R5 @ ( sset_rule @ Rules ) )
         => ( abstra4153889242452539894ist_fm @ Eff2 @ Rules @ S5 @ R5 ) )
     => ( abstra2778581409224112183ist_fm @ Eff2 @ Rules @ S5 ) ) ).

% PersistentRuleSystem_axioms.intro
thf(fact_295_PersistentRuleSystem__axioms__def,axiom,
    ( abstra2778581409224112183ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
        ! [R: rule] :
          ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
         => ( abstra4153889242452539894ist_fm @ Eff @ Rules2 @ S6 @ R ) ) ) ) ).

% PersistentRuleSystem_axioms_def
thf(fact_296_ipath__mkTree__saturated,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
              @ S2 )
            @ 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_297_ipath__mkTree__Saturated,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
              @ S2 )
            @ 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_298_szip_Ocode,axiom,
    ( szip_list_fm_list_fm
    = ( ^ [S12: stream_list_fm,S23: stream_list_fm] : ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ ( shd_list_fm @ S12 ) @ ( shd_list_fm @ S23 ) ) @ ( szip_list_fm_list_fm @ ( stl_list_fm @ S12 ) @ ( stl_list_fm @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_299_szip_Ocode,axiom,
    ( szip_P1977448745965526924m_rule
    = ( ^ [S12: stream8299795917829157543ist_fm,S23: stream_rule] : ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ ( shd_Pr772355297128350925ist_fm @ S12 ) @ ( shd_rule @ S23 ) ) @ ( szip_P1977448745965526924m_rule @ ( stl_Pr5027778045650968273ist_fm @ S12 ) @ ( stl_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_300_szip_Ocode,axiom,
    ( szip_P2499414959592755846m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S23: stream727092118206550309m_rule] : ( sCons_4186928284053167522m_rule @ ( produc5927390650430071747m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S23 ) ) @ ( szip_P2499414959592755846m_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_301_szip_Ocode,axiom,
    ( szip_P553291425819358518e_rule
    = ( ^ [S12: stream727092118206550309m_rule,S23: stream_rule] : ( sCons_6835861040388153298e_rule @ ( produc4383365014245906419e_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( shd_rule @ S23 ) ) @ ( szip_P553291425819358518e_rule @ ( stl_Pr950425576149878629m_rule @ S12 ) @ ( stl_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_302_szip_Ocode,axiom,
    ( szip_r8549736553299883584ist_fm
    = ( ^ [S12: stream_rule,S23: stream8299795917829157543ist_fm] : ( sCons_974865965546383930ist_fm @ ( produc7063755442767259331ist_fm @ ( shd_rule @ S12 ) @ ( shd_Pr772355297128350925ist_fm @ S23 ) ) @ ( szip_r8549736553299883584ist_fm @ ( stl_rule @ S12 ) @ ( stl_Pr5027778045650968273ist_fm @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_303_szip_Ocode,axiom,
    ( szip_r2254114822203693110m_rule
    = ( ^ [S12: stream_rule,S23: stream727092118206550309m_rule] : ( sCons_2207919253591517394m_rule @ ( produc6084188410630241011m_rule @ ( shd_rule @ S12 ) @ ( shd_Pr7235097944458474089m_rule @ S23 ) ) @ ( szip_r2254114822203693110m_rule @ ( stl_rule @ S12 ) @ ( stl_Pr950425576149878629m_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_304_szip_Ocode,axiom,
    ( szip_rule_rule
    = ( ^ [S12: stream_rule,S23: stream_rule] : ( sCons_7940479310257918722e_rule @ ( produc1769597910952761123e_rule @ ( shd_rule @ S12 ) @ ( shd_rule @ S23 ) ) @ ( szip_rule_rule @ ( stl_rule @ S12 ) @ ( stl_rule @ S23 ) ) ) ) ) ).

% szip.code
thf(fact_305_ipath__mkTree__sdrop,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,M2: nat] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
              @ S2 )
            @ 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 @ M2 @ Steps ) ) ) ) ) ) ).

% ipath_mkTree_sdrop
thf(fact_306_stream_Ocollapse,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_307_stream_Ocollapse,axiom,
    ! [Stream: stream_rule] :
      ( ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) )
      = Stream ) ).

% stream.collapse
thf(fact_308_RuleSystem__Defs_Ofair_Ocong,axiom,
    abstra1114877437032970289r_rule = abstra1114877437032970289r_rule ).

% RuleSystem_Defs.fair.cong
thf(fact_309_szip__unfold,axiom,
    ! [A: list_fm,S1: stream_list_fm,B: list_fm,S22: stream_list_fm] :
      ( ( szip_list_fm_list_fm @ ( sCons_list_fm @ A @ S1 ) @ ( sCons_list_fm @ B @ S22 ) )
      = ( sCons_307821682964077496ist_fm @ ( produc381145313068854617ist_fm @ A @ B ) @ ( szip_list_fm_list_fm @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_310_szip__unfold,axiom,
    ! [A: produc1996495991257130529ist_fm,S1: stream8299795917829157543ist_fm,B: rule,S22: stream_rule] :
      ( ( szip_P1977448745965526924m_rule @ ( sCons_307821682964077496ist_fm @ A @ S1 ) @ ( sCons_rule @ B @ S22 ) )
      = ( sCons_5731567480799343806m_rule @ ( produc491467635432902671m_rule @ A @ B ) @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_311_szip__unfold,axiom,
    ! [A: rule,S1: stream_rule,B: produc1996495991257130529ist_fm,S22: stream8299795917829157543ist_fm] :
      ( ( szip_r8549736553299883584ist_fm @ ( sCons_rule @ A @ S1 ) @ ( sCons_307821682964077496ist_fm @ B @ S22 ) )
      = ( sCons_974865965546383930ist_fm @ ( produc7063755442767259331ist_fm @ A @ B ) @ ( szip_r8549736553299883584ist_fm @ S1 @ S22 ) ) ) ).

% szip_unfold
thf(fact_312_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_313_stream_Osel_I1_J,axiom,
    ! [X1: rule,X2: stream_rule] :
      ( ( shd_rule @ ( sCons_rule @ X1 @ X2 ) )
      = X1 ) ).

% stream.sel(1)
thf(fact_314_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_315_stream_Osel_I2_J,axiom,
    ! [X1: rule,X2: stream_rule] :
      ( ( stl_rule @ ( sCons_rule @ X1 @ X2 ) )
      = X2 ) ).

% stream.sel(2)
thf(fact_316_stream_Oset__induct,axiom,
    ! [X4: fm,A: stream_fm,P2: fm > stream_fm > $o] :
      ( ( member_fm @ X4 @ ( sset_fm @ A ) )
     => ( ! [Z1: fm,Z22: stream_fm] : ( P2 @ Z1 @ ( sCons_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: fm,Z22: stream_fm,Xa: fm] :
              ( ( member_fm @ Xa @ ( sset_fm @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_fm @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X4 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_317_stream_Oset__induct,axiom,
    ! [X4: produc1996495991257130529ist_fm,A: stream8299795917829157543ist_fm,P2: produc1996495991257130529ist_fm > stream8299795917829157543ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X4 @ ( sset_P3431374123360452149ist_fm @ A ) )
     => ( ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm] : ( P2 @ Z1 @ ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) )
       => ( ! [Z1: produc1996495991257130529ist_fm,Z22: stream8299795917829157543ist_fm,Xa: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ Xa @ ( sset_P3431374123360452149ist_fm @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_307821682964077496ist_fm @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X4 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_318_stream_Oset__induct,axiom,
    ! [X4: rule,A: stream_rule,P2: rule > stream_rule > $o] :
      ( ( member_rule @ X4 @ ( sset_rule @ A ) )
     => ( ! [Z1: rule,Z22: stream_rule] : ( P2 @ Z1 @ ( sCons_rule @ Z1 @ Z22 ) )
       => ( ! [Z1: rule,Z22: stream_rule,Xa: rule] :
              ( ( member_rule @ Xa @ ( sset_rule @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_rule @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X4 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_319_stream_Oset__induct,axiom,
    ! [X4: produc164195504107695125m_rule,A: stream727092118206550309m_rule,P2: produc164195504107695125m_rule > stream727092118206550309m_rule > $o] :
      ( ( member4220325220686508332m_rule @ X4 @ ( sset_P4979403222752902913m_rule @ A ) )
     => ( ! [Z1: produc164195504107695125m_rule,Z22: stream727092118206550309m_rule] : ( P2 @ Z1 @ ( sCons_5731567480799343806m_rule @ Z1 @ Z22 ) )
       => ( ! [Z1: produc164195504107695125m_rule,Z22: stream727092118206550309m_rule,Xa: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ Xa @ ( sset_P4979403222752902913m_rule @ Z22 ) )
             => ( ( P2 @ Xa @ Z22 )
               => ( P2 @ Xa @ ( sCons_5731567480799343806m_rule @ Z1 @ Z22 ) ) ) )
         => ( P2 @ X4 @ A ) ) ) ) ).

% stream.set_induct
thf(fact_320_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_321_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_322_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_323_stream_Oset__cases,axiom,
    ! [E: produc164195504107695125m_rule,A: stream727092118206550309m_rule] :
      ( ( member4220325220686508332m_rule @ E @ ( sset_P4979403222752902913m_rule @ A ) )
     => ( ! [Z22: stream727092118206550309m_rule] :
            ( A
           != ( sCons_5731567480799343806m_rule @ E @ Z22 ) )
       => ~ ! [Z1: produc164195504107695125m_rule,Z22: stream727092118206550309m_rule] :
              ( ( A
                = ( sCons_5731567480799343806m_rule @ Z1 @ Z22 ) )
             => ~ ( member4220325220686508332m_rule @ E @ ( sset_P4979403222752902913m_rule @ Z22 ) ) ) ) ) ).

% stream.set_cases
thf(fact_324_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_325_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_326_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_327_stream_Oset__intros_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ X1 @ ( sset_P4979403222752902913m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) ) ) ).

% stream.set_intros(1)
thf(fact_328_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_329_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_330_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_331_stream_Oset__intros_I2_J,axiom,
    ! [Y4: produc164195504107695125m_rule,X2: stream727092118206550309m_rule,X1: produc164195504107695125m_rule] :
      ( ( member4220325220686508332m_rule @ Y4 @ ( sset_P4979403222752902913m_rule @ X2 ) )
     => ( member4220325220686508332m_rule @ Y4 @ ( sset_P4979403222752902913m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) ) ) ) ).

% stream.set_intros(2)
thf(fact_332_PersistentRuleSystem__def,axiom,
    ( abstra7887161495092839066ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
          ( ( abstra6455275554694158129ist_fm @ Eff @ Rules2 @ S6 )
          & ( abstra2778581409224112183ist_fm @ Eff @ Rules2 @ S6 ) ) ) ) ).

% PersistentRuleSystem_def
thf(fact_333_PersistentRuleSystem_Ointro,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra2778581409224112183ist_fm @ Eff2 @ Rules @ S5 )
       => ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S5 ) ) ) ).

% PersistentRuleSystem.intro
thf(fact_334_sdrop__fair,axiom,
    ! [Rs: stream_rule,M2: nat] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% sdrop_fair
thf(fact_335_fair__stl,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
     => ( abstra1114877437032970289r_rule @ rules @ ( stl_rule @ Rs ) ) ) ).

% fair_stl
thf(fact_336_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm,Rs: stream727092118206550309m_rule,Steps: stream5001952251264143308m_rule,M2: nat] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm @ S7 @ S5 )
                & ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_6203755225909610927m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_337_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,S2: rule,Rs: stream727092118206550309m_rule,Steps: stream1878305877988316353m_rule,M2: nat] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ? [N2: nat,S7: rule] :
                ( ( member_rule @ S7 @ S5 )
                & ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_1764812713384251940m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_338_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule,Steps: stream642806101564963573m_rule,M2: nat] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra7135721583251150454m_rule @ ( abstra2161496378874750759ist_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ? [N2: nat,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S5 )
                & ( abstra7135721583251150454m_rule @ ( abstra2161496378874750759ist_fm @ Eff2 @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_9094843686636319420m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_339_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule,M2: nat] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ? [N2: nat,S7: fm] :
                ( ( member_fm @ S7 @ S5 )
                & ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_432990894411838815m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_340_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule,M2: nat] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ? [N2: nat,S7: rule] :
                ( ( member_rule @ S7 @ S5 )
                & ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_2199339672054592340e_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_341_RuleSystem_Oipath__mkTree__sdrop,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,M2: nat] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ? [N2: nat,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S5 )
                & ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ ( sdrop_rule @ N2 @ Rs ) @ S7 ) @ ( sdrop_7224736112439592940m_rule @ M2 @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_sdrop
thf(fact_342_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_343_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_344_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_345_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_346_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_fm] : ( Eff2 @ X @ S2 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_347_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_rule] : ( Eff2 @ X @ S2 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_348_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ? [X: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ X @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ? [X_1: fset_fm] : ( Eff2 @ X @ S2 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_349_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,S2: rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ? [X: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ X @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ? [X_1: fset_rule] : ( Eff2 @ X @ S2 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_350_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ? [X: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ X @ ( sset_P4979403222752902913m_rule @ Rules ) )
            & ? [X_1: fset_P661503646757059847ist_fm] : ( Eff2 @ X @ S2 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_351_RuleSystem_Oenabled__R,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ? [X: rule] :
            ( ( member_rule @ X @ ( sset_rule @ Rules ) )
            & ? [X_1: fset_P661503646757059847ist_fm] : ( Eff2 @ X @ S2 @ X_1 ) ) ) ) ).

% RuleSystem.enabled_R
thf(fact_352_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M2: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( sdrop_7224736112439592940m_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_353_RuleSystem__Defs_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M2: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% RuleSystem_Defs.sdrop_fair
thf(fact_354_i_Osdrop__fair,axiom,
    ! [Rules: stream727092118206550309m_rule,Rs: stream727092118206550309m_rule,M2: nat] :
      ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
     => ( abstra4997497885011821313m_rule @ Rules @ ( sdrop_7224736112439592940m_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_355_i_Osdrop__fair,axiom,
    ! [Rules: stream_rule,Rs: stream_rule,M2: nat] :
      ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
     => ( abstra1114877437032970289r_rule @ Rules @ ( sdrop_rule @ M2 @ Rs ) ) ) ).

% i.sdrop_fair
thf(fact_356_PersistentRuleSystem_Oaxioms_I1_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S5 )
     => ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 ) ) ).

% PersistentRuleSystem.axioms(1)
thf(fact_357_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7152516257876272129ule_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S2 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_358_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,S2: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra2524698237922071414e_rule @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S2 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_359_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7200325324992684464ist_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N2 @ Rs ) ) @ S2 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_360_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra7038331712872434097ule_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S2 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_361_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3154316310710804774e_rule @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S2 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_362_RuleSystem_OminWait__ex,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ? [N2: nat] : ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S2 ) ) ) ) ).

% RuleSystem.minWait_ex
thf(fact_363_stream_Oexhaust__sel,axiom,
    ! [Stream: stream727092118206550309m_rule] :
      ( Stream
      = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ Stream ) @ ( stl_Pr950425576149878629m_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_364_stream_Oexhaust__sel,axiom,
    ! [Stream: stream_rule] :
      ( Stream
      = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ).

% stream.exhaust_sel
thf(fact_365_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( abstra7168897066996117381ule_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_366_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule] :
      ( ( abstra8115150628584006924e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( abstra4228102177642093818e_rule @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_367_PersistentRuleSystem_Oipath__mkTree__Saturated,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( abstra8647779336736900908ist_fm @ Eff2 @ Rules @ Steps ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_Saturated
thf(fact_368_RuleSystem_Oeff__S,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules: stream_fm,S5: set_fm,S2: fm,R2: fm,Sl3: fset_fm,S4: fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmember_fm @ S4 @ Sl3 )
             => ( member_fm @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_369_RuleSystem_Oeff__S,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules: stream_fm,S5: set_rule,S2: rule,R2: fm,Sl3: fset_rule,S4: rule] :
      ( ( abstra5167362717900046400m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmember_rule @ S4 @ Sl3 )
             => ( member_rule @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_370_RuleSystem_Oeff__S,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,R2: rule,Sl3: fset_fm,S4: fm] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmember_fm @ S4 @ Sl3 )
             => ( member_fm @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_371_RuleSystem_Oeff__S,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,R2: rule,Sl3: fset_rule,S4: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmember_rule @ S4 @ Sl3 )
             => ( member_rule @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_372_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S5: set_fm,S2: fm,R2: produc1996495991257130529ist_fm,Sl3: fset_fm,S4: fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmember_fm @ S4 @ Sl3 )
             => ( member_fm @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_373_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S5: set_rule,S2: rule,R2: produc1996495991257130529ist_fm,Sl3: fset_rule,S4: rule] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmember_rule @ S4 @ Sl3 )
             => ( member_rule @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_374_RuleSystem_Oeff__S,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,R2: fm,Sl3: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
      ( ( abstra7587511110199143014ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl3 )
             => ( member8102475879199740618ist_fm @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_375_RuleSystem_Oeff__S,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,R2: rule,Sl3: fset_P661503646757059847ist_fm,S4: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmembe3381613331217039976ist_fm @ S4 @ Sl3 )
             => ( member8102475879199740618ist_fm @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_376_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm,R2: produc164195504107695125m_rule,Sl3: fset_fm,S4: fm] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( member4220325220686508332m_rule @ R2 @ ( sset_P4979403222752902913m_rule @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmember_fm @ S4 @ Sl3 )
             => ( member_fm @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_377_RuleSystem_Oeff__S,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,S2: rule,R2: produc164195504107695125m_rule,Sl3: fset_rule,S4: rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( member4220325220686508332m_rule @ R2 @ ( sset_P4979403222752902913m_rule @ Rules ) )
         => ( ( Eff2 @ R2 @ S2 @ Sl3 )
           => ( ( fmember_rule @ S4 @ Sl3 )
             => ( member_rule @ S4 @ S5 ) ) ) ) ) ) ).

% RuleSystem.eff_S
thf(fact_378_RuleSystem_Ointro,axiom,
    ! [S5: set_fm,Rules: stream_fm,Eff2: fm > fm > fset_fm > $o] :
      ( ! [S3: fm] :
          ( ( member_fm @ S3 @ S5 )
         => ! [R5: fm] :
              ( ( member_fm @ R5 @ ( sset_fm @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: fm] :
            ( ( member_fm @ S3 @ S5 )
           => ? [X7: fm] :
                ( ( member_fm @ X7 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra886934874521163211_fm_fm @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_379_RuleSystem_Ointro,axiom,
    ! [S5: set_rule,Rules: stream_fm,Eff2: fm > rule > fset_rule > $o] :
      ( ! [S3: rule] :
          ( ( member_rule @ S3 @ S5 )
         => ! [R5: fm] :
              ( ( member_fm @ R5 @ ( sset_fm @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: rule] :
            ( ( member_rule @ S3 @ S5 )
           => ? [X7: fm] :
                ( ( member_fm @ X7 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra5167362717900046400m_rule @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_380_RuleSystem_Ointro,axiom,
    ! [S5: set_fm,Rules: stream_rule,Eff2: rule > fm > fset_fm > $o] :
      ( ! [S3: fm] :
          ( ( member_fm @ S3 @ S5 )
         => ! [R5: rule] :
              ( ( member_rule @ R5 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: fm] :
            ( ( member_fm @ S3 @ S5 )
           => ? [X7: rule] :
                ( ( member_rule @ X7 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_381_RuleSystem_Ointro,axiom,
    ! [S5: set_rule,Rules: stream_rule,Eff2: rule > rule > fset_rule > $o] :
      ( ! [S3: rule] :
          ( ( member_rule @ S3 @ S5 )
         => ! [R5: rule] :
              ( ( member_rule @ R5 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: rule] :
            ( ( member_rule @ S3 @ S5 )
           => ? [X7: rule] :
                ( ( member_rule @ X7 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_382_RuleSystem_Ointro,axiom,
    ! [S5: set_fm,Rules: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o] :
      ( ! [S3: fm] :
          ( ( member_fm @ S3 @ S5 )
         => ! [R5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R5 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: fm] :
            ( ( member_fm @ S3 @ S5 )
           => ? [X7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X7 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_383_RuleSystem_Ointro,axiom,
    ! [S5: set_rule,Rules: stream8299795917829157543ist_fm,Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o] :
      ( ! [S3: rule] :
          ( ( member_rule @ S3 @ S5 )
         => ! [R5: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ R5 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: rule] :
            ( ( member_rule @ S3 @ S5 )
           => ? [X7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X7 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_384_RuleSystem_Ointro,axiom,
    ! [S5: set_Pr7058068377845519745ist_fm,Rules: stream_fm,Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ! [S3: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ S3 @ S5 )
         => ! [R5: fm] :
              ( ( member_fm @ R5 @ ( sset_fm @ Rules ) )
             => ! [Sl2: fset_P661503646757059847ist_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: produc1996495991257130529ist_fm] :
                      ( ( fmembe3381613331217039976ist_fm @ S7 @ Sl2 )
                     => ( member8102475879199740618ist_fm @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ S3 @ S5 )
           => ? [X7: fm] :
                ( ( member_fm @ X7 @ ( sset_fm @ Rules ) )
                & ? [X_12: fset_P661503646757059847ist_fm] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra7587511110199143014ist_fm @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_385_RuleSystem_Ointro,axiom,
    ! [S5: set_Pr7058068377845519745ist_fm,Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ! [S3: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ S3 @ S5 )
         => ! [R5: rule] :
              ( ( member_rule @ R5 @ ( sset_rule @ Rules ) )
             => ! [Sl2: fset_P661503646757059847ist_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: produc1996495991257130529ist_fm] :
                      ( ( fmembe3381613331217039976ist_fm @ S7 @ Sl2 )
                     => ( member8102475879199740618ist_fm @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ S3 @ S5 )
           => ? [X7: rule] :
                ( ( member_rule @ X7 @ ( sset_rule @ Rules ) )
                & ? [X_12: fset_P661503646757059847ist_fm] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_386_RuleSystem_Ointro,axiom,
    ! [S5: set_fm,Rules: stream727092118206550309m_rule,Eff2: produc164195504107695125m_rule > fm > fset_fm > $o] :
      ( ! [S3: fm] :
          ( ( member_fm @ S3 @ S5 )
         => ! [R5: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ R5 @ ( sset_P4979403222752902913m_rule @ Rules ) )
             => ! [Sl2: fset_fm] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: fm] :
                      ( ( fmember_fm @ S7 @ Sl2 )
                     => ( member_fm @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: fm] :
            ( ( member_fm @ S3 @ S5 )
           => ? [X7: produc164195504107695125m_rule] :
                ( ( member4220325220686508332m_rule @ X7 @ ( sset_P4979403222752902913m_rule @ Rules ) )
                & ? [X_12: fset_fm] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_387_RuleSystem_Ointro,axiom,
    ! [S5: set_rule,Rules: stream727092118206550309m_rule,Eff2: produc164195504107695125m_rule > rule > fset_rule > $o] :
      ( ! [S3: rule] :
          ( ( member_rule @ S3 @ S5 )
         => ! [R5: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ R5 @ ( sset_P4979403222752902913m_rule @ Rules ) )
             => ! [Sl2: fset_rule] :
                  ( ( Eff2 @ R5 @ S3 @ Sl2 )
                 => ! [S7: rule] :
                      ( ( fmember_rule @ S7 @ Sl2 )
                     => ( member_rule @ S7 @ S5 ) ) ) ) )
     => ( ! [S3: rule] :
            ( ( member_rule @ S3 @ S5 )
           => ? [X7: produc164195504107695125m_rule] :
                ( ( member4220325220686508332m_rule @ X7 @ ( sset_P4979403222752902913m_rule @ Rules ) )
                & ? [X_12: fset_rule] : ( Eff2 @ X7 @ S3 @ X_12 ) ) )
       => ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 ) ) ) ).

% RuleSystem.intro
thf(fact_388_RuleSystem__def,axiom,
    ( abstra886934874521163211_fm_fm
    = ( ^ [Eff: fm > fm > fset_fm > $o,Rules2: stream_fm,S6: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S6 )
             => ! [R: fm] :
                  ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S6 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S6 )
             => ? [X3: fm] :
                  ( ( member_fm @ X3 @ ( sset_fm @ Rules2 ) )
                  & ? [X5: fset_fm] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_389_RuleSystem__def,axiom,
    ( abstra5167362717900046400m_rule
    = ( ^ [Eff: fm > rule > fset_rule > $o,Rules2: stream_fm,S6: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ! [R: fm] :
                  ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S6 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ? [X3: fm] :
                  ( ( member_fm @ X3 @ ( sset_fm @ Rules2 ) )
                  & ? [X5: fset_rule] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_390_RuleSystem__def,axiom,
    ( abstra916514217863227968ule_fm
    = ( ^ [Eff: rule > fm > fset_fm > $o,Rules2: stream_rule,S6: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S6 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S6 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S6 )
             => ? [X3: rule] :
                  ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
                  & ? [X5: fset_fm] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_391_RuleSystem__def,axiom,
    ( abstra2725590884229999157e_rule
    = ( ^ [Eff: rule > rule > fset_rule > $o,Rules2: stream_rule,S6: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S6 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ? [X3: rule] :
                  ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
                  & ? [X5: fset_rule] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_392_RuleSystem__def,axiom,
    ( abstra2919136520687232136_fm_fm
    = ( ^ [Eff: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules2: stream8299795917829157543ist_fm,S6: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S6 )
             => ! [R: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S6 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S6 )
             => ? [X3: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X3 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                  & ? [X5: fset_fm] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_393_RuleSystem__def,axiom,
    ( abstra9106359784214577277m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules2: stream8299795917829157543ist_fm,S6: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ! [R: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ R @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S6 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ? [X3: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X3 @ ( sset_P3431374123360452149ist_fm @ Rules2 ) )
                  & ? [X5: fset_rule] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_394_RuleSystem__def,axiom,
    ( abstra7587511110199143014ist_fm
    = ( ^ [Eff: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_fm,S6: set_Pr7058068377845519745ist_fm] :
          ( ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S6 )
             => ! [R: fm] :
                  ( ( member_fm @ R @ ( sset_fm @ Rules2 ) )
                 => ! [Sl: fset_P661503646757059847ist_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: produc1996495991257130529ist_fm] :
                          ( ( fmembe3381613331217039976ist_fm @ S8 @ Sl )
                         => ( member8102475879199740618ist_fm @ S8 @ S6 ) ) ) ) )
          & ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S6 )
             => ? [X3: fm] :
                  ( ( member_fm @ X3 @ ( sset_fm @ Rules2 ) )
                  & ? [X5: fset_P661503646757059847ist_fm] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_395_RuleSystem__def,axiom,
    ( abstra6455275554694158129ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,S6: set_Pr7058068377845519745ist_fm] :
          ( ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S6 )
             => ! [R: rule] :
                  ( ( member_rule @ R @ ( sset_rule @ Rules2 ) )
                 => ! [Sl: fset_P661503646757059847ist_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: produc1996495991257130529ist_fm] :
                          ( ( fmembe3381613331217039976ist_fm @ S8 @ Sl )
                         => ( member8102475879199740618ist_fm @ S8 @ S6 ) ) ) ) )
          & ! [S: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ S @ S6 )
             => ? [X3: rule] :
                  ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
                  & ? [X5: fset_P661503646757059847ist_fm] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_396_RuleSystem__def,axiom,
    ( abstra3764363818050485264ule_fm
    = ( ^ [Eff: produc164195504107695125m_rule > fm > fset_fm > $o,Rules2: stream727092118206550309m_rule,S6: set_fm] :
          ( ! [S: fm] :
              ( ( member_fm @ S @ S6 )
             => ! [R: produc164195504107695125m_rule] :
                  ( ( member4220325220686508332m_rule @ R @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
                 => ! [Sl: fset_fm] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: fm] :
                          ( ( fmember_fm @ S8 @ Sl )
                         => ( member_fm @ S8 @ S6 ) ) ) ) )
          & ! [S: fm] :
              ( ( member_fm @ S @ S6 )
             => ? [X3: produc164195504107695125m_rule] :
                  ( ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
                  & ? [X5: fset_fm] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_397_RuleSystem__def,axiom,
    ( abstra3778134268586021893e_rule
    = ( ^ [Eff: produc164195504107695125m_rule > rule > fset_rule > $o,Rules2: stream727092118206550309m_rule,S6: set_rule] :
          ( ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ! [R: produc164195504107695125m_rule] :
                  ( ( member4220325220686508332m_rule @ R @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
                 => ! [Sl: fset_rule] :
                      ( ( Eff @ R @ S @ Sl )
                     => ! [S8: rule] :
                          ( ( fmember_rule @ S8 @ Sl )
                         => ( member_rule @ S8 @ S6 ) ) ) ) )
          & ! [S: rule] :
              ( ( member_rule @ S @ S6 )
             => ? [X3: produc164195504107695125m_rule] :
                  ( ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
                  & ? [X5: fset_rule] : ( Eff @ X3 @ S @ X5 ) ) ) ) ) ) ).

% RuleSystem_def
thf(fact_398_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules: stream_fm,S5: set_fm,S2: fm,Rs: stream_fm,Steps: stream1375025385734440967_fm_fm,R2: fm] :
      ( ( abstra1990789280842237858_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra724614664783131952_fm_fm @ ( abstra5285091746773999237_fm_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra81079133870296496_fm_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_399_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules: stream_fm,S5: set_rule,S2: rule,Rs: stream_fm,Steps: stream1800043145262542076ule_fm,R2: fm] :
      ( ( abstra4039608574777915031m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra8780395760585530021ule_fm @ ( abstra8242076542743448570m_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra5359274141611169189m_rule @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_400_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule,R2: rule] :
      ( ( abstra9012132111595872407ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra1108425641574350757ule_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_401_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule,R2: rule] :
      ( ( abstra8115150628584006924e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra6372423936129899802e_rule @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_402_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S5: set_fm,S2: fm,Rs: stream8299795917829157543ist_fm,Steps: stream5068232101399153750ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra1665324712063988849_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( abstra230716855031908835_fm_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_403_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S5: set_rule,S2: rule,Rs: stream8299795917829157543ist_fm,Steps: stream1960312990768957601ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra1314873687758482406m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( abstra679888793666381912m_rule @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_404_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_fm,Steps: stream5128679079952569136_fm_fm,R2: fm] :
      ( ( abstra6333699301575899727ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra2316067621365718065_fm_fm @ ( abstra982595312256735276ist_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( abstra4899091444543819713ist_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_405_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( abstra7252176601000738572ist_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_406_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm,Rs: stream727092118206550309m_rule,Steps: stream5001952251264143308m_rule,R2: produc164195504107695125m_rule] :
      ( ( abstra9214187659791502823ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member4220325220686508332m_rule @ R2 @ ( sset_P4979403222752902913m_rule @ Rules ) )
             => ( abstra9161913875526226933ule_fm @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_407_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,S2: rule,Rs: stream727092118206550309m_rule,Steps: stream1878305877988316353m_rule,R2: produc164195504107695125m_rule] :
      ( ( abstra5324783417238284892e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra987325317039887594m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member4220325220686508332m_rule @ R2 @ ( sset_P4979403222752902913m_rule @ Rules ) )
             => ( abstra4932427653927946602e_rule @ Eff2 @ R2 @ Steps ) ) ) ) ) ) ).

% PersistentRuleSystem.ipath_mkTree_saturated
thf(fact_408_PersistentRuleSystem_Oaxioms_I2_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S5 )
     => ( abstra2778581409224112183ist_fm @ Eff2 @ Rules @ S5 ) ) ).

% PersistentRuleSystem.axioms(2)
thf(fact_409_RuleSystem_Opos,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( 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_410_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_411_trim__enabled,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
              @ S2 ) )
          @ S2 ) ) ) ).

% trim_enabled
thf(fact_412_trim__in__R,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
              @ S2 ) )
          @ ( sset_rule @ rules ) ) ) ) ).

% trim_in_R
thf(fact_413_fair__uu__def,axiom,
    abstra1913438439042302728u_rule = abstra1114877437032970289r_rule ).

% fair_uu_def
thf(fact_414_wf__mkTree,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
            @ S2 ) ) ) ) ).

% wf_mkTree
thf(fact_415_wf__ipath__epath,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ rules
        @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ 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_416_trim__fair,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
            @ S2 ) ) ) ) ).

% trim_fair
thf(fact_417_RuleSystem__Defs_Owf_Ocong,axiom,
    abstra4476735159789068068ist_fm = abstra4476735159789068068ist_fm ).

% RuleSystem_Defs.wf.cong
thf(fact_418_RuleSystem__Defs_Otrim_Ocong,axiom,
    abstra2363873759530810705ist_fm = abstra2363873759530810705ist_fm ).

% RuleSystem_Defs.trim.cong
thf(fact_419_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_420_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S2 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_421_RuleSystem_Otrim__fair,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra1114877437032970289r_rule @ Rules @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ).

% RuleSystem.trim_fair
thf(fact_422_RuleSystem__Defs_Owf__ipath__epath,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps )
       => ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ Steps ) ) ) ).

% RuleSystem_Defs.wf_ipath_epath
thf(fact_423_Stream_Osmember__def,axiom,
    ( smember_fm
    = ( ^ [X3: fm,S: stream_fm] : ( member_fm @ X3 @ ( sset_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_424_Stream_Osmember__def,axiom,
    ( smembe2449974647026456671ist_fm
    = ( ^ [X3: produc1996495991257130529ist_fm,S: stream8299795917829157543ist_fm] : ( member8102475879199740618ist_fm @ X3 @ ( sset_P3431374123360452149ist_fm @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_425_Stream_Osmember__def,axiom,
    ( smember_rule
    = ( ^ [X3: rule,S: stream_rule] : ( member_rule @ X3 @ ( sset_rule @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_426_Stream_Osmember__def,axiom,
    ( smembe6335822699099550295m_rule
    = ( ^ [X3: produc164195504107695125m_rule,S: stream727092118206550309m_rule] : ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ S ) ) ) ) ).

% Stream.smember_def
thf(fact_427_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra2794488811626936717ule_fm @ Eff2 @ Rules @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_428_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra5602818997816140546e_rule @ Eff2 @ Rules @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S2 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_429_RuleSystem_Owf__mkTree,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ).

% RuleSystem.wf_mkTree
thf(fact_430_i_Owf__ipath__epath,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T2 )
     => ( ( abstra2285467793620430502m_rule @ T2 @ Steps )
       => ( abstra5415202973855822231ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Steps ) ) ) ).

% i.wf_ipath_epath
thf(fact_431_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules: stream_fm,S5: set_fm,S2: fm,Rs: stream_fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra5942290089224017835_fm_fm @ Eff2 @ Rs @ S2 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_432_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules: stream_fm,S5: set_rule,S2: rule,Rs: stream_fm] :
      ( ( abstra5167362717900046400m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra8496492133263135264m_rule @ Eff2 @ Rs @ S2 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_433_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S2 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_434_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S2 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_435_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S5: set_fm,S2: fm,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( abstra5503639309725227688_fm_fm @ Eff2 @ Rs @ S2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_436_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S5: set_rule,S2: rule,Rs: stream8299795917829157543ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs @ S2 ) ) @ ( sset_P3431374123360452149ist_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_437_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_fm] :
      ( ( abstra7587511110199143014ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( member_fm @ ( shd_fm @ ( abstra948641862382362758ist_fm @ Eff2 @ Rs @ S2 ) ) @ ( sset_fm @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_438_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( member_rule @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 ) ) @ ( sset_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_439_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S2 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_440_RuleSystem_Otrim__in__R,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,S2: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S2 ) ) @ ( sset_P4979403222752902913m_rule @ Rules ) ) ) ) ) ).

% RuleSystem.trim_in_R
thf(fact_441_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra7152516257876272129ule_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S2 ) ) @ S2 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_442_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,S2: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra2524698237922071414e_rule @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S2 ) ) @ S2 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_443_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( abstra7200325324992684464ist_fm @ Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S2 ) ) @ S2 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_444_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra7038331712872434097ule_fm @ Eff2 @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S2 ) ) @ S2 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_445_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra3154316310710804774e_rule @ Eff2 @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S2 ) ) @ S2 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_446_RuleSystem_Otrim__enabled,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 ) ) @ S2 ) ) ) ) ).

% RuleSystem.trim_enabled
thf(fact_447_trim__alt,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
            @ S2 )
          = ( sdrop_rule
            @ ( abstra738319790293016300ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S2 )
            @ Rs ) ) ) ) ).

% trim_alt
thf(fact_448_in__cont__mkTree,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule,T3: abstra4572470701423150929m_rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
                @ S2 ) ) )
         => ? [Sl5: 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
                      @ S2 ) )
                  @ S2 )
                = ( some_f2162114139285431020ist_fm @ Sl5 ) )
              & ( fmembe3381613331217039976ist_fm @ S7 @ Sl5 )
              & ( 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
                      @ S2 ) )
                  @ S7 ) ) ) ) ) ) ).

% in_cont_mkTree
thf(fact_449_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S2 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra6093700014593756181ule_fm @ Eff2 @ Rs @ S2 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_450_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,S2: rule,Rs: stream727092118206550309m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S2 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra6412047235177443722e_rule @ Eff2 @ Rs @ S2 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_451_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S2 )
            = ( sdrop_7224736112439592940m_rule @ ( abstra5265644873707572892ist_fm @ Eff2 @ Rs @ S2 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_452_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S2 )
            = ( sdrop_rule @ ( abstra2294468009765627845ule_fm @ Eff2 @ Rs @ S2 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_453_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S2 )
            = ( sdrop_rule @ ( abstra5682536783049438522e_rule @ Eff2 @ Rs @ S2 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_454_RuleSystem_Otrim__alt,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 )
            = ( sdrop_rule @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S2 ) @ Rs ) ) ) ) ) ).

% RuleSystem.trim_alt
thf(fact_455_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm,Rs: stream727092118206550309m_rule,T3: abstra4575282202256230560m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe8691679578407899495m_rule @ T3 @ ( abstra2433431901973683783m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ Rs @ S2 ) ) )
           => ? [Sl5: fset_fm,S7: fm] :
                ( ( member_fm @ S7 @ S5 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S2 ) ) @ S2 @ Sl5 )
                & ( fmember_fm @ S7 @ Sl5 )
                & ( T3
                  = ( abstra1207708450097661130ule_fm @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S2 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_456_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,S2: rule,Rs: stream727092118206550309m_rule,T3: abstra4791331342562632597m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe1981849019698968796m_rule @ T3 @ ( abstra5403976286072314556m_rule @ ( abstra1231468025953361855e_rule @ Eff2 @ Rs @ S2 ) ) )
           => ? [Sl5: fset_rule,S7: rule] :
                ( ( member_rule @ S7 @ S5 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S2 ) ) @ S2 @ Sl5 )
                & ( fmember_rule @ S7 @ Sl5 )
                & ( T3
                  = ( abstra1231468025953361855e_rule @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S2 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_457_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream727092118206550309m_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream727092118206550309m_rule,T3: abstra4658698708444432161m_rule] :
      ( ( abstra3408323020314982753ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe2925761000581315610m_rule @ T3 @ ( abstra5924671108449608740m_rule @ ( abstra2161496378874750759ist_fm @ Eff2 @ Rs @ S2 ) ) )
           => ? [Sl5: fset_P661503646757059847ist_fm,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S5 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S2 ) ) @ S2 @ Sl5 )
                & ( fmembe3381613331217039976ist_fm @ S7 @ Sl5 )
                & ( T3
                  = ( abstra2161496378874750759ist_fm @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra6201285654312316801ist_fm @ Eff2 @ Rs @ S2 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_458_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: produc164195504107695125m_rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream727092118206550309m_rule,S5: set_Ab3744519486624674951m_rule,S2: abstra4572470701423150929m_rule,Rs: stream727092118206550309m_rule,T3: abstra5426939140630736801m_rule] :
      ( ( abstra6898608545798587409m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member1590433669991072232m_rule @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( fmembe8791921695509808616m_rule @ T3 @ ( abstra3071294317365906120m_rule @ ( abstra1631510761171731915m_rule @ Eff2 @ Rs @ S2 ) ) )
           => ? [Sl5: fset_A5396818216090053761m_rule,S7: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S7 @ S5 )
                & ( Eff2 @ ( shd_Pr7235097944458474089m_rule @ ( abstra7609338345072424433m_rule @ Eff2 @ Rs @ S2 ) ) @ S2 @ Sl5 )
                & ( fmembe1635983029874454474m_rule @ S7 @ Sl5 )
                & ( T3
                  = ( abstra1631510761171731915m_rule @ Eff2 @ ( stl_Pr950425576149878629m_rule @ ( abstra7609338345072424433m_rule @ Eff2 @ Rs @ S2 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_459_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule,T3: abstra4524330545568143696m_rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe2914889872492170903m_rule @ T3 @ ( abstra5134701107988495863m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S2 ) ) )
           => ? [Sl5: fset_fm,S7: fm] :
                ( ( member_fm @ S7 @ S5 )
                & ( Eff2 @ ( shd_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S2 ) ) @ S2 @ Sl5 )
                & ( fmember_fm @ S7 @ Sl5 )
                & ( T3
                  = ( abstra3991228042706630138ule_fm @ Eff2 @ ( stl_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S2 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_460_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule,T3: abstra1071693965251717573e_rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe5852619498312149900e_rule @ T3 @ ( abstra2651384284180525036e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S2 ) ) )
           => ? [Sl5: fset_rule,S7: rule] :
                ( ( member_rule @ S7 @ S5 )
                & ( Eff2 @ ( shd_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S2 ) ) @ S2 @ Sl5 )
                & ( fmember_rule @ S7 @ Sl5 )
                & ( T3
                  = ( abstra7443870075974203631e_rule @ Eff2 @ ( stl_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S2 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_461_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > abstra4572470701423150929m_rule > fset_A5396818216090053761m_rule > $o,Rules: stream_rule,S5: set_Ab3744519486624674951m_rule,S2: abstra4572470701423150929m_rule,Rs: stream_rule,T3: abstra4125161011840266193e_rule] :
      ( ( abstra6244190358723783233m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member1590433669991072232m_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe8010993199532995224e_rule @ T3 @ ( abstra6162512231279892984e_rule @ ( abstra2861652164185304827m_rule @ Eff2 @ Rs @ S2 ) ) )
           => ? [Sl5: fset_A5396818216090053761m_rule,S7: abstra4572470701423150929m_rule] :
                ( ( member1590433669991072232m_rule @ S7 @ S5 )
                & ( Eff2 @ ( shd_rule @ ( abstra1865002795007470625m_rule @ Eff2 @ Rs @ S2 ) ) @ S2 @ Sl5 )
                & ( fmembe1635983029874454474m_rule @ S7 @ Sl5 )
                & ( T3
                  = ( abstra2861652164185304827m_rule @ Eff2 @ ( stl_rule @ ( abstra1865002795007470625m_rule @ Eff2 @ Rs @ S2 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_462_RuleSystem_Oin__cont__mkTree,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule,T3: abstra4572470701423150929m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( fmembe1635983029874454474m_rule @ T3 @ ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S2 ) ) )
           => ? [Sl5: fset_P661503646757059847ist_fm,S7: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ S7 @ S5 )
                & ( Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 ) ) @ S2 @ Sl5 )
                & ( fmembe3381613331217039976ist_fm @ S7 @ Sl5 )
                & ( T3
                  = ( abstra178918799166095863ist_fm @ Eff2 @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 ) ) @ S7 ) ) ) ) ) ) ) ).

% RuleSystem.in_cont_mkTree
thf(fact_463_trim__def,axiom,
    ! [Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra2363873759530810705ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S2 )
      = ( sdrop_while_rule
        @ ^ [R: rule] :
            ~ ( abstra3055540779341473792ist_fm
              @ ^ [S: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ S @ T )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R
              @ S2 )
        @ Rs ) ) ).

% trim_def
thf(fact_464_mkTree_Osimps_I1_J,axiom,
    ! [Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S2 ) )
      = ( produc491467635432902671m_rule @ S2
        @ ( shd_rule
          @ ( abstra2363873759530810705ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ Rs
            @ S2 ) ) ) ) ).

% mkTree.simps(1)
thf(fact_465_epath__completeness__Saturated,axiom,
    ! [S2: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ S2 @ top_to728902761071950545ist_fm )
     => ( ? [T4: abstra4572470701423150929m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T4 ) )
              = S2 )
            & ( abstra4476735159789068068ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ rules
              @ T4 )
            & ( abstra3507476289341504315m_rule @ T4 ) )
        | ? [Steps4: stream727092118206550309m_rule] :
            ( ( ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps4 ) )
              = S2 )
            & ( 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_466_i_Otrim__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 )
      = ( sdrop_while_rule
        @ ^ [R: rule] :
            ~ ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S2 )
        @ Rs ) ) ).

% i.trim_def
thf(fact_467_tfinite_Ocases,axiom,
    ! [A: abstra4572470701423150929m_rule] :
      ( ( abstra3507476289341504315m_rule @ A )
     => ! [T5: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ A ) )
         => ( abstra3507476289341504315m_rule @ T5 ) ) ) ).

% tfinite.cases
thf(fact_468_tfinite_Osimps,axiom,
    ( abstra3507476289341504315m_rule
    = ( ^ [A2: abstra4572470701423150929m_rule] :
        ? [T: abstra4572470701423150929m_rule] :
          ( ( A2 = T )
          & ! [X3: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X3 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra3507476289341504315m_rule @ X3 ) ) ) ) ) ).

% tfinite.simps
thf(fact_469_tfinite,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ! [T6: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ T2 ) )
         => ( abstra3507476289341504315m_rule @ T6 ) )
     => ( abstra3507476289341504315m_rule @ T2 ) ) ).

% tfinite
thf(fact_470_RuleSystem_OminWait_Ocong,axiom,
    abstra738319790293016300ist_fm = abstra738319790293016300ist_fm ).

% RuleSystem.minWait.cong
thf(fact_471_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_472_ipath_Ocases,axiom,
    ! [A1: abstract_tree_rule,A22: stream_rule] :
      ( ( abstract_ipath_rule @ A1 @ A22 )
     => ~ ( ( ( abstract_root_rule @ A1 )
            = ( shd_rule @ A22 ) )
         => ! [T6: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T6 @ ( abstract_cont_rule @ A1 ) )
             => ~ ( abstract_ipath_rule @ T6 @ ( stl_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_473_ipath_Ocases,axiom,
    ! [A1: abstra4572470701423150929m_rule,A22: stream727092118206550309m_rule] :
      ( ( abstra2285467793620430502m_rule @ A1 @ A22 )
     => ~ ( ( ( abstra3503897793822247108m_rule @ A1 )
            = ( shd_Pr7235097944458474089m_rule @ A22 ) )
         => ! [T6: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ A1 ) )
             => ~ ( abstra2285467793620430502m_rule @ T6 @ ( stl_Pr950425576149878629m_rule @ A22 ) ) ) ) ) ).

% ipath.cases
thf(fact_474_ipath_Osimps,axiom,
    ( abstract_ipath_rule
    = ( ^ [A12: abstract_tree_rule,A23: stream_rule] :
        ? [T: abstract_tree_rule,Steps2: stream_rule,T7: abstract_tree_rule] :
          ( ( A12 = T )
          & ( A23 = Steps2 )
          & ( ( abstract_root_rule @ T )
            = ( shd_rule @ Steps2 ) )
          & ( fmembe7964111328986570234e_rule @ T7 @ ( abstract_cont_rule @ T ) )
          & ( abstract_ipath_rule @ T7 @ ( stl_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_475_ipath_Osimps,axiom,
    ( abstra2285467793620430502m_rule
    = ( ^ [A12: abstra4572470701423150929m_rule,A23: stream727092118206550309m_rule] :
        ? [T: abstra4572470701423150929m_rule,Steps2: stream727092118206550309m_rule,T7: abstra4572470701423150929m_rule] :
          ( ( A12 = T )
          & ( A23 = Steps2 )
          & ( ( abstra3503897793822247108m_rule @ T )
            = ( shd_Pr7235097944458474089m_rule @ Steps2 ) )
          & ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ T ) )
          & ( abstra2285467793620430502m_rule @ T7 @ ( stl_Pr950425576149878629m_rule @ Steps2 ) ) ) ) ) ).

% ipath.simps
thf(fact_476_ipath_Ointros,axiom,
    ! [T2: abstract_tree_rule,Steps: stream_rule,T3: abstract_tree_rule] :
      ( ( ( abstract_root_rule @ T2 )
        = ( shd_rule @ Steps ) )
     => ( ( fmembe7964111328986570234e_rule @ T3 @ ( abstract_cont_rule @ T2 ) )
       => ( ( abstract_ipath_rule @ T3 @ ( stl_rule @ Steps ) )
         => ( abstract_ipath_rule @ T2 @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_477_ipath_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule,T3: abstra4572470701423150929m_rule] :
      ( ( ( abstra3503897793822247108m_rule @ T2 )
        = ( shd_Pr7235097944458474089m_rule @ Steps ) )
     => ( ( fmembe1635983029874454474m_rule @ T3 @ ( abstra4962253831105676628m_rule @ T2 ) )
       => ( ( abstra2285467793620430502m_rule @ T3 @ ( stl_Pr950425576149878629m_rule @ Steps ) )
         => ( abstra2285467793620430502m_rule @ T2 @ Steps ) ) ) ) ).

% ipath.intros
thf(fact_478_ipath_Ocoinduct,axiom,
    ! [X6: abstract_tree_rule > stream_rule > $o,X1: abstract_tree_rule,X2: stream_rule] :
      ( ( X6 @ X1 @ X2 )
     => ( ! [X12: abstract_tree_rule,X23: stream_rule] :
            ( ( X6 @ X12 @ X23 )
           => ? [T8: abstract_tree_rule,Steps3: stream_rule,T5: abstract_tree_rule] :
                ( ( X12 = T8 )
                & ( X23 = Steps3 )
                & ( ( abstract_root_rule @ T8 )
                  = ( shd_rule @ Steps3 ) )
                & ( fmembe7964111328986570234e_rule @ T5 @ ( abstract_cont_rule @ T8 ) )
                & ( ( X6 @ T5 @ ( stl_rule @ Steps3 ) )
                  | ( abstract_ipath_rule @ T5 @ ( stl_rule @ Steps3 ) ) ) ) )
       => ( abstract_ipath_rule @ X1 @ X2 ) ) ) ).

% ipath.coinduct
thf(fact_479_ipath_Ocoinduct,axiom,
    ! [X6: abstra4572470701423150929m_rule > stream727092118206550309m_rule > $o,X1: abstra4572470701423150929m_rule,X2: stream727092118206550309m_rule] :
      ( ( X6 @ X1 @ X2 )
     => ( ! [X12: abstra4572470701423150929m_rule,X23: stream727092118206550309m_rule] :
            ( ( X6 @ X12 @ X23 )
           => ? [T8: abstra4572470701423150929m_rule,Steps3: stream727092118206550309m_rule,T5: abstra4572470701423150929m_rule] :
                ( ( X12 = T8 )
                & ( X23 = Steps3 )
                & ( ( abstra3503897793822247108m_rule @ T8 )
                  = ( shd_Pr7235097944458474089m_rule @ Steps3 ) )
                & ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ T8 ) )
                & ( ( X6 @ T5 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) )
                  | ( abstra2285467793620430502m_rule @ T5 @ ( stl_Pr950425576149878629m_rule @ Steps3 ) ) ) ) )
       => ( abstra2285467793620430502m_rule @ X1 @ X2 ) ) ) ).

% ipath.coinduct
thf(fact_480_sdrop__while__SCons,axiom,
    ! [P2: rule > $o,A: rule,S2: stream_rule] :
      ( ( ( P2 @ A )
       => ( ( sdrop_while_rule @ P2 @ ( sCons_rule @ A @ S2 ) )
          = ( sdrop_while_rule @ P2 @ S2 ) ) )
      & ( ~ ( P2 @ A )
       => ( ( sdrop_while_rule @ P2 @ ( sCons_rule @ A @ S2 ) )
          = ( sCons_rule @ A @ S2 ) ) ) ) ).

% sdrop_while_SCons
thf(fact_481_ftree__no__ipath,axiom,
    ! [T2: abstra4572470701423150929m_rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra3507476289341504315m_rule @ T2 )
     => ~ ( abstra2285467793620430502m_rule @ T2 @ Steps ) ) ).

% ftree_no_ipath
thf(fact_482_sdrop__while_Osimps,axiom,
    ( sdrop_5541336384294084785m_rule
    = ( ^ [P4: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( if_str8948254419368749791m_rule @ ( P4 @ ( shd_Pr7235097944458474089m_rule @ S ) ) @ ( sdrop_5541336384294084785m_rule @ P4 @ ( stl_Pr950425576149878629m_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_483_sdrop__while_Osimps,axiom,
    ( sdrop_while_rule
    = ( ^ [P4: rule > $o,S: stream_rule] : ( if_stream_rule @ ( P4 @ ( shd_rule @ S ) ) @ ( sdrop_while_rule @ P4 @ ( stl_rule @ S ) ) @ S ) ) ) ).

% sdrop_while.simps
thf(fact_484_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,S5: set_list_fm,S2: list_fm] :
      ( ( abstra1616580364776158018ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_list_fm @ S2 @ S5 )
       => ( ? [T4: abstra5042247676209264507ist_fm] :
              ( ( ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T4 ) )
                = S2 )
              & ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ T4 )
              & ( abstra5066959810000083067ist_fm @ T4 ) )
          | ? [Steps4: stream8299795917829157543ist_fm] :
              ( ( ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps4 ) )
                = S2 )
              & ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ Steps4 )
              & ( abstra8946172763262761776ist_fm @ Eff2 @ Rules @ Steps4 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_485_PersistentRuleSystem_Oepath__completeness__Saturated,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm] :
      ( ( abstra7887161495092839066ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ? [T4: abstra4572470701423150929m_rule] :
              ( ( ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T4 ) )
                = S2 )
              & ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T4 )
              & ( abstra3507476289341504315m_rule @ T4 ) )
          | ? [Steps4: stream727092118206550309m_rule] :
              ( ( ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps4 ) )
                = S2 )
              & ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ Steps4 )
              & ( abstra8647779336736900908ist_fm @ Eff2 @ Rules @ Steps4 ) ) ) ) ) ).

% PersistentRuleSystem.epath_completeness_Saturated
thf(fact_486_RuleSystem__Defs_Otrim__def,axiom,
    ( abstra2363873759530810705ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] :
          ( sdrop_while_rule
          @ ^ [R: rule] :
              ~ ( abstra3055540779341473792ist_fm @ Eff @ R @ S )
          @ Rs2 ) ) ) ).

% RuleSystem_Defs.trim_def
thf(fact_487_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rs: stream_list_fm,S2: list_fm] :
      ( ( abstra778364809706815794ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ Rs @ S2 ) )
      = ( produc381145313068854617ist_fm @ S2 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff2 @ Rs @ S2 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_488_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs: stream8299795917829157543ist_fm,S2: rule] :
      ( ( abstra7970568315424063040ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ Rs @ S2 ) )
      = ( produc7063755442767259331ist_fm @ S2 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs @ S2 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_489_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S2 ) )
      = ( produc491467635432902671m_rule @ S2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 ) ) ) ) ).

% RuleSystem_Defs.mkTree.simps(1)
thf(fact_490_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S2: list_fm] :
      ( ( abstra778364809706815794ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S2 ) )
      = ( produc381145313068854617ist_fm @ S2 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S2 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_491_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S2: rule] :
      ( ( abstra7970568315424063040ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S2 ) )
      = ( produc7063755442767259331ist_fm @ S2 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S2 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_492_i_OmkTree_Osimps_I1_J,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra3503897793822247108m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 ) )
      = ( produc491467635432902671m_rule @ S2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 ) ) ) ) ).

% i.mkTree.simps(1)
thf(fact_493_minWait__le__pos,axiom,
    ! [Rs: stream_rule,R2: rule,S2: 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
            @ S2 )
         => ( ord_less_eq_nat
            @ ( abstra738319790293016300ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S2 )
            @ ( abstract_pos_rule @ Rs @ R2 ) ) ) ) ) ).

% minWait_le_pos
thf(fact_494_mkTree_Osimps_I2_J,axiom,
    ! [Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule
        @ ( abstra178918799166095863ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S2 ) )
      = ( 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
              @ S2 ) ) )
        @ ( 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
              @ S2 ) )
          @ S2 ) ) ) ).

% mkTree.simps(2)
thf(fact_495_minWait__def,axiom,
    ! [Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra738319790293016300ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S2 )
      = ( 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 ) )
            @ S2 ) ) ) ).

% minWait_def
thf(fact_496_minWait__least,axiom,
    ! [N: nat,Rs: stream_rule,S2: 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 ) )
        @ S2 )
     => ( ord_less_eq_nat
        @ ( abstra738319790293016300ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S2 )
        @ N ) ) ).

% minWait_least
thf(fact_497_RuleSystem_OminWait__le__pos,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R2: rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
         => ( ( abstra3055540779341473792ist_fm @ Eff2 @ R2 @ S2 )
           => ( ord_less_eq_nat @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S2 ) @ ( abstract_pos_rule @ Rs @ R2 ) ) ) ) ) ) ).

% RuleSystem.minWait_le_pos
thf(fact_498_i_OmkTree_Osimps_I2_J,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 ) )
      = ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 ) ) ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 ) ) @ S2 ) ) ) ).

% i.mkTree.simps(2)
thf(fact_499_RuleSystem_OminWait__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] : ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) ) @ S2 ) ) ) ) ).

% RuleSystem.minWait_def
thf(fact_500_RuleSystem__Defs_OmkTree_Osimps_I2_J,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra4962253831105676628m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S2 ) )
      = ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 ) ) ) @ ( abstra458490714785879965ist_fm @ Eff2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 ) ) @ S2 ) ) ) ).

% RuleSystem_Defs.mkTree.simps(2)
thf(fact_501_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_502_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_503_LeastI,axiom,
    ! [P2: nat > $o,K2: nat] :
      ( ( P2 @ K2 )
     => ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).

% LeastI
thf(fact_504_Least__le,axiom,
    ! [P2: nat > $o,K2: nat] :
      ( ( P2 @ K2 )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ K2 ) ) ).

% Least_le
thf(fact_505_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_506_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_507_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
        ? [C3: nat] :
          ( B2
          = ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_508_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_509_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_510_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_511_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_512_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_513_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_514_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_515_top_Oextremum__uniqueI,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
     => ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_uniqueI
thf(fact_516_top_Oextremum__uniqueI,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A )
     => ( A = top_top_set_fm ) ) ).

% top.extremum_uniqueI
thf(fact_517_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_518_top_Oextremum__unique,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ top_to728902761071950545ist_fm @ A )
      = ( A = top_to728902761071950545ist_fm ) ) ).

% top.extremum_unique
thf(fact_519_top_Oextremum__unique,axiom,
    ! [A: set_fm] :
      ( ( ord_less_eq_set_fm @ top_top_set_fm @ A )
      = ( A = top_top_set_fm ) ) ).

% top.extremum_unique
thf(fact_520_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_521_top__greatest,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A @ top_to728902761071950545ist_fm ) ).

% top_greatest
thf(fact_522_top__greatest,axiom,
    ! [A: set_fm] : ( ord_less_eq_set_fm @ A @ top_top_set_fm ) ).

% top_greatest
thf(fact_523_top__greatest,axiom,
    ! [A: set_rule] : ( ord_less_eq_set_rule @ A @ top_top_set_rule ) ).

% top_greatest
thf(fact_524_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_525_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_526_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_527_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_528_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream727092118206550309m_rule,R2: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstra1882453022147397137m_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_529_RuleSystem_Opos__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
          = R2 )
       => ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R2 ) @ N ) ) ) ).

% RuleSystem.pos_least
thf(fact_530_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,Rs: stream727092118206550309m_rule,R2: produc164195504107695125m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra1882453022147397137m_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_531_RuleSystem_Opos__def,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,Rs: stream_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstract_pos_rule @ Rs @ R2 )
        = ( ord_Least_nat
          @ ^ [N3: nat] :
              ( ( shd_rule @ ( sdrop_rule @ N3 @ Rs ) )
              = R2 ) ) ) ) ).

% RuleSystem.pos_def
thf(fact_532_RuleSystem_OminWait__least,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,N: nat,Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra3055540779341473792ist_fm @ Eff2 @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S2 )
       => ( ord_less_eq_nat @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S2 ) @ N ) ) ) ).

% RuleSystem.minWait_least
thf(fact_533_fimage__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X4: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm] :
      ( ( B
        = ( F @ X4 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X4 @ A3 )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_534_fimage__eqI,axiom,
    ! [B: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,X4: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm] :
      ( ( B
        = ( F @ X4 ) )
     => ( ( fmembe3381613331217039976ist_fm @ X4 @ A3 )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_535_fimage__eqI,axiom,
    ! [B: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,X4: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule] :
      ( ( B
        = ( F @ X4 ) )
     => ( ( fmembe1635983029874454474m_rule @ X4 @ A3 )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_536_fimage__eqI,axiom,
    ! [B: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,X4: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule] :
      ( ( B
        = ( F @ X4 ) )
     => ( ( fmembe1635983029874454474m_rule @ X4 @ A3 )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F @ A3 ) ) ) ) ).

% fimage_eqI
thf(fact_537_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_538_mkTree_Ocode,axiom,
    ! [Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S2 )
      = ( abstra7467780926338743076m_rule
        @ ( produc491467635432902671m_rule @ S2
          @ ( shd_rule
            @ ( abstra2363873759530810705ist_fm
              @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ S )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ Rs
              @ S2 ) ) )
        @ ( 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
                @ S2 ) ) )
          @ ( 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
                @ S2 ) )
            @ S2 ) ) ) ) ).

% mkTree.code
thf(fact_539_i_OmkTree_Ocode,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S2: list_fm] :
      ( ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S2 )
      = ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S2 @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S2 ) ) ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ ( stl_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S2 ) ) ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S2 ) ) @ S2 ) ) ) ) ).

% i.mkTree.code
thf(fact_540_i_OmkTree_Ocode,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S2: rule] :
      ( ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S2 )
      = ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S2 @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S2 ) ) ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S2 ) ) ) @ ( abstra3109574944306299113m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S2 ) ) @ S2 ) ) ) ) ).

% i.mkTree.code
thf(fact_541_i_OmkTree_Ocode,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 )
      = ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S2 @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 ) ) ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 ) ) ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 ) ) @ S2 ) ) ) ) ).

% i.mkTree.code
thf(fact_542_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra3846921640779263781ist_fm
    = ( ^ [Eff: 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 @ Eff @ Rs2 @ S ) ) ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ Eff @ ( stl_list_fm @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S ) ) ) @ ( abstra1097029126060538623ist_fm @ Eff @ ( shd_list_fm @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_543_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra2830003028686515011m_rule
    = ( ^ [Eff: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rs2: stream8299795917829157543ist_fm,S: rule] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S ) ) ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ Eff @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S ) ) ) @ ( abstra3109574944306299113m_rule @ Eff @ ( shd_Pr772355297128350925ist_fm @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_544_RuleSystem__Defs_OmkTree_Ocode,axiom,
    ( abstra178918799166095863ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rs2: stream_rule,S: produc1996495991257130529ist_fm] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S ) ) ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ Eff @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S ) ) ) @ ( abstra458490714785879965ist_fm @ Eff @ ( shd_rule @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S ) ) @ S ) ) ) ) ) ).

% RuleSystem_Defs.mkTree.code
thf(fact_545_fsubsetI,axiom,
    ! [A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A3 )
         => ( fmembe3381613331217039976ist_fm @ X @ B3 ) )
     => ( ord_le2064643713053750439ist_fm @ A3 @ B3 ) ) ).

% fsubsetI
thf(fact_546_fsubsetI,axiom,
    ! [A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A3 )
         => ( fmembe1635983029874454474m_rule @ X @ B3 ) )
     => ( ord_le6951374404268211425m_rule @ A3 @ B3 ) ) ).

% fsubsetI
thf(fact_547_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_548_tree_Ocollapse,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) )
      = Tree ) ).

% tree.collapse
thf(fact_549_fsubsetD,axiom,
    ! [A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A3 @ B3 )
     => ( ( fmembe3381613331217039976ist_fm @ C @ A3 )
       => ( fmembe3381613331217039976ist_fm @ C @ B3 ) ) ) ).

% fsubsetD
thf(fact_550_fsubsetD,axiom,
    ! [A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A3 @ B3 )
     => ( ( fmembe1635983029874454474m_rule @ C @ A3 )
       => ( fmembe1635983029874454474m_rule @ C @ B3 ) ) ) ).

% fsubsetD
thf(fact_551_fin__mono,axiom,
    ! [A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm,X4: produc1996495991257130529ist_fm] :
      ( ( ord_le2064643713053750439ist_fm @ A3 @ B3 )
     => ( ( fmembe3381613331217039976ist_fm @ X4 @ A3 )
       => ( fmembe3381613331217039976ist_fm @ X4 @ B3 ) ) ) ).

% fin_mono
thf(fact_552_fin__mono,axiom,
    ! [A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule,X4: abstra4572470701423150929m_rule] :
      ( ( ord_le6951374404268211425m_rule @ A3 @ B3 )
     => ( ( fmembe1635983029874454474m_rule @ X4 @ A3 )
       => ( fmembe1635983029874454474m_rule @ X4 @ B3 ) ) ) ).

% fin_mono
thf(fact_553_pred__subset__eq2,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S5: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le1774903256965159270t_fm_o
        @ ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ R3 )
        @ ^ [X3: list_fm,Y5: list_fm] : ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ S5 ) )
      = ( ord_le2055072402069232929ist_fm @ R3 @ S5 ) ) ).

% pred_subset_eq2
thf(fact_554_pred__subset__eq2,axiom,
    ! [R3: set_Pr1008144964186165195m_rule,S5: set_Pr1008144964186165195m_rule] :
      ( ( ord_le4891813983447887534rule_o
        @ ^ [X3: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X3 @ Y5 ) @ R3 )
        @ ^ [X3: produc1996495991257130529ist_fm,Y5: rule] : ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X3 @ Y5 ) @ S5 ) )
      = ( ord_le5052251716133888043m_rule @ R3 @ S5 ) ) ).

% pred_subset_eq2
thf(fact_555_pred__subset__eq2,axiom,
    ! [R3: set_Pr2241365836748572487ist_fm,S5: set_Pr2241365836748572487ist_fm] :
      ( ( ord_le8079975967087607008t_fm_o
        @ ^ [X3: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y5 ) @ R3 )
        @ ^ [X3: rule,Y5: produc1996495991257130529ist_fm] : ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X3 @ Y5 ) @ S5 ) )
      = ( ord_le6285472588696295335ist_fm @ R3 @ S5 ) ) ).

% pred_subset_eq2
thf(fact_556_tree_Oexhaust,axiom,
    ! [Y4: abstra4572470701423150929m_rule] :
      ~ ! [X12: produc164195504107695125m_rule,X23: fset_A5396818216090053761m_rule] :
          ( Y4
         != ( abstra7467780926338743076m_rule @ X12 @ X23 ) ) ).

% tree.exhaust
thf(fact_557_subrelI,axiom,
    ! [R2: set_Pr7058068377845519745ist_fm,S2: set_Pr7058068377845519745ist_fm] :
      ( ! [X: list_fm,Y3: list_fm] :
          ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) @ R2 )
         => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X @ Y3 ) @ S2 ) )
     => ( ord_le2055072402069232929ist_fm @ R2 @ S2 ) ) ).

% subrelI
thf(fact_558_subrelI,axiom,
    ! [R2: set_Pr1008144964186165195m_rule,S2: set_Pr1008144964186165195m_rule] :
      ( ! [X: produc1996495991257130529ist_fm,Y3: rule] :
          ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X @ Y3 ) @ R2 )
         => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ X @ Y3 ) @ S2 ) )
     => ( ord_le5052251716133888043m_rule @ R2 @ S2 ) ) ).

% subrelI
thf(fact_559_subrelI,axiom,
    ! [R2: set_Pr2241365836748572487ist_fm,S2: set_Pr2241365836748572487ist_fm] :
      ( ! [X: rule,Y3: produc1996495991257130529ist_fm] :
          ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y3 ) @ R2 )
         => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ X @ Y3 ) @ S2 ) )
     => ( ord_le6285472588696295335ist_fm @ R2 @ S2 ) ) ).

% subrelI
thf(fact_560_subset__UNIV,axiom,
    ! [A3: set_Pr7058068377845519745ist_fm] : ( ord_le2055072402069232929ist_fm @ A3 @ top_to728902761071950545ist_fm ) ).

% subset_UNIV
thf(fact_561_subset__UNIV,axiom,
    ! [A3: set_fm] : ( ord_less_eq_set_fm @ A3 @ top_top_set_fm ) ).

% subset_UNIV
thf(fact_562_subset__UNIV,axiom,
    ! [A3: set_rule] : ( ord_less_eq_set_rule @ A3 @ top_top_set_rule ) ).

% subset_UNIV
thf(fact_563_fimage__fsubsetI,axiom,
    ! [A3: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A3 )
         => ( fmembe3381613331217039976ist_fm @ ( F @ X ) @ B3 ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage8674584286907012797ist_fm @ F @ A3 ) @ B3 ) ) ).

% fimage_fsubsetI
thf(fact_564_fimage__fsubsetI,axiom,
    ! [A3: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A3 )
         => ( fmembe1635983029874454474m_rule @ ( F @ X ) @ B3 ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage748043113761278389m_rule @ F @ A3 ) @ B3 ) ) ).

% fimage_fsubsetI
thf(fact_565_fimage__fsubsetI,axiom,
    ! [A3: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A3 )
         => ( fmembe3381613331217039976ist_fm @ ( F @ X ) @ B3 ) )
     => ( ord_le2064643713053750439ist_fm @ ( fimage9164954772359936849ist_fm @ F @ A3 ) @ B3 ) ) ).

% fimage_fsubsetI
thf(fact_566_fimage__fsubsetI,axiom,
    ! [A3: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A3 )
         => ( fmembe1635983029874454474m_rule @ ( F @ X ) @ B3 ) )
     => ( ord_le6951374404268211425m_rule @ ( fimage3349360726776475169m_rule @ F @ A3 ) @ B3 ) ) ).

% fimage_fsubsetI
thf(fact_567_tree_Osel_I1_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra3503897793822247108m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X2 ) )
      = X1 ) ).

% tree.sel(1)
thf(fact_568_tree_Osel_I2_J,axiom,
    ! [X1: produc164195504107695125m_rule,X2: fset_A5396818216090053761m_rule] :
      ( ( abstra4962253831105676628m_rule @ ( abstra7467780926338743076m_rule @ X1 @ X2 ) )
      = X2 ) ).

% tree.sel(2)
thf(fact_569_sset__sdrop,axiom,
    ! [N: nat,S2: stream727092118206550309m_rule] : ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) ) @ ( sset_P4979403222752902913m_rule @ S2 ) ) ).

% sset_sdrop
thf(fact_570_sset__sdrop,axiom,
    ! [N: nat,S2: stream_fm] : ( ord_less_eq_set_fm @ ( sset_fm @ ( sdrop_fm @ N @ S2 ) ) @ ( sset_fm @ S2 ) ) ).

% sset_sdrop
thf(fact_571_sset__sdrop,axiom,
    ! [N: nat,S2: stream_rule] : ( ord_less_eq_set_rule @ ( sset_rule @ ( sdrop_rule @ N @ S2 ) ) @ ( sset_rule @ S2 ) ) ).

% sset_sdrop
thf(fact_572_tree_Oexhaust__sel,axiom,
    ! [Tree: abstra4572470701423150929m_rule] :
      ( Tree
      = ( abstra7467780926338743076m_rule @ ( abstra3503897793822247108m_rule @ Tree ) @ ( abstra4962253831105676628m_rule @ Tree ) ) ) ).

% tree.exhaust_sel
thf(fact_573_treeB__sdrop,axiom,
    ! [N: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeB @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) @ ( treeB @ Steps ) ) ).

% treeB_sdrop
thf(fact_574_treeA__sdrop,axiom,
    ! [N: nat,Steps: stream727092118206550309m_rule] : ( ord_less_eq_set_fm @ ( treeA @ ( sdrop_7224736112439592940m_rule @ N @ Steps ) ) @ ( treeA @ Steps ) ) ).

% treeA_sdrop
thf(fact_575_fset__eqI,axiom,
    ! [A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ! [X: produc1996495991257130529ist_fm] :
          ( ( fmembe3381613331217039976ist_fm @ X @ A3 )
          = ( fmembe3381613331217039976ist_fm @ X @ B3 ) )
     => ( A3 = B3 ) ) ).

% fset_eqI
thf(fact_576_fset__eqI,axiom,
    ! [A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ! [X: abstra4572470701423150929m_rule] :
          ( ( fmembe1635983029874454474m_rule @ X @ A3 )
          = ( fmembe1635983029874454474m_rule @ X @ B3 ) )
     => ( A3 = B3 ) ) ).

% fset_eqI
thf(fact_577_fequalityCE,axiom,
    ! [A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( A3 = B3 )
     => ( ( ( fmembe3381613331217039976ist_fm @ C @ A3 )
         => ~ ( fmembe3381613331217039976ist_fm @ C @ B3 ) )
       => ~ ( ~ ( fmembe3381613331217039976ist_fm @ C @ A3 )
           => ( fmembe3381613331217039976ist_fm @ C @ B3 ) ) ) ) ).

% fequalityCE
thf(fact_578_fequalityCE,axiom,
    ! [A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( A3 = B3 )
     => ( ( ( fmembe1635983029874454474m_rule @ C @ A3 )
         => ~ ( fmembe1635983029874454474m_rule @ C @ B3 ) )
       => ~ ( ~ ( fmembe1635983029874454474m_rule @ C @ A3 )
           => ( fmembe1635983029874454474m_rule @ C @ B3 ) ) ) ) ).

% fequalityCE
thf(fact_579_eq__fmem__trans,axiom,
    ! [A: produc1996495991257130529ist_fm,B: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm] :
      ( ( A = B )
     => ( ( fmembe3381613331217039976ist_fm @ B @ A3 )
       => ( fmembe3381613331217039976ist_fm @ A @ A3 ) ) ) ).

% eq_fmem_trans
thf(fact_580_eq__fmem__trans,axiom,
    ! [A: abstra4572470701423150929m_rule,B: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule] :
      ( ( A = B )
     => ( ( fmembe1635983029874454474m_rule @ B @ A3 )
       => ( fmembe1635983029874454474m_rule @ A @ A3 ) ) ) ).

% eq_fmem_trans
thf(fact_581_eqfset__imp__iff,axiom,
    ! [A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm,X4: produc1996495991257130529ist_fm] :
      ( ( A3 = B3 )
     => ( ( fmembe3381613331217039976ist_fm @ X4 @ A3 )
        = ( fmembe3381613331217039976ist_fm @ X4 @ B3 ) ) ) ).

% eqfset_imp_iff
thf(fact_582_eqfset__imp__iff,axiom,
    ! [A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule,X4: abstra4572470701423150929m_rule] :
      ( ( A3 = B3 )
     => ( ( fmembe1635983029874454474m_rule @ X4 @ A3 )
        = ( fmembe1635983029874454474m_rule @ X4 @ B3 ) ) ) ).

% eqfset_imp_iff
thf(fact_583_if__split__fmem1,axiom,
    ! [Q2: $o,X4: produc1996495991257130529ist_fm,Y4: produc1996495991257130529ist_fm,B: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ ( if_Pro3930376587665744871ist_fm @ Q2 @ X4 @ Y4 ) @ B )
      = ( ( Q2
         => ( fmembe3381613331217039976ist_fm @ X4 @ B ) )
        & ( ~ Q2
         => ( fmembe3381613331217039976ist_fm @ Y4 @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_584_if__split__fmem1,axiom,
    ! [Q2: $o,X4: abstra4572470701423150929m_rule,Y4: abstra4572470701423150929m_rule,B: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ ( if_Abs4566652138351786251m_rule @ Q2 @ X4 @ Y4 ) @ B )
      = ( ( Q2
         => ( fmembe1635983029874454474m_rule @ X4 @ B ) )
        & ( ~ Q2
         => ( fmembe1635983029874454474m_rule @ Y4 @ B ) ) ) ) ).

% if_split_fmem1
thf(fact_585_if__split__fmem2,axiom,
    ! [A: produc1996495991257130529ist_fm,Q2: $o,X4: fset_P661503646757059847ist_fm,Y4: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ A @ ( if_fse3714977293194272717ist_fm @ Q2 @ X4 @ Y4 ) )
      = ( ( Q2
         => ( fmembe3381613331217039976ist_fm @ A @ X4 ) )
        & ( ~ Q2
         => ( fmembe3381613331217039976ist_fm @ A @ Y4 ) ) ) ) ).

% if_split_fmem2
thf(fact_586_if__split__fmem2,axiom,
    ! [A: abstra4572470701423150929m_rule,Q2: $o,X4: fset_A5396818216090053761m_rule,Y4: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ A @ ( if_fse3370020817956878651m_rule @ Q2 @ X4 @ Y4 ) )
      = ( ( Q2
         => ( fmembe1635983029874454474m_rule @ A @ X4 ) )
        & ( ~ Q2
         => ( fmembe1635983029874454474m_rule @ A @ Y4 ) ) ) ) ).

% if_split_fmem2
thf(fact_587_eqfelem__imp__iff,axiom,
    ! [X4: produc1996495991257130529ist_fm,Y4: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm] :
      ( ( X4 = Y4 )
     => ( ( fmembe3381613331217039976ist_fm @ X4 @ A3 )
        = ( fmembe3381613331217039976ist_fm @ Y4 @ A3 ) ) ) ).

% eqfelem_imp_iff
thf(fact_588_eqfelem__imp__iff,axiom,
    ! [X4: abstra4572470701423150929m_rule,Y4: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule] :
      ( ( X4 = Y4 )
     => ( ( fmembe1635983029874454474m_rule @ X4 @ A3 )
        = ( fmembe1635983029874454474m_rule @ Y4 @ A3 ) ) ) ).

% eqfelem_imp_iff
thf(fact_589_fimage__fimage,axiom,
    ! [F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A3: fset_P661503646757059847ist_fm] :
      ( ( fimage3349360726776475169m_rule @ F @ ( fimage748043113761278389m_rule @ G @ A3 ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( F @ ( G @ X3 ) )
        @ A3 ) ) ).

% fimage_fimage
thf(fact_590_fimage__fimage,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A3: fset_A5396818216090053761m_rule] :
      ( ( fimage8674584286907012797ist_fm @ F @ ( fimage9164954772359936849ist_fm @ G @ A3 ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X3: abstra4572470701423150929m_rule] : ( F @ ( G @ X3 ) )
        @ A3 ) ) ).

% fimage_fimage
thf(fact_591_fimage__fimage,axiom,
    ! [F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm] :
      ( ( fimage748043113761278389m_rule @ F @ ( fimage8674584286907012797ist_fm @ G @ A3 ) )
      = ( fimage748043113761278389m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( F @ ( G @ X3 ) )
        @ A3 ) ) ).

% fimage_fimage
thf(fact_592_fimage__fimage,axiom,
    ! [F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A3: fset_A5396818216090053761m_rule] :
      ( ( fimage748043113761278389m_rule @ F @ ( fimage9164954772359936849ist_fm @ G @ A3 ) )
      = ( fimage3349360726776475169m_rule
        @ ^ [X3: abstra4572470701423150929m_rule] : ( F @ ( G @ X3 ) )
        @ A3 ) ) ).

% fimage_fimage
thf(fact_593_fimage__fimage,axiom,
    ! [F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule] :
      ( ( fimage9164954772359936849ist_fm @ F @ ( fimage3349360726776475169m_rule @ G @ A3 ) )
      = ( fimage9164954772359936849ist_fm
        @ ^ [X3: abstra4572470701423150929m_rule] : ( F @ ( G @ X3 ) )
        @ A3 ) ) ).

% fimage_fimage
thf(fact_594_fimage__fimage,axiom,
    ! [F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A3: fset_P661503646757059847ist_fm] :
      ( ( fimage9164954772359936849ist_fm @ F @ ( fimage748043113761278389m_rule @ G @ A3 ) )
      = ( fimage8674584286907012797ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] : ( F @ ( G @ X3 ) )
        @ A3 ) ) ).

% fimage_fimage
thf(fact_595_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_596_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_597_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_598_add__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_le_mono1
thf(fact_599_add__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_600_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ K2 @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K2 @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_601_add__leD2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ( ord_less_eq_nat @ K2 @ N ) ) ).

% add_leD2
thf(fact_602_add__leD1,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% add_leD1
thf(fact_603_le__add2,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).

% le_add2
thf(fact_604_le__add1,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).

% le_add1
thf(fact_605_add__leE,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ~ ( ( ord_less_eq_nat @ M2 @ N )
         => ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).

% add_leE
thf(fact_606_fimageE,axiom,
    ! [B: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F @ A3 ) )
     => ~ ! [X: produc1996495991257130529ist_fm] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X @ A3 ) ) ) ).

% fimageE
thf(fact_607_fimageE,axiom,
    ! [B: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,A3: fset_A5396818216090053761m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A3 ) )
     => ~ ! [X: abstra4572470701423150929m_rule] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe1635983029874454474m_rule @ X @ A3 ) ) ) ).

% fimageE
thf(fact_608_fimageE,axiom,
    ! [B: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,A3: fset_P661503646757059847ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A3 ) )
     => ~ ! [X: produc1996495991257130529ist_fm] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe3381613331217039976ist_fm @ X @ A3 ) ) ) ).

% fimageE
thf(fact_609_fimageE,axiom,
    ! [B: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F @ A3 ) )
     => ~ ! [X: abstra4572470701423150929m_rule] :
            ( ( B
              = ( F @ X ) )
           => ~ ( fmembe1635983029874454474m_rule @ X @ A3 ) ) ) ).

% fimageE
thf(fact_610_fimageI,axiom,
    ! [X4: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X4 @ A3 )
     => ( fmembe3381613331217039976ist_fm @ ( F @ X4 ) @ ( fimage8674584286907012797ist_fm @ F @ A3 ) ) ) ).

% fimageI
thf(fact_611_fimageI,axiom,
    ! [X4: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X4 @ A3 )
     => ( fmembe1635983029874454474m_rule @ ( F @ X4 ) @ ( fimage748043113761278389m_rule @ F @ A3 ) ) ) ).

% fimageI
thf(fact_612_fimageI,axiom,
    ! [X4: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X4 @ A3 )
     => ( fmembe3381613331217039976ist_fm @ ( F @ X4 ) @ ( fimage9164954772359936849ist_fm @ F @ A3 ) ) ) ).

% fimageI
thf(fact_613_fimageI,axiom,
    ! [X4: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X4 @ A3 )
     => ( fmembe1635983029874454474m_rule @ ( F @ X4 ) @ ( fimage3349360726776475169m_rule @ F @ A3 ) ) ) ).

% fimageI
thf(fact_614_fimage__cong,axiom,
    ! [M4: fset_P661503646757059847ist_fm,N4: fset_P661503646757059847ist_fm,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule,G: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( M4 = N4 )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( fmembe3381613331217039976ist_fm @ X @ N4 )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( fimage748043113761278389m_rule @ F @ M4 )
          = ( fimage748043113761278389m_rule @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_615_fimage__cong,axiom,
    ! [M4: fset_A5396818216090053761m_rule,N4: fset_A5396818216090053761m_rule,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( M4 = N4 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( fmembe1635983029874454474m_rule @ X @ N4 )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( fimage9164954772359936849ist_fm @ F @ M4 )
          = ( fimage9164954772359936849ist_fm @ G @ N4 ) ) ) ) ).

% fimage_cong
thf(fact_616_rev__fimage__eqI,axiom,
    ! [X4: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm,B: produc1996495991257130529ist_fm,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X4 @ A3 )
     => ( ( B
          = ( F @ X4 ) )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage8674584286907012797ist_fm @ F @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_617_rev__fimage__eqI,axiom,
    ! [X4: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm,B: abstra4572470701423150929m_rule,F: produc1996495991257130529ist_fm > abstra4572470701423150929m_rule] :
      ( ( fmembe3381613331217039976ist_fm @ X4 @ A3 )
     => ( ( B
          = ( F @ X4 ) )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage748043113761278389m_rule @ F @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_618_rev__fimage__eqI,axiom,
    ! [X4: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule,B: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm] :
      ( ( fmembe1635983029874454474m_rule @ X4 @ A3 )
     => ( ( B
          = ( F @ X4 ) )
       => ( fmembe3381613331217039976ist_fm @ B @ ( fimage9164954772359936849ist_fm @ F @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_619_rev__fimage__eqI,axiom,
    ! [X4: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule,B: abstra4572470701423150929m_rule,F: abstra4572470701423150929m_rule > abstra4572470701423150929m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X4 @ A3 )
     => ( ( B
          = ( F @ X4 ) )
       => ( fmembe1635983029874454474m_rule @ B @ ( fimage3349360726776475169m_rule @ F @ A3 ) ) ) ) ).

% rev_fimage_eqI
thf(fact_620_mkTree__unfold,axiom,
    ! [Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm
        @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
            ( ( eff @ R @ S )
            = ( some_f2162114139285431020ist_fm @ Ss ) )
        @ Rs
        @ S2 )
      = ( case_s2494923709252486212m_rule
        @ ^ [R: rule,S8: stream_rule] :
            ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S2 @ R )
            @ ( fimage748043113761278389m_rule
              @ ( abstra178918799166095863ist_fm
                @ ^ [S: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ S8 )
              @ ( abstra458490714785879965ist_fm
                @ ^ [S: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ S @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R
                @ S2 ) ) )
        @ ( abstra2363873759530810705ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ Rs
          @ S2 ) ) ) ).

% mkTree_unfold
thf(fact_621_i_OmkTree__unfold,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,Rs: stream_list_fm,S2: list_fm] :
      ( ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S2 )
      = ( case_s5690291548269216621ist_fm
        @ ^ [R: list_fm,S8: stream_list_fm] : ( abstra6542830291470149330ist_fm @ ( produc381145313068854617ist_fm @ S2 @ R ) @ ( fimage27253841482530780ist_fm @ ( abstra3846921640779263781ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ S8 ) @ ( abstra1097029126060538623ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ R @ S2 ) ) )
        @ ( abstra2983386638146376267ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rs @ S2 ) ) ) ).

% i.mkTree_unfold
thf(fact_622_i_OmkTree__unfold,axiom,
    ! [Eff3: produc1996495991257130529ist_fm > rule > option_fset_rule,Rs: stream8299795917829157543ist_fm,S2: rule] :
      ( ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S2 )
      = ( case_s482378023405642050ist_fm
        @ ^ [R: produc1996495991257130529ist_fm,S8: stream8299795917829157543ist_fm] : ( abstra2711079411085783200ist_fm @ ( produc7063755442767259331ist_fm @ S2 @ R ) @ ( fimage4184475965855427473ist_fm @ ( abstra2830003028686515011m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ S8 ) @ ( abstra3109574944306299113m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ R @ S2 ) ) )
        @ ( abstra5014957989051229853m_rule @ ( abstra8110425618579670566m_rule @ Eff3 ) @ Rs @ S2 ) ) ) ).

% i.mkTree_unfold
thf(fact_623_i_OmkTree__unfold,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rs: stream_rule,S2: produc1996495991257130529ist_fm] :
      ( ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 )
      = ( case_s2494923709252486212m_rule
        @ ^ [R: rule,S8: stream_rule] : ( abstra7467780926338743076m_rule @ ( produc491467635432902671m_rule @ S2 @ R ) @ ( fimage748043113761278389m_rule @ ( abstra178918799166095863ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ S8 ) @ ( abstra458490714785879965ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R @ S2 ) ) )
        @ ( abstra2363873759530810705ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rs @ S2 ) ) ) ).

% i.mkTree_unfold
thf(fact_624_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra3846921640779263781ist_fm
    = ( ^ [Eff: 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 @ Eff @ S8 ) @ ( abstra1097029126060538623ist_fm @ Eff @ R @ S ) ) )
          @ ( abstra2983386638146376267ist_fm @ Eff @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_625_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra2830003028686515011m_rule
    = ( ^ [Eff: 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 @ Eff @ S8 ) @ ( abstra3109574944306299113m_rule @ Eff @ R @ S ) ) )
          @ ( abstra5014957989051229853m_rule @ Eff @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_626_RuleSystem__Defs_OmkTree__unfold,axiom,
    ( abstra178918799166095863ist_fm
    = ( ^ [Eff: 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 @ Eff @ S8 ) @ ( abstra458490714785879965ist_fm @ Eff @ R @ S ) ) )
          @ ( abstra2363873759530810705ist_fm @ Eff @ Rs2 @ S ) ) ) ) ).

% RuleSystem_Defs.mkTree_unfold
thf(fact_627_wf_Ocoinduct,axiom,
    ! [X6: abstra4572470701423150929m_rule > $o,X4: abstra4572470701423150929m_rule] :
      ( ( X6 @ X4 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X6 @ X )
           => ? [T8: abstra4572470701423150929m_rule] :
                ( ( X = T8 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( sset_rule @ rules ) )
                & ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) )
                  = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T8 ) ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T8 ) )
                   => ( ( X6 @ 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
          @ X4 ) ) ) ).

% wf.coinduct
thf(fact_628_wf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ rules ) )
     => ( ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) )
          = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) ) )
       => ( ! [T6: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm
                @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ S )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ rules
                @ T6 ) )
         => ( abstra4476735159789068068ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ rules
            @ T2 ) ) ) ) ).

% wf.intros
thf(fact_629_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 )
      = ( ? [T: abstra4572470701423150929m_rule] :
            ( ( A = T )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ rules ) )
            & ( ( eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) )
              = ( some_f2162114139285431020ist_fm @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) ) )
            & ! [X3: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X3 @ ( abstra4962253831105676628m_rule @ T ) )
               => ( abstra4476735159789068068ist_fm
                  @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                      ( ( eff @ R @ S )
                      = ( some_f2162114139285431020ist_fm @ Ss ) )
                  @ rules
                  @ X3 ) ) ) ) ) ).

% wf.simps
thf(fact_630_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 ) ) ) )
           => ~ ! [T5: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm
                    @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                        ( ( eff @ R @ S )
                        = ( some_f2162114139285431020ist_fm @ Ss ) )
                    @ rules
                    @ T5 ) ) ) ) ) ).

% wf.cases
thf(fact_631_pred__subset__eq,axiom,
    ! [R3: set_Pr7058068377845519745ist_fm,S5: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le8518180823940269308t_fm_o
        @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ R3 )
        @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ S5 ) )
      = ( ord_le2055072402069232929ist_fm @ R3 @ S5 ) ) ).

% pred_subset_eq
thf(fact_632_pred__subset__eq,axiom,
    ! [R3: set_fm,S5: set_fm] :
      ( ( ord_less_eq_fm_o
        @ ^ [X3: fm] : ( member_fm @ X3 @ R3 )
        @ ^ [X3: fm] : ( member_fm @ X3 @ S5 ) )
      = ( ord_less_eq_set_fm @ R3 @ S5 ) ) ).

% pred_subset_eq
thf(fact_633_pred__subset__eq,axiom,
    ! [R3: set_rule,S5: set_rule] :
      ( ( ord_less_eq_rule_o
        @ ^ [X3: rule] : ( member_rule @ X3 @ R3 )
        @ ^ [X3: rule] : ( member_rule @ X3 @ S5 ) )
      = ( ord_less_eq_set_rule @ R3 @ S5 ) ) ).

% pred_subset_eq
thf(fact_634_subset__CollectI,axiom,
    ! [B3: set_Pr7058068377845519745ist_fm,A3: set_Pr7058068377845519745ist_fm,Q2: produc1996495991257130529ist_fm > $o,P2: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B3 @ A3 )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X @ B3 )
           => ( ( Q2 @ X )
             => ( P2 @ X ) ) )
       => ( ord_le2055072402069232929ist_fm
          @ ( collec7231575776666803724ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ B3 )
                & ( Q2 @ X3 ) ) )
          @ ( collec7231575776666803724ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_635_subset__CollectI,axiom,
    ! [B3: set_fm,A3: set_fm,Q2: fm > $o,P2: fm > $o] :
      ( ( ord_less_eq_set_fm @ B3 @ A3 )
     => ( ! [X: fm] :
            ( ( member_fm @ X @ B3 )
           => ( ( Q2 @ X )
             => ( P2 @ X ) ) )
       => ( ord_less_eq_set_fm
          @ ( collect_fm
            @ ^ [X3: fm] :
                ( ( member_fm @ X3 @ B3 )
                & ( Q2 @ X3 ) ) )
          @ ( collect_fm
            @ ^ [X3: fm] :
                ( ( member_fm @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_636_subset__CollectI,axiom,
    ! [B3: set_rule,A3: set_rule,Q2: rule > $o,P2: rule > $o] :
      ( ( ord_less_eq_set_rule @ B3 @ A3 )
     => ( ! [X: rule] :
            ( ( member_rule @ X @ B3 )
           => ( ( Q2 @ X )
             => ( P2 @ X ) ) )
       => ( ord_less_eq_set_rule
          @ ( collect_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ B3 )
                & ( Q2 @ X3 ) ) )
          @ ( collect_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_637_subset__Collect__iff,axiom,
    ! [B3: set_Pr7058068377845519745ist_fm,A3: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ( ord_le2055072402069232929ist_fm @ B3 @ A3 )
     => ( ( ord_le2055072402069232929ist_fm @ B3
          @ ( collec7231575776666803724ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) )
        = ( ! [X3: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ X3 @ B3 )
             => ( P2 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_638_subset__Collect__iff,axiom,
    ! [B3: set_fm,A3: set_fm,P2: fm > $o] :
      ( ( ord_less_eq_set_fm @ B3 @ A3 )
     => ( ( ord_less_eq_set_fm @ B3
          @ ( collect_fm
            @ ^ [X3: fm] :
                ( ( member_fm @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) )
        = ( ! [X3: fm] :
              ( ( member_fm @ X3 @ B3 )
             => ( P2 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_639_subset__Collect__iff,axiom,
    ! [B3: set_rule,A3: set_rule,P2: rule > $o] :
      ( ( ord_less_eq_set_rule @ B3 @ A3 )
     => ( ( ord_less_eq_set_rule @ B3
          @ ( collect_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) )
        = ( ! [X3: rule] :
              ( ( member_rule @ X3 @ B3 )
             => ( P2 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_640_Collect__subset,axiom,
    ! [A3: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X3 @ A3 )
            & ( P2 @ X3 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_641_Collect__subset,axiom,
    ! [A3: set_fm,P2: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X3: fm] :
            ( ( member_fm @ X3 @ A3 )
            & ( P2 @ X3 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_642_Collect__subset,axiom,
    ! [A3: set_rule,P2: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X3: rule] :
            ( ( member_rule @ X3 @ A3 )
            & ( P2 @ X3 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_643_less__eq__set__def,axiom,
    ( ord_le2055072402069232929ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B7: set_Pr7058068377845519745ist_fm] :
          ( ord_le8518180823940269308t_fm_o
          @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ A7 )
          @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_644_less__eq__set__def,axiom,
    ( ord_less_eq_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( ord_less_eq_fm_o
          @ ^ [X3: fm] : ( member_fm @ X3 @ A7 )
          @ ^ [X3: fm] : ( member_fm @ X3 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_645_less__eq__set__def,axiom,
    ( ord_less_eq_set_rule
    = ( ^ [A7: set_rule,B7: set_rule] :
          ( ord_less_eq_rule_o
          @ ^ [X3: rule] : ( member_rule @ X3 @ A7 )
          @ ^ [X3: rule] : ( member_rule @ X3 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_646_comp__apply__eq,axiom,
    ! [F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > produc164195504107695125m_rule,X4: abstra4572470701423150929m_rule,H: produc164195504107695125m_rule > produc1996495991257130529ist_fm,K2: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( ( F @ ( G @ X4 ) )
        = ( H @ ( K2 @ X4 ) ) )
     => ( ( comp_P4378417847848123130m_rule @ F @ G @ X4 )
        = ( comp_P4378417847848123130m_rule @ H @ K2 @ X4 ) ) ) ).

% comp_apply_eq
thf(fact_647_K__record__comp,axiom,
    ! [C: produc1996495991257130529ist_fm,F: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P4378417847848123130m_rule
        @ ^ [X3: produc164195504107695125m_rule] : C
        @ F )
      = ( ^ [X3: abstra4572470701423150929m_rule] : C ) ) ).

% K_record_comp
thf(fact_648_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,X24: stream_rule] : ( H @ ( F @ X13 @ X24 ) )
        @ Stream ) ) ).

% stream.case_distrib
thf(fact_649_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_650_shd__def,axiom,
    ( shd_Pr7235097944458474089m_rule
    = ( case_s4599053733867249880m_rule
      @ ^ [X13: produc164195504107695125m_rule,X24: stream727092118206550309m_rule] : X13 ) ) ).

% shd_def
thf(fact_651_shd__def,axiom,
    ( shd_rule
    = ( case_s3980788006751379256e_rule
      @ ^ [X13: rule,X24: stream_rule] : X13 ) ) ).

% shd_def
thf(fact_652_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_653_stream_Osplit__sel,axiom,
    ! [P2: abstra4572470701423150929m_rule > $o,F: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( P2 @ ( case_s2494923709252486212m_rule @ F @ Stream ) )
      = ( ( Stream
          = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) )
       => ( P2 @ ( F @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ) ) ).

% stream.split_sel
thf(fact_654_stream_Osplit__sel__asm,axiom,
    ! [P2: abstra4572470701423150929m_rule > $o,F: rule > stream_rule > abstra4572470701423150929m_rule,Stream: stream_rule] :
      ( ( P2 @ ( case_s2494923709252486212m_rule @ F @ Stream ) )
      = ( ~ ( ( Stream
              = ( sCons_rule @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) )
            & ~ ( P2 @ ( F @ ( shd_rule @ Stream ) @ ( stl_rule @ Stream ) ) ) ) ) ) ).

% stream.split_sel_asm
thf(fact_655_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X6: abstra5042247676209264507ist_fm > $o,X4: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: abstra5042247676209264507ist_fm] :
            ( ( X6 @ X )
           => ? [T8: abstra5042247676209264507ist_fm] :
                ( ( X = T8 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T8 ) ) @ ( sset_list_fm @ Rules ) )
                & ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T8 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T8 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T8 ) ) )
                & ! [Xa: abstra5042247676209264507ist_fm] :
                    ( ( fmembe913231786053615938ist_fm @ Xa @ ( abstra1905332785217431714ist_fm @ T8 ) )
                   => ( ( X6 @ Xa )
                      | ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ Xa ) ) ) ) )
       => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_656_RuleSystem__Defs_Owf_Ocoinduct,axiom,
    ! [X6: abstra4572470701423150929m_rule > $o,X4: abstra4572470701423150929m_rule,Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X6 @ X )
           => ? [T8: abstra4572470701423150929m_rule] :
                ( ( X = T8 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( sset_rule @ Rules ) )
                & ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T8 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T8 ) )
                   => ( ( X6 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ X4 ) ) ) ).

% RuleSystem_Defs.wf.coinduct
thf(fact_657_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T2: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff2: list_fm > list_fm > fset_list_fm > $o] :
      ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( sset_list_fm @ Rules ) )
     => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T2 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T2 ) ) )
       => ( ! [T6: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ T6 @ ( abstra1905332785217431714ist_fm @ T2 ) )
             => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ T6 ) )
         => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ T2 ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_658_RuleSystem__Defs_Owf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Rules: stream_rule,Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) )
       => ( ! [T6: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T6 ) )
         => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T2 ) ) ) ) ).

% RuleSystem_Defs.wf.intros
thf(fact_659_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra8964572462466433336ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,Rules2: stream_list_fm,A2: abstra5042247676209264507ist_fm] :
        ? [T: abstra5042247676209264507ist_fm] :
          ( ( A2 = T )
          & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( sset_list_fm @ Rules2 ) )
          & ( Eff @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T ) ) )
          & ! [X3: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ X3 @ ( abstra1905332785217431714ist_fm @ T ) )
             => ( abstra8964572462466433336ist_fm @ Eff @ Rules2 @ X3 ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_660_RuleSystem__Defs_Owf_Osimps,axiom,
    ( abstra4476735159789068068ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules2: stream_rule,A2: abstra4572470701423150929m_rule] :
        ? [T: abstra4572470701423150929m_rule] :
          ( ( A2 = T )
          & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ Rules2 ) )
          & ( Eff @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) )
          & ! [X3: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ X3 @ ( abstra4962253831105676628m_rule @ T ) )
             => ( abstra4476735159789068068ist_fm @ Eff @ Rules2 @ X3 ) ) ) ) ) ).

% RuleSystem_Defs.wf.simps
thf(fact_661_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,A: abstra5042247676209264507ist_fm] :
      ( ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( sset_list_fm @ Rules ) )
         => ( ( Eff2 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ A ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ A ) ) )
           => ~ ! [T5: abstra5042247676209264507ist_fm] :
                  ( ( fmembe913231786053615938ist_fm @ T5 @ ( abstra1905332785217431714ist_fm @ A ) )
                 => ( abstra8964572462466433336ist_fm @ Eff2 @ Rules @ T5 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_662_RuleSystem__Defs_Owf_Ocases,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ A )
     => ~ ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( sset_rule @ Rules ) )
         => ( ( Eff2 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ A ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ A ) ) )
           => ~ ! [T5: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm @ Eff2 @ Rules @ T5 ) ) ) ) ) ).

% RuleSystem_Defs.wf.cases
thf(fact_663_i_Owf_Ocoinduct,axiom,
    ! [X6: abstra5042247676209264507ist_fm > $o,X4: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( X6 @ X4 )
     => ( ! [X: abstra5042247676209264507ist_fm] :
            ( ( X6 @ X )
           => ? [T8: abstra5042247676209264507ist_fm] :
                ( ( X = T8 )
                & ( member_list_fm @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T8 ) ) @ ( sset_list_fm @ Rules ) )
                & ( abstra256862739949234434ist_fm @ Eff3 @ ( produc4588648349897876871ist_fm @ ( abstra778364809706815794ist_fm @ T8 ) ) @ ( produc1501393135466168645ist_fm @ ( abstra778364809706815794ist_fm @ T8 ) ) @ ( fimage9094613911859171998ist_fm @ ( comp_P8076707111405491611ist_fm @ produc1501393135466168645ist_fm @ abstra778364809706815794ist_fm ) @ ( abstra1905332785217431714ist_fm @ T8 ) ) )
                & ! [Xa: abstra5042247676209264507ist_fm] :
                    ( ( fmembe913231786053615938ist_fm @ Xa @ ( abstra1905332785217431714ist_fm @ T8 ) )
                   => ( ( X6 @ Xa )
                      | ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ Xa ) ) ) ) )
       => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.wf.coinduct
thf(fact_664_i_Owf_Ocoinduct,axiom,
    ! [X6: abstra4572470701423150929m_rule > $o,X4: abstra4572470701423150929m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( X6 @ X4 )
     => ( ! [X: abstra4572470701423150929m_rule] :
            ( ( X6 @ X )
           => ? [T8: abstra4572470701423150929m_rule] :
                ( ( X = T8 )
                & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( sset_rule @ Rules ) )
                & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T8 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T8 ) ) )
                & ! [Xa: abstra4572470701423150929m_rule] :
                    ( ( fmembe1635983029874454474m_rule @ Xa @ ( abstra4962253831105676628m_rule @ T8 ) )
                   => ( ( X6 @ Xa )
                      | ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ Xa ) ) ) ) )
       => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X4 ) ) ) ).

% i.wf.coinduct
thf(fact_665_i_Owf_Ointros,axiom,
    ! [T2: abstra5042247676209264507ist_fm,Rules: stream_list_fm,Eff3: list_fm > list_fm > option_fset_list_fm] :
      ( ( 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 ) ) )
       => ( ! [T6: abstra5042247676209264507ist_fm] :
              ( ( fmembe913231786053615938ist_fm @ T6 @ ( abstra1905332785217431714ist_fm @ T2 ) )
             => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ T6 ) )
         => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ T2 ) ) ) ) ).

% i.wf.intros
thf(fact_666_i_Owf_Ointros,axiom,
    ! [T2: abstra4572470701423150929m_rule,Rules: stream_rule,Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm] :
      ( ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( sset_rule @ Rules ) )
     => ( ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T2 ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T2 ) ) )
       => ( ! [T6: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T6 @ ( abstra4962253831105676628m_rule @ T2 ) )
             => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T6 ) )
         => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T2 ) ) ) ) ).

% i.wf.intros
thf(fact_667_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 )
      = ( ? [T: abstra5042247676209264507ist_fm] :
            ( ( A = T )
            & ( 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 ) ) )
            & ! [X3: abstra5042247676209264507ist_fm] :
                ( ( fmembe913231786053615938ist_fm @ X3 @ ( abstra1905332785217431714ist_fm @ T ) )
               => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ) ) ).

% i.wf.simps
thf(fact_668_i_Owf_Osimps,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,Rules: stream_rule,A: abstra4572470701423150929m_rule] :
      ( ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ A )
      = ( ? [T: abstra4572470701423150929m_rule] :
            ( ( A = T )
            & ( member_rule @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( sset_rule @ Rules ) )
            & ( abstra5459341389059251418ist_fm @ Eff3 @ ( produc7165828336582415457m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( produc6879501374131015971m_rule @ ( abstra3503897793822247108m_rule @ T ) ) @ ( fimage9164954772359936849ist_fm @ ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ abstra3503897793822247108m_rule ) @ ( abstra4962253831105676628m_rule @ T ) ) )
            & ! [X3: abstra4572470701423150929m_rule] :
                ( ( fmembe1635983029874454474m_rule @ X3 @ ( abstra4962253831105676628m_rule @ T ) )
               => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ X3 ) ) ) ) ) ).

% i.wf.simps
thf(fact_669_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 ) ) )
           => ~ ! [T5: abstra5042247676209264507ist_fm] :
                  ( ( fmembe913231786053615938ist_fm @ T5 @ ( abstra1905332785217431714ist_fm @ A ) )
                 => ( abstra8964572462466433336ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ Rules @ T5 ) ) ) ) ) ).

% i.wf.cases
thf(fact_670_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 ) ) )
           => ~ ! [T5: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T5 @ ( abstra4962253831105676628m_rule @ A ) )
                 => ( abstra4476735159789068068ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ Rules @ T5 ) ) ) ) ) ).

% i.wf.cases
thf(fact_671_subset__code_I1_J,axiom,
    ! [Xs: list_P5616295576739893671ist_fm,B3: set_Pr7058068377845519745ist_fm] :
      ( ( ord_le2055072402069232929ist_fm @ ( set_Pr8767716839810916150ist_fm @ Xs ) @ B3 )
      = ( ! [X3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X3 @ ( set_Pr8767716839810916150ist_fm @ Xs ) )
           => ( member8102475879199740618ist_fm @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_672_subset__code_I1_J,axiom,
    ! [Xs: list_fm,B3: set_fm] :
      ( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B3 )
      = ( ! [X3: fm] :
            ( ( member_fm @ X3 @ ( set_fm2 @ Xs ) )
           => ( member_fm @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_673_subset__code_I1_J,axiom,
    ! [Xs: list_rule,B3: set_rule] :
      ( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ B3 )
      = ( ! [X3: rule] :
            ( ( member_rule @ X3 @ ( set_rule2 @ Xs ) )
           => ( member_rule @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_674_Konig,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ~ ( abstra3507476289341504315m_rule @ T2 )
     => ( abstra2285467793620430502m_rule @ T2 @ ( abstra3079604473823497664m_rule @ T2 ) ) ) ).

% Konig
thf(fact_675_sinterleave_Ocode,axiom,
    ( sinter8084399408526207096m_rule
    = ( ^ [S12: stream727092118206550309m_rule,S23: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S12 ) @ ( sinter8084399408526207096m_rule @ S23 @ ( stl_Pr950425576149878629m_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_676_sinterleave_Ocode,axiom,
    ( sinterleave_rule
    = ( ^ [S12: stream_rule,S23: stream_rule] : ( sCons_rule @ ( shd_rule @ S12 ) @ ( sinterleave_rule @ S23 @ ( stl_rule @ S12 ) ) ) ) ) ).

% sinterleave.code
thf(fact_677_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_678_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_679_fst__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc5894184038225292483ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X3 @ X3 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% fst_diag_fst
thf(fact_680_fst__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% fst_diag_fst
thf(fact_681_snd__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc6725306161561501521e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X3: rule] : ( produc1769597910952761123e_rule @ X3 @ X3 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% snd_diag_snd
thf(fact_682_snd__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% snd_diag_snd
thf(fact_683_fst__diag__snd,axiom,
    ( ( comp_P2548450393677710116m_rule @ produc7264528503596239631e_rule
      @ ( comp_r8929483813444710872m_rule
        @ ^ [X3: rule] : ( produc1769597910952761123e_rule @ X3 @ X3 )
        @ produc7165828336582415457m_rule ) )
    = produc7165828336582415457m_rule ) ).

% fst_diag_snd
thf(fact_684_fst__diag__snd,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ produc4588648349897876871ist_fm ) )
    = produc4588648349897876871ist_fm ) ).

% fst_diag_snd
thf(fact_685_snd__diag__fst,axiom,
    ( ( comp_P1085356462900927218m_rule @ produc1639249265332130309ist_fm
      @ ( comp_P2017078995931194366m_rule
        @ ^ [X3: produc1996495991257130529ist_fm] : ( produc2592314732767339223ist_fm @ X3 @ X3 )
        @ produc6879501374131015971m_rule ) )
    = produc6879501374131015971m_rule ) ).

% snd_diag_fst
thf(fact_686_snd__diag__fst,axiom,
    ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( comp_l2016392514137073923ist_fm
        @ ^ [X3: list_fm] : ( produc381145313068854617ist_fm @ X3 @ X3 )
        @ produc1501393135466168645ist_fm ) )
    = produc1501393135466168645ist_fm ) ).

% snd_diag_fst
thf(fact_687_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S22 ) )
      = ( shd_Pr7235097944458474089m_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_688_sinterleave_Osimps_I1_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( shd_rule @ ( sinterleave_rule @ S1 @ S22 ) )
      = ( shd_rule @ S1 ) ) ).

% sinterleave.simps(1)
thf(fact_689_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sinter8084399408526207096m_rule @ S1 @ S22 ) )
      = ( sinter8084399408526207096m_rule @ S22 @ ( stl_Pr950425576149878629m_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_690_sinterleave_Osimps_I2_J,axiom,
    ! [S1: stream_rule,S22: stream_rule] :
      ( ( stl_rule @ ( sinterleave_rule @ S1 @ S22 ) )
      = ( sinterleave_rule @ S22 @ ( stl_rule @ S1 ) ) ) ).

% sinterleave.simps(2)
thf(fact_691_stl__def,axiom,
    ( stl_Pr950425576149878629m_rule
    = ( case_s474603282864825704m_rule
      @ ^ [X13: produc164195504107695125m_rule,X24: stream727092118206550309m_rule] : X24 ) ) ).

% stl_def
thf(fact_692_stl__def,axiom,
    ( stl_rule
    = ( case_s5295311805034529096m_rule
      @ ^ [X13: rule,X24: stream_rule] : X24 ) ) ).

% stl_def
thf(fact_693_cont__def,axiom,
    ( abstra4962253831105676628m_rule
    = ( abstra1216198984563783442m_rule
      @ ^ [X13: produc164195504107695125m_rule,X24: fset_A5396818216090053761m_rule] : X24 ) ) ).

% cont_def
thf(fact_694_fset__of__list__elem,axiom,
    ! [X4: rule,Xs: list_rule] :
      ( ( fmember_rule @ X4 @ ( fset_of_list_rule @ Xs ) )
      = ( member_rule @ X4 @ ( set_rule2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_695_fset__of__list__elem,axiom,
    ! [X4: produc1996495991257130529ist_fm,Xs: list_P5616295576739893671ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ X4 @ ( fset_o3706400737857578983ist_fm @ Xs ) )
      = ( member8102475879199740618ist_fm @ X4 @ ( set_Pr8767716839810916150ist_fm @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_696_fset__of__list__elem,axiom,
    ! [X4: abstra4572470701423150929m_rule,Xs: list_A1849579841884561377m_rule] :
      ( ( fmembe1635983029874454474m_rule @ X4 @ ( fset_o8294491781805274379m_rule @ Xs ) )
      = ( member1590433669991072232m_rule @ X4 @ ( set_Ab5312821370972875900m_rule @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_697_fset__of__list__elem,axiom,
    ! [X4: fm,Xs: list_fm] :
      ( ( fmember_fm @ X4 @ ( fset_of_list_fm @ Xs ) )
      = ( member_fm @ X4 @ ( set_fm2 @ Xs ) ) ) ).

% fset_of_list_elem
thf(fact_698_tree_Oroot__def,axiom,
    ( abstra3503897793822247108m_rule
    = ( abstra6581154509029876902m_rule
      @ ^ [X13: produc164195504107695125m_rule,X24: fset_A5396818216090053761m_rule] : X13 ) ) ).

% tree.root_def
thf(fact_699_konig_Osimps_I1_J,axiom,
    ! [T2: abstract_tree_rule] :
      ( ( shd_rule @ ( abstract_konig_rule @ T2 ) )
      = ( abstract_root_rule @ T2 ) ) ).

% konig.simps(1)
thf(fact_700_konig_Osimps_I1_J,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( abstra3079604473823497664m_rule @ T2 ) )
      = ( abstra3503897793822247108m_rule @ T2 ) ) ).

% konig.simps(1)
thf(fact_701_konig_Ocode,axiom,
    ( abstra3079604473823497664m_rule
    = ( ^ [T: abstra4572470701423150929m_rule] :
          ( sCons_5731567480799343806m_rule @ ( abstra3503897793822247108m_rule @ T )
          @ ( abstra3079604473823497664m_rule
            @ ( fChoic7543385803022166781m_rule
              @ ^ [T7: abstra4572470701423150929m_rule] :
                  ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ T ) )
                  & ~ ( abstra3507476289341504315m_rule @ T7 ) ) ) ) ) ) ) ).

% konig.code
thf(fact_702_Bleast__def,axiom,
    ( bleast_nat
    = ( ^ [S6: set_nat,P4: nat > $o] :
          ( ord_Least_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ S6 )
              & ( P4 @ X3 ) ) ) ) ) ).

% Bleast_def
thf(fact_703_abort__Bleast__def,axiom,
    ( abort_Bleast_nat
    = ( ^ [S6: set_nat,P4: nat > $o] :
          ( ord_Least_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ S6 )
              & ( P4 @ X3 ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_704_konig_Osimps_I2_J,axiom,
    ! [T2: abstract_tree_rule] :
      ( ( stl_rule @ ( abstract_konig_rule @ T2 ) )
      = ( abstract_konig_rule
        @ ( fChoic8596484284512669613e_rule
          @ ^ [T7: abstract_tree_rule] :
              ( ( fmembe7964111328986570234e_rule @ T7 @ ( abstract_cont_rule @ T2 ) )
              & ~ ( abstra6797871900832723819e_rule @ T7 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_705_konig_Osimps_I2_J,axiom,
    ! [T2: abstra4572470701423150929m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( abstra3079604473823497664m_rule @ T2 ) )
      = ( abstra3079604473823497664m_rule
        @ ( fChoic7543385803022166781m_rule
          @ ^ [T7: abstra4572470701423150929m_rule] :
              ( ( fmembe1635983029874454474m_rule @ T7 @ ( abstra4962253831105676628m_rule @ T2 ) )
              & ~ ( abstra3507476289341504315m_rule @ T7 ) ) ) ) ) ).

% konig.simps(2)
thf(fact_706_sfilter_Ocode,axiom,
    ( sfilte7579662470208054517m_rule
    = ( ^ [P4: produc164195504107695125m_rule > $o,S: stream727092118206550309m_rule] : ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P4 ) @ S ) ) @ ( sfilte7579662470208054517m_rule @ P4 @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P4 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_707_sfilter_Ocode,axiom,
    ( sfilter_rule
    = ( ^ [P4: rule > $o,S: stream_rule] : ( sCons_rule @ ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P4 ) @ S ) ) @ ( sfilter_rule @ P4 @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P4 ) @ S ) ) ) ) ) ) ).

% sfilter.code
thf(fact_708_sfilter_Osimps_I1_J,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sfilte7579662470208054517m_rule @ P2 @ S2 ) )
      = ( shd_Pr7235097944458474089m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P2 ) @ S2 ) ) ) ).

% sfilter.simps(1)
thf(fact_709_sfilter_Osimps_I1_J,axiom,
    ! [P2: rule > $o,S2: stream_rule] :
      ( ( shd_rule @ ( sfilter_rule @ P2 @ S2 ) )
      = ( shd_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P2 ) @ S2 ) ) ) ).

% sfilter.simps(1)
thf(fact_710_sfilter_Osimps_I2_J,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sfilte7579662470208054517m_rule @ P2 @ S2 ) )
      = ( sfilte7579662470208054517m_rule @ P2 @ ( stl_Pr950425576149878629m_rule @ ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P2 ) @ S2 ) ) ) ) ).

% sfilter.simps(2)
thf(fact_711_sfilter_Osimps_I2_J,axiom,
    ! [P2: rule > $o,S2: stream_rule] :
      ( ( stl_rule @ ( sfilter_rule @ P2 @ S2 ) )
      = ( sfilter_rule @ P2 @ ( stl_rule @ ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P2 ) @ S2 ) ) ) ) ).

% sfilter.simps(2)
thf(fact_712_sfilter__P,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S2: stream727092118206550309m_rule] :
      ( ( P2 @ ( shd_Pr7235097944458474089m_rule @ S2 ) )
     => ( ( sfilte7579662470208054517m_rule @ P2 @ S2 )
        = ( sCons_5731567480799343806m_rule @ ( shd_Pr7235097944458474089m_rule @ S2 ) @ ( sfilte7579662470208054517m_rule @ P2 @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ) ) ).

% sfilter_P
thf(fact_713_sfilter__P,axiom,
    ! [P2: rule > $o,S2: stream_rule] :
      ( ( P2 @ ( shd_rule @ S2 ) )
     => ( ( sfilter_rule @ P2 @ S2 )
        = ( sCons_rule @ ( shd_rule @ S2 ) @ ( sfilter_rule @ P2 @ ( stl_rule @ S2 ) ) ) ) ) ).

% sfilter_P
thf(fact_714_sfilter__not__P,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S2: stream727092118206550309m_rule] :
      ( ~ ( P2 @ ( shd_Pr7235097944458474089m_rule @ S2 ) )
     => ( ( sfilte7579662470208054517m_rule @ P2 @ S2 )
        = ( sfilte7579662470208054517m_rule @ P2 @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ) ).

% sfilter_not_P
thf(fact_715_sfilter__not__P,axiom,
    ! [P2: rule > $o,S2: stream_rule] :
      ( ~ ( P2 @ ( shd_rule @ S2 ) )
     => ( ( sfilter_rule @ P2 @ S2 )
        = ( sfilter_rule @ P2 @ ( stl_rule @ S2 ) ) ) ) ).

% sfilter_not_P
thf(fact_716_some__sym__eq__trivial,axiom,
    ! [X4: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ( ^ [Y: fset_P661503646757059847ist_fm,Z: fset_P661503646757059847ist_fm] : ( Y = Z )
          @ X4 ) )
      = X4 ) ).

% some_sym_eq_trivial
thf(fact_717_some__eq__trivial,axiom,
    ! [X4: fset_P661503646757059847ist_fm] :
      ( ( fChoic3892542562059901019ist_fm
        @ ^ [Y5: fset_P661503646757059847ist_fm] : ( Y5 = X4 ) )
      = X4 ) ).

% some_eq_trivial
thf(fact_718_some__equality,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ( P2 @ A )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
           => ( X = A ) )
       => ( ( fChoic3892542562059901019ist_fm @ P2 )
          = A ) ) ) ).

% some_equality
thf(fact_719_someI2,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm,Q2: fset_P661503646757059847ist_fm > $o] :
      ( ( P2 @ A )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
           => ( Q2 @ X ) )
       => ( Q2 @ ( fChoic3892542562059901019ist_fm @ P2 ) ) ) ) ).

% someI2
thf(fact_720_someI__ex,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_12: fset_P661503646757059847ist_fm] : ( P2 @ X_12 )
     => ( P2 @ ( fChoic3892542562059901019ist_fm @ P2 ) ) ) ).

% someI_ex
thf(fact_721_someI2__ex,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,Q2: fset_P661503646757059847ist_fm > $o] :
      ( ? [X_12: fset_P661503646757059847ist_fm] : ( P2 @ X_12 )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
           => ( Q2 @ X ) )
       => ( Q2 @ ( fChoic3892542562059901019ist_fm @ P2 ) ) ) ) ).

% someI2_ex
thf(fact_722_someI2__bex,axiom,
    ! [A3: set_fm,P2: fm > $o,Q2: fm > $o] :
      ( ? [X7: fm] :
          ( ( member_fm @ X7 @ A3 )
          & ( P2 @ X7 ) )
     => ( ! [X: fm] :
            ( ( ( member_fm @ X @ A3 )
              & ( P2 @ X ) )
           => ( Q2 @ X ) )
       => ( Q2
          @ ( fChoice_fm
            @ ^ [X3: fm] :
                ( ( member_fm @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_723_someI2__bex,axiom,
    ! [A3: set_rule,P2: rule > $o,Q2: rule > $o] :
      ( ? [X7: rule] :
          ( ( member_rule @ X7 @ A3 )
          & ( P2 @ X7 ) )
     => ( ! [X: rule] :
            ( ( ( member_rule @ X @ A3 )
              & ( P2 @ X ) )
           => ( Q2 @ X ) )
       => ( Q2
          @ ( fChoice_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_724_someI2__bex,axiom,
    ! [A3: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > $o] :
      ( ? [X7: produc1996495991257130529ist_fm] :
          ( ( member8102475879199740618ist_fm @ X7 @ A3 )
          & ( P2 @ X7 ) )
     => ( ! [X: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ X @ A3 )
              & ( P2 @ X ) )
           => ( Q2 @ X ) )
       => ( Q2
          @ ( fChoic1980167584695909237ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_725_someI2__bex,axiom,
    ! [A3: set_fs7534060674568567527ist_fm,P2: fset_P661503646757059847ist_fm > $o,Q2: fset_P661503646757059847ist_fm > $o] :
      ( ? [X7: fset_P661503646757059847ist_fm] :
          ( ( member3493611713653719344ist_fm @ X7 @ A3 )
          & ( P2 @ X7 ) )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( ( member3493611713653719344ist_fm @ X @ A3 )
              & ( P2 @ X ) )
           => ( Q2 @ X ) )
       => ( Q2
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X3: fset_P661503646757059847ist_fm] :
                ( ( member3493611713653719344ist_fm @ X3 @ A3 )
                & ( P2 @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_726_some__eq__ex,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o] :
      ( ( P2 @ ( fChoic3892542562059901019ist_fm @ P2 ) )
      = ( ? [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) ) ) ).

% some_eq_ex
thf(fact_727_some1__equality,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A: fset_P661503646757059847ist_fm] :
      ( ? [X7: fset_P661503646757059847ist_fm] :
          ( ( P2 @ X7 )
          & ! [Y3: fset_P661503646757059847ist_fm] :
              ( ( P2 @ Y3 )
             => ( Y3 = X7 ) ) )
     => ( ( P2 @ A )
       => ( ( fChoic3892542562059901019ist_fm @ P2 )
          = A ) ) ) ).

% some1_equality
thf(fact_728_snd__sndOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q2: produc1996495991257130529ist_fm > rule > $o] :
      ( produc7165828336582415457m_rule
      = ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( bNF_sn6617447237779055481m_rule @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_729_snd__sndOp,axiom,
    ! [P2: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] :
      ( produc4588648349897876871ist_fm
      = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( bNF_sn5292590578203848342ist_fm @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_730_fst__fstOp,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o,Q2: rule > rule > $o] :
      ( produc6879501374131015971m_rule
      = ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( bNF_fs2143400664727635963e_rule @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_731_fst__fstOp,axiom,
    ! [P2: list_fm > list_fm > $o,Q2: list_fm > list_fm > $o] :
      ( produc1501393135466168645ist_fm
      = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( bNF_fs6361025107161210196ist_fm @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_732_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_733_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_734_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S2: stream727092118206550309m_rule] :
      ( ? [N5: nat] : ( P2 @ ( snth_P1712579846118164081m_rule @ S2 @ N5 ) )
     => ( ( sdrop_5541336384294084785m_rule @ ( comp_o5517589812586031224m_rule @ (~) @ P2 ) @ S2 )
        = ( sdrop_7224736112439592940m_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_P1712579846118164081m_rule @ S2 @ N3 ) ) )
          @ S2 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_735_sdrop__while__sdrop__LEAST,axiom,
    ! [P2: rule > $o,S2: stream_rule] :
      ( ? [N5: nat] : ( P2 @ ( snth_rule @ S2 @ N5 ) )
     => ( ( sdrop_while_rule @ ( comp_o_o_rule @ (~) @ P2 ) @ S2 )
        = ( sdrop_rule
          @ ( ord_Least_nat
            @ ^ [N3: nat] : ( P2 @ ( snth_rule @ S2 @ N3 ) ) )
          @ S2 ) ) ) ).

% sdrop_while_sdrop_LEAST
thf(fact_736_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_737_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_738_apfst__conv,axiom,
    ! [F: list_fm > list_fm,X4: list_fm,Y4: list_fm] :
      ( ( produc1130293176790411102ist_fm @ F @ ( produc381145313068854617ist_fm @ X4 @ Y4 ) )
      = ( produc381145313068854617ist_fm @ ( F @ X4 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_739_apfst__conv,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X4: produc1996495991257130529ist_fm,Y4: rule] :
      ( ( produc3731403639160201281m_rule @ F @ ( produc491467635432902671m_rule @ X4 @ Y4 ) )
      = ( produc491467635432902671m_rule @ ( F @ X4 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_740_apfst__conv,axiom,
    ! [F: rule > rule,X4: rule,Y4: produc1996495991257130529ist_fm] :
      ( ( produc5534506254562563289ist_fm @ F @ ( produc7063755442767259331ist_fm @ X4 @ Y4 ) )
      = ( produc7063755442767259331ist_fm @ ( F @ X4 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_741_apsnd__conv,axiom,
    ! [F: list_fm > list_fm,X4: list_fm,Y4: list_fm] :
      ( ( produc9047611296996456864ist_fm @ F @ ( produc381145313068854617ist_fm @ X4 @ Y4 ) )
      = ( produc381145313068854617ist_fm @ X4 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_742_apsnd__conv,axiom,
    ! [F: rule > rule,X4: produc1996495991257130529ist_fm,Y4: rule] :
      ( ( produc3063135074001183255ist_fm @ F @ ( produc491467635432902671m_rule @ X4 @ Y4 ) )
      = ( produc491467635432902671m_rule @ X4 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_743_apsnd__conv,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X4: rule,Y4: produc1996495991257130529ist_fm] :
      ( ( produc4332487942798490499m_rule @ F @ ( produc7063755442767259331ist_fm @ X4 @ Y4 ) )
      = ( produc7063755442767259331ist_fm @ X4 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_744_fst__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X4: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3731403639160201281m_rule @ F @ X4 ) )
      = ( F @ ( produc6879501374131015971m_rule @ X4 ) ) ) ).

% fst_apfst
thf(fact_745_fst__apfst,axiom,
    ! [F: list_fm > list_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc1130293176790411102ist_fm @ F @ X4 ) )
      = ( F @ ( produc1501393135466168645ist_fm @ X4 ) ) ) ).

% fst_apfst
thf(fact_746_snd__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X4: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F @ X4 ) )
      = ( produc7165828336582415457m_rule @ X4 ) ) ).

% snd_apfst
thf(fact_747_snd__apfst,axiom,
    ! [F: list_fm > list_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F @ X4 ) )
      = ( produc4588648349897876871ist_fm @ X4 ) ) ).

% snd_apfst
thf(fact_748_fst__apsnd,axiom,
    ! [F: rule > rule,X4: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F @ X4 ) )
      = ( produc6879501374131015971m_rule @ X4 ) ) ).

% fst_apsnd
thf(fact_749_fst__apsnd,axiom,
    ! [F: list_fm > list_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F @ X4 ) )
      = ( produc1501393135466168645ist_fm @ X4 ) ) ).

% fst_apsnd
thf(fact_750_snd__apsnd,axiom,
    ! [F: rule > rule,X4: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc3063135074001183255ist_fm @ F @ X4 ) )
      = ( F @ ( produc7165828336582415457m_rule @ X4 ) ) ) ).

% snd_apsnd
thf(fact_751_snd__apsnd,axiom,
    ! [F: list_fm > list_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc9047611296996456864ist_fm @ F @ X4 ) )
      = ( F @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ).

% snd_apsnd
thf(fact_752_sdrop__simps_I1_J,axiom,
    ! [N: nat,S2: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) )
      = ( snth_P1712579846118164081m_rule @ S2 @ N ) ) ).

% sdrop_simps(1)
thf(fact_753_sdrop__simps_I1_J,axiom,
    ! [N: nat,S2: stream_rule] :
      ( ( shd_rule @ ( sdrop_rule @ N @ S2 ) )
      = ( snth_rule @ S2 @ N ) ) ).

% sdrop_simps(1)
thf(fact_754_snth__szip,axiom,
    ! [S1: stream_list_fm,S22: stream_list_fm,N: nat] :
      ( ( snth_P2624749845746815685ist_fm @ ( szip_list_fm_list_fm @ S1 @ S22 ) @ N )
      = ( produc381145313068854617ist_fm @ ( snth_list_fm @ S1 @ N ) @ ( snth_list_fm @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_755_snth__szip,axiom,
    ! [S1: stream8299795917829157543ist_fm,S22: stream_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( szip_P1977448745965526924m_rule @ S1 @ S22 ) @ N )
      = ( produc491467635432902671m_rule @ ( snth_P2624749845746815685ist_fm @ S1 @ N ) @ ( snth_rule @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_756_snth__szip,axiom,
    ! [S1: stream_rule,S22: stream8299795917829157543ist_fm,N: nat] :
      ( ( snth_P6179250367719980013ist_fm @ ( szip_r8549736553299883584ist_fm @ S1 @ S22 ) @ N )
      = ( produc7063755442767259331ist_fm @ ( snth_rule @ S1 @ N ) @ ( snth_P2624749845746815685ist_fm @ S22 @ N ) ) ) ).

% snth_szip
thf(fact_757_snd__comp__apfst,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc3731403639160201281m_rule @ F ) )
      = produc7165828336582415457m_rule ) ).

% snd_comp_apfst
thf(fact_758_snd__comp__apfst,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc1130293176790411102ist_fm @ F ) )
      = produc4588648349897876871ist_fm ) ).

% snd_comp_apfst
thf(fact_759_fst__comp__apsnd,axiom,
    ! [F: rule > rule] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc3063135074001183255ist_fm @ F ) )
      = produc6879501374131015971m_rule ) ).

% fst_comp_apsnd
thf(fact_760_fst__comp__apsnd,axiom,
    ! [F: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc9047611296996456864ist_fm @ F ) )
      = produc1501393135466168645ist_fm ) ).

% fst_comp_apsnd
thf(fact_761_apsnd__apfst,axiom,
    ! [F: rule > list_fm,G: produc1996495991257130529ist_fm > list_fm,X4: produc164195504107695125m_rule] :
      ( ( produc7464737497110386437ist_fm @ F @ ( produc7276395001879883020m_rule @ G @ X4 ) )
      = ( produc381145313068854617ist_fm @ ( G @ ( produc6879501374131015971m_rule @ X4 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_762_apsnd__apfst,axiom,
    ! [F: rule > rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,X4: produc164195504107695125m_rule] :
      ( ( produc3063135074001183255ist_fm @ F @ ( produc3731403639160201281m_rule @ G @ X4 ) )
      = ( produc491467635432902671m_rule @ ( G @ ( produc6879501374131015971m_rule @ X4 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_763_apsnd__apfst,axiom,
    ! [F: rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > rule,X4: produc164195504107695125m_rule] :
      ( ( produc5714219303521602403m_rule @ F @ ( produc1151914582733876081e_rule @ G @ X4 ) )
      = ( produc7063755442767259331ist_fm @ ( G @ ( produc6879501374131015971m_rule @ X4 ) ) @ ( F @ ( produc7165828336582415457m_rule @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_764_apsnd__apfst,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc9047611296996456864ist_fm @ F @ ( produc1130293176790411102ist_fm @ G @ X4 ) )
      = ( produc381145313068854617ist_fm @ ( G @ ( produc1501393135466168645ist_fm @ X4 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_765_apsnd__apfst,axiom,
    ! [F: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc4802975340153367740ist_fm @ F @ ( produc147150953317395813ist_fm @ G @ X4 ) )
      = ( produc491467635432902671m_rule @ ( G @ ( produc1501393135466168645ist_fm @ X4 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_766_apsnd__apfst,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X4: produc1996495991257130529ist_fm] :
      ( ( produc7454059569673786888m_rule @ F @ ( produc8579845201498058179ist_fm @ G @ X4 ) )
      = ( produc7063755442767259331ist_fm @ ( G @ ( produc1501393135466168645ist_fm @ X4 ) ) @ ( F @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_767_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,X4: produc164195504107695125m_rule] :
      ( ( produc6153989603652719015ist_fm @ F @ ( produc2413407552729308604ist_fm @ G @ X4 ) )
      = ( produc381145313068854617ist_fm @ ( F @ ( produc6879501374131015971m_rule @ X4 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_768_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X4: produc164195504107695125m_rule] :
      ( ( produc3731403639160201281m_rule @ F @ ( produc3063135074001183255ist_fm @ G @ X4 ) )
      = ( produc491467635432902671m_rule @ ( F @ ( produc6879501374131015971m_rule @ X4 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_769_apfst__apsnd,axiom,
    ! [F: produc1996495991257130529ist_fm > rule,G: rule > produc1996495991257130529ist_fm,X4: produc164195504107695125m_rule] :
      ( ( produc1080319409639782133ist_fm @ F @ ( produc4729144337346781827ist_fm @ G @ X4 ) )
      = ( produc7063755442767259331ist_fm @ ( F @ ( produc6879501374131015971m_rule @ X4 ) ) @ ( G @ ( produc7165828336582415457m_rule @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_770_apfst__apsnd,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc1130293176790411102ist_fm @ F @ ( produc9047611296996456864ist_fm @ G @ X4 ) )
      = ( produc381145313068854617ist_fm @ ( F @ ( produc1501393135466168645ist_fm @ X4 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_771_apfst__apsnd,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X4: produc1996495991257130529ist_fm] :
      ( ( produc7806619372436685002m_rule @ F @ ( produc4688919482198968325ist_fm @ G @ X4 ) )
      = ( produc491467635432902671m_rule @ ( F @ ( produc1501393135466168645ist_fm @ X4 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_772_apfst__apsnd,axiom,
    ! [F: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc5155535142916265854ist_fm @ F @ ( produc2501461190424677027ist_fm @ G @ X4 ) )
      = ( produc7063755442767259331ist_fm @ ( F @ ( produc1501393135466168645ist_fm @ X4 ) ) @ ( G @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_773_snth__sset,axiom,
    ! [S2: stream_fm,N: nat] : ( member_fm @ ( snth_fm @ S2 @ N ) @ ( sset_fm @ S2 ) ) ).

% snth_sset
thf(fact_774_snth__sset,axiom,
    ! [S2: stream8299795917829157543ist_fm,N: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S2 @ N ) @ ( sset_P3431374123360452149ist_fm @ S2 ) ) ).

% snth_sset
thf(fact_775_snth__sset,axiom,
    ! [S2: stream_rule,N: nat] : ( member_rule @ ( snth_rule @ S2 @ N ) @ ( sset_rule @ S2 ) ) ).

% snth_sset
thf(fact_776_snth__sset,axiom,
    ! [S2: stream727092118206550309m_rule,N: nat] : ( member4220325220686508332m_rule @ ( snth_P1712579846118164081m_rule @ S2 @ N ) @ ( sset_P4979403222752902913m_rule @ S2 ) ) ).

% snth_sset
thf(fact_777_sdrop__snth,axiom,
    ! [N: nat,S2: stream727092118206550309m_rule,M2: nat] :
      ( ( snth_P1712579846118164081m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) @ M2 )
      = ( snth_P1712579846118164081m_rule @ S2 @ ( plus_plus_nat @ N @ M2 ) ) ) ).

% sdrop_snth
thf(fact_778_sdrop__snth,axiom,
    ! [N: nat,S2: stream_rule,M2: nat] :
      ( ( snth_rule @ ( sdrop_rule @ N @ S2 ) @ M2 )
      = ( snth_rule @ S2 @ ( plus_plus_nat @ N @ M2 ) ) ) ).

% sdrop_snth
thf(fact_779_sndOp__def,axiom,
    ( bNF_sn6617447237779055481m_rule
    = ( ^ [P4: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm > $o,Q3: produc1996495991257130529ist_fm > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( bNF_pi3438825987790746729m_rule @ P4 @ Q3 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_780_sndOp__def,axiom,
    ( bNF_sn5292590578203848342ist_fm
    = ( ^ [P4: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( bNF_pi3539088406881798022ist_fm @ P4 @ Q3 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_781_fstOp__def,axiom,
    ( bNF_fs2143400664727635963e_rule
    = ( ^ [P4: produc1996495991257130529ist_fm > rule > $o,Q3: rule > rule > $o,Ac: produc164195504107695125m_rule] : ( produc491467635432902671m_rule @ ( produc6879501374131015971m_rule @ Ac ) @ ( bNF_pi580457772512444745e_rule @ P4 @ Q3 @ ( produc6879501374131015971m_rule @ Ac ) @ ( produc7165828336582415457m_rule @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_782_fstOp__def,axiom,
    ( bNF_fs6361025107161210196ist_fm
    = ( ^ [P4: list_fm > list_fm > $o,Q3: list_fm > list_fm > $o,Ac: produc1996495991257130529ist_fm] : ( produc381145313068854617ist_fm @ ( produc1501393135466168645ist_fm @ Ac ) @ ( bNF_pi3539088406881798022ist_fm @ P4 @ Q3 @ ( produc1501393135466168645ist_fm @ Ac ) @ ( produc4588648349897876871ist_fm @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_783_apfst__convE,axiom,
    ! [Q: produc1996495991257130529ist_fm,F: list_fm > list_fm,P: produc1996495991257130529ist_fm] :
      ( ( Q
        = ( produc1130293176790411102ist_fm @ F @ P ) )
     => ~ ! [X: list_fm,Y3: list_fm] :
            ( ( P
              = ( produc381145313068854617ist_fm @ X @ Y3 ) )
           => ( Q
             != ( produc381145313068854617ist_fm @ ( F @ X ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_784_apfst__convE,axiom,
    ! [Q: produc164195504107695125m_rule,F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,P: produc164195504107695125m_rule] :
      ( ( Q
        = ( produc3731403639160201281m_rule @ F @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y3: rule] :
            ( ( P
              = ( produc491467635432902671m_rule @ X @ Y3 ) )
           => ( Q
             != ( produc491467635432902671m_rule @ ( F @ X ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_785_apfst__convE,axiom,
    ! [Q: produc4630866025709511057ist_fm,F: rule > rule,P: produc4630866025709511057ist_fm] :
      ( ( Q
        = ( produc5534506254562563289ist_fm @ F @ P ) )
     => ~ ! [X: rule,Y3: produc1996495991257130529ist_fm] :
            ( ( P
              = ( produc7063755442767259331ist_fm @ X @ Y3 ) )
           => ( Q
             != ( produc7063755442767259331ist_fm @ ( F @ X ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_786_snth__sset__smerge,axiom,
    ! [Ss2: stream_stream_fm,N: nat,M2: nat] : ( member_fm @ ( snth_fm @ ( snth_stream_fm @ Ss2 @ N ) @ M2 ) @ ( sset_fm @ ( smerge_fm @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_787_snth__sset__smerge,axiom,
    ! [Ss2: stream8183306005803988909ist_fm,N: nat,M2: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ ( snth_s4276058539312847179ist_fm @ Ss2 @ N ) @ M2 ) @ ( sset_P3431374123360452149ist_fm @ ( smerge8067294617398470191ist_fm @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_788_snth__sset__smerge,axiom,
    ! [Ss2: stream_stream_rule,N: nat,M2: nat] : ( member_rule @ ( snth_rule @ ( snth_stream_rule @ Ss2 @ N ) @ M2 ) @ ( sset_rule @ ( smerge_rule @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_789_snth__sset__smerge,axiom,
    ! [Ss2: stream3016090547575141045m_rule,N: nat,M2: nat] : ( member4220325220686508332m_rule @ ( snth_P1712579846118164081m_rule @ ( snth_s8878883179916095873m_rule @ Ss2 @ N ) @ M2 ) @ ( sset_P4979403222752902913m_rule @ ( smerge3869575255966183559m_rule @ Ss2 ) ) ) ).

% snth_sset_smerge
thf(fact_790_conj__subset__def,axiom,
    ! [A3: set_fm,P2: fm > $o,Q2: fm > $o] :
      ( ( ord_less_eq_set_fm @ A3
        @ ( collect_fm
          @ ^ [X3: fm] :
              ( ( P2 @ X3 )
              & ( Q2 @ X3 ) ) ) )
      = ( ( ord_less_eq_set_fm @ A3 @ ( collect_fm @ P2 ) )
        & ( ord_less_eq_set_fm @ A3 @ ( collect_fm @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_791_conj__subset__def,axiom,
    ! [A3: set_rule,P2: rule > $o,Q2: rule > $o] :
      ( ( ord_less_eq_set_rule @ A3
        @ ( collect_rule
          @ ^ [X3: rule] :
              ( ( P2 @ X3 )
              & ( Q2 @ X3 ) ) ) )
      = ( ( ord_less_eq_set_rule @ A3 @ ( collect_rule @ P2 ) )
        & ( ord_less_eq_set_rule @ A3 @ ( collect_rule @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_792_verit__sko__ex_H,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A3: $o] :
      ( ( ( P2 @ ( fChoic3892542562059901019ist_fm @ P2 ) )
        = A3 )
     => ( ( ? [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) )
        = A3 ) ) ).

% verit_sko_ex'
thf(fact_793_verit__sko__forall,axiom,
    ( ( ^ [P5: fset_P661503646757059847ist_fm > $o] :
        ! [X8: fset_P661503646757059847ist_fm] : ( P5 @ X8 ) )
    = ( ^ [P4: fset_P661503646757059847ist_fm > $o] :
          ( P4
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X3: fset_P661503646757059847ist_fm] :
                ~ ( P4 @ X3 ) ) ) ) ) ).

% verit_sko_forall
thf(fact_794_verit__sko__forall_H,axiom,
    ! [P2: fset_P661503646757059847ist_fm > $o,A3: $o] :
      ( ( ( P2
          @ ( fChoic3892542562059901019ist_fm
            @ ^ [X3: fset_P661503646757059847ist_fm] :
                ~ ( P2 @ X3 ) ) )
        = A3 )
     => ( ( ! [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) )
        = A3 ) ) ).

% verit_sko_forall'
thf(fact_795_verit__sko__forall_H_H,axiom,
    ! [B3: fset_P661503646757059847ist_fm,A3: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o] :
      ( ( B3 = A3 )
     => ( ( ( fChoic3892542562059901019ist_fm @ P2 )
          = A3 )
        = ( ( fChoic3892542562059901019ist_fm @ P2 )
          = B3 ) ) ) ).

% verit_sko_forall''
thf(fact_796_verit__sko__ex__indirect,axiom,
    ! [X4: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o] :
      ( ( X4
        = ( fChoic3892542562059901019ist_fm @ P2 ) )
     => ( ( ? [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) )
        = ( P2 @ X4 ) ) ) ).

% verit_sko_ex_indirect
thf(fact_797_verit__sko__forall__indirect2,axiom,
    ! [X4: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o,P6: fset_P661503646757059847ist_fm > $o] :
      ( ( X4
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X3: fset_P661503646757059847ist_fm] :
              ~ ( P2 @ X3 ) ) )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
            = ( P6 @ X ) )
       => ( ( ! [X5: fset_P661503646757059847ist_fm] : ( P6 @ X5 ) )
          = ( P2 @ X4 ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_798_verit__sko__forall__indirect,axiom,
    ! [X4: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o] :
      ( ( X4
        = ( fChoic3892542562059901019ist_fm
          @ ^ [X3: fset_P661503646757059847ist_fm] :
              ~ ( P2 @ X3 ) ) )
     => ( ( ! [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) )
        = ( P2 @ X4 ) ) ) ).

% verit_sko_forall_indirect
thf(fact_799_verit__sko__ex__indirect2,axiom,
    ! [X4: fset_P661503646757059847ist_fm,P2: fset_P661503646757059847ist_fm > $o,P6: fset_P661503646757059847ist_fm > $o] :
      ( ( X4
        = ( fChoic3892542562059901019ist_fm @ P2 ) )
     => ( ! [X: fset_P661503646757059847ist_fm] :
            ( ( P2 @ X )
            = ( P6 @ X ) )
       => ( ( ? [X5: fset_P661503646757059847ist_fm] : ( P6 @ X5 ) )
          = ( P2 @ X4 ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_800_Fair__Stream_Ofair__def,axiom,
    ( fair_fair_rule
    = ( ^ [S: stream_rule] :
        ! [X3: rule] :
          ( ( member_rule @ X3 @ ( sset_rule @ S ) )
         => ! [M3: nat] :
            ? [N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
              & ( ( snth_rule @ S @ N3 )
                = X3 ) ) ) ) ) ).

% Fair_Stream.fair_def
thf(fact_801_Fair__Stream_Ofair__def,axiom,
    ( fair_f6205644465234492970m_rule
    = ( ^ [S: stream727092118206550309m_rule] :
        ! [X3: produc164195504107695125m_rule] :
          ( ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ S ) )
         => ! [M3: nat] :
            ? [N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
              & ( ( snth_P1712579846118164081m_rule @ S @ N3 )
                = X3 ) ) ) ) ) ).

% Fair_Stream.fair_def
thf(fact_802_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_803_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_804_sdrop__smap2,axiom,
    ! [N: nat,F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S22 ) )
      = ( smap2_3865418324013074943m_rule @ F @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_805_sdrop__smap2,axiom,
    ! [N: nat,F: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( smap2_3092065398501458735m_rule @ F @ S1 @ S22 ) )
      = ( smap2_3092065398501458735m_rule @ F @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_806_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( smap2_7722561687528711727m_rule @ F @ S1 @ S22 ) )
      = ( smap2_7722561687528711727m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_807_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream_rule] :
      ( ( sdrop_7224736112439592940m_rule @ N @ ( smap2_4438185555641665375m_rule @ F @ S1 @ S22 ) )
      = ( smap2_4438185555641665375m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_808_sdrop__smap2,axiom,
    ! [N: nat,F: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( sdrop_rule @ N @ ( smap2_1391242002117124143e_rule @ F @ S1 @ S22 ) )
      = ( smap2_1391242002117124143e_rule @ F @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_809_sdrop__smap2,axiom,
    ! [N: nat,F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_190827247425732959e_rule @ F @ S1 @ S22 ) )
      = ( smap2_190827247425732959e_rule @ F @ ( sdrop_7224736112439592940m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_810_sdrop__smap2,axiom,
    ! [N: nat,F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( sdrop_rule @ N @ ( smap2_2737362159257330783e_rule @ F @ S1 @ S22 ) )
      = ( smap2_2737362159257330783e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_7224736112439592940m_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_811_sdrop__smap2,axiom,
    ! [N: nat,F: rule > rule > rule,S1: stream_rule,S22: stream_rule] :
      ( ( sdrop_rule @ N @ ( smap2_rule_rule_rule @ F @ S1 @ S22 ) )
      = ( smap2_rule_rule_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S22 ) ) ) ).

% sdrop_smap2
thf(fact_812_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S22 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_813_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_3092065398501458735m_rule @ F @ S1 @ S22 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_814_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_7722561687528711727m_rule @ F @ S1 @ S22 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_815_smap2_Osimps_I1_J,axiom,
    ! [F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream_rule] :
      ( ( shd_Pr7235097944458474089m_rule @ ( smap2_4438185555641665375m_rule @ F @ S1 @ S22 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_816_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_1391242002117124143e_rule @ F @ S1 @ S22 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_817_smap2_Osimps_I1_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( shd_rule @ ( smap2_190827247425732959e_rule @ F @ S1 @ S22 ) )
      = ( F @ ( shd_Pr7235097944458474089m_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_818_smap2_Osimps_I1_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( shd_rule @ ( smap2_2737362159257330783e_rule @ F @ S1 @ S22 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_Pr7235097944458474089m_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_819_smap2_Osimps_I1_J,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S22: stream_rule] :
      ( ( shd_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S22 ) )
      = ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S22 ) ) ) ).

% smap2.simps(1)
thf(fact_820_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3865418324013074943m_rule @ F @ S1 @ S22 ) )
      = ( smap2_3865418324013074943m_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_821_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > produc164195504107695125m_rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_3092065398501458735m_rule @ F @ S1 @ S22 ) )
      = ( smap2_3092065398501458735m_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_822_smap2_Osimps_I2_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_7722561687528711727m_rule @ F @ S1 @ S22 ) )
      = ( smap2_7722561687528711727m_rule @ F @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_823_smap2_Osimps_I2_J,axiom,
    ! [F: rule > rule > produc164195504107695125m_rule,S1: stream_rule,S22: stream_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( smap2_4438185555641665375m_rule @ F @ S1 @ S22 ) )
      = ( smap2_4438185555641665375m_rule @ F @ ( stl_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_824_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > produc164195504107695125m_rule > rule,S1: stream727092118206550309m_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_1391242002117124143e_rule @ F @ S1 @ S22 ) )
      = ( smap2_1391242002117124143e_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_825_smap2_Osimps_I2_J,axiom,
    ! [F: produc164195504107695125m_rule > rule > rule,S1: stream727092118206550309m_rule,S22: stream_rule] :
      ( ( stl_rule @ ( smap2_190827247425732959e_rule @ F @ S1 @ S22 ) )
      = ( smap2_190827247425732959e_rule @ F @ ( stl_Pr950425576149878629m_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_826_smap2_Osimps_I2_J,axiom,
    ! [F: rule > produc164195504107695125m_rule > rule,S1: stream_rule,S22: stream727092118206550309m_rule] :
      ( ( stl_rule @ ( smap2_2737362159257330783e_rule @ F @ S1 @ S22 ) )
      = ( smap2_2737362159257330783e_rule @ F @ ( stl_rule @ S1 ) @ ( stl_Pr950425576149878629m_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_827_smap2_Osimps_I2_J,axiom,
    ! [F: rule > rule > rule,S1: stream_rule,S22: stream_rule] :
      ( ( stl_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S22 ) )
      = ( smap2_rule_rule_rule @ F @ ( stl_rule @ S1 ) @ ( stl_rule @ S22 ) ) ) ).

% smap2.simps(2)
thf(fact_828_fst__convol,axiom,
    ! [F: abstra4572470701423150929m_rule > produc1996495991257130529ist_fm,G: abstra4572470701423150929m_rule > rule] :
      ( ( comp_P4378417847848123130m_rule @ produc6879501374131015971m_rule @ ( bNF_co2617238127527879852m_rule @ F @ G ) )
      = F ) ).

% fst_convol
thf(fact_829_prop__restrict,axiom,
    ! [X4: produc1996495991257130529ist_fm,Z3: set_Pr7058068377845519745ist_fm,X6: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X4 @ Z3 )
     => ( ( ord_le2055072402069232929ist_fm @ Z3
          @ ( collec7231575776666803724ist_fm
            @ ^ [X3: produc1996495991257130529ist_fm] :
                ( ( member8102475879199740618ist_fm @ X3 @ X6 )
                & ( P2 @ X3 ) ) ) )
       => ( P2 @ X4 ) ) ) ).

% prop_restrict
thf(fact_830_prop__restrict,axiom,
    ! [X4: fm,Z3: set_fm,X6: set_fm,P2: fm > $o] :
      ( ( member_fm @ X4 @ Z3 )
     => ( ( ord_less_eq_set_fm @ Z3
          @ ( collect_fm
            @ ^ [X3: fm] :
                ( ( member_fm @ X3 @ X6 )
                & ( P2 @ X3 ) ) ) )
       => ( P2 @ X4 ) ) ) ).

% prop_restrict
thf(fact_831_prop__restrict,axiom,
    ! [X4: rule,Z3: set_rule,X6: set_rule,P2: rule > $o] :
      ( ( member_rule @ X4 @ Z3 )
     => ( ( ord_less_eq_set_rule @ Z3
          @ ( collect_rule
            @ ^ [X3: rule] :
                ( ( member_rule @ X3 @ X6 )
                & ( P2 @ X3 ) ) ) )
       => ( P2 @ X4 ) ) ) ).

% prop_restrict
thf(fact_832_Collect__restrict,axiom,
    ! [X6: set_Pr7058068377845519745ist_fm,P2: produc1996495991257130529ist_fm > $o] :
      ( ord_le2055072402069232929ist_fm
      @ ( collec7231575776666803724ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm] :
            ( ( member8102475879199740618ist_fm @ X3 @ X6 )
            & ( P2 @ X3 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_833_Collect__restrict,axiom,
    ! [X6: set_fm,P2: fm > $o] :
      ( ord_less_eq_set_fm
      @ ( collect_fm
        @ ^ [X3: fm] :
            ( ( member_fm @ X3 @ X6 )
            & ( P2 @ X3 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_834_Collect__restrict,axiom,
    ! [X6: set_rule,P2: rule > $o] :
      ( ord_less_eq_set_rule
      @ ( collect_rule
        @ ^ [X3: rule] :
            ( ( member_rule @ X3 @ X6 )
            & ( P2 @ X3 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_835_fst__snd__flip,axiom,
    ( produc6879501374131015971m_rule
    = ( comp_P6646430376872094402m_rule @ produc4514744107061996309ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm,Y5: rule] : ( produc7063755442767259331ist_fm @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_836_fst__snd__flip,axiom,
    ( produc4228417144610596823ist_fm
    = ( comp_P4888803072877744336ist_fm @ produc7165828336582415457m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X3: rule,Y5: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_837_fst__snd__flip,axiom,
    ( produc1501393135466168645ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X3: list_fm,Y5: list_fm] : ( produc381145313068854617ist_fm @ Y5 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_838_snd__fst__flip,axiom,
    ( produc4514744107061996309ist_fm
    = ( comp_P2707267495825058490ist_fm @ produc6879501374131015971m_rule
      @ ( produc2185035559847904517m_rule
        @ ^ [X3: rule,Y5: produc1996495991257130529ist_fm] : ( produc491467635432902671m_rule @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_839_snd__fst__flip,axiom,
    ( produc7165828336582415457m_rule
    = ( comp_P3413117018928051920m_rule @ produc4228417144610596823ist_fm
      @ ( produc1775668667622460877ist_fm
        @ ^ [X3: produc1996495991257130529ist_fm,Y5: rule] : ( produc7063755442767259331ist_fm @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_840_snd__fst__flip,axiom,
    ( produc4588648349897876871ist_fm
    = ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm
      @ ( produc4544959197559615795ist_fm
        @ ^ [X3: list_fm,Y5: list_fm] : ( produc381145313068854617ist_fm @ Y5 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_841_case__prod__Pair__iden,axiom,
    ! [P: produc1996495991257130529ist_fm] :
      ( ( produc4544959197559615795ist_fm @ produc381145313068854617ist_fm @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_842_case__prod__Pair__iden,axiom,
    ! [P: produc164195504107695125m_rule] :
      ( ( produc6532370182875420753m_rule @ produc491467635432902671m_rule @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_843_case__prod__Pair__iden,axiom,
    ! [P: produc4630866025709511057ist_fm] :
      ( ( produc6651706081449720449ist_fm @ produc7063755442767259331ist_fm @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_844_fst__def,axiom,
    ( produc6879501374131015971m_rule
    = ( produc8807799049144449253ist_fm
      @ ^ [X13: produc1996495991257130529ist_fm,X24: rule] : X13 ) ) ).

% fst_def
thf(fact_845_fst__def,axiom,
    ( produc1501393135466168645ist_fm
    = ( produc4154610405322092878ist_fm
      @ ^ [X13: list_fm,X24: list_fm] : X13 ) ) ).

% fst_def
thf(fact_846_snd__def,axiom,
    ( produc7165828336582415457m_rule
    = ( produc1213517383330063745e_rule
      @ ^ [X13: produc1996495991257130529ist_fm,X24: rule] : X24 ) ) ).

% snd_def
thf(fact_847_snd__def,axiom,
    ( produc4588648349897876871ist_fm
    = ( produc4154610405322092878ist_fm
      @ ^ [X13: list_fm,X24: list_fm] : X24 ) ) ).

% snd_def
thf(fact_848_ssubst__Pair__rhs,axiom,
    ! [R2: list_fm,S2: list_fm,R3: set_Pr7058068377845519745ist_fm,S4: list_fm] :
      ( ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S2 ) @ R3 )
     => ( ( S4 = S2 )
       => ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ R2 @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_849_ssubst__Pair__rhs,axiom,
    ! [R2: produc1996495991257130529ist_fm,S2: rule,R3: set_Pr1008144964186165195m_rule,S4: rule] :
      ( ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R2 @ S2 ) @ R3 )
     => ( ( S4 = S2 )
       => ( member4220325220686508332m_rule @ ( produc491467635432902671m_rule @ R2 @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_850_ssubst__Pair__rhs,axiom,
    ! [R2: rule,S2: produc1996495991257130529ist_fm,R3: set_Pr2241365836748572487ist_fm,S4: produc1996495991257130529ist_fm] :
      ( ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S2 ) @ R3 )
     => ( ( S4 = S2 )
       => ( member8686995742288324264ist_fm @ ( produc7063755442767259331ist_fm @ R2 @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_851_stake__pos__minWait,axiom,
    ! [Rs: stream_rule,S2: 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
            @ S2 )
          @ ( abstract_pos_rule @ Rs @ R2 ) )
       => ( ( member_rule @ R2 @ ( sset_rule @ rules ) )
         => ( ( member8102475879199740618ist_fm @ S2 @ 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
                    @ S2 ) )
                @ 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
                    @ S2 ) ) ) ) ) ) ) ) ).

% stake_pos_minWait
thf(fact_852_snd__comp__map__prod,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule] :
      ( ( comp_P422132551275928404m_rule @ produc7165828336582415457m_rule @ ( produc7680103964935106114e_rule @ F @ G ) )
      = ( comp_r641804198332660996m_rule @ G @ produc7165828336582415457m_rule ) ) ).

% snd_comp_map_prod
thf(fact_853_snd__comp__map__prod,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule] :
      ( ( comp_P855734935590900386ist_fm @ produc7165828336582415457m_rule @ ( produc8124276167727720944m_rule @ F @ G ) )
      = ( comp_l6170393434051197847ist_fm @ G @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_map_prod
thf(fact_854_snd__comp__map__prod,axiom,
    ! [F: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm] :
      ( ( comp_P8020416857309042165m_rule @ produc4588648349897876871ist_fm @ ( produc1965614570530558770ist_fm @ F @ G ) )
      = ( comp_r4098878985462706207m_rule @ G @ produc7165828336582415457m_rule ) ) ).

% snd_comp_map_prod
thf(fact_855_snd__comp__map__prod,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc4588648349897876871ist_fm @ ( produc3421166768339006330ist_fm @ F @ G ) )
      = ( comp_l1868218281415653180ist_fm @ G @ produc4588648349897876871ist_fm ) ) ).

% snd_comp_map_prod
thf(fact_856_fst__comp__map__prod,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule] :
      ( ( comp_P7463969011078018366m_rule @ produc6879501374131015971m_rule @ ( produc7680103964935106114e_rule @ F @ G ) )
      = ( comp_P8245093624798776376m_rule @ F @ produc6879501374131015971m_rule ) ) ).

% fst_comp_map_prod
thf(fact_857_fst__comp__map__prod,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule] :
      ( ( comp_P695203452023487288ist_fm @ produc6879501374131015971m_rule @ ( produc8124276167727720944m_rule @ F @ G ) )
      = ( comp_l2016392514137073923ist_fm @ F @ produc1501393135466168645ist_fm ) ) ).

% fst_comp_map_prod
thf(fact_858_fst__comp__map__prod,axiom,
    ! [F: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm] :
      ( ( comp_P8020416857309042165m_rule @ produc1501393135466168645ist_fm @ ( produc1965614570530558770ist_fm @ F @ G ) )
      = ( comp_P8020416857309042165m_rule @ F @ produc6879501374131015971m_rule ) ) ).

% fst_comp_map_prod
thf(fact_859_fst__comp__map__prod,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm] :
      ( ( comp_P1426762558556845121ist_fm @ produc1501393135466168645ist_fm @ ( produc3421166768339006330ist_fm @ F @ G ) )
      = ( comp_l1868218281415653180ist_fm @ F @ produc1501393135466168645ist_fm ) ) ).

% fst_comp_map_prod
thf(fact_860_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_861_Eps__case__prod__eq,axiom,
    ! [X4: list_fm,Y4: list_fm] :
      ( ( fChoic1980167584695909237ist_fm
        @ ( produc2011568048573023402t_fm_o
          @ ^ [X9: list_fm,Y6: list_fm] :
              ( ( X4 = X9 )
              & ( Y4 = Y6 ) ) ) )
      = ( produc381145313068854617ist_fm @ X4 @ Y4 ) ) ).

% Eps_case_prod_eq
thf(fact_862_Eps__case__prod__eq,axiom,
    ! [X4: produc1996495991257130529ist_fm,Y4: rule] :
      ( ( fChoic9196029409387999681m_rule
        @ ( produc7369492558480804444rule_o
          @ ^ [X9: produc1996495991257130529ist_fm,Y6: rule] :
              ( ( X4 = X9 )
              & ( Y4 = Y6 ) ) ) )
      = ( produc491467635432902671m_rule @ X4 @ Y4 ) ) ).

% Eps_case_prod_eq
thf(fact_863_Eps__case__prod__eq,axiom,
    ! [X4: rule,Y4: produc1996495991257130529ist_fm] :
      ( ( fChoic4439327894135039805ist_fm
        @ ( produc1628407760719572648t_fm_o
          @ ^ [X9: rule,Y6: produc1996495991257130529ist_fm] :
              ( ( X4 = X9 )
              & ( Y4 = Y6 ) ) ) )
      = ( produc7063755442767259331ist_fm @ X4 @ Y4 ) ) ).

% Eps_case_prod_eq
thf(fact_864_case__prodI,axiom,
    ! [F: list_fm > list_fm > $o,A: list_fm,B: list_fm] :
      ( ( F @ A @ B )
     => ( produc2011568048573023402t_fm_o @ F @ ( produc381145313068854617ist_fm @ A @ B ) ) ) ).

% case_prodI
thf(fact_865_case__prodI,axiom,
    ! [F: produc1996495991257130529ist_fm > rule > $o,A: produc1996495991257130529ist_fm,B: rule] :
      ( ( F @ A @ B )
     => ( produc7369492558480804444rule_o @ F @ ( produc491467635432902671m_rule @ A @ B ) ) ) ).

% case_prodI
thf(fact_866_case__prodI,axiom,
    ! [F: rule > produc1996495991257130529ist_fm > $o,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( F @ A @ B )
     => ( produc1628407760719572648t_fm_o @ F @ ( produc7063755442767259331ist_fm @ A @ B ) ) ) ).

% case_prodI
thf(fact_867_case__prodI2,axiom,
    ! [P: produc1996495991257130529ist_fm,C: list_fm > list_fm > $o] :
      ( ! [A5: list_fm,B5: list_fm] :
          ( ( P
            = ( produc381145313068854617ist_fm @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc2011568048573023402t_fm_o @ C @ P ) ) ).

% case_prodI2
thf(fact_868_case__prodI2,axiom,
    ! [P: produc164195504107695125m_rule,C: produc1996495991257130529ist_fm > rule > $o] :
      ( ! [A5: produc1996495991257130529ist_fm,B5: rule] :
          ( ( P
            = ( produc491467635432902671m_rule @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc7369492558480804444rule_o @ C @ P ) ) ).

% case_prodI2
thf(fact_869_case__prodI2,axiom,
    ! [P: produc4630866025709511057ist_fm,C: rule > produc1996495991257130529ist_fm > $o] :
      ( ! [A5: rule,B5: produc1996495991257130529ist_fm] :
          ( ( P
            = ( produc7063755442767259331ist_fm @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc1628407760719572648t_fm_o @ C @ P ) ) ).

% case_prodI2
thf(fact_870_mem__case__prodI,axiom,
    ! [Z2: fm,C: list_fm > list_fm > set_fm,A: list_fm,B: list_fm] :
      ( ( member_fm @ Z2 @ ( C @ A @ B ) )
     => ( member_fm @ Z2 @ ( produc4802205653939627508set_fm @ C @ ( produc381145313068854617ist_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_871_mem__case__prodI,axiom,
    ! [Z2: rule,C: list_fm > list_fm > set_rule,A: list_fm,B: list_fm] :
      ( ( member_rule @ Z2 @ ( C @ A @ B ) )
     => ( member_rule @ Z2 @ ( produc9016151371058038505t_rule @ C @ ( produc381145313068854617ist_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_872_mem__case__prodI,axiom,
    ! [Z2: produc1996495991257130529ist_fm,C: list_fm > list_fm > set_Pr7058068377845519745ist_fm,A: list_fm,B: list_fm] :
      ( ( member8102475879199740618ist_fm @ Z2 @ ( C @ A @ B ) )
     => ( member8102475879199740618ist_fm @ Z2 @ ( produc3038944318576466067ist_fm @ C @ ( produc381145313068854617ist_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_873_mem__case__prodI,axiom,
    ! [Z2: fm,C: produc1996495991257130529ist_fm > rule > set_fm,A: produc1996495991257130529ist_fm,B: rule] :
      ( ( member_fm @ Z2 @ ( C @ A @ B ) )
     => ( member_fm @ Z2 @ ( produc966423478193074626set_fm @ C @ ( produc491467635432902671m_rule @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_874_mem__case__prodI,axiom,
    ! [Z2: rule,C: produc1996495991257130529ist_fm > rule > set_rule,A: produc1996495991257130529ist_fm,B: rule] :
      ( ( member_rule @ Z2 @ ( C @ A @ B ) )
     => ( member_rule @ Z2 @ ( produc6930014626940046263t_rule @ C @ ( produc491467635432902671m_rule @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_875_mem__case__prodI,axiom,
    ! [Z2: produc1996495991257130529ist_fm,C: produc1996495991257130529ist_fm > rule > set_Pr7058068377845519745ist_fm,A: produc1996495991257130529ist_fm,B: rule] :
      ( ( member8102475879199740618ist_fm @ Z2 @ ( C @ A @ B ) )
     => ( member8102475879199740618ist_fm @ Z2 @ ( produc2666912249037264965ist_fm @ C @ ( produc491467635432902671m_rule @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_876_mem__case__prodI,axiom,
    ! [Z2: fm,C: rule > produc1996495991257130529ist_fm > set_fm,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( member_fm @ Z2 @ ( C @ A @ B ) )
     => ( member_fm @ Z2 @ ( produc576982443051290230set_fm @ C @ ( produc7063755442767259331ist_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_877_mem__case__prodI,axiom,
    ! [Z2: rule,C: rule > produc1996495991257130529ist_fm > set_rule,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( member_rule @ Z2 @ ( C @ A @ B ) )
     => ( member_rule @ Z2 @ ( produc1332068493706956395t_rule @ C @ ( produc7063755442767259331ist_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_878_mem__case__prodI,axiom,
    ! [Z2: produc1996495991257130529ist_fm,C: rule > produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ Z2 @ ( C @ A @ B ) )
     => ( member8102475879199740618ist_fm @ Z2 @ ( produc3976105672090482833ist_fm @ C @ ( produc7063755442767259331ist_fm @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_879_mem__case__prodI2,axiom,
    ! [P: produc1996495991257130529ist_fm,Z2: fm,C: list_fm > list_fm > set_fm] :
      ( ! [A5: list_fm,B5: list_fm] :
          ( ( P
            = ( produc381145313068854617ist_fm @ A5 @ B5 ) )
         => ( member_fm @ Z2 @ ( C @ A5 @ B5 ) ) )
     => ( member_fm @ Z2 @ ( produc4802205653939627508set_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_880_mem__case__prodI2,axiom,
    ! [P: produc1996495991257130529ist_fm,Z2: rule,C: list_fm > list_fm > set_rule] :
      ( ! [A5: list_fm,B5: list_fm] :
          ( ( P
            = ( produc381145313068854617ist_fm @ A5 @ B5 ) )
         => ( member_rule @ Z2 @ ( C @ A5 @ B5 ) ) )
     => ( member_rule @ Z2 @ ( produc9016151371058038505t_rule @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_881_mem__case__prodI2,axiom,
    ! [P: produc1996495991257130529ist_fm,Z2: produc1996495991257130529ist_fm,C: list_fm > list_fm > set_Pr7058068377845519745ist_fm] :
      ( ! [A5: list_fm,B5: list_fm] :
          ( ( P
            = ( produc381145313068854617ist_fm @ A5 @ B5 ) )
         => ( member8102475879199740618ist_fm @ Z2 @ ( C @ A5 @ B5 ) ) )
     => ( member8102475879199740618ist_fm @ Z2 @ ( produc3038944318576466067ist_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_882_mem__case__prodI2,axiom,
    ! [P: produc164195504107695125m_rule,Z2: fm,C: produc1996495991257130529ist_fm > rule > set_fm] :
      ( ! [A5: produc1996495991257130529ist_fm,B5: rule] :
          ( ( P
            = ( produc491467635432902671m_rule @ A5 @ B5 ) )
         => ( member_fm @ Z2 @ ( C @ A5 @ B5 ) ) )
     => ( member_fm @ Z2 @ ( produc966423478193074626set_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_883_mem__case__prodI2,axiom,
    ! [P: produc164195504107695125m_rule,Z2: rule,C: produc1996495991257130529ist_fm > rule > set_rule] :
      ( ! [A5: produc1996495991257130529ist_fm,B5: rule] :
          ( ( P
            = ( produc491467635432902671m_rule @ A5 @ B5 ) )
         => ( member_rule @ Z2 @ ( C @ A5 @ B5 ) ) )
     => ( member_rule @ Z2 @ ( produc6930014626940046263t_rule @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_884_mem__case__prodI2,axiom,
    ! [P: produc164195504107695125m_rule,Z2: produc1996495991257130529ist_fm,C: produc1996495991257130529ist_fm > rule > set_Pr7058068377845519745ist_fm] :
      ( ! [A5: produc1996495991257130529ist_fm,B5: rule] :
          ( ( P
            = ( produc491467635432902671m_rule @ A5 @ B5 ) )
         => ( member8102475879199740618ist_fm @ Z2 @ ( C @ A5 @ B5 ) ) )
     => ( member8102475879199740618ist_fm @ Z2 @ ( produc2666912249037264965ist_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_885_mem__case__prodI2,axiom,
    ! [P: produc4630866025709511057ist_fm,Z2: fm,C: rule > produc1996495991257130529ist_fm > set_fm] :
      ( ! [A5: rule,B5: produc1996495991257130529ist_fm] :
          ( ( P
            = ( produc7063755442767259331ist_fm @ A5 @ B5 ) )
         => ( member_fm @ Z2 @ ( C @ A5 @ B5 ) ) )
     => ( member_fm @ Z2 @ ( produc576982443051290230set_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_886_mem__case__prodI2,axiom,
    ! [P: produc4630866025709511057ist_fm,Z2: rule,C: rule > produc1996495991257130529ist_fm > set_rule] :
      ( ! [A5: rule,B5: produc1996495991257130529ist_fm] :
          ( ( P
            = ( produc7063755442767259331ist_fm @ A5 @ B5 ) )
         => ( member_rule @ Z2 @ ( C @ A5 @ B5 ) ) )
     => ( member_rule @ Z2 @ ( produc1332068493706956395t_rule @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_887_mem__case__prodI2,axiom,
    ! [P: produc4630866025709511057ist_fm,Z2: produc1996495991257130529ist_fm,C: rule > produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm] :
      ( ! [A5: rule,B5: produc1996495991257130529ist_fm] :
          ( ( P
            = ( produc7063755442767259331ist_fm @ A5 @ B5 ) )
         => ( member8102475879199740618ist_fm @ Z2 @ ( C @ A5 @ B5 ) ) )
     => ( member8102475879199740618ist_fm @ Z2 @ ( produc3976105672090482833ist_fm @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_888_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_889_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_890_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_891_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_892_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_893_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_894_add__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc_right
thf(fact_895_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_896_diff__diff__left,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% diff_diff_left
thf(fact_897_map__prod__simp,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm,A: list_fm,B: list_fm] :
      ( ( produc3421166768339006330ist_fm @ F @ G @ ( produc381145313068854617ist_fm @ A @ B ) )
      = ( produc381145313068854617ist_fm @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_898_map__prod__simp,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,A: list_fm,B: list_fm] :
      ( ( produc8124276167727720944m_rule @ F @ G @ ( produc381145313068854617ist_fm @ A @ B ) )
      = ( produc491467635432902671m_rule @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_899_map__prod__simp,axiom,
    ! [F: list_fm > rule,G: list_fm > produc1996495991257130529ist_fm,A: list_fm,B: list_fm] :
      ( ( produc3613848521340044642ist_fm @ F @ G @ ( produc381145313068854617ist_fm @ A @ B ) )
      = ( produc7063755442767259331ist_fm @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_900_map__prod__simp,axiom,
    ! [F: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,A: produc1996495991257130529ist_fm,B: rule] :
      ( ( produc1965614570530558770ist_fm @ F @ G @ ( produc491467635432902671m_rule @ A @ B ) )
      = ( produc381145313068854617ist_fm @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_901_map__prod__simp,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,A: produc1996495991257130529ist_fm,B: rule] :
      ( ( produc7680103964935106114e_rule @ F @ G @ ( produc491467635432902671m_rule @ A @ B ) )
      = ( produc491467635432902671m_rule @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_902_map__prod__simp,axiom,
    ! [F: produc1996495991257130529ist_fm > rule,G: rule > produc1996495991257130529ist_fm,A: produc1996495991257130529ist_fm,B: rule] :
      ( ( produc2839323599909017514ist_fm @ F @ G @ ( produc491467635432902671m_rule @ A @ B ) )
      = ( produc7063755442767259331ist_fm @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_903_map__prod__simp,axiom,
    ! [F: rule > list_fm,G: produc1996495991257130529ist_fm > list_fm,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( produc1976628786251050148ist_fm @ F @ G @ ( produc7063755442767259331ist_fm @ A @ B ) )
      = ( produc381145313068854617ist_fm @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_904_map__prod__simp,axiom,
    ! [F: rule > produc1996495991257130529ist_fm,G: produc1996495991257130529ist_fm > rule,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( produc1545012558808995138m_rule @ F @ G @ ( produc7063755442767259331ist_fm @ A @ B ) )
      = ( produc491467635432902671m_rule @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_905_map__prod__simp,axiom,
    ! [F: rule > rule,G: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( produc1941668953357286466ist_fm @ F @ G @ ( produc7063755442767259331ist_fm @ A @ B ) )
      = ( produc7063755442767259331ist_fm @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_906_fst__map__prod,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X4: produc164195504107695125m_rule] :
      ( ( produc6879501374131015971m_rule @ ( produc7680103964935106114e_rule @ F @ G @ X4 ) )
      = ( F @ ( produc6879501374131015971m_rule @ X4 ) ) ) ).

% fst_map_prod
thf(fact_907_fst__map__prod,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X4: produc1996495991257130529ist_fm] :
      ( ( produc6879501374131015971m_rule @ ( produc8124276167727720944m_rule @ F @ G @ X4 ) )
      = ( F @ ( produc1501393135466168645ist_fm @ X4 ) ) ) ).

% fst_map_prod
thf(fact_908_fst__map__prod,axiom,
    ! [F: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,X4: produc164195504107695125m_rule] :
      ( ( produc1501393135466168645ist_fm @ ( produc1965614570530558770ist_fm @ F @ G @ X4 ) )
      = ( F @ ( produc6879501374131015971m_rule @ X4 ) ) ) ).

% fst_map_prod
thf(fact_909_fst__map__prod,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc1501393135466168645ist_fm @ ( produc3421166768339006330ist_fm @ F @ G @ X4 ) )
      = ( F @ ( produc1501393135466168645ist_fm @ X4 ) ) ) ).

% fst_map_prod
thf(fact_910_snd__map__prod,axiom,
    ! [F: produc1996495991257130529ist_fm > produc1996495991257130529ist_fm,G: rule > rule,X4: produc164195504107695125m_rule] :
      ( ( produc7165828336582415457m_rule @ ( produc7680103964935106114e_rule @ F @ G @ X4 ) )
      = ( G @ ( produc7165828336582415457m_rule @ X4 ) ) ) ).

% snd_map_prod
thf(fact_911_snd__map__prod,axiom,
    ! [F: list_fm > produc1996495991257130529ist_fm,G: list_fm > rule,X4: produc1996495991257130529ist_fm] :
      ( ( produc7165828336582415457m_rule @ ( produc8124276167727720944m_rule @ F @ G @ X4 ) )
      = ( G @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ).

% snd_map_prod
thf(fact_912_snd__map__prod,axiom,
    ! [F: produc1996495991257130529ist_fm > list_fm,G: rule > list_fm,X4: produc164195504107695125m_rule] :
      ( ( produc4588648349897876871ist_fm @ ( produc1965614570530558770ist_fm @ F @ G @ X4 ) )
      = ( G @ ( produc7165828336582415457m_rule @ X4 ) ) ) ).

% snd_map_prod
thf(fact_913_snd__map__prod,axiom,
    ! [F: list_fm > list_fm,G: list_fm > list_fm,X4: produc1996495991257130529ist_fm] :
      ( ( produc4588648349897876871ist_fm @ ( produc3421166768339006330ist_fm @ F @ G @ X4 ) )
      = ( G @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ).

% snd_map_prod
thf(fact_914_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).

% Nat.add_diff_assoc
thf(fact_915_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_916_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_917_sdrop__simps_I2_J,axiom,
    ! [N: nat,S2: stream727092118206550309m_rule] :
      ( ( stl_Pr950425576149878629m_rule @ ( sdrop_7224736112439592940m_rule @ N @ S2 ) )
      = ( sdrop_7224736112439592940m_rule @ ( suc @ N ) @ S2 ) ) ).

% sdrop_simps(2)
thf(fact_918_sdrop__simps_I2_J,axiom,
    ! [N: nat,S2: stream_rule] :
      ( ( stl_rule @ ( sdrop_rule @ N @ S2 ) )
      = ( sdrop_rule @ ( suc @ N ) @ S2 ) ) ).

% sdrop_simps(2)
thf(fact_919_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_920_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_921_mem__case__prodE,axiom,
    ! [Z2: fm,C: list_fm > list_fm > set_fm,P: produc1996495991257130529ist_fm] :
      ( ( member_fm @ Z2 @ ( produc4802205653939627508set_fm @ C @ P ) )
     => ~ ! [X: list_fm,Y3: list_fm] :
            ( ( P
              = ( produc381145313068854617ist_fm @ X @ Y3 ) )
           => ~ ( member_fm @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_922_mem__case__prodE,axiom,
    ! [Z2: rule,C: list_fm > list_fm > set_rule,P: produc1996495991257130529ist_fm] :
      ( ( member_rule @ Z2 @ ( produc9016151371058038505t_rule @ C @ P ) )
     => ~ ! [X: list_fm,Y3: list_fm] :
            ( ( P
              = ( produc381145313068854617ist_fm @ X @ Y3 ) )
           => ~ ( member_rule @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_923_mem__case__prodE,axiom,
    ! [Z2: produc1996495991257130529ist_fm,C: list_fm > list_fm > set_Pr7058068377845519745ist_fm,P: produc1996495991257130529ist_fm] :
      ( ( member8102475879199740618ist_fm @ Z2 @ ( produc3038944318576466067ist_fm @ C @ P ) )
     => ~ ! [X: list_fm,Y3: list_fm] :
            ( ( P
              = ( produc381145313068854617ist_fm @ X @ Y3 ) )
           => ~ ( member8102475879199740618ist_fm @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_924_mem__case__prodE,axiom,
    ! [Z2: fm,C: produc1996495991257130529ist_fm > rule > set_fm,P: produc164195504107695125m_rule] :
      ( ( member_fm @ Z2 @ ( produc966423478193074626set_fm @ C @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y3: rule] :
            ( ( P
              = ( produc491467635432902671m_rule @ X @ Y3 ) )
           => ~ ( member_fm @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_925_mem__case__prodE,axiom,
    ! [Z2: rule,C: produc1996495991257130529ist_fm > rule > set_rule,P: produc164195504107695125m_rule] :
      ( ( member_rule @ Z2 @ ( produc6930014626940046263t_rule @ C @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y3: rule] :
            ( ( P
              = ( produc491467635432902671m_rule @ X @ Y3 ) )
           => ~ ( member_rule @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_926_mem__case__prodE,axiom,
    ! [Z2: produc1996495991257130529ist_fm,C: produc1996495991257130529ist_fm > rule > set_Pr7058068377845519745ist_fm,P: produc164195504107695125m_rule] :
      ( ( member8102475879199740618ist_fm @ Z2 @ ( produc2666912249037264965ist_fm @ C @ P ) )
     => ~ ! [X: produc1996495991257130529ist_fm,Y3: rule] :
            ( ( P
              = ( produc491467635432902671m_rule @ X @ Y3 ) )
           => ~ ( member8102475879199740618ist_fm @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_927_mem__case__prodE,axiom,
    ! [Z2: fm,C: rule > produc1996495991257130529ist_fm > set_fm,P: produc4630866025709511057ist_fm] :
      ( ( member_fm @ Z2 @ ( produc576982443051290230set_fm @ C @ P ) )
     => ~ ! [X: rule,Y3: produc1996495991257130529ist_fm] :
            ( ( P
              = ( produc7063755442767259331ist_fm @ X @ Y3 ) )
           => ~ ( member_fm @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_928_mem__case__prodE,axiom,
    ! [Z2: rule,C: rule > produc1996495991257130529ist_fm > set_rule,P: produc4630866025709511057ist_fm] :
      ( ( member_rule @ Z2 @ ( produc1332068493706956395t_rule @ C @ P ) )
     => ~ ! [X: rule,Y3: produc1996495991257130529ist_fm] :
            ( ( P
              = ( produc7063755442767259331ist_fm @ X @ Y3 ) )
           => ~ ( member_rule @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_929_mem__case__prodE,axiom,
    ! [Z2: produc1996495991257130529ist_fm,C: rule > produc1996495991257130529ist_fm > set_Pr7058068377845519745ist_fm,P: produc4630866025709511057ist_fm] :
      ( ( member8102475879199740618ist_fm @ Z2 @ ( produc3976105672090482833ist_fm @ C @ P ) )
     => ~ ! [X: rule,Y3: produc1996495991257130529ist_fm] :
            ( ( P
              = ( produc7063755442767259331ist_fm @ X @ Y3 ) )
           => ~ ( member8102475879199740618ist_fm @ Z2 @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_930_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% Nat.diff_cancel
thf(fact_931_diff__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_cancel2
thf(fact_932_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
      = M2 ) ).

% diff_add_inverse
thf(fact_933_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
      = M2 ) ).

% diff_add_inverse2
thf(fact_934_nat__arith_Osuc1,axiom,
    ! [A3: nat,K2: nat,A: nat] :
      ( ( A3
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( suc @ A3 )
        = ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_935_add__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc
thf(fact_936_add__Suc__shift,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_937_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_938_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_939_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ? [K: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% less_imp_Suc_add
thf(fact_940_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_941_less__diff__conv,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).

% less_diff_conv
thf(fact_942_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).

% less_add_Suc2
thf(fact_943_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_944_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ~ ! [Q4: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M2 @ Q4 ) ) ) ) ).

% less_natE
thf(fact_945_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).

% less_diff_conv2
thf(fact_946_alw__sdrop,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule,N: nat] :
      ( ( linear5684299538428340377m_rule @ Phi @ Xs )
     => ( linear5684299538428340377m_rule @ Phi @ ( sdrop_7224736112439592940m_rule @ N @ Xs ) ) ) ).

% alw_sdrop
thf(fact_947_alw__sdrop,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule,N: nat] :
      ( ( linear7625967296368300489w_rule @ Phi @ Xs )
     => ( linear7625967296368300489w_rule @ Phi @ ( sdrop_rule @ N @ Xs ) ) ) ).

% alw_sdrop
thf(fact_948_alw__iff__sdrop,axiom,
    ( linear5684299538428340377m_rule
    = ( ^ [P4: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
        ! [M3: nat] : ( P4 @ ( sdrop_7224736112439592940m_rule @ M3 @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_949_alw__iff__sdrop,axiom,
    ( linear7625967296368300489w_rule
    = ( ^ [P4: stream_rule > $o,Omega: stream_rule] :
        ! [M3: nat] : ( P4 @ ( sdrop_rule @ M3 @ Omega ) ) ) ) ).

% alw_iff_sdrop
thf(fact_950_holds__aand,axiom,
    ! [P2: produc164195504107695125m_rule > $o,Steps: stream727092118206550309m_rule,Q2: produc164195504107695125m_rule > $o] :
      ( ( ( linear7082939723987557601m_rule @ P2 @ Steps )
        & ( linear7082939723987557601m_rule @ Q2 @ Steps ) )
      = ( linear7082939723987557601m_rule
        @ ^ [Step: produc164195504107695125m_rule] :
            ( ( P2 @ Step )
            & ( Q2 @ Step ) )
        @ Steps ) ) ).

% holds_aand
thf(fact_951_holds__aand,axiom,
    ! [P2: rule > $o,Steps: stream_rule,Q2: rule > $o] :
      ( ( ( linear4670012741531116049s_rule @ P2 @ Steps )
        & ( linear4670012741531116049s_rule @ Q2 @ Steps ) )
      = ( linear4670012741531116049s_rule
        @ ^ [Step: rule] :
            ( ( P2 @ Step )
            & ( Q2 @ Step ) )
        @ Steps ) ) ).

% holds_aand
thf(fact_952_alw__invar,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule] :
      ( ( Phi @ Xs )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs2: stream_rule] :
              ( ( Phi @ Xs2 )
             => ( linear860818938954496176rule_o @ Phi @ Xs2 ) )
          @ Xs )
       => ( linear7625967296368300489w_rule @ Phi @ Xs ) ) ) ).

% alw_invar
thf(fact_953_alw__invar,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule] :
      ( ( Phi @ Xs )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs2: stream727092118206550309m_rule] :
              ( ( Phi @ Xs2 )
             => ( linear7357459635800698592rule_o @ Phi @ Xs2 ) )
          @ Xs )
       => ( linear5684299538428340377m_rule @ Phi @ Xs ) ) ) ).

% alw_invar
thf(fact_954_alw__False,axiom,
    ! [Omega2: stream727092118206550309m_rule] :
      ~ ( linear5684299538428340377m_rule
        @ ^ [X3: stream727092118206550309m_rule] : $false
        @ Omega2 ) ).

% alw_False
thf(fact_955_alw__False,axiom,
    ! [Omega2: stream_rule] :
      ~ ( linear7625967296368300489w_rule
        @ ^ [X3: stream_rule] : $false
        @ Omega2 ) ).

% alw_False
thf(fact_956_alw__aand,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ( ( Phi @ Xs2 )
            & ( Psi @ Xs2 ) ) )
      = ( ^ [Xs2: stream727092118206550309m_rule] :
            ( ( linear5684299538428340377m_rule @ Phi @ Xs2 )
            & ( linear5684299538428340377m_rule @ Psi @ Xs2 ) ) ) ) ).

% alw_aand
thf(fact_957_alw__aand,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule
        @ ^ [Xs2: stream_rule] :
            ( ( Phi @ Xs2 )
            & ( Psi @ Xs2 ) ) )
      = ( ^ [Xs2: stream_rule] :
            ( ( linear7625967296368300489w_rule @ Phi @ Xs2 )
            & ( linear7625967296368300489w_rule @ Psi @ Xs2 ) ) ) ) ).

% alw_aand
thf(fact_958_alw__mp,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ Phi @ Xs )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs2: stream727092118206550309m_rule] :
              ( ( Phi @ Xs2 )
             => ( Psi @ Xs2 ) )
          @ Xs )
       => ( linear5684299538428340377m_rule @ Psi @ Xs ) ) ) ).

% alw_mp
thf(fact_959_alw__mp,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule,Psi: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ Phi @ Xs )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs2: stream_rule] :
              ( ( Phi @ Xs2 )
             => ( Psi @ Xs2 ) )
          @ Xs )
       => ( linear7625967296368300489w_rule @ Psi @ Xs ) ) ) ).

% alw_mp
thf(fact_960_alw__holds2,axiom,
    ! [P2: produc164195504107695125m_rule > $o,Ss2: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear7082939723987557601m_rule @ P2 ) @ Ss2 )
      = ( ( P2 @ ( shd_Pr7235097944458474089m_rule @ Ss2 ) )
        & ( linear5684299538428340377m_rule @ ( linear7082939723987557601m_rule @ P2 ) @ ( stl_Pr950425576149878629m_rule @ Ss2 ) ) ) ) ).

% alw_holds2
thf(fact_961_alw__holds2,axiom,
    ! [P2: rule > $o,Ss2: stream_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear4670012741531116049s_rule @ P2 ) @ Ss2 )
      = ( ( P2 @ ( shd_rule @ Ss2 ) )
        & ( linear7625967296368300489w_rule @ ( linear4670012741531116049s_rule @ P2 ) @ ( stl_rule @ Ss2 ) ) ) ) ).

% alw_holds2
thf(fact_962_alw__coinduct,axiom,
    ! [X6: stream727092118206550309m_rule > $o,X4: stream727092118206550309m_rule,Phi: stream727092118206550309m_rule > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X6 @ X )
           => ( Phi @ X ) )
       => ( ! [X: stream727092118206550309m_rule] :
              ( ( X6 @ X )
             => ( ~ ( linear5684299538428340377m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ X ) )
               => ( X6 @ ( stl_Pr950425576149878629m_rule @ X ) ) ) )
         => ( linear5684299538428340377m_rule @ Phi @ X4 ) ) ) ) ).

% alw_coinduct
thf(fact_963_alw__coinduct,axiom,
    ! [X6: stream_rule > $o,X4: stream_rule,Phi: stream_rule > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream_rule] :
            ( ( X6 @ X )
           => ( Phi @ X ) )
       => ( ! [X: stream_rule] :
              ( ( X6 @ X )
             => ( ~ ( linear7625967296368300489w_rule @ Phi @ ( stl_rule @ X ) )
               => ( X6 @ ( stl_rule @ X ) ) ) )
         => ( linear7625967296368300489w_rule @ Phi @ X4 ) ) ) ) ).

% alw_coinduct
thf(fact_964_alw_Ocoinduct,axiom,
    ! [X6: stream727092118206550309m_rule > $o,X4: stream727092118206550309m_rule,Phi: stream727092118206550309m_rule > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream727092118206550309m_rule] :
            ( ( X6 @ X )
           => ? [Xs3: stream727092118206550309m_rule] :
                ( ( X = Xs3 )
                & ( Phi @ Xs3 )
                & ( ( X6 @ ( stl_Pr950425576149878629m_rule @ Xs3 ) )
                  | ( linear5684299538428340377m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ Xs3 ) ) ) ) )
       => ( linear5684299538428340377m_rule @ Phi @ X4 ) ) ) ).

% alw.coinduct
thf(fact_965_alw_Ocoinduct,axiom,
    ! [X6: stream_rule > $o,X4: stream_rule,Phi: stream_rule > $o] :
      ( ( X6 @ X4 )
     => ( ! [X: stream_rule] :
            ( ( X6 @ X )
           => ? [Xs3: stream_rule] :
                ( ( X = Xs3 )
                & ( Phi @ Xs3 )
                & ( ( X6 @ ( stl_rule @ Xs3 ) )
                  | ( linear7625967296368300489w_rule @ Phi @ ( stl_rule @ Xs3 ) ) ) ) )
       => ( linear7625967296368300489w_rule @ Phi @ X4 ) ) ) ).

% alw.coinduct
thf(fact_966_alw_Ointros,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule] :
      ( ( Phi @ Xs )
     => ( ( linear5684299538428340377m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ Xs ) )
       => ( linear5684299538428340377m_rule @ Phi @ Xs ) ) ) ).

% alw.intros
thf(fact_967_alw_Ointros,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule] :
      ( ( Phi @ Xs )
     => ( ( linear7625967296368300489w_rule @ Phi @ ( stl_rule @ Xs ) )
       => ( linear7625967296368300489w_rule @ Phi @ Xs ) ) ) ).

% alw.intros
thf(fact_968_alw_Osimps,axiom,
    ( linear5684299538428340377m_rule
    = ( ^ [Phi2: stream727092118206550309m_rule > $o,A2: stream727092118206550309m_rule] :
        ? [Xs2: stream727092118206550309m_rule] :
          ( ( A2 = Xs2 )
          & ( Phi2 @ Xs2 )
          & ( linear5684299538428340377m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ Xs2 ) ) ) ) ) ).

% alw.simps
thf(fact_969_alw_Osimps,axiom,
    ( linear7625967296368300489w_rule
    = ( ^ [Phi2: stream_rule > $o,A2: stream_rule] :
        ? [Xs2: stream_rule] :
          ( ( A2 = Xs2 )
          & ( Phi2 @ Xs2 )
          & ( linear7625967296368300489w_rule @ Phi2 @ ( stl_rule @ Xs2 ) ) ) ) ) ).

% alw.simps
thf(fact_970_alw_Ocases,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,A: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ Phi @ A )
     => ~ ( ( Phi @ A )
         => ~ ( linear5684299538428340377m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ).

% alw.cases
thf(fact_971_alw_Ocases,axiom,
    ! [Phi: stream_rule > $o,A: stream_rule] :
      ( ( linear7625967296368300489w_rule @ Phi @ A )
     => ~ ( ( Phi @ A )
         => ~ ( linear7625967296368300489w_rule @ Phi @ ( stl_rule @ A ) ) ) ) ).

% alw.cases
thf(fact_972_top_Oextremum__strict,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ~ ( ord_le5784518313675440685ist_fm @ top_to728902761071950545ist_fm @ A ) ).

% top.extremum_strict
thf(fact_973_top_Oextremum__strict,axiom,
    ! [A: set_rule] :
      ~ ( ord_less_set_rule @ top_top_set_rule @ A ) ).

% top.extremum_strict
thf(fact_974_top_Onot__eq__extremum,axiom,
    ! [A: set_Pr7058068377845519745ist_fm] :
      ( ( A != top_to728902761071950545ist_fm )
      = ( ord_le5784518313675440685ist_fm @ A @ top_to728902761071950545ist_fm ) ) ).

% top.not_eq_extremum
thf(fact_975_top_Onot__eq__extremum,axiom,
    ! [A: set_rule] :
      ( ( A != top_top_set_rule )
      = ( ord_less_set_rule @ A @ top_top_set_rule ) ) ).

% top.not_eq_extremum
thf(fact_976_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_977_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_978_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_979_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_980_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_981_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_982_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_983_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_984_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_985_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_986_add__lessD1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
     => ( ord_less_nat @ I @ K2 ) ) ).

% add_lessD1
thf(fact_987_add__less__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K2 @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_988_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_989_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_990_add__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_less_mono1
thf(fact_991_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_992_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_993_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% less_add_eq_less
thf(fact_994_map__prod__compose,axiom,
    ! [F1: produc164195504107695125m_rule > produc1996495991257130529ist_fm,F22: abstra4572470701423150929m_rule > produc164195504107695125m_rule,G1: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G2: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( produc4449957851134572790ist_fm @ ( comp_P4378417847848123130m_rule @ F1 @ F22 ) @ ( comp_P4378417847848123130m_rule @ G1 @ G2 ) )
      = ( comp_P8819851879376895428m_rule @ ( produc602106178381476086ist_fm @ F1 @ G1 ) @ ( produc6213034867950506950m_rule @ F22 @ G2 ) ) ) ).

% map_prod_compose
thf(fact_995_map__prod_Ocompositionality,axiom,
    ! [F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,I: abstra4572470701423150929m_rule > produc164195504107695125m_rule,Prod: produc2413811689340339075m_rule] :
      ( ( produc602106178381476086ist_fm @ F @ G @ ( produc6213034867950506950m_rule @ H @ I @ Prod ) )
      = ( produc4449957851134572790ist_fm @ ( comp_P4378417847848123130m_rule @ F @ H ) @ ( comp_P4378417847848123130m_rule @ G @ I ) @ Prod ) ) ).

% map_prod.compositionality
thf(fact_996_map__prod_Ocomp,axiom,
    ! [F: produc164195504107695125m_rule > produc1996495991257130529ist_fm,G: produc164195504107695125m_rule > produc1996495991257130529ist_fm,H: abstra4572470701423150929m_rule > produc164195504107695125m_rule,I: abstra4572470701423150929m_rule > produc164195504107695125m_rule] :
      ( ( comp_P8819851879376895428m_rule @ ( produc602106178381476086ist_fm @ F @ G ) @ ( produc6213034867950506950m_rule @ H @ I ) )
      = ( produc4449957851134572790ist_fm @ ( comp_P4378417847848123130m_rule @ F @ H ) @ ( comp_P4378417847848123130m_rule @ G @ I ) ) ) ).

% map_prod.comp
thf(fact_997_split__paired__Eps,axiom,
    ( fChoic1980167584695909237ist_fm
    = ( ^ [P4: produc1996495991257130529ist_fm > $o] :
          ( fChoic1980167584695909237ist_fm
          @ ( produc2011568048573023402t_fm_o
            @ ^ [A2: list_fm,B2: list_fm] : ( P4 @ ( produc381145313068854617ist_fm @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_998_split__paired__Eps,axiom,
    ( fChoic9196029409387999681m_rule
    = ( ^ [P4: produc164195504107695125m_rule > $o] :
          ( fChoic9196029409387999681m_rule
          @ ( produc7369492558480804444rule_o
            @ ^ [A2: produc1996495991257130529ist_fm,B2: rule] : ( P4 @ ( produc491467635432902671m_rule @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_999_split__paired__Eps,axiom,
    ( fChoic4439327894135039805ist_fm
    = ( ^ [P4: produc4630866025709511057ist_fm > $o] :
          ( fChoic4439327894135039805ist_fm
          @ ( produc1628407760719572648t_fm_o
            @ ^ [A2: rule,B2: produc1996495991257130529ist_fm] : ( P4 @ ( produc7063755442767259331ist_fm @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_1000_case__prodD,axiom,
    ! [F: list_fm > list_fm > $o,A: list_fm,B: list_fm] :
      ( ( produc2011568048573023402t_fm_o @ F @ ( produc381145313068854617ist_fm @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_1001_case__prodD,axiom,
    ! [F: produc1996495991257130529ist_fm > rule > $o,A: produc1996495991257130529ist_fm,B: rule] :
      ( ( produc7369492558480804444rule_o @ F @ ( produc491467635432902671m_rule @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_1002_case__prodD,axiom,
    ! [F: rule > produc1996495991257130529ist_fm > $o,A: rule,B: produc1996495991257130529ist_fm] :
      ( ( produc1628407760719572648t_fm_o @ F @ ( produc7063755442767259331ist_fm @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_1003_case__prodE,axiom,
    ! [C: list_fm > list_fm > $o,P: produc1996495991257130529ist_fm] :
      ( ( produc2011568048573023402t_fm_o @ C @ P )
     => ~ ! [X: list_fm,Y3: list_fm] :
            ( ( P
              = ( produc381145313068854617ist_fm @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_1004_case__prodE,axiom,
    ! [C: produc1996495991257130529ist_fm > rule > $o,P: produc164195504107695125m_rule] :
      ( ( produc7369492558480804444rule_o @ C @ P )
     => ~ ! [X: produc1996495991257130529ist_fm,Y3: rule] :
            ( ( P
              = ( produc491467635432902671m_rule @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_1005_case__prodE,axiom,
    ! [C: rule > produc1996495991257130529ist_fm > $o,P: produc4630866025709511057ist_fm] :
      ( ( produc1628407760719572648t_fm_o @ C @ P )
     => ~ ! [X: rule,Y3: produc1996495991257130529ist_fm] :
            ( ( P
              = ( produc7063755442767259331ist_fm @ X @ Y3 ) )
           => ~ ( C @ X @ Y3 ) ) ) ).

% case_prodE
thf(fact_1006_alw__inv,axiom,
    ! [F: stream727092118206550309m_rule > stream727092118206550309m_rule,P2: stream727092118206550309m_rule > $o,S2: stream727092118206550309m_rule] :
      ( ! [S3: stream727092118206550309m_rule] :
          ( ( F @ ( stl_Pr950425576149878629m_rule @ S3 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F @ S3 ) ) )
     => ( ( linear5684299538428340377m_rule @ P2 @ ( F @ S2 ) )
        = ( linear5684299538428340377m_rule
          @ ^ [X3: stream727092118206550309m_rule] : ( P2 @ ( F @ X3 ) )
          @ S2 ) ) ) ).

% alw_inv
thf(fact_1007_alw__inv,axiom,
    ! [F: stream_rule > stream727092118206550309m_rule,P2: stream727092118206550309m_rule > $o,S2: stream_rule] :
      ( ! [S3: stream_rule] :
          ( ( F @ ( stl_rule @ S3 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F @ S3 ) ) )
     => ( ( linear5684299538428340377m_rule @ P2 @ ( F @ S2 ) )
        = ( linear7625967296368300489w_rule
          @ ^ [X3: stream_rule] : ( P2 @ ( F @ X3 ) )
          @ S2 ) ) ) ).

% alw_inv
thf(fact_1008_alw__inv,axiom,
    ! [F: stream727092118206550309m_rule > stream_rule,P2: stream_rule > $o,S2: stream727092118206550309m_rule] :
      ( ! [S3: stream727092118206550309m_rule] :
          ( ( F @ ( stl_Pr950425576149878629m_rule @ S3 ) )
          = ( stl_rule @ ( F @ S3 ) ) )
     => ( ( linear7625967296368300489w_rule @ P2 @ ( F @ S2 ) )
        = ( linear5684299538428340377m_rule
          @ ^ [X3: stream727092118206550309m_rule] : ( P2 @ ( F @ X3 ) )
          @ S2 ) ) ) ).

% alw_inv
thf(fact_1009_alw__inv,axiom,
    ! [F: stream_rule > stream_rule,P2: stream_rule > $o,S2: stream_rule] :
      ( ! [S3: stream_rule] :
          ( ( F @ ( stl_rule @ S3 ) )
          = ( stl_rule @ ( F @ S3 ) ) )
     => ( ( linear7625967296368300489w_rule @ P2 @ ( F @ S2 ) )
        = ( linear7625967296368300489w_rule
          @ ^ [X3: stream_rule] : ( P2 @ ( F @ X3 ) )
          @ S2 ) ) ) ).

% alw_inv
thf(fact_1010_holds_Oelims_I3_J,axiom,
    ! [X4: produc164195504107695125m_rule > $o,Xa2: stream727092118206550309m_rule] :
      ( ~ ( linear7082939723987557601m_rule @ X4 @ Xa2 )
     => ~ ( X4 @ ( shd_Pr7235097944458474089m_rule @ Xa2 ) ) ) ).

% holds.elims(3)
thf(fact_1011_holds_Oelims_I3_J,axiom,
    ! [X4: rule > $o,Xa2: stream_rule] :
      ( ~ ( linear4670012741531116049s_rule @ X4 @ Xa2 )
     => ~ ( X4 @ ( shd_rule @ Xa2 ) ) ) ).

% holds.elims(3)
thf(fact_1012_holds_Oelims_I2_J,axiom,
    ! [X4: produc164195504107695125m_rule > $o,Xa2: stream727092118206550309m_rule] :
      ( ( linear7082939723987557601m_rule @ X4 @ Xa2 )
     => ( X4 @ ( shd_Pr7235097944458474089m_rule @ Xa2 ) ) ) ).

% holds.elims(2)
thf(fact_1013_holds_Oelims_I2_J,axiom,
    ! [X4: rule > $o,Xa2: stream_rule] :
      ( ( linear4670012741531116049s_rule @ X4 @ Xa2 )
     => ( X4 @ ( shd_rule @ Xa2 ) ) ) ).

% holds.elims(2)
thf(fact_1014_holds_Oelims_I1_J,axiom,
    ! [X4: produc164195504107695125m_rule > $o,Xa2: stream727092118206550309m_rule,Y4: $o] :
      ( ( ( linear7082939723987557601m_rule @ X4 @ Xa2 )
        = Y4 )
     => ( Y4
        = ( X4 @ ( shd_Pr7235097944458474089m_rule @ Xa2 ) ) ) ) ).

% holds.elims(1)
thf(fact_1015_holds_Oelims_I1_J,axiom,
    ! [X4: rule > $o,Xa2: stream_rule,Y4: $o] :
      ( ( ( linear4670012741531116049s_rule @ X4 @ Xa2 )
        = Y4 )
     => ( Y4
        = ( X4 @ ( shd_rule @ Xa2 ) ) ) ) ).

% holds.elims(1)
thf(fact_1016_holds_Osimps,axiom,
    ( linear7082939723987557601m_rule
    = ( ^ [P4: produc164195504107695125m_rule > $o,Xs2: stream727092118206550309m_rule] : ( P4 @ ( shd_Pr7235097944458474089m_rule @ Xs2 ) ) ) ) ).

% holds.simps
thf(fact_1017_holds_Osimps,axiom,
    ( linear4670012741531116049s_rule
    = ( ^ [P4: rule > $o,Xs2: stream_rule] : ( P4 @ ( shd_rule @ Xs2 ) ) ) ) ).

% holds.simps
thf(fact_1018_not__less__Least,axiom,
    ! [K2: nat,P2: nat > $o] :
      ( ( ord_less_nat @ K2 @ ( ord_Least_nat @ P2 ) )
     => ~ ( P2 @ K2 ) ) ).

% not_less_Least
thf(fact_1019_Product__Type_OCollect__case__prodD,axiom,
    ! [X4: produc164195504107695125m_rule,A3: produc1996495991257130529ist_fm > rule > $o] :
      ( ( member4220325220686508332m_rule @ X4 @ ( collec2069506698588392810m_rule @ ( produc7369492558480804444rule_o @ A3 ) ) )
     => ( A3 @ ( produc6879501374131015971m_rule @ X4 ) @ ( produc7165828336582415457m_rule @ X4 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_1020_Product__Type_OCollect__case__prodD,axiom,
    ! [X4: produc1996495991257130529ist_fm,A3: list_fm > list_fm > $o] :
      ( ( member8102475879199740618ist_fm @ X4 @ ( collec7231575776666803724ist_fm @ ( produc2011568048573023402t_fm_o @ A3 ) ) )
     => ( A3 @ ( produc1501393135466168645ist_fm @ X4 ) @ ( produc4588648349897876871ist_fm @ X4 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_1021_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% diff_add
thf(fact_1022_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_1023_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1024_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1025_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1026_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1027_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1028_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1029_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1030_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1031_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1032_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1033_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1034_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1035_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).

% le_diff_conv
thf(fact_1036_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1037_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1038_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1039_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K2 )
        = ( J
          = ( plus_plus_nat @ K2 @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1040_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K2: nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_nat @ M @ N2 )
         => ( ord_less_nat @ ( F @ M ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1041_Eps__case__prod,axiom,
    ! [P2: produc1996495991257130529ist_fm > rule > $o] :
      ( ( fChoic9196029409387999681m_rule @ ( produc7369492558480804444rule_o @ P2 ) )
      = ( fChoic9196029409387999681m_rule
        @ ^ [Xy: produc164195504107695125m_rule] : ( P2 @ ( produc6879501374131015971m_rule @ Xy ) @ ( produc7165828336582415457m_rule @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_1042_Eps__case__prod,axiom,
    ! [P2: list_fm > list_fm > $o] :
      ( ( fChoic1980167584695909237ist_fm @ ( produc2011568048573023402t_fm_o @ P2 ) )
      = ( fChoic1980167584695909237ist_fm
        @ ^ [Xy: produc1996495991257130529ist_fm] : ( P2 @ ( produc1501393135466168645ist_fm @ Xy ) @ ( produc4588648349897876871ist_fm @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_1043_sdrop_Osimps_I2_J,axiom,
    ! [N: nat,S2: stream727092118206550309m_rule] :
      ( ( sdrop_7224736112439592940m_rule @ ( suc @ N ) @ S2 )
      = ( sdrop_7224736112439592940m_rule @ N @ ( stl_Pr950425576149878629m_rule @ S2 ) ) ) ).

% sdrop.simps(2)
thf(fact_1044_sdrop_Osimps_I2_J,axiom,
    ! [N: nat,S2: stream_rule] :
      ( ( sdrop_rule @ ( suc @ N ) @ S2 )
      = ( sdrop_rule @ N @ ( stl_rule @ S2 ) ) ) ).

% sdrop.simps(2)
thf(fact_1045_snth_Osimps_I2_J,axiom,
    ! [S2: stream727092118206550309m_rule,N: nat] :
      ( ( snth_P1712579846118164081m_rule @ S2 @ ( suc @ N ) )
      = ( snth_P1712579846118164081m_rule @ ( stl_Pr950425576149878629m_rule @ S2 ) @ N ) ) ).

% snth.simps(2)
thf(fact_1046_snth_Osimps_I2_J,axiom,
    ! [S2: stream_rule,N: nat] :
      ( ( snth_rule @ S2 @ ( suc @ N ) )
      = ( snth_rule @ ( stl_rule @ S2 ) @ N ) ) ).

% snth.simps(2)
thf(fact_1047_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules: stream_fm,S5: set_fm,Rs: stream_fm,S2: fm,R2: fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra3888632272101688784_fm_fm @ Eff2 @ Rs @ S2 ) @ ( abstract_pos_fm @ Rs @ R2 ) )
         => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
           => ( ( member_fm @ S2 @ S5 )
             => ( ( abstract_pos_fm @ ( stl_fm @ ( abstra5942290089224017835_fm_fm @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstract_pos_fm @ Rs @ R2 ) @ ( suc @ ( abstra3888632272101688784_fm_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1048_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules: stream_fm,S5: set_rule,Rs: stream_fm,S2: rule,R2: fm] :
      ( ( abstra5167362717900046400m_rule @ Eff2 @ Rules @ S5 )
     => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra6545316509802446277m_rule @ Eff2 @ Rs @ S2 ) @ ( abstract_pos_fm @ Rs @ R2 ) )
         => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
           => ( ( member_rule @ S2 @ S5 )
             => ( ( abstract_pos_fm @ ( stl_fm @ ( abstra8496492133263135264m_rule @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstract_pos_fm @ Rs @ R2 ) @ ( suc @ ( abstra6545316509802446277m_rule @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1049_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,Rs: stream_rule,S2: fm,R2: rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra2294468009765627845ule_fm @ Eff2 @ Rs @ S2 ) @ ( abstract_pos_rule @ Rs @ R2 ) )
         => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
           => ( ( member_fm @ S2 @ S5 )
             => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra4245643633226316832ule_fm @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R2 ) @ ( suc @ ( abstra2294468009765627845ule_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1050_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,Rs: stream_rule,S2: rule,R2: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra5682536783049438522e_rule @ Eff2 @ Rs @ S2 ) @ ( abstract_pos_rule @ Rs @ R2 ) )
         => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
           => ( ( member_rule @ S2 @ S5 )
             => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra7824469560916757525e_rule @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R2 ) @ ( suc @ ( abstra5682536783049438522e_rule @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1051_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S5: set_Pr7058068377845519745ist_fm,Rs: stream_fm,S2: produc1996495991257130529ist_fm,R2: fm] :
      ( ( abstra7587511110199143014ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra4328824187026956193ist_fm @ Eff2 @ Rs @ S2 ) @ ( abstract_pos_fm @ Rs @ R2 ) )
         => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
           => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
             => ( ( abstract_pos_fm @ ( stl_fm @ ( abstra948641862382362758ist_fm @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstract_pos_fm @ Rs @ R2 ) @ ( suc @ ( abstra4328824187026956193ist_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1052_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S5: set_fm,Rs: stream8299795917829157543ist_fm,S2: fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra8883821634369821123_fm_fm @ Eff2 @ Rs @ S2 ) @ ( abstra3513618393749879589ist_fm @ Rs @ R2 ) )
         => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
           => ( ( member_fm @ S2 @ S5 )
             => ( ( abstra3513618393749879589ist_fm @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5503639309725227688_fm_fm @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstra3513618393749879589ist_fm @ Rs @ R2 ) @ ( suc @ ( abstra8883821634369821123_fm_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1053_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S5: set_rule,Rs: stream8299795917829157543ist_fm,S2: rule,R2: produc1996495991257130529ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S5 )
     => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra3389404019813435448m_rule @ Eff2 @ Rs @ S2 ) @ ( abstra3513618393749879589ist_fm @ Rs @ R2 ) )
         => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
           => ( ( member_rule @ S2 @ S5 )
             => ( ( abstra3513618393749879589ist_fm @ ( stl_Pr5027778045650968273ist_fm @ ( abstra5014957989051229853m_rule @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstra3513618393749879589ist_fm @ Rs @ R2 ) @ ( suc @ ( abstra3389404019813435448m_rule @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1054_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,Rs: stream_rule,S2: produc1996495991257130529ist_fm,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S2 ) @ ( abstract_pos_rule @ Rs @ R2 ) )
         => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
           => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
             => ( ( abstract_pos_rule @ ( stl_rule @ ( abstra2363873759530810705ist_fm @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstract_pos_rule @ Rs @ R2 ) @ ( suc @ ( abstra738319790293016300ist_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1055_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,Rs: stream727092118206550309m_rule,S2: fm,R2: produc164195504107695125m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra6093700014593756181ule_fm @ Eff2 @ Rs @ S2 ) @ ( abstra1882453022147397137m_rule @ Rs @ R2 ) )
         => ( ( member4220325220686508332m_rule @ R2 @ ( sset_P4979403222752902913m_rule @ Rules ) )
           => ( ( member_fm @ S2 @ S5 )
             => ( ( abstra1882453022147397137m_rule @ ( stl_Pr950425576149878629m_rule @ ( abstra2315704231547878896ule_fm @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstra1882453022147397137m_rule @ Rs @ R2 ) @ ( suc @ ( abstra6093700014593756181ule_fm @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1056_RuleSystem_Ostake__pos__minWait,axiom,
    ! [Eff2: produc164195504107695125m_rule > rule > fset_rule > $o,Rules: stream727092118206550309m_rule,S5: set_rule,Rs: stream727092118206550309m_rule,S2: rule,R2: produc164195504107695125m_rule] :
      ( ( abstra3778134268586021893e_rule @ Eff2 @ Rules @ S5 )
     => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
       => ( ( ord_less_nat @ ( abstra6412047235177443722e_rule @ Eff2 @ Rs @ S2 ) @ ( abstra1882453022147397137m_rule @ Rs @ R2 ) )
         => ( ( member4220325220686508332m_rule @ R2 @ ( sset_P4979403222752902913m_rule @ Rules ) )
           => ( ( member_rule @ S2 @ S5 )
             => ( ( abstra1882453022147397137m_rule @ ( stl_Pr950425576149878629m_rule @ ( abstra5289413328258868709e_rule @ Eff2 @ Rs @ S2 ) ) @ R2 )
                = ( minus_minus_nat @ ( abstra1882453022147397137m_rule @ Rs @ R2 ) @ ( suc @ ( abstra6412047235177443722e_rule @ Eff2 @ Rs @ S2 ) ) ) ) ) ) ) ) ) ).

% RuleSystem.stake_pos_minWait
thf(fact_1057_RuleSystem_Oper__alw,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,S5: set_list_fm,R2: list_fm,Steps: stream8299795917829157543ist_fm] :
      ( ( abstra12045985430712427ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra6160584476764717798ist_fm @ Eff2 @ Rules @ S5 @ R2 )
       => ( ( ( abstra6579460235205289861ist_fm @ Eff2 @ Rules @ Steps )
            & ( member_list_fm @ ( produc1501393135466168645ist_fm @ ( shd_Pr772355297128350925ist_fm @ Steps ) ) @ S5 ) )
         => ( linear1269638131612670621ist_fm
            @ ^ [Xs2: stream8299795917829157543ist_fm] :
                ( ( linear7000341289576568917ist_fm
                  @ ^ [Step: produc1996495991257130529ist_fm] : ( abstra360940532588453724ist_fm @ Eff2 @ R2 @ ( produc1501393135466168645ist_fm @ Step ) )
                  @ Xs2 )
               => ( ( linear7000341289576568917ist_fm
                    @ ^ [Step: produc1996495991257130529ist_fm] :
                        ( ( produc4588648349897876871ist_fm @ Step )
                        = R2 )
                    @ Xs2 )
                  | ( linear6153005567242706520t_fm_o
                    @ ( linear7000341289576568917ist_fm
                      @ ^ [Step: produc1996495991257130529ist_fm] : ( abstra360940532588453724ist_fm @ Eff2 @ R2 @ ( produc1501393135466168645ist_fm @ Step ) ) )
                    @ Xs2 ) ) )
            @ Steps ) ) ) ) ).

% RuleSystem.per_alw
thf(fact_1058_RuleSystem_Oper__alw,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,R2: rule,Steps: stream727092118206550309m_rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( abstra4153889242452539894ist_fm @ Eff2 @ Rules @ S5 @ R2 )
       => ( ( ( abstra5415202973855822231ist_fm @ Eff2 @ Rules @ Steps )
            & ( member8102475879199740618ist_fm @ ( produc6879501374131015971m_rule @ ( shd_Pr7235097944458474089m_rule @ Steps ) ) @ S5 ) )
         => ( linear5684299538428340377m_rule
            @ ^ [Xs2: stream727092118206550309m_rule] :
                ( ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] : ( abstra3055540779341473792ist_fm @ Eff2 @ R2 @ ( produc6879501374131015971m_rule @ Step ) )
                  @ Xs2 )
               => ( ( linear7082939723987557601m_rule
                    @ ^ [Step: produc164195504107695125m_rule] :
                        ( ( produc7165828336582415457m_rule @ Step )
                        = R2 )
                    @ Xs2 )
                  | ( linear7357459635800698592rule_o
                    @ ( linear7082939723987557601m_rule
                      @ ^ [Step: produc164195504107695125m_rule] : ( abstra3055540779341473792ist_fm @ Eff2 @ R2 @ ( produc6879501374131015971m_rule @ Step ) ) )
                    @ Xs2 ) ) )
            @ Steps ) ) ) ) ).

% RuleSystem.per_alw
thf(fact_1059_nxt_Oelims,axiom,
    ! [X4: stream727092118206550309m_rule > $o,Xa2: stream727092118206550309m_rule,Y4: $o] :
      ( ( ( linear7357459635800698592rule_o @ X4 @ Xa2 )
        = Y4 )
     => ( Y4
        = ( X4 @ ( stl_Pr950425576149878629m_rule @ Xa2 ) ) ) ) ).

% nxt.elims
thf(fact_1060_nxt_Osimps,axiom,
    ( linear7357459635800698592rule_o
    = ( ^ [Phi2: stream727092118206550309m_rule > $o,Xs2: stream727092118206550309m_rule] : ( Phi2 @ ( stl_Pr950425576149878629m_rule @ Xs2 ) ) ) ) ).

% nxt.simps
thf(fact_1061_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_1062_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_1063_ipath__mkTree__ev,axiom,
    ! [S2: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( member8102475879199740618ist_fm @ S2 @ 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
              @ S2 )
            @ 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_1064_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: fm > fm > fset_fm > $o,Rules: stream_fm,S5: set_fm,S2: fm,Rs: stream_fm,Steps: stream1375025385734440967_fm_fm,R2: fm] :
      ( ( abstra886934874521163211_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra724614664783131952_fm_fm @ ( abstra5285091746773999237_fm_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( ( linear5567832576463481085_fm_fm
                  @ ( linear3538412811542617781_fm_fm
                    @ ^ [Step: product_prod_fm_fm] : ( abstra6247030358587465148_fm_fm @ Eff2 @ R2 @ ( product_fst_fm_fm @ Step ) ) )
                  @ Steps )
               => ( linear7981529059021590662_fm_fm
                  @ ( linear3538412811542617781_fm_fm
                    @ ^ [Step: product_prod_fm_fm] :
                        ( ( product_snd_fm_fm @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1065_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: fm > rule > fset_rule > $o,Rules: stream_fm,S5: set_rule,S2: rule,Rs: stream_fm,Steps: stream1800043145262542076ule_fm,R2: fm] :
      ( ( abstra5167362717900046400m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra8780395760585530021ule_fm @ ( abstra8242076542743448570m_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( ( linear2894481046950705138ule_fm
                  @ ( linear543948517440606634ule_fm
                    @ ^ [Step: product_prod_rule_fm] : ( abstra2065808176054476721m_rule @ Eff2 @ R2 @ ( product_fst_rule_fm @ Step ) ) )
                  @ Steps )
               => ( linear6327378476458846971ule_fm
                  @ ( linear543948517440606634ule_fm
                    @ ^ [Step: product_prod_rule_fm] :
                        ( ( product_snd_rule_fm @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1066_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: rule > fm > fset_fm > $o,Rules: stream_rule,S5: set_fm,S2: fm,Rs: stream_rule,Steps: stream8579757915198507644m_rule,R2: rule] :
      ( ( abstra916514217863227968ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra7716921463347034149m_rule @ ( abstra3991228042706630138ule_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( ( linear1831006749712209266m_rule
                  @ ( linear8703846257056886570m_rule
                    @ ^ [Step: product_prod_fm_rule] : ( abstra7038331712872434097ule_fm @ Eff2 @ R2 @ ( product_fst_fm_rule @ Step ) ) )
                  @ Steps )
               => ( linear5263904179220351099m_rule
                  @ ( linear8703846257056886570m_rule
                    @ ^ [Step: product_prod_fm_rule] :
                        ( ( product_snd_fm_rule @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1067_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: rule > rule > fset_rule > $o,Rules: stream_rule,S5: set_rule,S2: rule,Rs: stream_rule,Steps: stream3588992565182678257e_rule,R2: rule] :
      ( ( abstra2725590884229999157e_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra3969007714708337434e_rule @ ( abstra7443870075974203631e_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( ( linear6469244230239975911e_rule
                  @ ( linear7974216874357323679e_rule
                    @ ^ [Step: produc780299800278082923e_rule] : ( abstra3154316310710804774e_rule @ Eff2 @ R2 @ ( produc7264528503596239631e_rule @ Step ) ) )
                  @ Steps )
               => ( linear7219246396390979696e_rule
                  @ ( linear7974216874357323679e_rule
                    @ ^ [Step: produc780299800278082923e_rule] :
                        ( ( produc6725306161561501521e_rule @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1068_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: list_fm > list_fm > fset_list_fm > $o,Rules: stream_list_fm,S5: set_list_fm,S2: list_fm,Rs: stream_list_fm,Steps: stream8299795917829157543ist_fm,R2: list_fm] :
      ( ( abstra12045985430712427ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_list_fm @ S2 @ S5 )
       => ( ( abstra274702546776099020ist_fm @ Rules @ Rs )
         => ( ( abstra5863983821914536912ist_fm @ ( abstra3846921640779263781ist_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_list_fm @ R2 @ ( sset_list_fm @ Rules ) )
             => ( ( linear1269638131612670621ist_fm
                  @ ( linear7000341289576568917ist_fm
                    @ ^ [Step: produc1996495991257130529ist_fm] : ( abstra360940532588453724ist_fm @ Eff2 @ R2 @ ( produc1501393135466168645ist_fm @ Step ) ) )
                  @ Steps )
               => ( linear1231927432584843558ist_fm
                  @ ( linear7000341289576568917ist_fm
                    @ ^ [Step: produc1996495991257130529ist_fm] :
                        ( ( produc4588648349897876871ist_fm @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1069_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > fm > fset_fm > $o,Rules: stream8299795917829157543ist_fm,S5: set_fm,S2: fm,Rs: stream8299795917829157543ist_fm,Steps: stream5068232101399153750ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra2919136520687232136_fm_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra8787874164724549719ist_fm @ ( abstra5537592759599600206_fm_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( ( linear4909073038792100810ist_fm
                  @ ( linear7877637260900181010ist_fm
                    @ ^ [Step: produc292114620442053958ist_fm] : ( abstra8628512393757682903_fm_fm @ Eff2 @ R2 @ ( produc5885873064403807756ist_fm @ Step ) ) )
                  @ Steps )
               => ( linear1768337326564784769ist_fm
                  @ ( linear7877637260900181010ist_fm
                    @ ^ [Step: produc292114620442053958ist_fm] :
                        ( ( produc2183440689561710666ist_fm @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1070_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: produc1996495991257130529ist_fm > rule > fset_rule > $o,Rules: stream8299795917829157543ist_fm,S5: set_rule,S2: rule,Rs: stream8299795917829157543ist_fm,Steps: stream1960312990768957601ist_fm,R2: produc1996495991257130529ist_fm] :
      ( ( abstra9106359784214577277m_rule @ Eff2 @ Rules @ S5 )
     => ( ( member_rule @ S2 @ S5 )
       => ( ( abstra7454370078233524789ist_fm @ Rules @ Rs )
         => ( ( abstra6752138315222246434ist_fm @ ( abstra2830003028686515011m_rule @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member8102475879199740618ist_fm @ R2 @ ( sset_P3431374123360452149ist_fm @ Rules ) )
             => ( ( linear927598023175380501ist_fm
                  @ ( linear2326238208734597725ist_fm
                    @ ^ [Step: produc4630866025709511057ist_fm] : ( abstra5706625008861892940m_rule @ Eff2 @ R2 @ ( produc4228417144610596823ist_fm @ Step ) ) )
                  @ Steps )
               => ( linear7189418855485006156ist_fm
                  @ ( linear2326238208734597725ist_fm
                    @ ^ [Step: produc4630866025709511057ist_fm] :
                        ( ( produc4514744107061996309ist_fm @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1071_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: fm > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_fm,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_fm,Steps: stream5128679079952569136_fm_fm,R2: fm] :
      ( ( abstra7587511110199143014ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra5119165595139683900air_fm @ Rules @ Rs )
         => ( ( abstra2316067621365718065_fm_fm @ ( abstra982595312256735276ist_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_fm @ R2 @ ( sset_fm @ Rules ) )
             => ( ( linear7660638532288044964_fm_fm
                  @ ( linear1405830717541349356_fm_fm
                    @ ^ [Step: produc3043680113937998112_fm_fm] : ( abstra4073514946414817973ist_fm @ Eff2 @ R2 @ ( produc1217498474891896878_fm_fm @ Step ) ) )
                  @ Steps )
               => ( linear4519902820060728923_fm_fm
                  @ ( linear1405830717541349356_fm_fm
                    @ ^ [Step: produc3043680113937998112_fm_fm] :
                        ( ( produc6738438136904575596_fm_fm @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1072_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,Rules: stream_rule,S5: set_Pr7058068377845519745ist_fm,S2: produc1996495991257130529ist_fm,Rs: stream_rule,Steps: stream727092118206550309m_rule,R2: rule] :
      ( ( abstra6455275554694158129ist_fm @ Eff2 @ Rules @ S5 )
     => ( ( member8102475879199740618ist_fm @ S2 @ S5 )
       => ( ( abstra1114877437032970289r_rule @ Rules @ Rs )
         => ( ( abstra2285467793620430502m_rule @ ( abstra178918799166095863ist_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member_rule @ R2 @ ( sset_rule @ Rules ) )
             => ( ( linear5684299538428340377m_rule
                  @ ( linear7082939723987557601m_rule
                    @ ^ [Step: produc164195504107695125m_rule] : ( abstra3055540779341473792ist_fm @ Eff2 @ R2 @ ( produc6879501374131015971m_rule @ Step ) ) )
                  @ Steps )
               => ( linear2722748333883190224m_rule
                  @ ( linear7082939723987557601m_rule
                    @ ^ [Step: produc164195504107695125m_rule] :
                        ( ( produc7165828336582415457m_rule @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1073_RuleSystem_Oipath__mkTree__ev,axiom,
    ! [Eff2: produc164195504107695125m_rule > fm > fset_fm > $o,Rules: stream727092118206550309m_rule,S5: set_fm,S2: fm,Rs: stream727092118206550309m_rule,Steps: stream5001952251264143308m_rule,R2: produc164195504107695125m_rule] :
      ( ( abstra3764363818050485264ule_fm @ Eff2 @ Rules @ S5 )
     => ( ( member_fm @ S2 @ S5 )
       => ( ( abstra4997497885011821313m_rule @ Rules @ Rs )
         => ( ( abstra3470856493162581877m_rule @ ( abstra1207708450097661130ule_fm @ Eff2 @ Rs @ S2 ) @ Steps )
           => ( ( member4220325220686508332m_rule @ R2 @ ( sset_P4979403222752902913m_rule @ Rules ) )
             => ( ( linear6754809376179413826m_rule
                  @ ( linear5564667394171607290m_rule
                    @ ^ [Step: produc4153787046269522630m_rule] : ( abstra7152516257876272129ule_fm @ Eff2 @ R2 @ ( produc767568915850739050m_rule @ Step ) ) )
                  @ Steps )
               => ( linear8837502450296302283m_rule
                  @ ( linear5564667394171607290m_rule
                    @ ^ [Step: produc4153787046269522630m_rule] :
                        ( ( produc7704281967196386220m_rule @ Step )
                        = R2 ) )
                  @ Steps ) ) ) ) ) ) ) ).

% RuleSystem.ipath_mkTree_ev
thf(fact_1074_fminus__iff,axiom,
    ! [C: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A3 @ B3 ) )
      = ( ( fmembe3381613331217039976ist_fm @ C @ A3 )
        & ~ ( fmembe3381613331217039976ist_fm @ C @ B3 ) ) ) ).

% fminus_iff
thf(fact_1075_fminus__iff,axiom,
    ! [C: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ C @ ( minus_1606610846927913178m_rule @ A3 @ B3 ) )
      = ( ( fmembe1635983029874454474m_rule @ C @ A3 )
        & ~ ( fmembe1635983029874454474m_rule @ C @ B3 ) ) ) ).

% fminus_iff
thf(fact_1076_fminusI,axiom,
    ! [C: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ A3 )
     => ( ~ ( fmembe3381613331217039976ist_fm @ C @ B3 )
       => ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A3 @ B3 ) ) ) ) ).

% fminusI
thf(fact_1077_fminusI,axiom,
    ! [C: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ C @ A3 )
     => ( ~ ( fmembe1635983029874454474m_rule @ C @ B3 )
       => ( fmembe1635983029874454474m_rule @ C @ ( minus_1606610846927913178m_rule @ A3 @ B3 ) ) ) ) ).

% fminusI
thf(fact_1078_not__ev__not,axiom,
    ! [Phi: stream727092118206550309m_rule > $o] :
      ( ( ^ [Xs2: stream727092118206550309m_rule] :
            ~ ( linear2722748333883190224m_rule
              @ ^ [Xt: stream727092118206550309m_rule] :
                  ~ ( Phi @ Xt )
              @ Xs2 ) )
      = ( linear5684299538428340377m_rule @ Phi ) ) ).

% not_ev_not
thf(fact_1079_not__ev__not,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( ^ [Xs2: stream_rule] :
            ~ ( linear6831045468625226112v_rule
              @ ^ [Xt: stream_rule] :
                  ~ ( Phi @ Xt )
              @ Xs2 ) )
      = ( linear7625967296368300489w_rule @ Phi ) ) ).

% not_ev_not
thf(fact_1080_not__alw__not,axiom,
    ! [Phi: stream727092118206550309m_rule > $o] :
      ( ( ^ [Xs2: stream727092118206550309m_rule] :
            ~ ( linear5684299538428340377m_rule
              @ ^ [Xt: stream727092118206550309m_rule] :
                  ~ ( Phi @ Xt )
              @ Xs2 ) )
      = ( linear2722748333883190224m_rule @ Phi ) ) ).

% not_alw_not
thf(fact_1081_not__alw__not,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( ^ [Xs2: stream_rule] :
            ~ ( linear7625967296368300489w_rule
              @ ^ [Xt: stream_rule] :
                  ~ ( Phi @ Xt )
              @ Xs2 ) )
      = ( linear6831045468625226112v_rule @ Phi ) ) ).

% not_alw_not
thf(fact_1082_ev__alw__stl,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,X4: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi ) @ ( stl_Pr950425576149878629m_rule @ X4 ) )
      = ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi ) @ X4 ) ) ).

% ev_alw_stl
thf(fact_1083_ev__alw__stl,axiom,
    ! [Phi: stream_rule > $o,X4: stream_rule] :
      ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi ) @ ( stl_rule @ X4 ) )
      = ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi ) @ X4 ) ) ).

% ev_alw_stl
thf(fact_1084_pfsubsetD,axiom,
    ! [A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm,C: produc1996495991257130529ist_fm] :
      ( ( ord_le8344328325585982387ist_fm @ A3 @ B3 )
     => ( ( fmembe3381613331217039976ist_fm @ C @ A3 )
       => ( fmembe3381613331217039976ist_fm @ C @ B3 ) ) ) ).

% pfsubsetD
thf(fact_1085_pfsubsetD,axiom,
    ! [A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule,C: abstra4572470701423150929m_rule] :
      ( ( ord_le4463057631884877013m_rule @ A3 @ B3 )
     => ( ( fmembe1635983029874454474m_rule @ C @ A3 )
       => ( fmembe1635983029874454474m_rule @ C @ B3 ) ) ) ).

% pfsubsetD
thf(fact_1086_pfsubset__imp__ex__fmem,axiom,
    ! [A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ( ord_le8344328325585982387ist_fm @ A3 @ B3 )
     => ? [B5: produc1996495991257130529ist_fm] : ( fmembe3381613331217039976ist_fm @ B5 @ ( minus_8437252545254675822ist_fm @ B3 @ A3 ) ) ) ).

% pfsubset_imp_ex_fmem
thf(fact_1087_pfsubset__imp__ex__fmem,axiom,
    ! [A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ( ord_le4463057631884877013m_rule @ A3 @ B3 )
     => ? [B5: abstra4572470701423150929m_rule] : ( fmembe1635983029874454474m_rule @ B5 @ ( minus_1606610846927913178m_rule @ B3 @ A3 ) ) ) ).

% pfsubset_imp_ex_fmem
thf(fact_1088_fminusE,axiom,
    ! [C: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A3 @ B3 ) )
     => ~ ( ( fmembe3381613331217039976ist_fm @ C @ A3 )
         => ( fmembe3381613331217039976ist_fm @ C @ B3 ) ) ) ).

% fminusE
thf(fact_1089_fminusE,axiom,
    ! [C: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ C @ ( minus_1606610846927913178m_rule @ A3 @ B3 ) )
     => ~ ( ( fmembe1635983029874454474m_rule @ C @ A3 )
         => ( fmembe1635983029874454474m_rule @ C @ B3 ) ) ) ).

% fminusE
thf(fact_1090_fminusD1,axiom,
    ! [C: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A3 @ B3 ) )
     => ( fmembe3381613331217039976ist_fm @ C @ A3 ) ) ).

% fminusD1
thf(fact_1091_fminusD1,axiom,
    ! [C: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ C @ ( minus_1606610846927913178m_rule @ A3 @ B3 ) )
     => ( fmembe1635983029874454474m_rule @ C @ A3 ) ) ).

% fminusD1
thf(fact_1092_fminusD2,axiom,
    ! [C: produc1996495991257130529ist_fm,A3: fset_P661503646757059847ist_fm,B3: fset_P661503646757059847ist_fm] :
      ( ( fmembe3381613331217039976ist_fm @ C @ ( minus_8437252545254675822ist_fm @ A3 @ B3 ) )
     => ~ ( fmembe3381613331217039976ist_fm @ C @ B3 ) ) ).

% fminusD2
thf(fact_1093_fminusD2,axiom,
    ! [C: abstra4572470701423150929m_rule,A3: fset_A5396818216090053761m_rule,B3: fset_A5396818216090053761m_rule] :
      ( ( fmembe1635983029874454474m_rule @ C @ ( minus_1606610846927913178m_rule @ A3 @ B3 ) )
     => ~ ( fmembe1635983029874454474m_rule @ C @ B3 ) ) ).

% fminusD2
thf(fact_1094_ev_Ostep,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ Xs ) )
     => ( linear2722748333883190224m_rule @ Phi @ Xs ) ) ).

% ev.step
thf(fact_1095_ev_Ostep,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule] :
      ( ( linear6831045468625226112v_rule @ Phi @ ( stl_rule @ Xs ) )
     => ( linear6831045468625226112v_rule @ Phi @ Xs ) ) ).

% ev.step
thf(fact_1096_ev_Ocases,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,A: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ Phi @ A )
     => ( ~ ( Phi @ A )
       => ( linear2722748333883190224m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ A ) ) ) ) ).

% ev.cases
thf(fact_1097_ev_Ocases,axiom,
    ! [Phi: stream_rule > $o,A: stream_rule] :
      ( ( linear6831045468625226112v_rule @ Phi @ A )
     => ( ~ ( Phi @ A )
       => ( linear6831045468625226112v_rule @ Phi @ ( stl_rule @ A ) ) ) ) ).

% ev.cases
thf(fact_1098_ev_Osimps,axiom,
    ( linear2722748333883190224m_rule
    = ( ^ [Phi2: stream727092118206550309m_rule > $o,A2: stream727092118206550309m_rule] :
          ( ? [Xs2: stream727092118206550309m_rule] :
              ( ( A2 = Xs2 )
              & ( Phi2 @ Xs2 ) )
          | ? [Xs2: stream727092118206550309m_rule] :
              ( ( A2 = Xs2 )
              & ( linear2722748333883190224m_rule @ Phi2 @ ( stl_Pr950425576149878629m_rule @ Xs2 ) ) ) ) ) ) ).

% ev.simps
thf(fact_1099_ev_Osimps,axiom,
    ( linear6831045468625226112v_rule
    = ( ^ [Phi2: stream_rule > $o,A2: stream_rule] :
          ( ? [Xs2: stream_rule] :
              ( ( A2 = Xs2 )
              & ( Phi2 @ Xs2 ) )
          | ? [Xs2: stream_rule] :
              ( ( A2 = Xs2 )
              & ( linear6831045468625226112v_rule @ Phi2 @ ( stl_rule @ Xs2 ) ) ) ) ) ) ).

% ev.simps
thf(fact_1100_ev__induct__strong,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,X4: stream727092118206550309m_rule,P2: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ Phi @ X4 )
     => ( ! [Xs4: stream727092118206550309m_rule] :
            ( ( Phi @ Xs4 )
           => ( P2 @ Xs4 ) )
       => ( ! [Xs4: stream727092118206550309m_rule] :
              ( ( linear2722748333883190224m_rule @ Phi @ ( stl_Pr950425576149878629m_rule @ Xs4 ) )
             => ( ~ ( Phi @ Xs4 )
               => ( ( P2 @ ( stl_Pr950425576149878629m_rule @ Xs4 ) )
                 => ( P2 @ Xs4 ) ) ) )
         => ( P2 @ X4 ) ) ) ) ).

% ev_induct_strong
thf(fact_1101_ev__induct__strong,axiom,
    ! [Phi: stream_rule > $o,X4: stream_rule,P2: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ Phi @ X4 )
     => ( ! [Xs4: stream_rule] :
            ( ( Phi @ Xs4 )
           => ( P2 @ Xs4 ) )
       => ( ! [Xs4: stream_rule] :
              ( ( linear6831045468625226112v_rule @ Phi @ ( stl_rule @ Xs4 ) )
             => ( ~ ( Phi @ Xs4 )
               => ( ( P2 @ ( stl_rule @ Xs4 ) )
                 => ( P2 @ Xs4 ) ) ) )
         => ( P2 @ X4 ) ) ) ) ).

% ev_induct_strong
thf(fact_1102_ev__or,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ( ( Phi @ Xs2 )
            | ( Psi @ Xs2 ) ) )
      = ( ^ [Xs2: stream727092118206550309m_rule] :
            ( ( linear2722748333883190224m_rule @ Phi @ Xs2 )
            | ( linear2722748333883190224m_rule @ Psi @ Xs2 ) ) ) ) ).

% ev_or
thf(fact_1103_ev__or,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule
        @ ^ [Xs2: stream_rule] :
            ( ( Phi @ Xs2 )
            | ( Psi @ Xs2 ) ) )
      = ( ^ [Xs2: stream_rule] :
            ( ( linear6831045468625226112v_rule @ Phi @ Xs2 )
            | ( linear6831045468625226112v_rule @ Psi @ Xs2 ) ) ) ) ).

% ev_or
thf(fact_1104_ev__False,axiom,
    ! [Omega2: stream727092118206550309m_rule] :
      ~ ( linear2722748333883190224m_rule
        @ ^ [X3: stream727092118206550309m_rule] : $false
        @ Omega2 ) ).

% ev_False
thf(fact_1105_ev__False,axiom,
    ! [Omega2: stream_rule] :
      ~ ( linear6831045468625226112v_rule
        @ ^ [X3: stream_rule] : $false
        @ Omega2 ) ).

% ev_False
thf(fact_1106_ev__iff__sdrop,axiom,
    ( linear2722748333883190224m_rule
    = ( ^ [P4: stream727092118206550309m_rule > $o,Omega: stream727092118206550309m_rule] :
        ? [M3: nat] : ( P4 @ ( sdrop_7224736112439592940m_rule @ M3 @ Omega ) ) ) ) ).

% ev_iff_sdrop
thf(fact_1107_ev__iff__sdrop,axiom,
    ( linear6831045468625226112v_rule
    = ( ^ [P4: stream_rule > $o,Omega: stream_rule] :
        ? [M3: nat] : ( P4 @ ( sdrop_rule @ M3 @ Omega ) ) ) ) ).

% ev_iff_sdrop
thf(fact_1108_ev__inv,axiom,
    ! [F: stream727092118206550309m_rule > stream727092118206550309m_rule,P2: stream727092118206550309m_rule > $o,S2: stream727092118206550309m_rule] :
      ( ! [S3: stream727092118206550309m_rule] :
          ( ( F @ ( stl_Pr950425576149878629m_rule @ S3 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F @ S3 ) ) )
     => ( ( linear2722748333883190224m_rule @ P2 @ ( F @ S2 ) )
        = ( linear2722748333883190224m_rule
          @ ^ [X3: stream727092118206550309m_rule] : ( P2 @ ( F @ X3 ) )
          @ S2 ) ) ) ).

% ev_inv
thf(fact_1109_ev__inv,axiom,
    ! [F: stream_rule > stream727092118206550309m_rule,P2: stream727092118206550309m_rule > $o,S2: stream_rule] :
      ( ! [S3: stream_rule] :
          ( ( F @ ( stl_rule @ S3 ) )
          = ( stl_Pr950425576149878629m_rule @ ( F @ S3 ) ) )
     => ( ( linear2722748333883190224m_rule @ P2 @ ( F @ S2 ) )
        = ( linear6831045468625226112v_rule
          @ ^ [X3: stream_rule] : ( P2 @ ( F @ X3 ) )
          @ S2 ) ) ) ).

% ev_inv
thf(fact_1110_ev__inv,axiom,
    ! [F: stream727092118206550309m_rule > stream_rule,P2: stream_rule > $o,S2: stream727092118206550309m_rule] :
      ( ! [S3: stream727092118206550309m_rule] :
          ( ( F @ ( stl_Pr950425576149878629m_rule @ S3 ) )
          = ( stl_rule @ ( F @ S3 ) ) )
     => ( ( linear6831045468625226112v_rule @ P2 @ ( F @ S2 ) )
        = ( linear2722748333883190224m_rule
          @ ^ [X3: stream727092118206550309m_rule] : ( P2 @ ( F @ X3 ) )
          @ S2 ) ) ) ).

% ev_inv
thf(fact_1111_ev__inv,axiom,
    ! [F: stream_rule > stream_rule,P2: stream_rule > $o,S2: stream_rule] :
      ( ! [S3: stream_rule] :
          ( ( F @ ( stl_rule @ S3 ) )
          = ( stl_rule @ ( F @ S3 ) ) )
     => ( ( linear6831045468625226112v_rule @ P2 @ ( F @ S2 ) )
        = ( linear6831045468625226112v_rule
          @ ^ [X3: stream_rule] : ( P2 @ ( F @ X3 ) )
          @ S2 ) ) ) ).

% ev_inv
thf(fact_1112_not__ev,axiom,
    ! [Phi: stream727092118206550309m_rule > $o] :
      ( ( ^ [Xs2: stream727092118206550309m_rule] :
            ~ ( linear2722748333883190224m_rule @ Phi @ Xs2 ) )
      = ( linear5684299538428340377m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ~ ( Phi @ Xs2 ) ) ) ).

% not_ev
thf(fact_1113_not__ev,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( ^ [Xs2: stream_rule] :
            ~ ( linear6831045468625226112v_rule @ Phi @ Xs2 ) )
      = ( linear7625967296368300489w_rule
        @ ^ [Xs2: stream_rule] :
            ~ ( Phi @ Xs2 ) ) ) ).

% not_ev
thf(fact_1114_not__alw,axiom,
    ! [Phi: stream727092118206550309m_rule > $o] :
      ( ( ^ [Xs2: stream727092118206550309m_rule] :
            ~ ( linear5684299538428340377m_rule @ Phi @ Xs2 ) )
      = ( linear2722748333883190224m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ~ ( Phi @ Xs2 ) ) ) ).

% not_alw
thf(fact_1115_not__alw,axiom,
    ! [Phi: stream_rule > $o] :
      ( ( ^ [Xs2: stream_rule] :
            ~ ( linear7625967296368300489w_rule @ Phi @ Xs2 ) )
      = ( linear6831045468625226112v_rule
        @ ^ [Xs2: stream_rule] :
            ~ ( Phi @ Xs2 ) ) ) ).

% not_alw
thf(fact_1116_not__ev__iff,axiom,
    ! [P2: stream727092118206550309m_rule > $o,Omega2: stream727092118206550309m_rule] :
      ( ( ~ ( linear2722748333883190224m_rule @ P2 @ Omega2 ) )
      = ( linear5684299538428340377m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ~ ( P2 @ Xs2 )
        @ Omega2 ) ) ).

% not_ev_iff
thf(fact_1117_not__ev__iff,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( ~ ( linear6831045468625226112v_rule @ P2 @ Omega2 ) )
      = ( linear7625967296368300489w_rule
        @ ^ [Xs2: stream_rule] :
            ~ ( P2 @ Xs2 )
        @ Omega2 ) ) ).

% not_ev_iff
thf(fact_1118_ev__alw__aand,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi ) @ Xs )
     => ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Psi ) @ Xs )
       => ( linear2722748333883190224m_rule
          @ ( linear5684299538428340377m_rule
            @ ^ [Xs2: stream727092118206550309m_rule] :
                ( ( Phi @ Xs2 )
                & ( Psi @ Xs2 ) ) )
          @ Xs ) ) ) ).

% ev_alw_aand
thf(fact_1119_ev__alw__aand,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi ) @ Xs )
     => ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Psi ) @ Xs )
       => ( linear6831045468625226112v_rule
          @ ( linear7625967296368300489w_rule
            @ ^ [Xs2: stream_rule] :
                ( ( Phi @ Xs2 )
                & ( Psi @ Xs2 ) ) )
          @ Xs ) ) ) ).

% ev_alw_aand
thf(fact_1120_ev__alw__impl,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ Phi @ Xs )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs2: stream727092118206550309m_rule] :
              ( ( Phi @ Xs2 )
             => ( Psi @ Xs2 ) )
          @ Xs )
       => ( linear2722748333883190224m_rule @ Psi @ Xs ) ) ) ).

% ev_alw_impl
thf(fact_1121_ev__alw__impl,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ Phi @ Xs )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs2: stream_rule] :
              ( ( Phi @ Xs2 )
             => ( Psi @ Xs2 ) )
          @ Xs )
       => ( linear6831045468625226112v_rule @ Psi @ Xs ) ) ) ).

% ev_alw_impl
thf(fact_1122_not__alw__iff,axiom,
    ! [P2: stream727092118206550309m_rule > $o,Omega2: stream727092118206550309m_rule] :
      ( ( ~ ( linear5684299538428340377m_rule @ P2 @ Omega2 ) )
      = ( linear2722748333883190224m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ~ ( P2 @ Xs2 )
        @ Omega2 ) ) ).

% not_alw_iff
thf(fact_1123_not__alw__iff,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( ~ ( linear7625967296368300489w_rule @ P2 @ Omega2 ) )
      = ( linear6831045468625226112v_rule
        @ ^ [Xs2: stream_rule] :
            ~ ( P2 @ Xs2 )
        @ Omega2 ) ) ).

% not_alw_iff
thf(fact_1124_ev__alw__impl__ev,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ Phi @ Xs )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs2: stream727092118206550309m_rule] :
              ( ( Phi @ Xs2 )
             => ( linear2722748333883190224m_rule @ Psi @ Xs2 ) )
          @ Xs )
       => ( linear2722748333883190224m_rule @ Psi @ Xs ) ) ) ).

% ev_alw_impl_ev
thf(fact_1125_ev__alw__impl__ev,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ Phi @ Xs )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs2: stream_rule] :
              ( ( Phi @ Xs2 )
             => ( linear6831045468625226112v_rule @ Psi @ Xs2 ) )
          @ Xs )
       => ( linear6831045468625226112v_rule @ Psi @ Xs ) ) ) ).

% ev_alw_impl_ev
thf(fact_1126_alw__alw__impl__ev,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ( ( linear5684299538428340377m_rule @ Phi @ Xs2 )
           => ( linear2722748333883190224m_rule @ Psi @ Xs2 ) ) )
      = ( ^ [Xs2: stream727092118206550309m_rule] :
            ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi ) @ Xs2 )
           => ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ Psi ) @ Xs2 ) ) ) ) ).

% alw_alw_impl_ev
thf(fact_1127_alw__alw__impl__ev,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule
        @ ^ [Xs2: stream_rule] :
            ( ( linear7625967296368300489w_rule @ Phi @ Xs2 )
           => ( linear6831045468625226112v_rule @ Psi @ Xs2 ) ) )
      = ( ^ [Xs2: stream_rule] :
            ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi ) @ Xs2 )
           => ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ Psi ) @ Xs2 ) ) ) ) ).

% alw_alw_impl_ev
thf(fact_1128_alw__impl__ev__alw,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Psi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ( ( Phi @ Xs2 )
           => ( linear2722748333883190224m_rule @ Psi @ Xs2 ) )
        @ Xs )
     => ( linear5684299538428340377m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ( ( linear2722748333883190224m_rule @ Phi @ Xs2 )
           => ( linear2722748333883190224m_rule @ Psi @ Xs2 ) )
        @ Xs ) ) ).

% alw_impl_ev_alw
thf(fact_1129_alw__impl__ev__alw,axiom,
    ! [Phi: stream_rule > $o,Psi: stream_rule > $o,Xs: stream_rule] :
      ( ( linear7625967296368300489w_rule
        @ ^ [Xs2: stream_rule] :
            ( ( Phi @ Xs2 )
           => ( linear6831045468625226112v_rule @ Psi @ Xs2 ) )
        @ Xs )
     => ( linear7625967296368300489w_rule
        @ ^ [Xs2: stream_rule] :
            ( ( linear6831045468625226112v_rule @ Phi @ Xs2 )
           => ( linear6831045468625226112v_rule @ Psi @ Xs2 ) )
        @ Xs ) ) ).

% alw_impl_ev_alw
thf(fact_1130_ev__alw__alw__impl,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi ) @ Xs )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs2: stream727092118206550309m_rule] :
              ( ( linear5684299538428340377m_rule @ Phi @ Xs2 )
             => ( linear2722748333883190224m_rule @ Psi @ Xs2 ) )
          @ Xs )
       => ( linear2722748333883190224m_rule @ Psi @ Xs ) ) ) ).

% ev_alw_alw_impl
thf(fact_1131_ev__alw__alw__impl,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule,Psi: stream_rule > $o] :
      ( ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi ) @ Xs )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs2: stream_rule] :
              ( ( linear7625967296368300489w_rule @ Phi @ Xs2 )
             => ( linear6831045468625226112v_rule @ Psi @ Xs2 ) )
          @ Xs )
       => ( linear6831045468625226112v_rule @ Psi @ Xs ) ) ) ).

% ev_alw_alw_impl
thf(fact_1132_alw__ev__imp__ev__alw,axiom,
    ! [P2: stream727092118206550309m_rule > $o,Omega2: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ P2 ) @ Omega2 )
     => ( linear2722748333883190224m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ( ( P2 @ Xs2 )
            & ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ P2 ) @ Xs2 ) )
        @ Omega2 ) ) ).

% alw_ev_imp_ev_alw
thf(fact_1133_alw__ev__imp__ev__alw,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ P2 ) @ Omega2 )
     => ( linear6831045468625226112v_rule
        @ ^ [Xs2: stream_rule] :
            ( ( P2 @ Xs2 )
            & ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ P2 ) @ Xs2 ) )
        @ Omega2 ) ) ).

% alw_ev_imp_ev_alw
thf(fact_1134_alw__ev__stl,axiom,
    ! [P2: stream727092118206550309m_rule > $o,Omega2: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ P2 ) @ ( stl_Pr950425576149878629m_rule @ Omega2 ) )
      = ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ P2 ) @ Omega2 ) ) ).

% alw_ev_stl
thf(fact_1135_alw__ev__stl,axiom,
    ! [P2: stream_rule > $o,Omega2: stream_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ P2 ) @ ( stl_rule @ Omega2 ) )
      = ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ P2 ) @ Omega2 ) ) ).

% alw_ev_stl
thf(fact_1136_alw__ev__sdrop,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,M2: nat,Xs: stream727092118206550309m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ Phi ) @ ( sdrop_7224736112439592940m_rule @ M2 @ Xs ) )
     => ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ Phi ) @ Xs ) ) ).

% alw_ev_sdrop
thf(fact_1137_alw__ev__sdrop,axiom,
    ! [Phi: stream_rule > $o,M2: nat,Xs: stream_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ Phi ) @ ( sdrop_rule @ M2 @ Xs ) )
     => ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ Phi ) @ Xs ) ) ).

% alw_ev_sdrop
thf(fact_1138_ev__holds__sset,axiom,
    ! [P2: produc164195504107695125m_rule > $o,Xs: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ ( linear7082939723987557601m_rule @ P2 ) @ Xs )
      = ( ? [X3: produc164195504107695125m_rule] :
            ( ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ Xs ) )
            & ( P2 @ X3 ) ) ) ) ).

% ev_holds_sset
thf(fact_1139_ev__holds__sset,axiom,
    ! [P2: rule > $o,Xs: stream_rule] :
      ( ( linear6831045468625226112v_rule @ ( linear4670012741531116049s_rule @ P2 ) @ Xs )
      = ( ? [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ Xs ) )
            & ( P2 @ X3 ) ) ) ) ).

% ev_holds_sset
thf(fact_1140_variance,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule,Psi: stream_rule > $o] :
      ( ( Phi @ Xs )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs2: stream_rule] :
              ( ( Phi @ Xs2 )
             => ( ( Psi @ Xs2 )
                | ( linear860818938954496176rule_o @ Phi @ Xs2 ) ) )
          @ Xs )
       => ( ( linear7625967296368300489w_rule @ Phi @ Xs )
          | ( linear6831045468625226112v_rule @ Psi @ Xs ) ) ) ) ).

% variance
thf(fact_1141_variance,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule,Psi: stream727092118206550309m_rule > $o] :
      ( ( Phi @ Xs )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs2: stream727092118206550309m_rule] :
              ( ( Phi @ Xs2 )
             => ( ( Psi @ Xs2 )
                | ( linear7357459635800698592rule_o @ Phi @ Xs2 ) ) )
          @ Xs )
       => ( ( linear5684299538428340377m_rule @ Phi @ Xs )
          | ( linear2722748333883190224m_rule @ Psi @ Xs ) ) ) ) ).

% variance
thf(fact_1142_ev__alw__imp__nxt,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule] :
      ( ( linear6831045468625226112v_rule @ Phi @ Xs )
     => ( ( linear7625967296368300489w_rule
          @ ^ [Xs2: stream_rule] :
              ( ( Phi @ Xs2 )
             => ( linear860818938954496176rule_o @ Phi @ Xs2 ) )
          @ Xs )
       => ( linear6831045468625226112v_rule @ ( linear7625967296368300489w_rule @ Phi ) @ Xs ) ) ) ).

% ev_alw_imp_nxt
thf(fact_1143_ev__alw__imp__nxt,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ Phi @ Xs )
     => ( ( linear5684299538428340377m_rule
          @ ^ [Xs2: stream727092118206550309m_rule] :
              ( ( Phi @ Xs2 )
             => ( linear7357459635800698592rule_o @ Phi @ Xs2 ) )
          @ Xs )
       => ( linear2722748333883190224m_rule @ ( linear5684299538428340377m_rule @ Phi ) @ Xs ) ) ) ).

% ev_alw_imp_nxt
thf(fact_1144_alw__sfilter,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S2: stream727092118206550309m_rule,Q2: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ ( linear7082939723987557601m_rule @ P2 ) ) @ S2 )
     => ( ( linear5684299538428340377m_rule @ Q2 @ ( sfilte7579662470208054517m_rule @ P2 @ S2 ) )
        = ( linear5684299538428340377m_rule
          @ ^ [X3: stream727092118206550309m_rule] : ( Q2 @ ( sfilte7579662470208054517m_rule @ P2 @ X3 ) )
          @ S2 ) ) ) ).

% alw_sfilter
thf(fact_1145_alw__sfilter,axiom,
    ! [P2: rule > $o,S2: stream_rule,Q2: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ ( linear4670012741531116049s_rule @ P2 ) ) @ S2 )
     => ( ( linear7625967296368300489w_rule @ Q2 @ ( sfilter_rule @ P2 @ S2 ) )
        = ( linear7625967296368300489w_rule
          @ ^ [X3: stream_rule] : ( Q2 @ ( sfilter_rule @ P2 @ X3 ) )
          @ S2 ) ) ) ).

% alw_sfilter
thf(fact_1146_ev__sfilter,axiom,
    ! [P2: produc164195504107695125m_rule > $o,S2: stream727092118206550309m_rule,Q2: stream727092118206550309m_rule > $o] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ ( linear7082939723987557601m_rule @ P2 ) ) @ S2 )
     => ( ( linear2722748333883190224m_rule @ Q2 @ ( sfilte7579662470208054517m_rule @ P2 @ S2 ) )
        = ( linear2722748333883190224m_rule
          @ ^ [X3: stream727092118206550309m_rule] : ( Q2 @ ( sfilte7579662470208054517m_rule @ P2 @ X3 ) )
          @ S2 ) ) ) ).

% ev_sfilter
thf(fact_1147_ev__sfilter,axiom,
    ! [P2: rule > $o,S2: stream_rule,Q2: stream_rule > $o] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ ( linear4670012741531116049s_rule @ P2 ) ) @ S2 )
     => ( ( linear6831045468625226112v_rule @ Q2 @ ( sfilter_rule @ P2 @ S2 ) )
        = ( linear6831045468625226112v_rule
          @ ^ [X3: stream_rule] : ( Q2 @ ( sfilter_rule @ P2 @ X3 ) )
          @ S2 ) ) ) ).

% ev_sfilter
thf(fact_1148_RuleSystem__Defs_Ofair__def,axiom,
    ( abstra5119165595139683900air_fm
    = ( ^ [Rules2: stream_fm,Rs2: stream_fm] :
          ( ( ord_less_eq_set_fm @ ( sset_fm @ Rs2 ) @ ( sset_fm @ Rules2 ) )
          & ! [X3: fm] :
              ( ( member_fm @ X3 @ ( sset_fm @ Rules2 ) )
             => ( linear3923485788533267412alw_fm
                @ ( linear5758214424494685963_ev_fm
                  @ ( linear2965097450329772060lds_fm
                    @ ( ^ [Y: fm,Z: fm] : ( Y = Z )
                      @ X3 ) ) )
                @ Rs2 ) ) ) ) ) ).

% RuleSystem_Defs.fair_def
thf(fact_1149_RuleSystem__Defs_Ofair__def,axiom,
    ( abstra4997497885011821313m_rule
    = ( ^ [Rules2: stream727092118206550309m_rule,Rs2: stream727092118206550309m_rule] :
          ( ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ Rs2 ) @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
          & ! [X3: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
             => ( linear5684299538428340377m_rule
                @ ( linear2722748333883190224m_rule
                  @ ( linear7082939723987557601m_rule
                    @ ( ^ [Y: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y = Z )
                      @ X3 ) ) )
                @ Rs2 ) ) ) ) ) ).

% RuleSystem_Defs.fair_def
thf(fact_1150_RuleSystem__Defs_Ofair__def,axiom,
    ( abstra1114877437032970289r_rule
    = ( ^ [Rules2: stream_rule,Rs2: stream_rule] :
          ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs2 ) @ ( sset_rule @ Rules2 ) )
          & ! [X3: rule] :
              ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
             => ( linear7625967296368300489w_rule
                @ ( linear6831045468625226112v_rule
                  @ ( linear4670012741531116049s_rule
                    @ ( ^ [Y: rule,Z: rule] : ( Y = Z )
                      @ X3 ) ) )
                @ Rs2 ) ) ) ) ) ).

% RuleSystem_Defs.fair_def
thf(fact_1151_i_Ofair__def,axiom,
    ( abstra5119165595139683900air_fm
    = ( ^ [Rules2: stream_fm,Rs2: stream_fm] :
          ( ( ord_less_eq_set_fm @ ( sset_fm @ Rs2 ) @ ( sset_fm @ Rules2 ) )
          & ! [X3: fm] :
              ( ( member_fm @ X3 @ ( sset_fm @ Rules2 ) )
             => ( linear3923485788533267412alw_fm
                @ ( linear5758214424494685963_ev_fm
                  @ ( linear2965097450329772060lds_fm
                    @ ( ^ [Y: fm,Z: fm] : ( Y = Z )
                      @ X3 ) ) )
                @ Rs2 ) ) ) ) ) ).

% i.fair_def
thf(fact_1152_i_Ofair__def,axiom,
    ( abstra4997497885011821313m_rule
    = ( ^ [Rules2: stream727092118206550309m_rule,Rs2: stream727092118206550309m_rule] :
          ( ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ Rs2 ) @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
          & ! [X3: produc164195504107695125m_rule] :
              ( ( member4220325220686508332m_rule @ X3 @ ( sset_P4979403222752902913m_rule @ Rules2 ) )
             => ( linear5684299538428340377m_rule
                @ ( linear2722748333883190224m_rule
                  @ ( linear7082939723987557601m_rule
                    @ ( ^ [Y: produc164195504107695125m_rule,Z: produc164195504107695125m_rule] : ( Y = Z )
                      @ X3 ) ) )
                @ Rs2 ) ) ) ) ) ).

% i.fair_def
thf(fact_1153_i_Ofair__def,axiom,
    ( abstra1114877437032970289r_rule
    = ( ^ [Rules2: stream_rule,Rs2: stream_rule] :
          ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs2 ) @ ( sset_rule @ Rules2 ) )
          & ! [X3: rule] :
              ( ( member_rule @ X3 @ ( sset_rule @ Rules2 ) )
             => ( linear7625967296368300489w_rule
                @ ( linear6831045468625226112v_rule
                  @ ( linear4670012741531116049s_rule
                    @ ( ^ [Y: rule,Z: rule] : ( Y = Z )
                      @ X3 ) ) )
                @ Rs2 ) ) ) ) ) ).

% i.fair_def
thf(fact_1154_RuleSystem__Defs_Osaturated__def,axiom,
    ( abstra4290540667760458576ist_fm
    = ( ^ [Eff: list_fm > list_fm > fset_list_fm > $o,R: list_fm] :
          ( linear1269638131612670621ist_fm
          @ ^ [Xs2: stream8299795917829157543ist_fm] :
              ( ( linear7000341289576568917ist_fm
                @ ^ [Step: produc1996495991257130529ist_fm] : ( abstra360940532588453724ist_fm @ Eff @ R @ ( produc1501393135466168645ist_fm @ Step ) )
                @ Xs2 )
             => ( linear1231927432584843558ist_fm
                @ ( linear7000341289576568917ist_fm
                  @ ^ [Step: produc1996495991257130529ist_fm] :
                      ( ( produc4588648349897876871ist_fm @ Step )
                      = R ) )
                @ Xs2 ) ) ) ) ) ).

% RuleSystem_Defs.saturated_def
thf(fact_1155_RuleSystem__Defs_Osaturated__def,axiom,
    ( abstra7252176601000738572ist_fm
    = ( ^ [Eff: rule > produc1996495991257130529ist_fm > fset_P661503646757059847ist_fm > $o,R: rule] :
          ( linear5684299538428340377m_rule
          @ ^ [Xs2: stream727092118206550309m_rule] :
              ( ( linear7082939723987557601m_rule
                @ ^ [Step: produc164195504107695125m_rule] : ( abstra3055540779341473792ist_fm @ Eff @ R @ ( produc6879501374131015971m_rule @ Step ) )
                @ Xs2 )
             => ( linear2722748333883190224m_rule
                @ ( linear7082939723987557601m_rule
                  @ ^ [Step: produc164195504107695125m_rule] :
                      ( ( produc7165828336582415457m_rule @ Step )
                      = R ) )
                @ Xs2 ) ) ) ) ) ).

% RuleSystem_Defs.saturated_def
thf(fact_1156_i_Osaturated__def,axiom,
    ! [Eff3: list_fm > list_fm > option_fset_list_fm,R2: list_fm] :
      ( ( abstra4290540667760458576ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ R2 )
      = ( linear1269638131612670621ist_fm
        @ ^ [Xs2: stream8299795917829157543ist_fm] :
            ( ( linear7000341289576568917ist_fm
              @ ^ [Step: produc1996495991257130529ist_fm] : ( abstra360940532588453724ist_fm @ ( abstra256862739949234434ist_fm @ Eff3 ) @ R2 @ ( produc1501393135466168645ist_fm @ Step ) )
              @ Xs2 )
           => ( linear1231927432584843558ist_fm
              @ ( linear7000341289576568917ist_fm
                @ ^ [Step: produc1996495991257130529ist_fm] :
                    ( ( produc4588648349897876871ist_fm @ Step )
                    = R2 ) )
              @ Xs2 ) ) ) ) ).

% i.saturated_def
thf(fact_1157_i_Osaturated__def,axiom,
    ! [Eff3: rule > produc1996495991257130529ist_fm > option6029102822600979917ist_fm,R2: rule] :
      ( ( abstra7252176601000738572ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 )
      = ( linear5684299538428340377m_rule
        @ ^ [Xs2: stream727092118206550309m_rule] :
            ( ( linear7082939723987557601m_rule
              @ ^ [Step: produc164195504107695125m_rule] : ( abstra3055540779341473792ist_fm @ ( abstra5459341389059251418ist_fm @ Eff3 ) @ R2 @ ( produc6879501374131015971m_rule @ Step ) )
              @ Xs2 )
           => ( linear2722748333883190224m_rule
              @ ( linear7082939723987557601m_rule
                @ ^ [Step: produc164195504107695125m_rule] :
                    ( ( produc7165828336582415457m_rule @ Step )
                    = R2 ) )
              @ Xs2 ) ) ) ) ).

% i.saturated_def
thf(fact_1158_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_1159_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K2: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1160_add__le__imp__le__diff,axiom,
    ! [I: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_1161_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1162_sdrop__wait__least,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule,N: nat] :
      ( ( linear2722748333883190224m_rule @ Phi @ Xs )
     => ( ( Phi @ ( sdrop_7224736112439592940m_rule @ N @ Xs ) )
       => ( ord_less_eq_nat @ ( linear2497734134072303348m_rule @ Phi @ Xs ) @ N ) ) ) ).

% sdrop_wait_least
thf(fact_1163_sdrop__wait__least,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule,N: nat] :
      ( ( linear6831045468625226112v_rule @ Phi @ Xs )
     => ( ( Phi @ ( sdrop_rule @ N @ Xs ) )
       => ( ord_less_eq_nat @ ( linear25361906725448356t_rule @ Phi @ Xs ) @ N ) ) ) ).

% sdrop_wait_least
thf(fact_1164_Prover_Ofair__def,axiom,
    ! [Rs: stream_rule] :
      ( ( abstra1114877437032970289r_rule @ rules @ Rs )
      = ( ( ord_less_eq_set_rule @ ( sset_rule @ Rs ) @ ( sset_rule @ rules ) )
        & ! [X3: rule] :
            ( ( member_rule @ X3 @ ( sset_rule @ rules ) )
           => ( linear7625967296368300489w_rule
              @ ( linear6831045468625226112v_rule
                @ ( linear4670012741531116049s_rule
                  @ ( ^ [Y: rule,Z: rule] : ( Y = Z )
                    @ X3 ) ) )
              @ Rs ) ) ) ) ).

% Prover.fair_def
thf(fact_1165_sdrop__wait,axiom,
    ! [Phi: stream727092118206550309m_rule > $o,Xs: stream727092118206550309m_rule] :
      ( ( linear2722748333883190224m_rule @ Phi @ Xs )
     => ( Phi @ ( sdrop_7224736112439592940m_rule @ ( linear2497734134072303348m_rule @ Phi @ Xs ) @ Xs ) ) ) ).

% sdrop_wait
thf(fact_1166_sdrop__wait,axiom,
    ! [Phi: stream_rule > $o,Xs: stream_rule] :
      ( ( linear6831045468625226112v_rule @ Phi @ Xs )
     => ( Phi @ ( sdrop_rule @ ( linear25361906725448356t_rule @ Phi @ Xs ) @ Xs ) ) ) ).

% sdrop_wait
thf(fact_1167_sfilter__streams,axiom,
    ! [P2: fm > $o,Omega2: stream_fm,A3: set_fm] :
      ( ( linear3923485788533267412alw_fm @ ( linear5758214424494685963_ev_fm @ ( linear2965097450329772060lds_fm @ P2 ) ) @ Omega2 )
     => ( ( member_stream_fm @ Omega2 @ ( streams_fm @ A3 ) )
       => ( member_stream_fm @ ( sfilter_fm @ P2 @ Omega2 )
          @ ( streams_fm
            @ ( collect_fm
              @ ^ [X3: fm] :
                  ( ( member_fm @ X3 @ A3 )
                  & ( P2 @ X3 ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_1168_sfilter__streams,axiom,
    ! [P2: produc1996495991257130529ist_fm > $o,Omega2: stream8299795917829157543ist_fm,A3: set_Pr7058068377845519745ist_fm] :
      ( ( linear1269638131612670621ist_fm @ ( linear1231927432584843558ist_fm @ ( linear7000341289576568917ist_fm @ P2 ) ) @ Omega2 )
     => ( ( member5006235884692811216ist_fm @ Omega2 @ ( stream8099348070969791641ist_fm @ A3 ) )
       => ( member5006235884692811216ist_fm @ ( sfilte646686417537525057ist_fm @ P2 @ Omega2 )
          @ ( stream8099348070969791641ist_fm
            @ ( collec7231575776666803724ist_fm
              @ ^ [X3: produc1996495991257130529ist_fm] :
                  ( ( member8102475879199740618ist_fm @ X3 @ A3 )
                  & ( P2 @ X3 ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_1169_sfilter__streams,axiom,
    ! [P2: produc164195504107695125m_rule > $o,Omega2: stream727092118206550309m_rule,A3: set_Pr1008144964186165195m_rule] :
      ( ( linear5684299538428340377m_rule @ ( linear2722748333883190224m_rule @ ( linear7082939723987557601m_rule @ P2 ) ) @ Omega2 )
     => ( ( member6117863459850561468m_rule @ Omega2 @ ( stream1749396743056113565m_rule @ A3 ) )
       => ( member6117863459850561468m_rule @ ( sfilte7579662470208054517m_rule @ P2 @ Omega2 )
          @ ( stream1749396743056113565m_rule
            @ ( collec2069506698588392810m_rule
              @ ^ [X3: produc164195504107695125m_rule] :
                  ( ( member4220325220686508332m_rule @ X3 @ A3 )
                  & ( P2 @ X3 ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_1170_sfilter__streams,axiom,
    ! [P2: rule > $o,Omega2: stream_rule,A3: set_rule] :
      ( ( linear7625967296368300489w_rule @ ( linear6831045468625226112v_rule @ ( linear4670012741531116049s_rule @ P2 ) ) @ Omega2 )
     => ( ( member_stream_rule @ Omega2 @ ( streams_rule @ A3 ) )
       => ( member_stream_rule @ ( sfilter_rule @ P2 @ Omega2 )
          @ ( streams_rule
            @ ( collect_rule
              @ ^ [X3: rule] :
                  ( ( member_rule @ X3 @ A3 )
                  & ( P2 @ X3 ) ) ) ) ) ) ) ).

% sfilter_streams
thf(fact_1171_mlex__eq,axiom,
    ( mlex_prod_list_fm
    = ( ^ [F2: list_fm > nat,R6: set_Pr7058068377845519745ist_fm] :
          ( collec7231575776666803724ist_fm
          @ ( produc2011568048573023402t_fm_o
            @ ^ [X3: list_fm,Y5: list_fm] :
                ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) )
                | ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) )
                  & ( member8102475879199740618ist_fm @ ( produc381145313068854617ist_fm @ X3 @ Y5 ) @ R6 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_1172_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,R1: rule,Sl6: fset_P661503646757059847ist_fm,S8: produc1996495991257130529ist_fm] :
            ( ( ( member8102475879199740618ist_fm @ S @ top_to728902761071950545ist_fm )
              & ( abstra3055540779341473792ist_fm
                @ ^ [R: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                    ( ( eff @ R @ T )
                    = ( some_f2162114139285431020ist_fm @ Ss ) )
                @ R2
                @ S )
              & ( member_rule @ R1 @ ( minus_minus_set_rule @ ( sset_rule @ rules ) @ ( insert_rule @ R2 @ bot_bot_set_rule ) ) )
              & ( ( eff @ R1 @ S )
                = ( some_f2162114139285431020ist_fm @ Sl6 ) )
              & ( fmembe3381613331217039976ist_fm @ S8 @ Sl6 ) )
           => ( abstra3055540779341473792ist_fm
              @ ^ [R: rule,T: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                  ( ( eff @ R @ T )
                  = ( some_f2162114139285431020ist_fm @ Ss ) )
              @ R2
              @ S8 ) ) ) ) ).

% per_def
thf(fact_1173_streams__UNIV,axiom,
    ( ( stream8099348070969791641ist_fm @ top_to728902761071950545ist_fm )
    = top_to3943610876133046487ist_fm ) ).

% streams_UNIV
thf(fact_1174_streams__UNIV,axiom,
    ( ( streams_rule @ top_top_set_rule )
    = top_to3705917391389534779m_rule ) ).

% streams_UNIV
thf(fact_1175_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec7231575776666803724ist_fm
            @ ^ [S: produc1996495991257130529ist_fm] : P2 )
          = top_to728902761071950545ist_fm ) )
      & ( ~ P2
       => ( ( collec7231575776666803724ist_fm
            @ ^ [S: produc1996495991257130529ist_fm] : P2 )
          = bot_bo4530011010606209517ist_fm ) ) ) ).

% Collect_const
thf(fact_1176_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collect_rule
            @ ^ [S: rule] : P2 )
          = top_top_set_rule ) )
      & ( ~ P2
       => ( ( collect_rule
            @ ^ [S: rule] : P2 )
          = bot_bot_set_rule ) ) ) ).

% Collect_const
thf(fact_1177_singleton__conv,axiom,
    ! [A: rule] :
      ( ( collect_rule
        @ ^ [X3: rule] : ( X3 = A ) )
      = ( insert_rule @ A @ bot_bot_set_rule ) ) ).

% singleton_conv
thf(fact_1178_singleton__conv2,axiom,
    ! [A: rule] :
      ( ( collect_rule
        @ ( ^ [Y: rule,Z: rule] : ( Y = Z )
          @ A ) )
      = ( insert_rule @ A @ bot_bot_set_rule ) ) ).

% singleton_conv2
thf(fact_1179_Diff__UNIV,axiom,
    ! [A3: set_Pr7058068377845519745ist_fm] :
      ( ( minus_5615909706921521512ist_fm @ A3 @ top_to728902761071950545ist_fm )
      = bot_bo4530011010606209517ist_fm ) ).

% Diff_UNIV
thf(fact_1180_Diff__UNIV,axiom,
    ! [A3: set_rule] :
      ( ( minus_minus_set_rule @ A3 @ top_top_set_rule )
      = bot_bot_set_rule ) ).

% Diff_UNIV
thf(fact_1181_stream_Oset,axiom,
    ! [X1: rule,X2: stream_rule] :
      ( ( sset_rule @ ( sCons_rule @ X1 @ X2 ) )
      = ( insert_rule @ X1 @ ( sset_rule @ X2 ) ) ) ).

% stream.set
thf(fact_1182_stream_Oset,axiom,
    ! [X1: produc164195504107695125m_rule,X2: stream727092118206550309m_rule] :
      ( ( sset_P4979403222752902913m_rule @ ( sCons_5731567480799343806m_rule @ X1 @ X2 ) )
      = ( insert8332210346540479045m_rule @ X1 @ ( sset_P4979403222752902913m_rule @ X2 ) ) ) ).

% stream.set
thf(fact_1183_minus__set__def,axiom,
    ( minus_minus_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( collect_fm
          @ ( minus_minus_fm_o
            @ ^ [X3: fm] : ( member_fm @ X3 @ A7 )
            @ ^ [X3: fm] : ( member_fm @ X3 @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_1184_minus__set__def,axiom,
    ( minus_5615909706921521512ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B7: set_Pr7058068377845519745ist_fm] :
          ( collec7231575776666803724ist_fm
          @ ( minus_297532591554960245t_fm_o
            @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ A7 )
            @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_1185_minus__set__def,axiom,
    ( minus_minus_set_rule
    = ( ^ [A7: set_rule,B7: set_rule] :
          ( collect_rule
          @ ( minus_minus_rule_o
            @ ^ [X3: rule] : ( member_rule @ X3 @ A7 )
            @ ^ [X3: rule] : ( member_rule @ X3 @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_1186_set__diff__eq,axiom,
    ( minus_minus_set_fm
    = ( ^ [A7: set_fm,B7: set_fm] :
          ( collect_fm
          @ ^ [X3: fm] :
              ( ( member_fm @ X3 @ A7 )
              & ~ ( member_fm @ X3 @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1187_set__diff__eq,axiom,
    ( minus_5615909706921521512ist_fm
    = ( ^ [A7: set_Pr7058068377845519745ist_fm,B7: set_Pr7058068377845519745ist_fm] :
          ( collec7231575776666803724ist_fm
          @ ^ [X3: produc1996495991257130529ist_fm] :
              ( ( member8102475879199740618ist_fm @ X3 @ A7 )
              & ~ ( member8102475879199740618ist_fm @ X3 @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1188_set__diff__eq,axiom,
    ( minus_minus_set_rule
    = ( ^ [A7: set_rule,B7: set_rule] :
          ( collect_rule
          @ ^ [X3: rule] :
              ( ( member_rule @ X3 @ A7 )
              & ~ ( member_rule @ X3 @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1189_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules: stream727092118206550309m_rule] :
      ( ( sset_P4979403222752902913m_rule @ Rules )
     != bot_bo2182007826548605023m_rule ) ).

% RuleSystem_Defs.NE_R
thf(fact_1190_RuleSystem__Defs_ONE__R,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ Rules )
     != bot_bot_set_rule ) ).

% RuleSystem_Defs.NE_R
thf(fact_1191_i_ONE__R,axiom,
    ! [Rules: stream727092118206550309m_rule] :
      ( ( sset_P4979403222752902913m_rule @ Rules )
     != bot_bo2182007826548605023m_rule ) ).

% i.NE_R
thf(fact_1192_i_ONE__R,axiom,
    ! [Rules: stream_rule] :
      ( ( sset_rule @ Rules )
     != bot_bot_set_rule ) ).

% i.NE_R
thf(fact_1193_empty__not__UNIV,axiom,
    bot_bo4530011010606209517ist_fm != top_to728902761071950545ist_fm ).

% empty_not_UNIV
thf(fact_1194_empty__not__UNIV,axiom,
    bot_bot_set_rule != top_top_set_rule ).

% empty_not_UNIV
thf(fact_1195_insert__UNIV,axiom,
    ! [X4: produc1996495991257130529ist_fm] :
      ( ( insert589278167076416497ist_fm @ X4 @ top_to728902761071950545ist_fm )
      = top_to728902761071950545ist_fm ) ).

% insert_UNIV
thf(fact_1196_insert__UNIV,axiom,
    ! [X4: rule] :
      ( ( insert_rule @ X4 @ top_top_set_rule )
      = top_top_set_rule ) ).

% insert_UNIV
thf(fact_1197_empty__def,axiom,
    ( bot_bot_set_rule
    = ( collect_rule
      @ ^ [X3: rule] : $false ) ) ).

% empty_def
thf(fact_1198_streams__empty,axiom,
    ( ( streams_rule @ bot_bot_set_rule )
    = bot_bo8401694146965224479m_rule ) ).

% streams_empty
thf(fact_1199_streams__empty__iff,axiom,
    ! [S5: set_rule] :
      ( ( ( streams_rule @ S5 )
        = bot_bo8401694146965224479m_rule )
      = ( S5 = bot_bot_set_rule ) ) ).

% streams_empty_iff
thf(fact_1200_insert__compr,axiom,
    ( insert_fm
    = ( ^ [A2: fm,B7: set_fm] :
          ( collect_fm
          @ ^ [X3: fm] :
              ( ( X3 = A2 )
              | ( member_fm @ X3 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_1201_insert__compr,axiom,
    ( insert_rule
    = ( ^ [A2: rule,B7: set_rule] :
          ( collect_rule
          @ ^ [X3: rule] :
              ( ( X3 = A2 )
              | ( member_rule @ X3 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_1202_insert__compr,axiom,
    ( insert589278167076416497ist_fm
    = ( ^ [A2: produc1996495991257130529ist_fm,B7: set_Pr7058068377845519745ist_fm] :
          ( collec7231575776666803724ist_fm
          @ ^ [X3: produc1996495991257130529ist_fm] :
              ( ( X3 = A2 )
              | ( member8102475879199740618ist_fm @ X3 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_1203_insert__Collect,axiom,
    ! [A: rule,P2: rule > $o] :
      ( ( insert_rule @ A @ ( collect_rule @ P2 ) )
      = ( collect_rule
        @ ^ [U: rule] :
            ( ( U != A )
           => ( P2 @ U ) ) ) ) ).

% insert_Collect
thf(fact_1204_Collect__conv__if,axiom,
    ! [P2: rule > $o,A: rule] :
      ( ( ( P2 @ A )
       => ( ( collect_rule
            @ ^ [X3: rule] :
                ( ( X3 = A )
                & ( P2 @ X3 ) ) )
          = ( insert_rule @ A @ bot_bot_set_rule ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_rule
            @ ^ [X3: rule] :
                ( ( X3 = A )
                & ( P2 @ X3 ) ) )
          = bot_bot_set_rule ) ) ) ).

% Collect_conv_if
thf(fact_1205_Collect__conv__if2,axiom,
    ! [P2: rule > $o,A: rule] :
      ( ( ( P2 @ A )
       => ( ( collect_rule
            @ ^ [X3: rule] :
                ( ( A = X3 )
                & ( P2 @ X3 ) ) )
          = ( insert_rule @ A @ bot_bot_set_rule ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_rule
            @ ^ [X3: rule] :
                ( ( A = X3 )
                & ( P2 @ X3 ) ) )
          = bot_bot_set_rule ) ) ) ).

% Collect_conv_if2
thf(fact_1206_some__in__eq,axiom,
    ! [A3: set_fm] :
      ( ( member_fm
        @ ( fChoice_fm
          @ ^ [X3: fm] : ( member_fm @ X3 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_fm ) ) ).

% some_in_eq
thf(fact_1207_some__in__eq,axiom,
    ! [A3: set_Pr7058068377845519745ist_fm] :
      ( ( member8102475879199740618ist_fm
        @ ( fChoic1980167584695909237ist_fm
          @ ^ [X3: produc1996495991257130529ist_fm] : ( member8102475879199740618ist_fm @ X3 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bo4530011010606209517ist_fm ) ) ).

% some_in_eq
thf(fact_1208_some__in__eq,axiom,
    ! [A3: set_fs7534060674568567527ist_fm] :
      ( ( member3493611713653719344ist_fm
        @ ( fChoic3892542562059901019ist_fm
          @ ^ [X3: fset_P661503646757059847ist_fm] : ( member3493611713653719344ist_fm @ X3 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bo8667300132079229267ist_fm ) ) ).

% some_in_eq
thf(fact_1209_some__in__eq,axiom,
    ! [A3: set_rule] :
      ( ( member_rule
        @ ( fChoice_rule
          @ ^ [X3: rule] : ( member_rule @ X3 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_rule ) ) ).

% some_in_eq
thf(fact_1210_streams__iff__snth,axiom,
    ! [S2: stream_fm,X6: set_fm] :
      ( ( member_stream_fm @ S2 @ ( streams_fm @ X6 ) )
      = ( ! [N3: nat] : ( member_fm @ ( snth_fm @ S2 @ N3 ) @ X6 ) ) ) ).

% streams_iff_snth
thf(fact_1211_streams__iff__snth,axiom,
    ! [S2: stream_rule,X6: set_rule] :
      ( ( member_stream_rule @ S2 @ ( streams_rule @ X6 ) )
      = ( ! [N3: nat] : ( member_rule @ ( snth_rule @ S2 @ N3 ) @ X6 ) ) ) ).

% streams_iff_snth
thf(fact_1212_streams__iff__snth,axiom,
    ! [S2: stream8299795917829157543ist_fm,X6: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ S2 @ ( stream8099348070969791641ist_fm @ X6 ) )
      = ( ! [N3: nat] : ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S2 @ N3 ) @ X6 ) ) ) ).

% streams_iff_snth
thf(fact_1213_snth__in,axiom,
    ! [S2: stream_fm,X6: set_fm,N: nat] :
      ( ( member_stream_fm @ S2 @ ( streams_fm @ X6 ) )
     => ( member_fm @ ( snth_fm @ S2 @ N ) @ X6 ) ) ).

% snth_in
thf(fact_1214_snth__in,axiom,
    ! [S2: stream_rule,X6: set_rule,N: nat] :
      ( ( member_stream_rule @ S2 @ ( streams_rule @ X6 ) )
     => ( member_rule @ ( snth_rule @ S2 @ N ) @ X6 ) ) ).

% snth_in
thf(fact_1215_snth__in,axiom,
    ! [S2: stream8299795917829157543ist_fm,X6: set_Pr7058068377845519745ist_fm,N: nat] :
      ( ( member5006235884692811216ist_fm @ S2 @ ( stream8099348070969791641ist_fm @ X6 ) )
     => ( member8102475879199740618ist_fm @ ( snth_P2624749845746815685ist_fm @ S2 @ N ) @ X6 ) ) ).

% snth_in
thf(fact_1216_streams__Stream,axiom,
    ! [X4: fm,S2: stream_fm,A3: set_fm] :
      ( ( member_stream_fm @ ( sCons_fm @ X4 @ S2 ) @ ( streams_fm @ A3 ) )
      = ( ( member_fm @ X4 @ A3 )
        & ( member_stream_fm @ S2 @ ( streams_fm @ A3 ) ) ) ) ).

% streams_Stream
thf(fact_1217_streams__Stream,axiom,
    ! [X4: rule,S2: stream_rule,A3: set_rule] :
      ( ( member_stream_rule @ ( sCons_rule @ X4 @ S2 ) @ ( streams_rule @ A3 ) )
      = ( ( member_rule @ X4 @ A3 )
        & ( member_stream_rule @ S2 @ ( streams_rule @ A3 ) ) ) ) ).

% streams_Stream
thf(fact_1218_streams__Stream,axiom,
    ! [X4: produc1996495991257130529ist_fm,S2: stream8299795917829157543ist_fm,A3: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ ( sCons_307821682964077496ist_fm @ X4 @ S2 ) @ ( stream8099348070969791641ist_fm @ A3 ) )
      = ( ( member8102475879199740618ist_fm @ X4 @ A3 )
        & ( member5006235884692811216ist_fm @ S2 @ ( stream8099348070969791641ist_fm @ A3 ) ) ) ) ).

% streams_Stream
thf(fact_1219_streams_Ocoinduct,axiom,
    ! [X6: stream_fm > $o,X4: stream_fm,A3: set_fm] :
      ( ( X6 @ X4 )
     => ( ! [X: stream_fm] :
            ( ( X6 @ X )
           => ? [A8: fm,S9: stream_fm] :
                ( ( X
                  = ( sCons_fm @ A8 @ S9 ) )
                & ( member_fm @ A8 @ A3 )
                & ( ( X6 @ S9 )
                  | ( member_stream_fm @ S9 @ ( streams_fm @ A3 ) ) ) ) )
       => ( member_stream_fm @ X4 @ ( streams_fm @ A3 ) ) ) ) ).

% streams.coinduct
thf(fact_1220_streams_Ocoinduct,axiom,
    ! [X6: stream_rule > $o,X4: stream_rule,A3: set_rule] :
      ( ( X6 @ X4 )
     => ( ! [X: stream_rule] :
            ( ( X6 @ X )
           => ? [A8: rule,S9: stream_rule] :
                ( ( X
                  = ( sCons_rule @ A8 @ S9 ) )
                & ( member_rule @ A8 @ A3 )
                & ( ( X6 @ S9 )
                  | ( member_stream_rule @ S9 @ ( streams_rule @ A3 ) ) ) ) )
       => ( member_stream_rule @ X4 @ ( streams_rule @ A3 ) ) ) ) ).

% streams.coinduct
thf(fact_1221_streams_Ocoinduct,axiom,
    ! [X6: stream8299795917829157543ist_fm > $o,X4: stream8299795917829157543ist_fm,A3: set_Pr7058068377845519745ist_fm] :
      ( ( X6 @ X4 )
     => ( ! [X: stream8299795917829157543ist_fm] :
            ( ( X6 @ X )
           => ? [A8: produc1996495991257130529ist_fm,S9: stream8299795917829157543ist_fm] :
                ( ( X
                  = ( sCons_307821682964077496ist_fm @ A8 @ S9 ) )
                & ( member8102475879199740618ist_fm @ A8 @ A3 )
                & ( ( X6 @ S9 )
                  | ( member5006235884692811216ist_fm @ S9 @ ( stream8099348070969791641ist_fm @ A3 ) ) ) ) )
       => ( member5006235884692811216ist_fm @ X4 @ ( stream8099348070969791641ist_fm @ A3 ) ) ) ) ).

% streams.coinduct
thf(fact_1222_streams_Osimps,axiom,
    ! [A: stream_fm,A3: set_fm] :
      ( ( member_stream_fm @ A @ ( streams_fm @ A3 ) )
      = ( ? [A2: fm,S: stream_fm] :
            ( ( A
              = ( sCons_fm @ A2 @ S ) )
            & ( member_fm @ A2 @ A3 )
            & ( member_stream_fm @ S @ ( streams_fm @ A3 ) ) ) ) ) ).

% streams.simps
thf(fact_1223_streams_Osimps,axiom,
    ! [A: stream_rule,A3: set_rule] :
      ( ( member_stream_rule @ A @ ( streams_rule @ A3 ) )
      = ( ? [A2: rule,S: stream_rule] :
            ( ( A
              = ( sCons_rule @ A2 @ S ) )
            & ( member_rule @ A2 @ A3 )
            & ( member_stream_rule @ S @ ( streams_rule @ A3 ) ) ) ) ) ).

% streams.simps
thf(fact_1224_streams_Osimps,axiom,
    ! [A: stream8299795917829157543ist_fm,A3: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ A @ ( stream8099348070969791641ist_fm @ A3 ) )
      = ( ? [A2: produc1996495991257130529ist_fm,S: stream8299795917829157543ist_fm] :
            ( ( A
              = ( sCons_307821682964077496ist_fm @ A2 @ S ) )
            & ( member8102475879199740618ist_fm @ A2 @ A3 )
            & ( member5006235884692811216ist_fm @ S @ ( stream8099348070969791641ist_fm @ A3 ) ) ) ) ) ).

% streams.simps
thf(fact_1225_streams_Ocases,axiom,
    ! [A: stream_fm,A3: set_fm] :
      ( ( member_stream_fm @ A @ ( streams_fm @ A3 ) )
     => ~ ! [A5: fm,S3: stream_fm] :
            ( ( A
              = ( sCons_fm @ A5 @ S3 ) )
           => ( ( member_fm @ A5 @ A3 )
             => ~ ( member_stream_fm @ S3 @ ( streams_fm @ A3 ) ) ) ) ) ).

% streams.cases
thf(fact_1226_streams_Ocases,axiom,
    ! [A: stream_rule,A3: set_rule] :
      ( ( member_stream_rule @ A @ ( streams_rule @ A3 ) )
     => ~ ! [A5: rule,S3: stream_rule] :
            ( ( A
              = ( sCons_rule @ A5 @ S3 ) )
           => ( ( member_rule @ A5 @ A3 )
             => ~ ( member_stream_rule @ S3 @ ( streams_rule @ A3 ) ) ) ) ) ).

% streams.cases
thf(fact_1227_streams_Ocases,axiom,
    ! [A: stream8299795917829157543ist_fm,A3: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ A @ ( stream8099348070969791641ist_fm @ A3 ) )
     => ~ ! [A5: produc1996495991257130529ist_fm,S3: stream8299795917829157543ist_fm] :
            ( ( A
              = ( sCons_307821682964077496ist_fm @ A5 @ S3 ) )
           => ( ( member8102475879199740618ist_fm @ A5 @ A3 )
             => ~ ( member5006235884692811216ist_fm @ S3 @ ( stream8099348070969791641ist_fm @ A3 ) ) ) ) ) ).

% streams.cases
thf(fact_1228_streams__stl,axiom,
    ! [S2: stream727092118206550309m_rule,A3: set_Pr1008144964186165195m_rule] :
      ( ( member6117863459850561468m_rule @ S2 @ ( stream1749396743056113565m_rule @ A3 ) )
     => ( member6117863459850561468m_rule @ ( stl_Pr950425576149878629m_rule @ S2 ) @ ( stream1749396743056113565m_rule @ A3 ) ) ) ).

% streams_stl
thf(fact_1229_streams__stl,axiom,
    ! [S2: stream_rule,A3: set_rule] :
      ( ( member_stream_rule @ S2 @ ( streams_rule @ A3 ) )
     => ( member_stream_rule @ ( stl_rule @ S2 ) @ ( streams_rule @ A3 ) ) ) ).

% streams_stl
thf(fact_1230_streams__shd,axiom,
    ! [S2: stream_fm,A3: set_fm] :
      ( ( member_stream_fm @ S2 @ ( streams_fm @ A3 ) )
     => ( member_fm @ ( shd_fm @ S2 ) @ A3 ) ) ).

% streams_shd
thf(fact_1231_streams__shd,axiom,
    ! [S2: stream8299795917829157543ist_fm,A3: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ S2 @ ( stream8099348070969791641ist_fm @ A3 ) )
     => ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ S2 ) @ A3 ) ) ).

% streams_shd
thf(fact_1232_streams__shd,axiom,
    ! [S2: stream727092118206550309m_rule,A3: set_Pr1008144964186165195m_rule] :
      ( ( member6117863459850561468m_rule @ S2 @ ( stream1749396743056113565m_rule @ A3 ) )
     => ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ S2 ) @ A3 ) ) ).

% streams_shd
thf(fact_1233_streams__shd,axiom,
    ! [S2: stream_rule,A3: set_rule] :
      ( ( member_stream_rule @ S2 @ ( streams_rule @ A3 ) )
     => ( member_rule @ ( shd_rule @ S2 ) @ A3 ) ) ).

% streams_shd
thf(fact_1234_streams__mono2,axiom,
    ! [S5: set_fm,T9: set_fm] :
      ( ( ord_less_eq_set_fm @ S5 @ T9 )
     => ( ord_le4225877452037527798eam_fm @ ( streams_fm @ S5 ) @ ( streams_fm @ T9 ) ) ) ).

% streams_mono2
thf(fact_1235_streams__mono2,axiom,
    ! [S5: set_rule,T9: set_rule] :
      ( ( ord_less_eq_set_rule @ S5 @ T9 )
     => ( ord_le5832554575816053739m_rule @ ( streams_rule @ S5 ) @ ( streams_rule @ T9 ) ) ) ).

% streams_mono2
thf(fact_1236_streams__mono,axiom,
    ! [S2: stream_fm,A3: set_fm,B3: set_fm] :
      ( ( member_stream_fm @ S2 @ ( streams_fm @ A3 ) )
     => ( ( ord_less_eq_set_fm @ A3 @ B3 )
       => ( member_stream_fm @ S2 @ ( streams_fm @ B3 ) ) ) ) ).

% streams_mono
thf(fact_1237_streams__mono,axiom,
    ! [S2: stream_rule,A3: set_rule,B3: set_rule] :
      ( ( member_stream_rule @ S2 @ ( streams_rule @ A3 ) )
     => ( ( ord_less_eq_set_rule @ A3 @ B3 )
       => ( member_stream_rule @ S2 @ ( streams_rule @ B3 ) ) ) ) ).

% streams_mono
thf(fact_1238_sset__streams,axiom,
    ! [S2: stream727092118206550309m_rule,A3: set_Pr1008144964186165195m_rule] :
      ( ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ S2 ) @ A3 )
     => ( member6117863459850561468m_rule @ S2 @ ( stream1749396743056113565m_rule @ A3 ) ) ) ).

% sset_streams
thf(fact_1239_sset__streams,axiom,
    ! [S2: stream_fm,A3: set_fm] :
      ( ( ord_less_eq_set_fm @ ( sset_fm @ S2 ) @ A3 )
     => ( member_stream_fm @ S2 @ ( streams_fm @ A3 ) ) ) ).

% sset_streams
thf(fact_1240_sset__streams,axiom,
    ! [S2: stream_rule,A3: set_rule] :
      ( ( ord_less_eq_set_rule @ ( sset_rule @ S2 ) @ A3 )
     => ( member_stream_rule @ S2 @ ( streams_rule @ A3 ) ) ) ).

% sset_streams
thf(fact_1241_streams__sset,axiom,
    ! [S2: stream727092118206550309m_rule,A3: set_Pr1008144964186165195m_rule] :
      ( ( member6117863459850561468m_rule @ S2 @ ( stream1749396743056113565m_rule @ A3 ) )
     => ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ S2 ) @ A3 ) ) ).

% streams_sset
thf(fact_1242_streams__sset,axiom,
    ! [S2: stream_fm,A3: set_fm] :
      ( ( member_stream_fm @ S2 @ ( streams_fm @ A3 ) )
     => ( ord_less_eq_set_fm @ ( sset_fm @ S2 ) @ A3 ) ) ).

% streams_sset
thf(fact_1243_streams__sset,axiom,
    ! [S2: stream_rule,A3: set_rule] :
      ( ( member_stream_rule @ S2 @ ( streams_rule @ A3 ) )
     => ( ord_less_eq_set_rule @ ( sset_rule @ S2 ) @ A3 ) ) ).

% streams_sset
thf(fact_1244_streams__iff__sset,axiom,
    ! [S2: stream727092118206550309m_rule,A3: set_Pr1008144964186165195m_rule] :
      ( ( member6117863459850561468m_rule @ S2 @ ( stream1749396743056113565m_rule @ A3 ) )
      = ( ord_le5052251716133888043m_rule @ ( sset_P4979403222752902913m_rule @ S2 ) @ A3 ) ) ).

% streams_iff_sset
thf(fact_1245_streams__iff__sset,axiom,
    ! [S2: stream_fm,A3: set_fm] :
      ( ( member_stream_fm @ S2 @ ( streams_fm @ A3 ) )
      = ( ord_less_eq_set_fm @ ( sset_fm @ S2 ) @ A3 ) ) ).

% streams_iff_sset
thf(fact_1246_streams__iff__sset,axiom,
    ! [S2: stream_rule,A3: set_rule] :
      ( ( member_stream_rule @ S2 @ ( streams_rule @ A3 ) )
      = ( ord_less_eq_set_rule @ ( sset_rule @ S2 ) @ A3 ) ) ).

% streams_iff_sset
thf(fact_1247_in__streams,axiom,
    ! [S2: stream_fm,S5: set_fm] :
      ( ( member_stream_fm @ ( stl_fm @ S2 ) @ ( streams_fm @ S5 ) )
     => ( ( member_fm @ ( shd_fm @ S2 ) @ S5 )
       => ( member_stream_fm @ S2 @ ( streams_fm @ S5 ) ) ) ) ).

% in_streams
thf(fact_1248_in__streams,axiom,
    ! [S2: stream8299795917829157543ist_fm,S5: set_Pr7058068377845519745ist_fm] :
      ( ( member5006235884692811216ist_fm @ ( stl_Pr5027778045650968273ist_fm @ S2 ) @ ( stream8099348070969791641ist_fm @ S5 ) )
     => ( ( member8102475879199740618ist_fm @ ( shd_Pr772355297128350925ist_fm @ S2 ) @ S5 )
       => ( member5006235884692811216ist_fm @ S2 @ ( stream8099348070969791641ist_fm @ S5 ) ) ) ) ).

% in_streams
thf(fact_1249_in__streams,axiom,
    ! [S2: stream727092118206550309m_rule,S5: set_Pr1008144964186165195m_rule] :
      ( ( member6117863459850561468m_rule @ ( stl_Pr950425576149878629m_rule @ S2 ) @ ( stream1749396743056113565m_rule @ S5 ) )
     => ( ( member4220325220686508332m_rule @ ( shd_Pr7235097944458474089m_rule @ S2 ) @ S5 )
       => ( member6117863459850561468m_rule @ S2 @ ( stream1749396743056113565m_rule @ S5 ) ) ) ) ).

% in_streams
thf(fact_1250_in__streams,axiom,
    ! [S2: stream_rule,S5: set_rule] :
      ( ( member_stream_rule @ ( stl_rule @ S2 ) @ ( streams_rule @ S5 ) )
     => ( ( member_rule @ ( shd_rule @ S2 ) @ S5 )
       => ( member_stream_rule @ S2 @ ( streams_rule @ S5 ) ) ) ) ).

% in_streams
thf(fact_1251_streamsE,axiom,
    ! [S2: stream_rule,A3: set_rule] :
      ( ( member_stream_rule @ S2 @ ( streams_rule @ A3 ) )
     => ~ ( ( member_rule @ ( shd_rule @ S2 ) @ A3 )
         => ~ ( member_stream_rule @ ( stl_rule @ S2 ) @ ( streams_rule @ A3 ) ) ) ) ).

% streamsE
thf(fact_1252_NE__R,axiom,
    ( ( sset_rule @ rules )
   != bot_bot_set_rule ) ).

% NE_R
thf(fact_1253_eff_Osimps_I2_J,axiom,
    ! [P2: nat,Ts: list_tm,A3: list_fm,B3: list_fm] :
      ( ( ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A3 ) )
          & ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B3 ) ) )
       => ( ( eff @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
          = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
      & ( ~ ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A3 ) )
            & ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B3 ) ) )
       => ( ( eff @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(2)
thf(fact_1254_pickEff__def,axiom,
    ! [R2: rule,S2: produc1996495991257130529ist_fm] :
      ( ( ( abstra3055540779341473792ist_fm
          @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
              ( ( eff @ R @ S )
              = ( some_f2162114139285431020ist_fm @ Ss ) )
          @ R2
          @ S2 )
       => ( ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S2 )
          = ( fChoic3892542562059901019ist_fm
            @ ^ [Sl: fset_P661503646757059847ist_fm] :
                ( ( eff @ R2 @ S2 )
                = ( some_f2162114139285431020ist_fm @ Sl ) ) ) ) )
      & ( ~ ( abstra3055540779341473792ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S2 )
       => ( ( abstra458490714785879965ist_fm
            @ ^ [R: rule,S: produc1996495991257130529ist_fm,Ss: fset_P661503646757059847ist_fm] :
                ( ( eff @ R @ S )
                = ( some_f2162114139285431020ist_fm @ Ss ) )
            @ R2
            @ S2 )
          = ( the_fs400605564276472219ist_fm @ none_f2701453242279119720ist_fm ) ) ) ) ).

% pickEff_def
thf(fact_1255_eff_Osimps_I1_J,axiom,
    ! [A3: list_fm,B3: list_fm] :
      ( ( eff @ idle @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
      = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ bot_bo2367426573206113139ist_fm ) ) ) ).

% eff.simps(1)
thf(fact_1256_rule_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_tm] :
      ( idle
     != ( axiom @ X21 @ X22 ) ) ).

% rule.distinct(1)
thf(fact_1257_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1258_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1259_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1260_eff_Osimps_I4_J,axiom,
    ! [B3: list_fm,A3: list_fm] :
      ( ( ( member_fm @ falsity @ ( set_fm2 @ B3 ) )
       => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
          = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A3 @ ( removeAll_fm @ falsity @ B3 ) ) @ bot_bo2367426573206113139ist_fm ) ) ) )
      & ( ~ ( member_fm @ falsity @ ( set_fm2 @ B3 ) )
       => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(4)
thf(fact_1261_rule_Odistinct_I17_J,axiom,
    ! [X21: nat,X22: list_tm] :
      ( ( axiom @ X21 @ X22 )
     != flsR ) ).

% rule.distinct(17)
thf(fact_1262_fm_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_tm] :
      ( falsity
     != ( pre @ X21 @ X22 ) ) ).

% fm.distinct(1)
thf(fact_1263_rule_Odistinct_I5_J,axiom,
    idle != flsR ).

% rule.distinct(5)
thf(fact_1264_eff_Osimps_I3_J,axiom,
    ! [A3: list_fm,B3: list_fm] :
      ( ( ( member_fm @ falsity @ ( set_fm2 @ A3 ) )
       => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
          = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
      & ( ~ ( member_fm @ falsity @ ( set_fm2 @ A3 ) )
       => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
          = none_f2701453242279119720ist_fm ) ) ) ).

% eff.simps(3)
thf(fact_1265_eff_Opsimps_I4_J,axiom,
    ! [A3: list_fm,B3: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ flsR @ ( produc381145313068854617ist_fm @ A3 @ B3 ) ) )
     => ( ( ( member_fm @ falsity @ ( set_fm2 @ B3 ) )
         => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
            = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A3 @ ( removeAll_fm @ falsity @ B3 ) ) @ bot_bo2367426573206113139ist_fm ) ) ) )
        & ( ~ ( member_fm @ falsity @ ( set_fm2 @ B3 ) )
         => ( ( eff @ flsR @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(4)
thf(fact_1266_rule_Odistinct_I27_J,axiom,
    flsL != flsR ).

% rule.distinct(27)
thf(fact_1267_rule_Odistinct_I3_J,axiom,
    idle != flsL ).

% rule.distinct(3)
thf(fact_1268_rule_Odistinct_I15_J,axiom,
    ! [X21: nat,X22: list_tm] :
      ( ( axiom @ X21 @ X22 )
     != flsL ) ).

% rule.distinct(15)
thf(fact_1269_eff_Opsimps_I3_J,axiom,
    ! [A3: list_fm,B3: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ flsL @ ( produc381145313068854617ist_fm @ A3 @ B3 ) ) )
     => ( ( ( member_fm @ falsity @ ( set_fm2 @ A3 ) )
         => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
            = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
        & ( ~ ( member_fm @ falsity @ ( set_fm2 @ A3 ) )
         => ( ( eff @ flsL @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(3)
thf(fact_1270_eff_Opsimps_I1_J,axiom,
    ! [A3: list_fm,B3: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ idle @ ( produc381145313068854617ist_fm @ A3 @ B3 ) ) )
     => ( ( eff @ idle @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
        = ( some_f2162114139285431020ist_fm @ ( finser3446675674286072169ist_fm @ ( produc381145313068854617ist_fm @ A3 @ B3 ) @ bot_bo2367426573206113139ist_fm ) ) ) ) ).

% eff.psimps(1)
thf(fact_1271_eff_Opsimps_I2_J,axiom,
    ! [P2: nat,Ts: list_tm,A3: list_fm,B3: list_fm] :
      ( ( accp_P2380921852949180872ist_fm @ eff_rel @ ( produc7063755442767259331ist_fm @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A3 @ B3 ) ) )
     => ( ( ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A3 ) )
            & ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B3 ) ) )
         => ( ( eff @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
            = ( some_f2162114139285431020ist_fm @ bot_bo2367426573206113139ist_fm ) ) )
        & ( ~ ( ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ A3 ) )
              & ( member_fm @ ( pre @ P2 @ Ts ) @ ( set_fm2 @ B3 ) ) )
         => ( ( eff @ ( axiom @ P2 @ Ts ) @ ( produc381145313068854617ist_fm @ A3 @ B3 ) )
            = none_f2701453242279119720ist_fm ) ) ) ) ).

% eff.psimps(2)
thf(fact_1272_treeA__def,axiom,
    ( treeA
    = ( ^ [Steps2: stream727092118206550309m_rule] :
          ( comple7677669044296858969set_fm
          @ ( image_5698427363735666338set_fm
            @ ^ [S: produc164195504107695125m_rule] : ( set_fm2 @ ( produc1501393135466168645ist_fm @ ( produc6879501374131015971m_rule @ S ) ) )
            @ ( sset_P4979403222752902913m_rule @ Steps2 ) ) ) ) ) ).

% treeA_def
thf(fact_1273_treeB__def,axiom,
    ( treeB
    = ( ^ [Steps2: stream727092118206550309m_rule] :
          ( comple7677669044296858969set_fm
          @ ( image_5698427363735666338set_fm
            @ ^ [S: produc164195504107695125m_rule] : ( set_fm2 @ ( produc4588648349897876871ist_fm @ ( produc6879501374131015971m_rule @ S ) ) )
            @ ( sset_P4979403222752902913m_rule @ Steps2 ) ) ) ) ) ).

% treeB_def

% Helper facts (21)
thf(help_fChoice_1_1_fChoice_001t__Syntax__Ofm_T,axiom,
    ! [P2: fm > $o] :
      ( ( P2 @ ( fChoice_fm @ P2 ) )
      = ( ? [X5: fm] : ( P2 @ X5 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Syntax__Orule_T,axiom,
    ! [P2: rule > $o] :
      ( ( P2 @ ( fChoice_rule @ P2 ) )
      = ( ? [X5: rule] : ( P2 @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X4: stream_rule,Y4: stream_rule] :
      ( ( if_stream_rule @ $false @ X4 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Stream__Ostream_It__Syntax__Orule_J_T,axiom,
    ! [X4: stream_rule,Y4: stream_rule] :
      ( ( if_stream_rule @ $true @ X4 @ Y4 )
      = X4 ) ).

thf(help_fChoice_1_1_fChoice_001t__Abstract____Completeness__Otree_It__Syntax__Orule_J_T,axiom,
    ! [P2: abstract_tree_rule > $o] :
      ( ( P2 @ ( fChoic8596484284512669613e_rule @ P2 ) )
      = ( ? [X5: abstract_tree_rule] : ( P2 @ X5 ) ) ) ).

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,
    ! [X4: produc1996495991257130529ist_fm,Y4: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $false @ X4 @ 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,
    ! [X4: produc1996495991257130529ist_fm,Y4: produc1996495991257130529ist_fm] :
      ( ( if_Pro3930376587665744871ist_fm @ $true @ X4 @ Y4 )
      = X4 ) ).

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,
    ! [P2: produc1996495991257130529ist_fm > $o] :
      ( ( P2 @ ( fChoic1980167584695909237ist_fm @ P2 ) )
      = ( ? [X5: produc1996495991257130529ist_fm] : ( P2 @ X5 ) ) ) ).

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,
    ! [X4: fset_P661503646757059847ist_fm,Y4: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $false @ X4 @ 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,
    ! [X4: fset_P661503646757059847ist_fm,Y4: fset_P661503646757059847ist_fm] :
      ( ( if_fse3714977293194272717ist_fm @ $true @ X4 @ Y4 )
      = X4 ) ).

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,
    ! [P2: fset_P661503646757059847ist_fm > $o] :
      ( ( P2 @ ( fChoic3892542562059901019ist_fm @ P2 ) )
      = ( ? [X5: fset_P661503646757059847ist_fm] : ( P2 @ X5 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Ofm_J_Mt__List__Olist_It__Syntax__Ofm_J_J_Mt__Syntax__Orule_J_T,axiom,
    ! [P2: produc164195504107695125m_rule > $o] :
      ( ( P2 @ ( fChoic9196029409387999681m_rule @ P2 ) )
      = ( ? [X5: produc164195504107695125m_rule] : ( P2 @ X5 ) ) ) ).

thf(help_fChoice_1_1_fChoice_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_T,axiom,
    ! [P2: produc4630866025709511057ist_fm > $o] :
      ( ( P2 @ ( fChoic4439327894135039805ist_fm @ P2 ) )
      = ( ? [X5: produc4630866025709511057ist_fm] : ( P2 @ X5 ) ) ) ).

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,
    ! [X4: stream727092118206550309m_rule,Y4: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $false @ X4 @ 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,
    ! [X4: stream727092118206550309m_rule,Y4: stream727092118206550309m_rule] :
      ( ( if_str8948254419368749791m_rule @ $true @ X4 @ Y4 )
      = 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,
    ! [X4: abstra4572470701423150929m_rule,Y4: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $false @ X4 @ 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,
    ! [X4: abstra4572470701423150929m_rule,Y4: abstra4572470701423150929m_rule] :
      ( ( if_Abs4566652138351786251m_rule @ $true @ X4 @ Y4 )
      = X4 ) ).

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,
    ! [P2: abstra4572470701423150929m_rule > $o] :
      ( ( P2 @ ( fChoic7543385803022166781m_rule @ P2 ) )
      = ( ? [X5: abstra4572470701423150929m_rule] : ( P2 @ X5 ) ) ) ).

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,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $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,
    ! [X4: fset_A5396818216090053761m_rule,Y4: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $false @ X4 @ 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,
    ! [X4: fset_A5396818216090053761m_rule,Y4: fset_A5396818216090053761m_rule] :
      ( ( if_fse3370020817956878651m_rule @ $true @ X4 @ Y4 )
      = X4 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [J2: nat] :
      ( ( ( produc7165828336582415457m_rule @ ( shd_Pr7235097944458474089m_rule @ ( sdrop_7224736112439592940m_rule @ J2 @ steps ) ) )
        = ( axiom @ p @ ts ) )
     => thesis ) ).

thf(conj_1,conjecture,
    thesis ).

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