%------------------------------------------------------------------------------
% 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    : FSM_Tests/0036_FSM/prob_02427_095311__19301704_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2363 ( 570 unt;1094 typ;   0 def)
%            Number of atoms       : 4160 (2226 equ;   0 cnn)
%            Maximal formula atoms :   15 (   3 avg)
%            Number of connectives : 17239 ( 872   ~;  28   |; 449   &;14114   @)
%                                         (   0 <=>;1776  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   7 avg)
%            Number of types       :  237 ( 236 usr)
%            Number of type conns  : 2361 (2361   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  861 ( 858 usr;  48 con; 0-5 aty)
%            Number of variables   : 4960 ( 557   ^;4087   !; 316   ?;4960   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:27:44.918
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thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    member4558601409277499449od_c_a: produc6253095604734789392od_c_a > set_Pr6352661749802654064od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    member8587115632268026449od_c_a: produc5283743357988929576od_c_a > set_Pr3884473935879652232od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    member5750183862411248520od_c_a: produc871828282075189343od_c_a > set_Pr5816981679309595967od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__a_J_J_J,type,
    member4187420457662959969od_b_a: produc1355312390766300472od_b_a > set_Pr172305270015935128od_b_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    member5502040268208861279od_c_a: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    member5418238279316946185od_c_a: produc7112732474774236128od_c_a > set_Pr1212000722004284992od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    member4343506560499364960od_c_a: produc1040134286220268087od_c_a > set_Pr4299858919719910935od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    member4823365305010683256od_c_a: produc9168381761529399887od_c_a > set_Pr5338671032280311599od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J,type,
    member1160383752307813046od_c_a: produc2133359284458116493od_c_a > set_Pr272002107461315949od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    member3093337089038594223od_c_a: produc9013336851614273158od_c_a > set_Pr4854443895542973158od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    member558622287379480070od_c_a: produc4903638743898196701od_c_a > set_Pr1963666890401038525od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    member7958828110252711198od_c_a: produc5126720043356051701od_c_a > set_Pr6912582227606077397od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    member6622777490300016861od_c_a: produc3790669423403357364od_c_a > set_Pr6327249630596369428od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mtf__a_J,type,
    member5074992350434858958od_c_a: product_prod_c_a > set_Product_prod_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mtf__c_J,type,
    member5074992359041316560od_c_c: product_prod_c_c > set_Product_prod_c_c > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_c_member_001tf__c,type,
    member_c: c > set_c > $o ).

thf(sy_v_M,type,
    m: fsm_a_b_c ).

thf(sy_v_p____,type,
    p: list_P6327159017948738492od_c_a ).

thf(sy_v_q,type,
    q: a ).

thf(sy_v_x____,type,
    x: b ).

thf(sy_v_y____,type,
    y: c ).

% Relevant facts (1259)
thf(fact_0__092_060open_062path_AM_A_IFSM_Oinitial_AM_J_Ap_092_060close_062,axiom,
    path_a_b_c @ m @ ( initial_a_b_c @ m ) @ p ).

% \<open>path M (FSM.initial M) p\<close>
thf(fact_1_assms_I1_J,axiom,
    acyclic_a_b_c @ m ).

% assms(1)
thf(fact_2_assms_I2_J,axiom,
    member_a @ q @ ( reacha1620305530751930115_a_b_c @ m ) ).

% assms(2)
thf(fact_3_path__prefix,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P1: list_P1562850763667305982od_c_a,P2: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ M @ Q @ ( append311360319448987923od_c_a @ P1 @ P2 ) )
     => ( path_P1107610407223809260_a_a_b @ M @ Q @ P1 ) ) ).

% path_prefix
thf(fact_4_path__prefix,axiom,
    ! [M: fsm_a_c_a,Q: a,P1: list_P6647267739079350397od_a_a,P2: list_P6647267739079350397od_a_a] :
      ( ( path_a_c_a @ M @ Q @ ( append6470145159784622866od_a_a @ P1 @ P2 ) )
     => ( path_a_c_a @ M @ Q @ P1 ) ) ).

% path_prefix
thf(fact_5_path__prefix,axiom,
    ! [M: fsm_a_4970532336856918856od_c_a,Q: a,P1: list_P3747170188271196375_c_a_a,P2: list_P3747170188271196375_c_a_a] :
      ( ( path_a4972803899755839402od_c_a @ M @ Q @ ( append461839767961090156_c_a_a @ P1 @ P2 ) )
     => ( path_a4972803899755839402od_c_a @ M @ Q @ P1 ) ) ).

% path_prefix
thf(fact_6_path__prefix,axiom,
    ! [M: fsm_a_1173558889237794545od_c_a,Q: a,P1: list_P8924682930347365808_c_a_a,P2: list_P8924682930347365808_c_a_a] :
      ( ( path_a4575340523007214163od_c_a @ M @ Q @ ( append3109463570867311045_c_a_a @ P1 @ P2 ) )
     => ( path_a4575340523007214163od_c_a @ M @ Q @ P1 ) ) ).

% path_prefix
thf(fact_7_path__prefix,axiom,
    ! [M: fsm_a_b_c,Q: a,P1: list_P6327159017948738492od_c_a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P1 @ P2 ) )
     => ( path_a_b_c @ M @ Q @ P1 ) ) ).

% path_prefix
thf(fact_8__092_060open_062path_AM_A_IFSM_Oinitial_AM_J_A_Ip_A_064_A_091_Iq_M_Ax_M_Ay_M_Aq_J_093_J_092_060close_062,axiom,
    path_a_b_c @ m @ ( initial_a_b_c @ m ) @ ( append3884164603879732689od_c_a @ p @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ q @ ( produc3440865914000881661od_c_a @ x @ ( product_Pair_c_a @ y @ q ) ) ) @ nil_Pr1342775757158464060od_c_a ) ) ).

% \<open>path M (FSM.initial M) (p @ [(q, x, y, q)])\<close>
thf(fact_9__092_060open_062_092_060not_062_Adistinct_A_Ivisited__states_A_IFSM_Oinitial_AM_J_A_Ip_A_064_A_091_Iq_M_Ax_M_Ay_M_Aq_J_093_J_J_092_060close_062,axiom,
    ~ ( distinct_a @ ( visited_states_a_b_c @ ( initial_a_b_c @ m ) @ ( append3884164603879732689od_c_a @ p @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ q @ ( produc3440865914000881661od_c_a @ x @ ( product_Pair_c_a @ y @ q ) ) ) @ nil_Pr1342775757158464060od_c_a ) ) ) ) ).

% \<open>\<not> distinct (visited_states (FSM.initial M) (p @ [(q, x, y, q)]))\<close>
thf(fact_10__092_060open_062target_A_IFSM_Oinitial_AM_J_Ap_A_061_Aq_092_060close_062,axiom,
    ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ p )
    = q ) ).

% \<open>target (FSM.initial M) p = q\<close>
thf(fact_11__092_060open_062_092_060exists_062x_Ay_O_A_Iq_M_Ax_M_Ay_M_Aq_J_A_092_060in_062_AFSM_Otransitions_AM_092_060close_062,axiom,
    ? [X: b,Y: c] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ q @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ q ) ) ) @ ( transitions_a_b_c @ m ) ) ).

% \<open>\<exists>x y. (q, x, y, q) \<in> FSM.transitions M\<close>
thf(fact_12_reachable__states__initial,axiom,
    ! [M: fsm_Pr6425204305901329755_a_b_c] : ( member5502040268208861279od_c_a @ ( initia3341036835946388646_a_b_c @ M ) @ ( reacha1389639598965321787_a_b_c @ M ) ) ).

% reachable_states_initial
thf(fact_13_reachable__states__initial,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b] : ( member5074992350434858958od_c_a @ ( initia681974817753967189_a_a_b @ M ) @ ( reacha5035603111973943274_a_a_b @ M ) ) ).

% reachable_states_initial
thf(fact_14_reachable__states__initial,axiom,
    ! [M: fsm_Pr9212659883570776987_a_a_b] : ( member5502040268208861279od_c_a @ ( initia6128492417919064678_a_a_b @ M ) @ ( reacha4177095180937997819_a_a_b @ M ) ) ).

% reachable_states_initial
thf(fact_15_reachable__states__initial,axiom,
    ! [M: fsm_a_a_b] : ( member_a @ ( initial_a_a_b @ M ) @ ( reacha4407761112724606147_a_a_b @ M ) ) ).

% reachable_states_initial
thf(fact_16_reachable__states__initial,axiom,
    ! [M: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ).

% reachable_states_initial
thf(fact_17__092_060open_062_Iq_M_Ax_M_Ay_M_Aq_J_A_092_060in_062_AFSM_Otransitions_AM_092_060close_062,axiom,
    member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ q @ ( produc3440865914000881661od_c_a @ x @ ( product_Pair_c_a @ y @ q ) ) ) @ ( transitions_a_b_c @ m ) ).

% \<open>(q, x, y, q) \<in> FSM.transitions M\<close>
thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_O_A_092_060lbrakk_062path_AM_A_IFSM_Oinitial_AM_J_Ap_059_Atarget_A_IFSM_Oinitial_AM_J_Ap_A_061_Aq_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [P: list_P6327159017948738492od_c_a] :
        ( ( path_a_b_c @ m @ ( initial_a_b_c @ m ) @ P )
       => ( ( target_a_b_c @ ( initial_a_b_c @ m ) @ P )
         != q ) ) ).

% \<open>\<And>thesis. (\<And>p. \<lbrakk>path M (FSM.initial M) p; target (FSM.initial M) p = q\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_19__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Ay_O_A_Iq_M_Ax_M_Ay_M_Aq_J_A_092_060in_062_AFSM_Otransitions_AM_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [X: b,Y: c] :
        ~ ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ q @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ q ) ) ) @ ( transitions_a_b_c @ m ) ) ).

% \<open>\<And>thesis. (\<And>x y. (q, x, y, q) \<in> FSM.transitions M \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_20_append_Oassoc,axiom,
    ! [A: list_P125642481956313003od_c_a,B: list_P125642481956313003od_c_a,C: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ ( append8983669691956257088od_c_a @ A @ B ) @ C )
      = ( append8983669691956257088od_c_a @ A @ ( append8983669691956257088od_c_a @ B @ C ) ) ) ).

% append.assoc
thf(fact_21_append_Oassoc,axiom,
    ! [A: list_P5767727266746070163od_c_a,B: list_P5767727266746070163od_c_a,C: list_P5767727266746070163od_c_a] :
      ( ( append2814445079200870952od_c_a @ ( append2814445079200870952od_c_a @ A @ B ) @ C )
      = ( append2814445079200870952od_c_a @ A @ ( append2814445079200870952od_c_a @ B @ C ) ) ) ).

% append.assoc
thf(fact_22_append_Oassoc,axiom,
    ! [A: list_P903359562653991662od_b_c,B: list_P903359562653991662od_b_c,C: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ ( append2547753245680614915od_b_c @ A @ B ) @ C )
      = ( append2547753245680614915od_b_c @ A @ ( append2547753245680614915od_b_c @ B @ C ) ) ) ).

% append.assoc
thf(fact_23_append_Oassoc,axiom,
    ! [A: list_l2826717958560394946od_c_a,B: list_l2826717958560394946od_c_a,C: list_l2826717958560394946od_c_a] :
      ( ( append2943488786620085207od_c_a @ ( append2943488786620085207od_c_a @ A @ B ) @ C )
      = ( append2943488786620085207od_c_a @ A @ ( append2943488786620085207od_c_a @ B @ C ) ) ) ).

% append.assoc
thf(fact_24_append_Oassoc,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( append_a @ ( append_a @ A @ B ) @ C )
      = ( append_a @ A @ ( append_a @ B @ C ) ) ) ).

% append.assoc
thf(fact_25_append_Oassoc,axiom,
    ! [A: list_P6327159017948738492od_c_a,B: list_P6327159017948738492od_c_a,C: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ ( append3884164603879732689od_c_a @ A @ B ) @ C )
      = ( append3884164603879732689od_c_a @ A @ ( append3884164603879732689od_c_a @ B @ C ) ) ) ).

% append.assoc
thf(fact_26_append__assoc,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ ( append8983669691956257088od_c_a @ Xs @ Ys ) @ Zs )
      = ( append8983669691956257088od_c_a @ Xs @ ( append8983669691956257088od_c_a @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_27_append__assoc,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,Ys: list_P5767727266746070163od_c_a,Zs: list_P5767727266746070163od_c_a] :
      ( ( append2814445079200870952od_c_a @ ( append2814445079200870952od_c_a @ Xs @ Ys ) @ Zs )
      = ( append2814445079200870952od_c_a @ Xs @ ( append2814445079200870952od_c_a @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_28_append__assoc,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ ( append2547753245680614915od_b_c @ Xs @ Ys ) @ Zs )
      = ( append2547753245680614915od_b_c @ Xs @ ( append2547753245680614915od_b_c @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_29_append__assoc,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a,Zs: list_l2826717958560394946od_c_a] :
      ( ( append2943488786620085207od_c_a @ ( append2943488786620085207od_c_a @ Xs @ Ys ) @ Zs )
      = ( append2943488786620085207od_c_a @ Xs @ ( append2943488786620085207od_c_a @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_30_append__assoc,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
      = ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_31_append__assoc,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) @ Zs )
      = ( append3884164603879732689od_c_a @ Xs @ ( append3884164603879732689od_c_a @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_32_append__same__eq,axiom,
    ! [Ys: list_P125642481956313003od_c_a,Xs: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Ys @ Xs )
        = ( append8983669691956257088od_c_a @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_33_append__same__eq,axiom,
    ! [Ys: list_P5767727266746070163od_c_a,Xs: list_P5767727266746070163od_c_a,Zs: list_P5767727266746070163od_c_a] :
      ( ( ( append2814445079200870952od_c_a @ Ys @ Xs )
        = ( append2814445079200870952od_c_a @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_34_append__same__eq,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Ys @ Xs )
        = ( append2547753245680614915od_b_c @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_35_append__same__eq,axiom,
    ! [Ys: list_l2826717958560394946od_c_a,Xs: list_l2826717958560394946od_c_a,Zs: list_l2826717958560394946od_c_a] :
      ( ( ( append2943488786620085207od_c_a @ Ys @ Xs )
        = ( append2943488786620085207od_c_a @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_36_append__same__eq,axiom,
    ! [Ys: list_a,Xs: list_a,Zs: list_a] :
      ( ( ( append_a @ Ys @ Xs )
        = ( append_a @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_37_append__same__eq,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Ys @ Xs )
        = ( append3884164603879732689od_c_a @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_38_same__append__eq,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys: list_P125642481956313003od_c_a,Zs: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ Ys )
        = ( append8983669691956257088od_c_a @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_39_same__append__eq,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,Ys: list_P5767727266746070163od_c_a,Zs: list_P5767727266746070163od_c_a] :
      ( ( ( append2814445079200870952od_c_a @ Xs @ Ys )
        = ( append2814445079200870952od_c_a @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_40_same__append__eq,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = ( append2547753245680614915od_b_c @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_41_same__append__eq,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a,Zs: list_l2826717958560394946od_c_a] :
      ( ( ( append2943488786620085207od_c_a @ Xs @ Ys )
        = ( append2943488786620085207od_c_a @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_42_same__append__eq,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = ( append_a @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_43_same__append__eq,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Xs @ Ys )
        = ( append3884164603879732689od_c_a @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_44_prod_Oinject,axiom,
    ! [X1: a,X2: list_b,Y1: a,Y2: list_b] :
      ( ( ( produc6670463076781050526list_b @ X1 @ X2 )
        = ( produc6670463076781050526list_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_45_prod_Oinject,axiom,
    ! [X1: a,X2: set_list_b,Y1: a,Y2: set_list_b] :
      ( ( ( produc9038752348217253374list_b @ X1 @ X2 )
        = ( produc9038752348217253374list_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_46_prod_Oinject,axiom,
    ! [X1: a,X2: nat,Y1: a,Y2: nat] :
      ( ( ( product_Pair_a_nat @ X1 @ X2 )
        = ( product_Pair_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_47_prod_Oinject,axiom,
    ! [X1: a,X2: b,Y1: a,Y2: b] :
      ( ( ( product_Pair_a_b @ X1 @ X2 )
        = ( product_Pair_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_48_prod_Oinject,axiom,
    ! [X1: a,X2: a,Y1: a,Y2: a] :
      ( ( ( product_Pair_a_a @ X1 @ X2 )
        = ( product_Pair_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_49_prod_Oinject,axiom,
    ! [X1: a,X2: produc2133359284458116493od_c_a,Y1: a,Y2: produc2133359284458116493od_c_a] :
      ( ( ( produc5762801557034676134od_c_a @ X1 @ X2 )
        = ( produc5762801557034676134od_c_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_50_prod_Oinject,axiom,
    ! [X1: b,X2: product_prod_c_a,Y1: b,Y2: product_prod_c_a] :
      ( ( ( produc3440865914000881661od_c_a @ X1 @ X2 )
        = ( produc3440865914000881661od_c_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_51_prod_Oinject,axiom,
    ! [X1: c,X2: a,Y1: c,Y2: a] :
      ( ( ( product_Pair_c_a @ X1 @ X2 )
        = ( product_Pair_c_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_52_old_Oprod_Oinject,axiom,
    ! [A: a,B: list_b,A2: a,B2: list_b] :
      ( ( ( produc6670463076781050526list_b @ A @ B )
        = ( produc6670463076781050526list_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_53_old_Oprod_Oinject,axiom,
    ! [A: a,B: set_list_b,A2: a,B2: set_list_b] :
      ( ( ( produc9038752348217253374list_b @ A @ B )
        = ( produc9038752348217253374list_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_54_old_Oprod_Oinject,axiom,
    ! [A: a,B: nat,A2: a,B2: nat] :
      ( ( ( product_Pair_a_nat @ A @ B )
        = ( product_Pair_a_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_55_old_Oprod_Oinject,axiom,
    ! [A: a,B: b,A2: a,B2: b] :
      ( ( ( product_Pair_a_b @ A @ B )
        = ( product_Pair_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_56_old_Oprod_Oinject,axiom,
    ! [A: a,B: a,A2: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_57_old_Oprod_Oinject,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A2: a,B2: produc2133359284458116493od_c_a] :
      ( ( ( produc5762801557034676134od_c_a @ A @ B )
        = ( produc5762801557034676134od_c_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_58_old_Oprod_Oinject,axiom,
    ! [A: b,B: product_prod_c_a,A2: b,B2: product_prod_c_a] :
      ( ( ( produc3440865914000881661od_c_a @ A @ B )
        = ( produc3440865914000881661od_c_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_59_old_Oprod_Oinject,axiom,
    ! [A: c,B: a,A2: c,B2: a] :
      ( ( ( product_Pair_c_a @ A @ B )
        = ( product_Pair_c_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_60_acyclic_Oelims_I3_J,axiom,
    ! [X3: fsm_Pr6425204305901329755_a_b_c] :
      ( ~ ( acycli6127326990137245968_a_b_c @ X3 )
     => ~ ! [P: list_P7591818816154907580od_c_a] :
            ( ( path_P6052804182842616125_a_b_c @ X3 @ ( initia3341036835946388646_a_b_c @ X3 ) @ P )
           => ( distin811446047591323055od_c_a @ ( visite2969974120305769310_a_b_c @ ( initia3341036835946388646_a_b_c @ X3 ) @ P ) ) ) ) ).

% acyclic.elims(3)
thf(fact_61_acyclic_Oelims_I3_J,axiom,
    ! [X3: fsm_Pr9212659883570776987_a_a_b] :
      ( ~ ( acycli8914782572109922000_a_a_b @ X3 )
     => ~ ! [P: list_P1501825136581950910od_c_a] :
            ( ( path_P8840259764815292157_a_a_b @ X3 @ ( initia6128492417919064678_a_a_b @ X3 ) @ P )
           => ( distin811446047591323055od_c_a @ ( visite5757429702278445342_a_a_b @ ( initia6128492417919064678_a_a_b @ X3 ) @ P ) ) ) ) ).

% acyclic.elims(3)
thf(fact_62_acyclic_Oelims_I3_J,axiom,
    ! [X3: fsm_a_a_b] :
      ( ~ ( acyclic_a_a_b @ X3 )
     => ~ ! [P: list_P7365212274028606398od_b_a] :
            ( ( path_a_a_b @ X3 @ ( initial_a_a_b @ X3 ) @ P )
           => ( distinct_a @ ( visited_states_a_a_b @ ( initial_a_a_b @ X3 ) @ P ) ) ) ) ).

% acyclic.elims(3)
thf(fact_63_acyclic_Oelims_I3_J,axiom,
    ! [X3: fsm_a_c_a] :
      ( ~ ( acyclic_a_c_a @ X3 )
     => ~ ! [P: list_P6647267739079350397od_a_a] :
            ( ( path_a_c_a @ X3 @ ( initial_a_c_a @ X3 ) @ P )
           => ( distinct_a @ ( visited_states_a_c_a @ ( initial_a_c_a @ X3 ) @ P ) ) ) ) ).

% acyclic.elims(3)
thf(fact_64_acyclic_Oelims_I3_J,axiom,
    ! [X3: fsm_a_4970532336856918856od_c_a] :
      ( ~ ( acycli5244099224745504125od_c_a @ X3 )
     => ~ ! [P: list_P3747170188271196375_c_a_a] :
            ( ( path_a4972803899755839402od_c_a @ X3 @ ( initia4547168310285997331od_c_a @ X3 ) @ P )
           => ( distinct_a @ ( visite1724772098679956427od_c_a @ ( initia4547168310285997331od_c_a @ X3 ) @ P ) ) ) ) ).

% acyclic.elims(3)
thf(fact_65_acyclic_Oelims_I3_J,axiom,
    ! [X3: fsm_a_1173558889237794545od_c_a] :
      ( ~ ( acycli1429210738163115302od_c_a @ X3 )
     => ~ ! [P: list_P8924682930347365808_c_a_a] :
            ( ( path_a4575340523007214163od_c_a @ X3 @ ( initia8238001472959153852od_c_a @ X3 ) @ P )
           => ( distinct_a @ ( visite4103323964629638516od_c_a @ ( initia8238001472959153852od_c_a @ X3 ) @ P ) ) ) ) ).

% acyclic.elims(3)
thf(fact_66_acyclic_Oelims_I3_J,axiom,
    ! [X3: fsm_Pr1886043760914421514_a_a_b] :
      ( ~ ( acycli1378905732213473983_a_a_b @ X3 )
     => ~ ! [P: list_P1562850763667305982od_c_a] :
            ( ( path_P1107610407223809260_a_a_b @ X3 @ ( initia681974817753967189_a_a_b @ X3 ) @ P )
           => ( distin3780794742951484702od_c_a @ ( visite7082950643002702093_a_a_b @ ( initia681974817753967189_a_a_b @ X3 ) @ P ) ) ) ) ).

% acyclic.elims(3)
thf(fact_67_acyclic_Oelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( acyclic_a_b_c @ X3 )
     => ~ ! [P: list_P6327159017948738492od_c_a] :
            ( ( path_a_b_c @ X3 @ ( initial_a_b_c @ X3 ) @ P )
           => ( distinct_a @ ( visited_states_a_b_c @ ( initial_a_b_c @ X3 ) @ P ) ) ) ) ).

% acyclic.elims(3)
thf(fact_68_acyclic_Oelims_I2_J,axiom,
    ! [X3: fsm_Pr6425204305901329755_a_b_c] :
      ( ( acycli6127326990137245968_a_b_c @ X3 )
     => ! [P3: list_P7591818816154907580od_c_a] :
          ( ( path_P6052804182842616125_a_b_c @ X3 @ ( initia3341036835946388646_a_b_c @ X3 ) @ P3 )
         => ( distin811446047591323055od_c_a @ ( visite2969974120305769310_a_b_c @ ( initia3341036835946388646_a_b_c @ X3 ) @ P3 ) ) ) ) ).

% acyclic.elims(2)
thf(fact_69_acyclic_Oelims_I2_J,axiom,
    ! [X3: fsm_Pr9212659883570776987_a_a_b] :
      ( ( acycli8914782572109922000_a_a_b @ X3 )
     => ! [P3: list_P1501825136581950910od_c_a] :
          ( ( path_P8840259764815292157_a_a_b @ X3 @ ( initia6128492417919064678_a_a_b @ X3 ) @ P3 )
         => ( distin811446047591323055od_c_a @ ( visite5757429702278445342_a_a_b @ ( initia6128492417919064678_a_a_b @ X3 ) @ P3 ) ) ) ) ).

% acyclic.elims(2)
thf(fact_70_acyclic_Oelims_I2_J,axiom,
    ! [X3: fsm_a_a_b] :
      ( ( acyclic_a_a_b @ X3 )
     => ! [P3: list_P7365212274028606398od_b_a] :
          ( ( path_a_a_b @ X3 @ ( initial_a_a_b @ X3 ) @ P3 )
         => ( distinct_a @ ( visited_states_a_a_b @ ( initial_a_a_b @ X3 ) @ P3 ) ) ) ) ).

% acyclic.elims(2)
thf(fact_71_acyclic_Oelims_I2_J,axiom,
    ! [X3: fsm_a_c_a] :
      ( ( acyclic_a_c_a @ X3 )
     => ! [P3: list_P6647267739079350397od_a_a] :
          ( ( path_a_c_a @ X3 @ ( initial_a_c_a @ X3 ) @ P3 )
         => ( distinct_a @ ( visited_states_a_c_a @ ( initial_a_c_a @ X3 ) @ P3 ) ) ) ) ).

% acyclic.elims(2)
thf(fact_72_acyclic_Oelims_I2_J,axiom,
    ! [X3: fsm_a_4970532336856918856od_c_a] :
      ( ( acycli5244099224745504125od_c_a @ X3 )
     => ! [P3: list_P3747170188271196375_c_a_a] :
          ( ( path_a4972803899755839402od_c_a @ X3 @ ( initia4547168310285997331od_c_a @ X3 ) @ P3 )
         => ( distinct_a @ ( visite1724772098679956427od_c_a @ ( initia4547168310285997331od_c_a @ X3 ) @ P3 ) ) ) ) ).

% acyclic.elims(2)
thf(fact_73_acyclic_Oelims_I2_J,axiom,
    ! [X3: fsm_a_1173558889237794545od_c_a] :
      ( ( acycli1429210738163115302od_c_a @ X3 )
     => ! [P3: list_P8924682930347365808_c_a_a] :
          ( ( path_a4575340523007214163od_c_a @ X3 @ ( initia8238001472959153852od_c_a @ X3 ) @ P3 )
         => ( distinct_a @ ( visite4103323964629638516od_c_a @ ( initia8238001472959153852od_c_a @ X3 ) @ P3 ) ) ) ) ).

% acyclic.elims(2)
thf(fact_74_acyclic_Oelims_I2_J,axiom,
    ! [X3: fsm_Pr1886043760914421514_a_a_b] :
      ( ( acycli1378905732213473983_a_a_b @ X3 )
     => ! [P3: list_P1562850763667305982od_c_a] :
          ( ( path_P1107610407223809260_a_a_b @ X3 @ ( initia681974817753967189_a_a_b @ X3 ) @ P3 )
         => ( distin3780794742951484702od_c_a @ ( visite7082950643002702093_a_a_b @ ( initia681974817753967189_a_a_b @ X3 ) @ P3 ) ) ) ) ).

% acyclic.elims(2)
thf(fact_75_acyclic_Oelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( acyclic_a_b_c @ X3 )
     => ! [P3: list_P6327159017948738492od_c_a] :
          ( ( path_a_b_c @ X3 @ ( initial_a_b_c @ X3 ) @ P3 )
         => ( distinct_a @ ( visited_states_a_b_c @ ( initial_a_b_c @ X3 ) @ P3 ) ) ) ) ).

% acyclic.elims(2)
thf(fact_76_acyclic_Oelims_I1_J,axiom,
    ! [X3: fsm_Pr6425204305901329755_a_b_c,Y3: $o] :
      ( ( ( acycli6127326990137245968_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [P4: list_P7591818816154907580od_c_a] :
              ( ( path_P6052804182842616125_a_b_c @ X3 @ ( initia3341036835946388646_a_b_c @ X3 ) @ P4 )
             => ( distin811446047591323055od_c_a @ ( visite2969974120305769310_a_b_c @ ( initia3341036835946388646_a_b_c @ X3 ) @ P4 ) ) ) ) ) ) ).

% acyclic.elims(1)
thf(fact_77_acyclic_Oelims_I1_J,axiom,
    ! [X3: fsm_Pr9212659883570776987_a_a_b,Y3: $o] :
      ( ( ( acycli8914782572109922000_a_a_b @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [P4: list_P1501825136581950910od_c_a] :
              ( ( path_P8840259764815292157_a_a_b @ X3 @ ( initia6128492417919064678_a_a_b @ X3 ) @ P4 )
             => ( distin811446047591323055od_c_a @ ( visite5757429702278445342_a_a_b @ ( initia6128492417919064678_a_a_b @ X3 ) @ P4 ) ) ) ) ) ) ).

% acyclic.elims(1)
thf(fact_78_acyclic_Oelims_I1_J,axiom,
    ! [X3: fsm_a_a_b,Y3: $o] :
      ( ( ( acyclic_a_a_b @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [P4: list_P7365212274028606398od_b_a] :
              ( ( path_a_a_b @ X3 @ ( initial_a_a_b @ X3 ) @ P4 )
             => ( distinct_a @ ( visited_states_a_a_b @ ( initial_a_a_b @ X3 ) @ P4 ) ) ) ) ) ) ).

% acyclic.elims(1)
thf(fact_79_acyclic_Oelims_I1_J,axiom,
    ! [X3: fsm_a_c_a,Y3: $o] :
      ( ( ( acyclic_a_c_a @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [P4: list_P6647267739079350397od_a_a] :
              ( ( path_a_c_a @ X3 @ ( initial_a_c_a @ X3 ) @ P4 )
             => ( distinct_a @ ( visited_states_a_c_a @ ( initial_a_c_a @ X3 ) @ P4 ) ) ) ) ) ) ).

% acyclic.elims(1)
thf(fact_80_acyclic_Oelims_I1_J,axiom,
    ! [X3: fsm_a_4970532336856918856od_c_a,Y3: $o] :
      ( ( ( acycli5244099224745504125od_c_a @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [P4: list_P3747170188271196375_c_a_a] :
              ( ( path_a4972803899755839402od_c_a @ X3 @ ( initia4547168310285997331od_c_a @ X3 ) @ P4 )
             => ( distinct_a @ ( visite1724772098679956427od_c_a @ ( initia4547168310285997331od_c_a @ X3 ) @ P4 ) ) ) ) ) ) ).

% acyclic.elims(1)
thf(fact_81_acyclic_Oelims_I1_J,axiom,
    ! [X3: fsm_a_1173558889237794545od_c_a,Y3: $o] :
      ( ( ( acycli1429210738163115302od_c_a @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [P4: list_P8924682930347365808_c_a_a] :
              ( ( path_a4575340523007214163od_c_a @ X3 @ ( initia8238001472959153852od_c_a @ X3 ) @ P4 )
             => ( distinct_a @ ( visite4103323964629638516od_c_a @ ( initia8238001472959153852od_c_a @ X3 ) @ P4 ) ) ) ) ) ) ).

% acyclic.elims(1)
thf(fact_82_acyclic_Oelims_I1_J,axiom,
    ! [X3: fsm_Pr1886043760914421514_a_a_b,Y3: $o] :
      ( ( ( acycli1378905732213473983_a_a_b @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [P4: list_P1562850763667305982od_c_a] :
              ( ( path_P1107610407223809260_a_a_b @ X3 @ ( initia681974817753967189_a_a_b @ X3 ) @ P4 )
             => ( distin3780794742951484702od_c_a @ ( visite7082950643002702093_a_a_b @ ( initia681974817753967189_a_a_b @ X3 ) @ P4 ) ) ) ) ) ) ).

% acyclic.elims(1)
thf(fact_83_acyclic_Oelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( acyclic_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [P4: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ X3 @ ( initial_a_b_c @ X3 ) @ P4 )
             => ( distinct_a @ ( visited_states_a_b_c @ ( initial_a_b_c @ X3 ) @ P4 ) ) ) ) ) ) ).

% acyclic.elims(1)
thf(fact_84_list_Oinject,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c,Y21: product_prod_b_c,Y22: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X21 @ X22 )
        = ( cons_P4529483553340347422od_b_c @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_85_list_Oinject,axiom,
    ! [X21: list_P6327159017948738492od_c_a,X22: list_l2826717958560394946od_c_a,Y21: list_P6327159017948738492od_c_a,Y22: list_l2826717958560394946od_c_a] :
      ( ( ( cons_l7906879460341455474od_c_a @ X21 @ X22 )
        = ( cons_l7906879460341455474od_c_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_86_list_Oinject,axiom,
    ! [X21: c,X22: list_c,Y21: c,Y22: list_c] :
      ( ( ( cons_c @ X21 @ X22 )
        = ( cons_c @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_87_list_Oinject,axiom,
    ! [X21: b,X22: list_b,Y21: b,Y22: list_b] :
      ( ( ( cons_b @ X21 @ X22 )
        = ( cons_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_88_list_Oinject,axiom,
    ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
      ( ( ( cons_a @ X21 @ X22 )
        = ( cons_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_89_list_Oinject,axiom,
    ! [X21: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a,Y21: produc2669932201312201782od_c_a,Y22: list_P6327159017948738492od_c_a] :
      ( ( ( cons_P8659537020926305772od_c_a @ X21 @ X22 )
        = ( cons_P8659537020926305772od_c_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_90_append__is__Nil__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ Ys )
        = nil_Product_prod_c_a )
      = ( ( Xs = nil_Product_prod_c_a )
        & ( Ys = nil_Product_prod_c_a ) ) ) ).

% append_is_Nil_conv
thf(fact_91_append__is__Nil__conv,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( ( append2814445079200870952od_c_a @ Xs @ Ys )
        = nil_Pr5380409136740620691od_c_a )
      = ( ( Xs = nil_Pr5380409136740620691od_c_a )
        & ( Ys = nil_Pr5380409136740620691od_c_a ) ) ) ).

% append_is_Nil_conv
thf(fact_92_append__is__Nil__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = nil_Product_prod_b_c )
      = ( ( Xs = nil_Product_prod_b_c )
        & ( Ys = nil_Product_prod_b_c ) ) ) ).

% append_is_Nil_conv
thf(fact_93_append__is__Nil__conv,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a] :
      ( ( ( append2943488786620085207od_c_a @ Xs @ Ys )
        = nil_li2790768921375229634od_c_a )
      = ( ( Xs = nil_li2790768921375229634od_c_a )
        & ( Ys = nil_li2790768921375229634od_c_a ) ) ) ).

% append_is_Nil_conv
thf(fact_94_append__is__Nil__conv,axiom,
    ! [Xs: list_s3765944269294497308od_c_a,Ys: list_s3765944269294497308od_c_a] :
      ( ( ( append8623952414185676849od_c_a @ Xs @ Ys )
        = nil_se4123578418339231900od_c_a )
      = ( ( Xs = nil_se4123578418339231900od_c_a )
        & ( Ys = nil_se4123578418339231900od_c_a ) ) ) ).

% append_is_Nil_conv
thf(fact_95_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_96_append__is__Nil__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Xs @ Ys )
        = nil_Pr1342775757158464060od_c_a )
      = ( ( Xs = nil_Pr1342775757158464060od_c_a )
        & ( Ys = nil_Pr1342775757158464060od_c_a ) ) ) ).

% append_is_Nil_conv
thf(fact_97_Nil__is__append__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys: list_P125642481956313003od_c_a] :
      ( ( nil_Product_prod_c_a
        = ( append8983669691956257088od_c_a @ Xs @ Ys ) )
      = ( ( Xs = nil_Product_prod_c_a )
        & ( Ys = nil_Product_prod_c_a ) ) ) ).

% Nil_is_append_conv
thf(fact_98_Nil__is__append__conv,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( nil_Pr5380409136740620691od_c_a
        = ( append2814445079200870952od_c_a @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr5380409136740620691od_c_a )
        & ( Ys = nil_Pr5380409136740620691od_c_a ) ) ) ).

% Nil_is_append_conv
thf(fact_99_Nil__is__append__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( nil_Product_prod_b_c
        = ( append2547753245680614915od_b_c @ Xs @ Ys ) )
      = ( ( Xs = nil_Product_prod_b_c )
        & ( Ys = nil_Product_prod_b_c ) ) ) ).

% Nil_is_append_conv
thf(fact_100_Nil__is__append__conv,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a] :
      ( ( nil_li2790768921375229634od_c_a
        = ( append2943488786620085207od_c_a @ Xs @ Ys ) )
      = ( ( Xs = nil_li2790768921375229634od_c_a )
        & ( Ys = nil_li2790768921375229634od_c_a ) ) ) ).

% Nil_is_append_conv
thf(fact_101_Nil__is__append__conv,axiom,
    ! [Xs: list_s3765944269294497308od_c_a,Ys: list_s3765944269294497308od_c_a] :
      ( ( nil_se4123578418339231900od_c_a
        = ( append8623952414185676849od_c_a @ Xs @ Ys ) )
      = ( ( Xs = nil_se4123578418339231900od_c_a )
        & ( Ys = nil_se4123578418339231900od_c_a ) ) ) ).

% Nil_is_append_conv
thf(fact_102_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys ) )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_103_Nil__is__append__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( nil_Pr1342775757158464060od_c_a
        = ( append3884164603879732689od_c_a @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr1342775757158464060od_c_a )
        & ( Ys = nil_Pr1342775757158464060od_c_a ) ) ) ).

% Nil_is_append_conv
thf(fact_104_self__append__conv2,axiom,
    ! [Y3: list_P125642481956313003od_c_a,Xs: list_P125642481956313003od_c_a] :
      ( ( Y3
        = ( append8983669691956257088od_c_a @ Xs @ Y3 ) )
      = ( Xs = nil_Product_prod_c_a ) ) ).

% self_append_conv2
thf(fact_105_self__append__conv2,axiom,
    ! [Y3: list_P5767727266746070163od_c_a,Xs: list_P5767727266746070163od_c_a] :
      ( ( Y3
        = ( append2814445079200870952od_c_a @ Xs @ Y3 ) )
      = ( Xs = nil_Pr5380409136740620691od_c_a ) ) ).

% self_append_conv2
thf(fact_106_self__append__conv2,axiom,
    ! [Y3: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( Y3
        = ( append2547753245680614915od_b_c @ Xs @ Y3 ) )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% self_append_conv2
thf(fact_107_self__append__conv2,axiom,
    ! [Y3: list_l2826717958560394946od_c_a,Xs: list_l2826717958560394946od_c_a] :
      ( ( Y3
        = ( append2943488786620085207od_c_a @ Xs @ Y3 ) )
      = ( Xs = nil_li2790768921375229634od_c_a ) ) ).

% self_append_conv2
thf(fact_108_self__append__conv2,axiom,
    ! [Y3: list_s3765944269294497308od_c_a,Xs: list_s3765944269294497308od_c_a] :
      ( ( Y3
        = ( append8623952414185676849od_c_a @ Xs @ Y3 ) )
      = ( Xs = nil_se4123578418339231900od_c_a ) ) ).

% self_append_conv2
thf(fact_109_self__append__conv2,axiom,
    ! [Y3: list_a,Xs: list_a] :
      ( ( Y3
        = ( append_a @ Xs @ Y3 ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_110_self__append__conv2,axiom,
    ! [Y3: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( Y3
        = ( append3884164603879732689od_c_a @ Xs @ Y3 ) )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% self_append_conv2
thf(fact_111_append__self__conv2,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Product_prod_c_a ) ) ).

% append_self_conv2
thf(fact_112_append__self__conv2,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( ( append2814445079200870952od_c_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr5380409136740620691od_c_a ) ) ).

% append_self_conv2
thf(fact_113_append__self__conv2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% append_self_conv2
thf(fact_114_append__self__conv2,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a] :
      ( ( ( append2943488786620085207od_c_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_li2790768921375229634od_c_a ) ) ).

% append_self_conv2
thf(fact_115_append__self__conv2,axiom,
    ! [Xs: list_s3765944269294497308od_c_a,Ys: list_s3765944269294497308od_c_a] :
      ( ( ( append8623952414185676849od_c_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_se4123578418339231900od_c_a ) ) ).

% append_self_conv2
thf(fact_116_append__self__conv2,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_117_append__self__conv2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% append_self_conv2
thf(fact_118_self__append__conv,axiom,
    ! [Y3: list_P125642481956313003od_c_a,Ys: list_P125642481956313003od_c_a] :
      ( ( Y3
        = ( append8983669691956257088od_c_a @ Y3 @ Ys ) )
      = ( Ys = nil_Product_prod_c_a ) ) ).

% self_append_conv
thf(fact_119_self__append__conv,axiom,
    ! [Y3: list_P5767727266746070163od_c_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( Y3
        = ( append2814445079200870952od_c_a @ Y3 @ Ys ) )
      = ( Ys = nil_Pr5380409136740620691od_c_a ) ) ).

% self_append_conv
thf(fact_120_self__append__conv,axiom,
    ! [Y3: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( Y3
        = ( append2547753245680614915od_b_c @ Y3 @ Ys ) )
      = ( Ys = nil_Product_prod_b_c ) ) ).

% self_append_conv
thf(fact_121_self__append__conv,axiom,
    ! [Y3: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a] :
      ( ( Y3
        = ( append2943488786620085207od_c_a @ Y3 @ Ys ) )
      = ( Ys = nil_li2790768921375229634od_c_a ) ) ).

% self_append_conv
thf(fact_122_self__append__conv,axiom,
    ! [Y3: list_s3765944269294497308od_c_a,Ys: list_s3765944269294497308od_c_a] :
      ( ( Y3
        = ( append8623952414185676849od_c_a @ Y3 @ Ys ) )
      = ( Ys = nil_se4123578418339231900od_c_a ) ) ).

% self_append_conv
thf(fact_123_self__append__conv,axiom,
    ! [Y3: list_a,Ys: list_a] :
      ( ( Y3
        = ( append_a @ Y3 @ Ys ) )
      = ( Ys = nil_a ) ) ).

% self_append_conv
thf(fact_124_self__append__conv,axiom,
    ! [Y3: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( Y3
        = ( append3884164603879732689od_c_a @ Y3 @ Ys ) )
      = ( Ys = nil_Pr1342775757158464060od_c_a ) ) ).

% self_append_conv
thf(fact_125_append__self__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys: list_P125642481956313003od_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Product_prod_c_a ) ) ).

% append_self_conv
thf(fact_126_append__self__conv,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( ( append2814445079200870952od_c_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr5380409136740620691od_c_a ) ) ).

% append_self_conv
thf(fact_127_append__self__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Product_prod_b_c ) ) ).

% append_self_conv
thf(fact_128_append__self__conv,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a] :
      ( ( ( append2943488786620085207od_c_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_li2790768921375229634od_c_a ) ) ).

% append_self_conv
thf(fact_129_append__self__conv,axiom,
    ! [Xs: list_s3765944269294497308od_c_a,Ys: list_s3765944269294497308od_c_a] :
      ( ( ( append8623952414185676849od_c_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_se4123578418339231900od_c_a ) ) ).

% append_self_conv
thf(fact_130_append__self__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_a ) ) ).

% append_self_conv
thf(fact_131_append__self__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr1342775757158464060od_c_a ) ) ).

% append_self_conv
thf(fact_132_append__Nil2,axiom,
    ! [Xs: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ Xs @ nil_Product_prod_c_a )
      = Xs ) ).

% append_Nil2
thf(fact_133_append__Nil2,axiom,
    ! [Xs: list_P5767727266746070163od_c_a] :
      ( ( append2814445079200870952od_c_a @ Xs @ nil_Pr5380409136740620691od_c_a )
      = Xs ) ).

% append_Nil2
thf(fact_134_append__Nil2,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ Xs @ nil_Product_prod_b_c )
      = Xs ) ).

% append_Nil2
thf(fact_135_append__Nil2,axiom,
    ! [Xs: list_l2826717958560394946od_c_a] :
      ( ( append2943488786620085207od_c_a @ Xs @ nil_li2790768921375229634od_c_a )
      = Xs ) ).

% append_Nil2
thf(fact_136_append__Nil2,axiom,
    ! [Xs: list_s3765944269294497308od_c_a] :
      ( ( append8623952414185676849od_c_a @ Xs @ nil_se4123578418339231900od_c_a )
      = Xs ) ).

% append_Nil2
thf(fact_137_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_138_append__Nil2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ Xs @ nil_Pr1342775757158464060od_c_a )
      = Xs ) ).

% append_Nil2
thf(fact_139_append_Oright__neutral,axiom,
    ! [A: list_P125642481956313003od_c_a] :
      ( ( append8983669691956257088od_c_a @ A @ nil_Product_prod_c_a )
      = A ) ).

% append.right_neutral
thf(fact_140_append_Oright__neutral,axiom,
    ! [A: list_P5767727266746070163od_c_a] :
      ( ( append2814445079200870952od_c_a @ A @ nil_Pr5380409136740620691od_c_a )
      = A ) ).

% append.right_neutral
thf(fact_141_append_Oright__neutral,axiom,
    ! [A: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ A @ nil_Product_prod_b_c )
      = A ) ).

% append.right_neutral
thf(fact_142_append_Oright__neutral,axiom,
    ! [A: list_l2826717958560394946od_c_a] :
      ( ( append2943488786620085207od_c_a @ A @ nil_li2790768921375229634od_c_a )
      = A ) ).

% append.right_neutral
thf(fact_143_append_Oright__neutral,axiom,
    ! [A: list_s3765944269294497308od_c_a] :
      ( ( append8623952414185676849od_c_a @ A @ nil_se4123578418339231900od_c_a )
      = A ) ).

% append.right_neutral
thf(fact_144_append_Oright__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ A @ nil_a )
      = A ) ).

% append.right_neutral
thf(fact_145_append_Oright__neutral,axiom,
    ! [A: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ A @ nil_Pr1342775757158464060od_c_a )
      = A ) ).

% append.right_neutral
thf(fact_146_target__nil,axiom,
    ! [Q: product_prod_c_a] :
      ( ( target6303326113586172192_a_a_b @ Q @ nil_Pr7913020553331505022od_c_a )
      = Q ) ).

% target_nil
thf(fact_147_target__nil,axiom,
    ! [Q: a] :
      ( ( target_a_b_c @ Q @ nil_Pr1342775757158464060od_c_a )
      = Q ) ).

% target_nil
thf(fact_148_append1__eq__conv,axiom,
    ! [Xs: list_P125642481956313003od_c_a,X3: product_prod_c_a,Ys: list_P125642481956313003od_c_a,Y3: product_prod_c_a] :
      ( ( ( append8983669691956257088od_c_a @ Xs @ ( cons_P1742027962761213787od_c_a @ X3 @ nil_Product_prod_c_a ) )
        = ( append8983669691956257088od_c_a @ Ys @ ( cons_P1742027962761213787od_c_a @ Y3 @ nil_Product_prod_c_a ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_149_append1__eq__conv,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,X3: produc2133359284458116493od_c_a,Ys: list_P5767727266746070163od_c_a,Y3: produc2133359284458116493od_c_a] :
      ( ( ( append2814445079200870952od_c_a @ Xs @ ( cons_P7376644893796496707od_c_a @ X3 @ nil_Pr5380409136740620691od_c_a ) )
        = ( append2814445079200870952od_c_a @ Ys @ ( cons_P7376644893796496707od_c_a @ Y3 @ nil_Pr5380409136740620691od_c_a ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_150_append1__eq__conv,axiom,
    ! [Xs: list_s3765944269294497308od_c_a,X3: set_Pr8357624050790843030od_c_a,Ys: list_s3765944269294497308od_c_a,Y3: set_Pr8357624050790843030od_c_a] :
      ( ( ( append8623952414185676849od_c_a @ Xs @ ( cons_s8716283482848191564od_c_a @ X3 @ nil_se4123578418339231900od_c_a ) )
        = ( append8623952414185676849od_c_a @ Ys @ ( cons_s8716283482848191564od_c_a @ Y3 @ nil_se4123578418339231900od_c_a ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_151_append1__eq__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c,X3: product_prod_b_c,Ys: list_P903359562653991662od_b_c,Y3: product_prod_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
        = ( append2547753245680614915od_b_c @ Ys @ ( cons_P4529483553340347422od_b_c @ Y3 @ nil_Product_prod_b_c ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_152_append1__eq__conv,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,X3: list_P6327159017948738492od_c_a,Ys: list_l2826717958560394946od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( append2943488786620085207od_c_a @ Xs @ ( cons_l7906879460341455474od_c_a @ X3 @ nil_li2790768921375229634od_c_a ) )
        = ( append2943488786620085207od_c_a @ Ys @ ( cons_l7906879460341455474od_c_a @ Y3 @ nil_li2790768921375229634od_c_a ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_153_append1__eq__conv,axiom,
    ! [Xs: list_c,X3: c,Ys: list_c,Y3: c] :
      ( ( ( append_c @ Xs @ ( cons_c @ X3 @ nil_c ) )
        = ( append_c @ Ys @ ( cons_c @ Y3 @ nil_c ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_154_append1__eq__conv,axiom,
    ! [Xs: list_b,X3: b,Ys: list_b,Y3: b] :
      ( ( ( append_b @ Xs @ ( cons_b @ X3 @ nil_b ) )
        = ( append_b @ Ys @ ( cons_b @ Y3 @ nil_b ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_155_append1__eq__conv,axiom,
    ! [Xs: list_a,X3: a,Ys: list_a,Y3: a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) )
        = ( append_a @ Ys @ ( cons_a @ Y3 @ nil_a ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_156_append1__eq__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) )
        = ( append3884164603879732689od_c_a @ Ys @ ( cons_P8659537020926305772od_c_a @ Y3 @ nil_Pr1342775757158464060od_c_a ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_157_path__append,axiom,
    ! [M: fsm_a_c_a,Q: a,P1: list_P6647267739079350397od_a_a,P2: list_P6647267739079350397od_a_a] :
      ( ( path_a_c_a @ M @ Q @ P1 )
     => ( ( path_a_c_a @ M @ ( target_a_c_a @ Q @ P1 ) @ P2 )
       => ( path_a_c_a @ M @ Q @ ( append6470145159784622866od_a_a @ P1 @ P2 ) ) ) ) ).

% path_append
thf(fact_158_path__append,axiom,
    ! [M: fsm_a_4970532336856918856od_c_a,Q: a,P1: list_P3747170188271196375_c_a_a,P2: list_P3747170188271196375_c_a_a] :
      ( ( path_a4972803899755839402od_c_a @ M @ Q @ P1 )
     => ( ( path_a4972803899755839402od_c_a @ M @ ( target945147569263426526od_c_a @ Q @ P1 ) @ P2 )
       => ( path_a4972803899755839402od_c_a @ M @ Q @ ( append461839767961090156_c_a_a @ P1 @ P2 ) ) ) ) ).

% path_append
thf(fact_159_path__append,axiom,
    ! [M: fsm_a_1173558889237794545od_c_a,Q: a,P1: list_P8924682930347365808_c_a_a,P2: list_P8924682930347365808_c_a_a] :
      ( ( path_a4575340523007214163od_c_a @ M @ Q @ P1 )
     => ( ( path_a4575340523007214163od_c_a @ M @ ( target3796347768218495623od_c_a @ Q @ P1 ) @ P2 )
       => ( path_a4575340523007214163od_c_a @ M @ Q @ ( append3109463570867311045_c_a_a @ P1 @ P2 ) ) ) ) ).

% path_append
thf(fact_160_path__append,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P1: list_P1562850763667305982od_c_a,P2: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ M @ Q @ P1 )
     => ( ( path_P1107610407223809260_a_a_b @ M @ ( target6303326113586172192_a_a_b @ Q @ P1 ) @ P2 )
       => ( path_P1107610407223809260_a_a_b @ M @ Q @ ( append311360319448987923od_c_a @ P1 @ P2 ) ) ) ) ).

% path_append
thf(fact_161_path__append,axiom,
    ! [M: fsm_a_b_c,Q: a,P1: list_P6327159017948738492od_c_a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P1 )
     => ( ( path_a_b_c @ M @ ( target_a_b_c @ Q @ P1 ) @ P2 )
       => ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P1 @ P2 ) ) ) ) ).

% path_append
thf(fact_162_reachable__states__intro,axiom,
    ! [M: fsm_a_c_a,P5: list_P6647267739079350397od_a_a] :
      ( ( path_a_c_a @ M @ ( initial_a_c_a @ M ) @ P5 )
     => ( member_a @ ( target_a_c_a @ ( initial_a_c_a @ M ) @ P5 ) @ ( reacha8056221985634029888_a_c_a @ M ) ) ) ).

% reachable_states_intro
thf(fact_163_reachable__states__intro,axiom,
    ! [M: fsm_a_4970532336856918856od_c_a,P5: list_P3747170188271196375_c_a_a] :
      ( ( path_a4972803899755839402od_c_a @ M @ ( initia4547168310285997331od_c_a @ M ) @ P5 )
     => ( member_a @ ( target945147569263426526od_c_a @ ( initia4547168310285997331od_c_a @ M ) @ P5 ) @ ( reacha8900796604505973416od_c_a @ M ) ) ) ).

% reachable_states_intro
thf(fact_164_reachable__states__intro,axiom,
    ! [M: fsm_a_1173558889237794545od_c_a,P5: list_P8924682930347365808_c_a_a] :
      ( ( path_a4575340523007214163od_c_a @ M @ ( initia8238001472959153852od_c_a @ M ) @ P5 )
     => ( member_a @ ( target3796347768218495623od_c_a @ ( initia8238001472959153852od_c_a @ M ) @ P5 ) @ ( reacha8031885885039376209od_c_a @ M ) ) ) ).

% reachable_states_intro
thf(fact_165_reachable__states__intro,axiom,
    ! [M: fsm_Pr6425204305901329755_a_b_c,P5: list_P7591818816154907580od_c_a] :
      ( ( path_P6052804182842616125_a_b_c @ M @ ( initia3341036835946388646_a_b_c @ M ) @ P5 )
     => ( member5502040268208861279od_c_a @ ( target7519567265479654769_a_b_c @ ( initia3341036835946388646_a_b_c @ M ) @ P5 ) @ ( reacha1389639598965321787_a_b_c @ M ) ) ) ).

% reachable_states_intro
thf(fact_166_reachable__states__intro,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,P5: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ M @ ( initia681974817753967189_a_a_b @ M ) @ P5 )
     => ( member5074992350434858958od_c_a @ ( target6303326113586172192_a_a_b @ ( initia681974817753967189_a_a_b @ M ) @ P5 ) @ ( reacha5035603111973943274_a_a_b @ M ) ) ) ).

% reachable_states_intro
thf(fact_167_reachable__states__intro,axiom,
    ! [M: fsm_Pr9212659883570776987_a_a_b,P5: list_P1501825136581950910od_c_a] :
      ( ( path_P8840259764815292157_a_a_b @ M @ ( initia6128492417919064678_a_a_b @ M ) @ P5 )
     => ( member5502040268208861279od_c_a @ ( target1083650810597554993_a_a_b @ ( initia6128492417919064678_a_a_b @ M ) @ P5 ) @ ( reacha4177095180937997819_a_a_b @ M ) ) ) ).

% reachable_states_intro
thf(fact_168_reachable__states__intro,axiom,
    ! [M: fsm_a_a_b,P5: list_P7365212274028606398od_b_a] :
      ( ( path_a_a_b @ M @ ( initial_a_a_b @ M ) @ P5 )
     => ( member_a @ ( target_a_a_b @ ( initial_a_a_b @ M ) @ P5 ) @ ( reacha4407761112724606147_a_a_b @ M ) ) ) ).

% reachable_states_intro
thf(fact_169_reachable__states__intro,axiom,
    ! [M: fsm_a_b_c,P5: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ P5 )
     => ( member_a @ ( target_a_b_c @ ( initial_a_b_c @ M ) @ P5 ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ) ).

% reachable_states_intro
thf(fact_170_splice_Ocases,axiom,
    ! [X3: produc5684054305078322407od_c_a] :
      ( ! [Ys2: list_P5767727266746070163od_c_a] :
          ( X3
         != ( produc3287727441117448279od_c_a @ nil_Pr5380409136740620691od_c_a @ Ys2 ) )
     => ~ ! [X: produc2133359284458116493od_c_a,Xs2: list_P5767727266746070163od_c_a,Ys2: list_P5767727266746070163od_c_a] :
            ( X3
           != ( produc3287727441117448279od_c_a @ ( cons_P7376644893796496707od_c_a @ X @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_171_splice_Ocases,axiom,
    ! [X3: produc6973795807838804519od_c_a] :
      ( ! [Ys2: list_s3765944269294497308od_c_a] :
          ( X3
         != ( produc2434387664047349783od_c_a @ nil_se4123578418339231900od_c_a @ Ys2 ) )
     => ~ ! [X: set_Pr8357624050790843030od_c_a,Xs2: list_s3765944269294497308od_c_a,Ys2: list_s3765944269294497308od_c_a] :
            ( X3
           != ( produc2434387664047349783od_c_a @ ( cons_s8716283482848191564od_c_a @ X @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_172_splice_Ocases,axiom,
    ! [X3: produc5878855887281848999od_b_c] :
      ( ! [Ys2: list_P903359562653991662od_b_c] :
          ( X3
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys2 ) )
     => ~ ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
            ( X3
           != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_173_splice_Ocases,axiom,
    ! [X3: produc7462261154759727143od_c_a] :
      ( ! [Ys2: list_l2826717958560394946od_c_a] :
          ( X3
         != ( produc1996976070648013847od_c_a @ nil_li2790768921375229634od_c_a @ Ys2 ) )
     => ~ ! [X: list_P6327159017948738492od_c_a,Xs2: list_l2826717958560394946od_c_a,Ys2: list_l2826717958560394946od_c_a] :
            ( X3
           != ( produc1996976070648013847od_c_a @ ( cons_l7906879460341455474od_c_a @ X @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_174_splice_Ocases,axiom,
    ! [X3: produc7985223085803476903list_c] :
      ( ! [Ys2: list_c] :
          ( X3
         != ( produc5515445818230282391list_c @ nil_c @ Ys2 ) )
     => ~ ! [X: c,Xs2: list_c,Ys2: list_c] :
            ( X3
           != ( produc5515445818230282391list_c @ ( cons_c @ X @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_175_splice_Ocases,axiom,
    ! [X3: produc3963297410138542439list_b] :
      ( ! [Ys2: list_b] :
          ( X3
         != ( produc1564554178308465111list_b @ nil_b @ Ys2 ) )
     => ~ ! [X: b,Xs2: list_b,Ys2: list_b] :
            ( X3
           != ( produc1564554178308465111list_b @ ( cons_b @ X @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_176_splice_Ocases,axiom,
    ! [X3: produc9164743771328383783list_a] :
      ( ! [Ys2: list_a] :
          ( X3
         != ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
     => ~ ! [X: a,Xs2: list_a,Ys2: list_a] :
            ( X3
           != ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_177_splice_Ocases,axiom,
    ! [X3: produc8143030500101326375od_c_a] :
      ( ! [Ys2: list_P6327159017948738492od_c_a] :
          ( X3
         != ( produc8447801329164731415od_c_a @ nil_Pr1342775757158464060od_c_a @ Ys2 ) )
     => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Ys2: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_178_shuffles_Ocases,axiom,
    ! [X3: produc5684054305078322407od_c_a] :
      ( ! [Ys2: list_P5767727266746070163od_c_a] :
          ( X3
         != ( produc3287727441117448279od_c_a @ nil_Pr5380409136740620691od_c_a @ Ys2 ) )
     => ( ! [Xs2: list_P5767727266746070163od_c_a] :
            ( X3
           != ( produc3287727441117448279od_c_a @ Xs2 @ nil_Pr5380409136740620691od_c_a ) )
       => ~ ! [X: produc2133359284458116493od_c_a,Xs2: list_P5767727266746070163od_c_a,Y: produc2133359284458116493od_c_a,Ys2: list_P5767727266746070163od_c_a] :
              ( X3
             != ( produc3287727441117448279od_c_a @ ( cons_P7376644893796496707od_c_a @ X @ Xs2 ) @ ( cons_P7376644893796496707od_c_a @ Y @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_179_shuffles_Ocases,axiom,
    ! [X3: produc6973795807838804519od_c_a] :
      ( ! [Ys2: list_s3765944269294497308od_c_a] :
          ( X3
         != ( produc2434387664047349783od_c_a @ nil_se4123578418339231900od_c_a @ Ys2 ) )
     => ( ! [Xs2: list_s3765944269294497308od_c_a] :
            ( X3
           != ( produc2434387664047349783od_c_a @ Xs2 @ nil_se4123578418339231900od_c_a ) )
       => ~ ! [X: set_Pr8357624050790843030od_c_a,Xs2: list_s3765944269294497308od_c_a,Y: set_Pr8357624050790843030od_c_a,Ys2: list_s3765944269294497308od_c_a] :
              ( X3
             != ( produc2434387664047349783od_c_a @ ( cons_s8716283482848191564od_c_a @ X @ Xs2 ) @ ( cons_s8716283482848191564od_c_a @ Y @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_180_shuffles_Ocases,axiom,
    ! [X3: produc5878855887281848999od_b_c] :
      ( ! [Ys2: list_P903359562653991662od_b_c] :
          ( X3
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys2 ) )
     => ( ! [Xs2: list_P903359562653991662od_b_c] :
            ( X3
           != ( produc3076307724246486423od_b_c @ Xs2 @ nil_Product_prod_b_c ) )
       => ~ ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
              ( X3
             != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_181_shuffles_Ocases,axiom,
    ! [X3: produc7462261154759727143od_c_a] :
      ( ! [Ys2: list_l2826717958560394946od_c_a] :
          ( X3
         != ( produc1996976070648013847od_c_a @ nil_li2790768921375229634od_c_a @ Ys2 ) )
     => ( ! [Xs2: list_l2826717958560394946od_c_a] :
            ( X3
           != ( produc1996976070648013847od_c_a @ Xs2 @ nil_li2790768921375229634od_c_a ) )
       => ~ ! [X: list_P6327159017948738492od_c_a,Xs2: list_l2826717958560394946od_c_a,Y: list_P6327159017948738492od_c_a,Ys2: list_l2826717958560394946od_c_a] :
              ( X3
             != ( produc1996976070648013847od_c_a @ ( cons_l7906879460341455474od_c_a @ X @ Xs2 ) @ ( cons_l7906879460341455474od_c_a @ Y @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_182_shuffles_Ocases,axiom,
    ! [X3: produc7985223085803476903list_c] :
      ( ! [Ys2: list_c] :
          ( X3
         != ( produc5515445818230282391list_c @ nil_c @ Ys2 ) )
     => ( ! [Xs2: list_c] :
            ( X3
           != ( produc5515445818230282391list_c @ Xs2 @ nil_c ) )
       => ~ ! [X: c,Xs2: list_c,Y: c,Ys2: list_c] :
              ( X3
             != ( produc5515445818230282391list_c @ ( cons_c @ X @ Xs2 ) @ ( cons_c @ Y @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_183_shuffles_Ocases,axiom,
    ! [X3: produc3963297410138542439list_b] :
      ( ! [Ys2: list_b] :
          ( X3
         != ( produc1564554178308465111list_b @ nil_b @ Ys2 ) )
     => ( ! [Xs2: list_b] :
            ( X3
           != ( produc1564554178308465111list_b @ Xs2 @ nil_b ) )
       => ~ ! [X: b,Xs2: list_b,Y: b,Ys2: list_b] :
              ( X3
             != ( produc1564554178308465111list_b @ ( cons_b @ X @ Xs2 ) @ ( cons_b @ Y @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_184_shuffles_Ocases,axiom,
    ! [X3: produc9164743771328383783list_a] :
      ( ! [Ys2: list_a] :
          ( X3
         != ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
     => ( ! [Xs2: list_a] :
            ( X3
           != ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
       => ~ ! [X: a,Xs2: list_a,Y: a,Ys2: list_a] :
              ( X3
             != ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_185_shuffles_Ocases,axiom,
    ! [X3: produc8143030500101326375od_c_a] :
      ( ! [Ys2: list_P6327159017948738492od_c_a] :
          ( X3
         != ( produc8447801329164731415od_c_a @ nil_Pr1342775757158464060od_c_a @ Ys2 ) )
     => ( ! [Xs2: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc8447801329164731415od_c_a @ Xs2 @ nil_Pr1342775757158464060od_c_a ) )
       => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a] :
              ( X3
             != ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_186_sorted__wrt_Ocases,axiom,
    ! [X3: produc3125590402920344460od_c_a] :
      ( ! [P6: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a > $o] :
          ( X3
         != ( produc1633839590730437500od_c_a @ P6 @ nil_Pr5380409136740620691od_c_a ) )
     => ~ ! [P6: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a > $o,X: produc2133359284458116493od_c_a,Ys2: list_P5767727266746070163od_c_a] :
            ( X3
           != ( produc1633839590730437500od_c_a @ P6 @ ( cons_P7376644893796496707od_c_a @ X @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_187_sorted__wrt_Ocases,axiom,
    ! [X3: produc5179877164081724373od_c_a] :
      ( ! [P6: set_Pr8357624050790843030od_c_a > set_Pr8357624050790843030od_c_a > $o] :
          ( X3
         != ( produc3362519565722104645od_c_a @ P6 @ nil_se4123578418339231900od_c_a ) )
     => ~ ! [P6: set_Pr8357624050790843030od_c_a > set_Pr8357624050790843030od_c_a > $o,X: set_Pr8357624050790843030od_c_a,Ys2: list_s3765944269294497308od_c_a] :
            ( X3
           != ( produc3362519565722104645od_c_a @ P6 @ ( cons_s8716283482848191564od_c_a @ X @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_188_sorted__wrt_Ocases,axiom,
    ! [X3: produc2416979464461413415od_b_c] :
      ( ! [P6: product_prod_b_c > product_prod_b_c > $o] :
          ( X3
         != ( produc6092892548594537111od_b_c @ P6 @ nil_Product_prod_b_c ) )
     => ~ ! [P6: product_prod_b_c > product_prod_b_c > $o,X: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
            ( X3
           != ( produc6092892548594537111od_b_c @ P6 @ ( cons_P4529483553340347422od_b_c @ X @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_189_sorted__wrt_Ocases,axiom,
    ! [X3: produc8878234039744503803od_c_a] :
      ( ! [P6: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o] :
          ( X3
         != ( produc270945891827043819od_c_a @ P6 @ nil_li2790768921375229634od_c_a ) )
     => ~ ! [P6: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o,X: list_P6327159017948738492od_c_a,Ys2: list_l2826717958560394946od_c_a] :
            ( X3
           != ( produc270945891827043819od_c_a @ P6 @ ( cons_l7906879460341455474od_c_a @ X @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_190_sorted__wrt_Ocases,axiom,
    ! [X3: produc5337753222811372479list_c] :
      ( ! [P6: c > c > $o] :
          ( X3
         != ( produc8274703458617474863list_c @ P6 @ nil_c ) )
     => ~ ! [P6: c > c > $o,X: c,Ys2: list_c] :
            ( X3
           != ( produc8274703458617474863list_c @ P6 @ ( cons_c @ X @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_191_sorted__wrt_Ocases,axiom,
    ! [X3: produc5185152304234826110list_b] :
      ( ! [P6: b > b > $o] :
          ( X3
         != ( produc8193136575784045678list_b @ P6 @ nil_b ) )
     => ~ ! [P6: b > b > $o,X: b,Ys2: list_b] :
            ( X3
           != ( produc8193136575784045678list_b @ P6 @ ( cons_b @ X @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_192_sorted__wrt_Ocases,axiom,
    ! [X3: produc5032551385658279741list_a] :
      ( ! [P6: a > a > $o] :
          ( X3
         != ( produc8111569692950616493list_a @ P6 @ nil_a ) )
     => ~ ! [P6: a > a > $o,X: a,Ys2: list_a] :
            ( X3
           != ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_193_sorted__wrt_Ocases,axiom,
    ! [X3: produc5759257467515349365od_c_a] :
      ( ! [P6: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o] :
          ( X3
         != ( produc390224926211134693od_c_a @ P6 @ nil_Pr1342775757158464060od_c_a ) )
     => ~ ! [P6: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc390224926211134693od_c_a @ P6 @ ( cons_P8659537020926305772od_c_a @ X @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_194_successively_Ocases,axiom,
    ! [X3: produc3125590402920344460od_c_a] :
      ( ! [P6: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a > $o] :
          ( X3
         != ( produc1633839590730437500od_c_a @ P6 @ nil_Pr5380409136740620691od_c_a ) )
     => ( ! [P6: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a > $o,X: produc2133359284458116493od_c_a] :
            ( X3
           != ( produc1633839590730437500od_c_a @ P6 @ ( cons_P7376644893796496707od_c_a @ X @ nil_Pr5380409136740620691od_c_a ) ) )
       => ~ ! [P6: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a > $o,X: produc2133359284458116493od_c_a,Y: produc2133359284458116493od_c_a,Xs2: list_P5767727266746070163od_c_a] :
              ( X3
             != ( produc1633839590730437500od_c_a @ P6 @ ( cons_P7376644893796496707od_c_a @ X @ ( cons_P7376644893796496707od_c_a @ Y @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_195_successively_Ocases,axiom,
    ! [X3: produc5179877164081724373od_c_a] :
      ( ! [P6: set_Pr8357624050790843030od_c_a > set_Pr8357624050790843030od_c_a > $o] :
          ( X3
         != ( produc3362519565722104645od_c_a @ P6 @ nil_se4123578418339231900od_c_a ) )
     => ( ! [P6: set_Pr8357624050790843030od_c_a > set_Pr8357624050790843030od_c_a > $o,X: set_Pr8357624050790843030od_c_a] :
            ( X3
           != ( produc3362519565722104645od_c_a @ P6 @ ( cons_s8716283482848191564od_c_a @ X @ nil_se4123578418339231900od_c_a ) ) )
       => ~ ! [P6: set_Pr8357624050790843030od_c_a > set_Pr8357624050790843030od_c_a > $o,X: set_Pr8357624050790843030od_c_a,Y: set_Pr8357624050790843030od_c_a,Xs2: list_s3765944269294497308od_c_a] :
              ( X3
             != ( produc3362519565722104645od_c_a @ P6 @ ( cons_s8716283482848191564od_c_a @ X @ ( cons_s8716283482848191564od_c_a @ Y @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_196_successively_Ocases,axiom,
    ! [X3: produc2416979464461413415od_b_c] :
      ( ! [P6: product_prod_b_c > product_prod_b_c > $o] :
          ( X3
         != ( produc6092892548594537111od_b_c @ P6 @ nil_Product_prod_b_c ) )
     => ( ! [P6: product_prod_b_c > product_prod_b_c > $o,X: product_prod_b_c] :
            ( X3
           != ( produc6092892548594537111od_b_c @ P6 @ ( cons_P4529483553340347422od_b_c @ X @ nil_Product_prod_b_c ) ) )
       => ~ ! [P6: product_prod_b_c > product_prod_b_c > $o,X: product_prod_b_c,Y: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( X3
             != ( produc6092892548594537111od_b_c @ P6 @ ( cons_P4529483553340347422od_b_c @ X @ ( cons_P4529483553340347422od_b_c @ Y @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_197_successively_Ocases,axiom,
    ! [X3: produc8878234039744503803od_c_a] :
      ( ! [P6: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o] :
          ( X3
         != ( produc270945891827043819od_c_a @ P6 @ nil_li2790768921375229634od_c_a ) )
     => ( ! [P6: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o,X: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc270945891827043819od_c_a @ P6 @ ( cons_l7906879460341455474od_c_a @ X @ nil_li2790768921375229634od_c_a ) ) )
       => ~ ! [P6: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o,X: list_P6327159017948738492od_c_a,Y: list_P6327159017948738492od_c_a,Xs2: list_l2826717958560394946od_c_a] :
              ( X3
             != ( produc270945891827043819od_c_a @ P6 @ ( cons_l7906879460341455474od_c_a @ X @ ( cons_l7906879460341455474od_c_a @ Y @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_198_successively_Ocases,axiom,
    ! [X3: produc5337753222811372479list_c] :
      ( ! [P6: c > c > $o] :
          ( X3
         != ( produc8274703458617474863list_c @ P6 @ nil_c ) )
     => ( ! [P6: c > c > $o,X: c] :
            ( X3
           != ( produc8274703458617474863list_c @ P6 @ ( cons_c @ X @ nil_c ) ) )
       => ~ ! [P6: c > c > $o,X: c,Y: c,Xs2: list_c] :
              ( X3
             != ( produc8274703458617474863list_c @ P6 @ ( cons_c @ X @ ( cons_c @ Y @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_199_successively_Ocases,axiom,
    ! [X3: produc5185152304234826110list_b] :
      ( ! [P6: b > b > $o] :
          ( X3
         != ( produc8193136575784045678list_b @ P6 @ nil_b ) )
     => ( ! [P6: b > b > $o,X: b] :
            ( X3
           != ( produc8193136575784045678list_b @ P6 @ ( cons_b @ X @ nil_b ) ) )
       => ~ ! [P6: b > b > $o,X: b,Y: b,Xs2: list_b] :
              ( X3
             != ( produc8193136575784045678list_b @ P6 @ ( cons_b @ X @ ( cons_b @ Y @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_200_successively_Ocases,axiom,
    ! [X3: produc5032551385658279741list_a] :
      ( ! [P6: a > a > $o] :
          ( X3
         != ( produc8111569692950616493list_a @ P6 @ nil_a ) )
     => ( ! [P6: a > a > $o,X: a] :
            ( X3
           != ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X @ nil_a ) ) )
       => ~ ! [P6: a > a > $o,X: a,Y: a,Xs2: list_a] :
              ( X3
             != ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X @ ( cons_a @ Y @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_201_successively_Ocases,axiom,
    ! [X3: produc5759257467515349365od_c_a] :
      ( ! [P6: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o] :
          ( X3
         != ( produc390224926211134693od_c_a @ P6 @ nil_Pr1342775757158464060od_c_a ) )
     => ( ! [P6: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a] :
            ( X3
           != ( produc390224926211134693od_c_a @ P6 @ ( cons_P8659537020926305772od_c_a @ X @ nil_Pr1342775757158464060od_c_a ) ) )
       => ~ ! [P6: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a,Y: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( X3
             != ( produc390224926211134693od_c_a @ P6 @ ( cons_P8659537020926305772od_c_a @ X @ ( cons_P8659537020926305772od_c_a @ Y @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_202_distinct__length__2__or__more,axiom,
    ! [A: list_a,B: list_a,Xs: list_list_a] :
      ( ( distinct_list_a @ ( cons_list_a @ A @ ( cons_list_a @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distinct_list_a @ ( cons_list_a @ A @ Xs ) )
        & ( distinct_list_a @ ( cons_list_a @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_203_distinct__length__2__or__more,axiom,
    ! [A: set_a,B: set_a,Xs: list_set_a] :
      ( ( distinct_set_a @ ( cons_set_a @ A @ ( cons_set_a @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distinct_set_a @ ( cons_set_a @ A @ Xs ) )
        & ( distinct_set_a @ ( cons_set_a @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_204_distinct__length__2__or__more,axiom,
    ! [A: product_prod_b_c,B: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( distin6568250333530618337od_b_c @ ( cons_P4529483553340347422od_b_c @ A @ ( cons_P4529483553340347422od_b_c @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distin6568250333530618337od_b_c @ ( cons_P4529483553340347422od_b_c @ A @ Xs ) )
        & ( distin6568250333530618337od_b_c @ ( cons_P4529483553340347422od_b_c @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_205_distinct__length__2__or__more,axiom,
    ! [A: list_P6327159017948738492od_c_a,B: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a] :
      ( ( distin5812642747228818613od_c_a @ ( cons_l7906879460341455474od_c_a @ A @ ( cons_l7906879460341455474od_c_a @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distin5812642747228818613od_c_a @ ( cons_l7906879460341455474od_c_a @ A @ Xs ) )
        & ( distin5812642747228818613od_c_a @ ( cons_l7906879460341455474od_c_a @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_206_distinct__length__2__or__more,axiom,
    ! [A: c,B: c,Xs: list_c] :
      ( ( distinct_c @ ( cons_c @ A @ ( cons_c @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distinct_c @ ( cons_c @ A @ Xs ) )
        & ( distinct_c @ ( cons_c @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_207_distinct__length__2__or__more,axiom,
    ! [A: b,B: b,Xs: list_b] :
      ( ( distinct_b @ ( cons_b @ A @ ( cons_b @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distinct_b @ ( cons_b @ A @ Xs ) )
        & ( distinct_b @ ( cons_b @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_208_distinct__length__2__or__more,axiom,
    ! [A: a,B: a,Xs: list_a] :
      ( ( distinct_a @ ( cons_a @ A @ ( cons_a @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distinct_a @ ( cons_a @ A @ Xs ) )
        & ( distinct_a @ ( cons_a @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_209_distinct__length__2__or__more,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( distin811446047591323055od_c_a @ ( cons_P8659537020926305772od_c_a @ A @ ( cons_P8659537020926305772od_c_a @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distin811446047591323055od_c_a @ ( cons_P8659537020926305772od_c_a @ A @ Xs ) )
        & ( distin811446047591323055od_c_a @ ( cons_P8659537020926305772od_c_a @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_210_mem__Collect__eq,axiom,
    ! [A: list_b,P7: list_b > $o] :
      ( ( member_list_b @ A @ ( collect_list_b @ P7 ) )
      = ( P7 @ A ) ) ).

% mem_Collect_eq
thf(fact_211_mem__Collect__eq,axiom,
    ! [A: list_a,P7: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P7 ) )
      = ( P7 @ A ) ) ).

% mem_Collect_eq
thf(fact_212_mem__Collect__eq,axiom,
    ! [A: nat,P7: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P7 ) )
      = ( P7 @ A ) ) ).

% mem_Collect_eq
thf(fact_213_mem__Collect__eq,axiom,
    ! [A: c,P7: c > $o] :
      ( ( member_c @ A @ ( collect_c @ P7 ) )
      = ( P7 @ A ) ) ).

% mem_Collect_eq
thf(fact_214_mem__Collect__eq,axiom,
    ! [A: b,P7: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P7 ) )
      = ( P7 @ A ) ) ).

% mem_Collect_eq
thf(fact_215_mem__Collect__eq,axiom,
    ! [A: a,P7: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P7 ) )
      = ( P7 @ A ) ) ).

% mem_Collect_eq
thf(fact_216_mem__Collect__eq,axiom,
    ! [A: produc2669932201312201782od_c_a,P7: produc2669932201312201782od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ A @ ( collec8405073525749625505od_c_a @ P7 ) )
      = ( P7 @ A ) ) ).

% mem_Collect_eq
thf(fact_217_Collect__mem__eq,axiom,
    ! [A3: set_list_b] :
      ( ( collect_list_b
        @ ^ [X4: list_b] : ( member_list_b @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_218_Collect__mem__eq,axiom,
    ! [A3: set_list_a] :
      ( ( collect_list_a
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_219_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_220_Collect__mem__eq,axiom,
    ! [A3: set_c] :
      ( ( collect_c
        @ ^ [X4: c] : ( member_c @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_221_Collect__mem__eq,axiom,
    ! [A3: set_b] :
      ( ( collect_b
        @ ^ [X4: b] : ( member_b @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_222_Collect__mem__eq,axiom,
    ! [A3: set_a] :
      ( ( collect_a
        @ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_223_Collect__mem__eq,axiom,
    ! [A3: set_Pr8357624050790843030od_c_a] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_224_Collect__cong,axiom,
    ! [P7: list_a > $o,Q2: list_a > $o] :
      ( ! [X: list_a] :
          ( ( P7 @ X )
          = ( Q2 @ X ) )
     => ( ( collect_list_a @ P7 )
        = ( collect_list_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_225_Collect__cong,axiom,
    ! [P7: nat > $o,Q2: nat > $o] :
      ( ! [X: nat] :
          ( ( P7 @ X )
          = ( Q2 @ X ) )
     => ( ( collect_nat @ P7 )
        = ( collect_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_226_Collect__cong,axiom,
    ! [P7: c > $o,Q2: c > $o] :
      ( ! [X: c] :
          ( ( P7 @ X )
          = ( Q2 @ X ) )
     => ( ( collect_c @ P7 )
        = ( collect_c @ Q2 ) ) ) ).

% Collect_cong
thf(fact_227_Collect__cong,axiom,
    ! [P7: b > $o,Q2: b > $o] :
      ( ! [X: b] :
          ( ( P7 @ X )
          = ( Q2 @ X ) )
     => ( ( collect_b @ P7 )
        = ( collect_b @ Q2 ) ) ) ).

% Collect_cong
thf(fact_228_Collect__cong,axiom,
    ! [P7: a > $o,Q2: a > $o] :
      ( ! [X: a] :
          ( ( P7 @ X )
          = ( Q2 @ X ) )
     => ( ( collect_a @ P7 )
        = ( collect_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_229_list__nonempty__induct,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,P7: list_P5767727266746070163od_c_a > $o] :
      ( ( Xs != nil_Pr5380409136740620691od_c_a )
     => ( ! [X: produc2133359284458116493od_c_a] : ( P7 @ ( cons_P7376644893796496707od_c_a @ X @ nil_Pr5380409136740620691od_c_a ) )
       => ( ! [X: produc2133359284458116493od_c_a,Xs2: list_P5767727266746070163od_c_a] :
              ( ( Xs2 != nil_Pr5380409136740620691od_c_a )
             => ( ( P7 @ Xs2 )
               => ( P7 @ ( cons_P7376644893796496707od_c_a @ X @ Xs2 ) ) ) )
         => ( P7 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_230_list__nonempty__induct,axiom,
    ! [Xs: list_s3765944269294497308od_c_a,P7: list_s3765944269294497308od_c_a > $o] :
      ( ( Xs != nil_se4123578418339231900od_c_a )
     => ( ! [X: set_Pr8357624050790843030od_c_a] : ( P7 @ ( cons_s8716283482848191564od_c_a @ X @ nil_se4123578418339231900od_c_a ) )
       => ( ! [X: set_Pr8357624050790843030od_c_a,Xs2: list_s3765944269294497308od_c_a] :
              ( ( Xs2 != nil_se4123578418339231900od_c_a )
             => ( ( P7 @ Xs2 )
               => ( P7 @ ( cons_s8716283482848191564od_c_a @ X @ Xs2 ) ) ) )
         => ( P7 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_231_list__nonempty__induct,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P7: list_P903359562653991662od_b_c > $o] :
      ( ( Xs != nil_Product_prod_b_c )
     => ( ! [X: product_prod_b_c] : ( P7 @ ( cons_P4529483553340347422od_b_c @ X @ nil_Product_prod_b_c ) )
       => ( ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( ( Xs2 != nil_Product_prod_b_c )
             => ( ( P7 @ Xs2 )
               => ( P7 @ ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) ) ) )
         => ( P7 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_232_list__nonempty__induct,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,P7: list_l2826717958560394946od_c_a > $o] :
      ( ( Xs != nil_li2790768921375229634od_c_a )
     => ( ! [X: list_P6327159017948738492od_c_a] : ( P7 @ ( cons_l7906879460341455474od_c_a @ X @ nil_li2790768921375229634od_c_a ) )
       => ( ! [X: list_P6327159017948738492od_c_a,Xs2: list_l2826717958560394946od_c_a] :
              ( ( Xs2 != nil_li2790768921375229634od_c_a )
             => ( ( P7 @ Xs2 )
               => ( P7 @ ( cons_l7906879460341455474od_c_a @ X @ Xs2 ) ) ) )
         => ( P7 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_233_list__nonempty__induct,axiom,
    ! [Xs: list_c,P7: list_c > $o] :
      ( ( Xs != nil_c )
     => ( ! [X: c] : ( P7 @ ( cons_c @ X @ nil_c ) )
       => ( ! [X: c,Xs2: list_c] :
              ( ( Xs2 != nil_c )
             => ( ( P7 @ Xs2 )
               => ( P7 @ ( cons_c @ X @ Xs2 ) ) ) )
         => ( P7 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_234_list__nonempty__induct,axiom,
    ! [Xs: list_b,P7: list_b > $o] :
      ( ( Xs != nil_b )
     => ( ! [X: b] : ( P7 @ ( cons_b @ X @ nil_b ) )
       => ( ! [X: b,Xs2: list_b] :
              ( ( Xs2 != nil_b )
             => ( ( P7 @ Xs2 )
               => ( P7 @ ( cons_b @ X @ Xs2 ) ) ) )
         => ( P7 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_235_list__nonempty__induct,axiom,
    ! [Xs: list_a,P7: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X: a] : ( P7 @ ( cons_a @ X @ nil_a ) )
       => ( ! [X: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( ( P7 @ Xs2 )
               => ( P7 @ ( cons_a @ X @ Xs2 ) ) ) )
         => ( P7 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_236_list__nonempty__induct,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: list_P6327159017948738492od_c_a > $o] :
      ( ( Xs != nil_Pr1342775757158464060od_c_a )
     => ( ! [X: produc2669932201312201782od_c_a] : ( P7 @ ( cons_P8659537020926305772od_c_a @ X @ nil_Pr1342775757158464060od_c_a ) )
       => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( ( Xs2 != nil_Pr1342775757158464060od_c_a )
             => ( ( P7 @ Xs2 )
               => ( P7 @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) ) )
         => ( P7 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_237_distinct__singleton,axiom,
    ! [X3: list_a] : ( distinct_list_a @ ( cons_list_a @ X3 @ nil_list_a ) ) ).

% distinct_singleton
thf(fact_238_distinct__singleton,axiom,
    ! [X3: set_a] : ( distinct_set_a @ ( cons_set_a @ X3 @ nil_set_a ) ) ).

% distinct_singleton
thf(fact_239_distinct__singleton,axiom,
    ! [X3: produc2133359284458116493od_c_a] : ( distin2128679875739541510od_c_a @ ( cons_P7376644893796496707od_c_a @ X3 @ nil_Pr5380409136740620691od_c_a ) ) ).

% distinct_singleton
thf(fact_240_distinct__singleton,axiom,
    ! [X3: set_Pr8357624050790843030od_c_a] : ( distin2796594386987124751od_c_a @ ( cons_s8716283482848191564od_c_a @ X3 @ nil_se4123578418339231900od_c_a ) ) ).

% distinct_singleton
thf(fact_241_distinct__singleton,axiom,
    ! [X3: product_prod_b_c] : ( distin6568250333530618337od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) ).

% distinct_singleton
thf(fact_242_distinct__singleton,axiom,
    ! [X3: list_P6327159017948738492od_c_a] : ( distin5812642747228818613od_c_a @ ( cons_l7906879460341455474od_c_a @ X3 @ nil_li2790768921375229634od_c_a ) ) ).

% distinct_singleton
thf(fact_243_distinct__singleton,axiom,
    ! [X3: c] : ( distinct_c @ ( cons_c @ X3 @ nil_c ) ) ).

% distinct_singleton
thf(fact_244_distinct__singleton,axiom,
    ! [X3: b] : ( distinct_b @ ( cons_b @ X3 @ nil_b ) ) ).

% distinct_singleton
thf(fact_245_distinct__singleton,axiom,
    ! [X3: a] : ( distinct_a @ ( cons_a @ X3 @ nil_a ) ) ).

% distinct_singleton
thf(fact_246_distinct__singleton,axiom,
    ! [X3: produc2669932201312201782od_c_a] : ( distin811446047591323055od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) ).

% distinct_singleton
thf(fact_247_not__Cons__self2,axiom,
    ! [X3: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( cons_P4529483553340347422od_b_c @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_248_not__Cons__self2,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a] :
      ( ( cons_l7906879460341455474od_c_a @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_249_not__Cons__self2,axiom,
    ! [X3: c,Xs: list_c] :
      ( ( cons_c @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_250_not__Cons__self2,axiom,
    ! [X3: b,Xs: list_b] :
      ( ( cons_b @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_251_not__Cons__self2,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( cons_a @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_252_not__Cons__self2,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( cons_P8659537020926305772od_c_a @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_253_list__induct2_H,axiom,
    ! [P7: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( P7 @ nil_Pr1342775757158464060od_c_a @ nil_Pr1342775757158464060od_c_a )
     => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] : ( P7 @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ nil_Pr1342775757158464060od_c_a )
       => ( ! [Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a] : ( P7 @ nil_Pr1342775757158464060od_c_a @ ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) )
         => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_254_list__induct2_H,axiom,
    ! [P7: list_c > list_c > $o,Xs: list_c,Ys: list_c] :
      ( ( P7 @ nil_c @ nil_c )
     => ( ! [X: c,Xs2: list_c] : ( P7 @ ( cons_c @ X @ Xs2 ) @ nil_c )
       => ( ! [Y: c,Ys2: list_c] : ( P7 @ nil_c @ ( cons_c @ Y @ Ys2 ) )
         => ( ! [X: c,Xs2: list_c,Y: c,Ys2: list_c] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_c @ X @ Xs2 ) @ ( cons_c @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_255_list__induct2_H,axiom,
    ! [P7: list_c > list_b > $o,Xs: list_c,Ys: list_b] :
      ( ( P7 @ nil_c @ nil_b )
     => ( ! [X: c,Xs2: list_c] : ( P7 @ ( cons_c @ X @ Xs2 ) @ nil_b )
       => ( ! [Y: b,Ys2: list_b] : ( P7 @ nil_c @ ( cons_b @ Y @ Ys2 ) )
         => ( ! [X: c,Xs2: list_c,Y: b,Ys2: list_b] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_c @ X @ Xs2 ) @ ( cons_b @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_256_list__induct2_H,axiom,
    ! [P7: list_c > list_a > $o,Xs: list_c,Ys: list_a] :
      ( ( P7 @ nil_c @ nil_a )
     => ( ! [X: c,Xs2: list_c] : ( P7 @ ( cons_c @ X @ Xs2 ) @ nil_a )
       => ( ! [Y: a,Ys2: list_a] : ( P7 @ nil_c @ ( cons_a @ Y @ Ys2 ) )
         => ( ! [X: c,Xs2: list_c,Y: a,Ys2: list_a] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_c @ X @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_257_list__induct2_H,axiom,
    ! [P7: list_b > list_c > $o,Xs: list_b,Ys: list_c] :
      ( ( P7 @ nil_b @ nil_c )
     => ( ! [X: b,Xs2: list_b] : ( P7 @ ( cons_b @ X @ Xs2 ) @ nil_c )
       => ( ! [Y: c,Ys2: list_c] : ( P7 @ nil_b @ ( cons_c @ Y @ Ys2 ) )
         => ( ! [X: b,Xs2: list_b,Y: c,Ys2: list_c] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_b @ X @ Xs2 ) @ ( cons_c @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_258_list__induct2_H,axiom,
    ! [P7: list_b > list_b > $o,Xs: list_b,Ys: list_b] :
      ( ( P7 @ nil_b @ nil_b )
     => ( ! [X: b,Xs2: list_b] : ( P7 @ ( cons_b @ X @ Xs2 ) @ nil_b )
       => ( ! [Y: b,Ys2: list_b] : ( P7 @ nil_b @ ( cons_b @ Y @ Ys2 ) )
         => ( ! [X: b,Xs2: list_b,Y: b,Ys2: list_b] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_b @ X @ Xs2 ) @ ( cons_b @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_259_list__induct2_H,axiom,
    ! [P7: list_b > list_a > $o,Xs: list_b,Ys: list_a] :
      ( ( P7 @ nil_b @ nil_a )
     => ( ! [X: b,Xs2: list_b] : ( P7 @ ( cons_b @ X @ Xs2 ) @ nil_a )
       => ( ! [Y: a,Ys2: list_a] : ( P7 @ nil_b @ ( cons_a @ Y @ Ys2 ) )
         => ( ! [X: b,Xs2: list_b,Y: a,Ys2: list_a] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_b @ X @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_260_list__induct2_H,axiom,
    ! [P7: list_a > list_c > $o,Xs: list_a,Ys: list_c] :
      ( ( P7 @ nil_a @ nil_c )
     => ( ! [X: a,Xs2: list_a] : ( P7 @ ( cons_a @ X @ Xs2 ) @ nil_c )
       => ( ! [Y: c,Ys2: list_c] : ( P7 @ nil_a @ ( cons_c @ Y @ Ys2 ) )
         => ( ! [X: a,Xs2: list_a,Y: c,Ys2: list_c] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_a @ X @ Xs2 ) @ ( cons_c @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_261_list__induct2_H,axiom,
    ! [P7: list_a > list_b > $o,Xs: list_a,Ys: list_b] :
      ( ( P7 @ nil_a @ nil_b )
     => ( ! [X: a,Xs2: list_a] : ( P7 @ ( cons_a @ X @ Xs2 ) @ nil_b )
       => ( ! [Y: b,Ys2: list_b] : ( P7 @ nil_a @ ( cons_b @ Y @ Ys2 ) )
         => ( ! [X: a,Xs2: list_a,Y: b,Ys2: list_b] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_262_list__induct2_H,axiom,
    ! [P7: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
      ( ( P7 @ nil_a @ nil_a )
     => ( ! [X: a,Xs2: list_a] : ( P7 @ ( cons_a @ X @ Xs2 ) @ nil_a )
       => ( ! [Y: a,Ys2: list_a] : ( P7 @ nil_a @ ( cons_a @ Y @ Ys2 ) )
         => ( ! [X: a,Xs2: list_a,Y: a,Ys2: list_a] :
                ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys2 ) ) )
           => ( P7 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_263_neq__Nil__conv,axiom,
    ! [Xs: list_P5767727266746070163od_c_a] :
      ( ( Xs != nil_Pr5380409136740620691od_c_a )
      = ( ? [Y4: produc2133359284458116493od_c_a,Ys3: list_P5767727266746070163od_c_a] :
            ( Xs
            = ( cons_P7376644893796496707od_c_a @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_264_neq__Nil__conv,axiom,
    ! [Xs: list_s3765944269294497308od_c_a] :
      ( ( Xs != nil_se4123578418339231900od_c_a )
      = ( ? [Y4: set_Pr8357624050790843030od_c_a,Ys3: list_s3765944269294497308od_c_a] :
            ( Xs
            = ( cons_s8716283482848191564od_c_a @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_265_neq__Nil__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( Xs != nil_Product_prod_b_c )
      = ( ? [Y4: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] :
            ( Xs
            = ( cons_P4529483553340347422od_b_c @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_266_neq__Nil__conv,axiom,
    ! [Xs: list_l2826717958560394946od_c_a] :
      ( ( Xs != nil_li2790768921375229634od_c_a )
      = ( ? [Y4: list_P6327159017948738492od_c_a,Ys3: list_l2826717958560394946od_c_a] :
            ( Xs
            = ( cons_l7906879460341455474od_c_a @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_267_neq__Nil__conv,axiom,
    ! [Xs: list_c] :
      ( ( Xs != nil_c )
      = ( ? [Y4: c,Ys3: list_c] :
            ( Xs
            = ( cons_c @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_268_neq__Nil__conv,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
      = ( ? [Y4: b,Ys3: list_b] :
            ( Xs
            = ( cons_b @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_269_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
      = ( ? [Y4: a,Ys3: list_a] :
            ( Xs
            = ( cons_a @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_270_neq__Nil__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( Xs != nil_Pr1342775757158464060od_c_a )
      = ( ? [Y4: produc2669932201312201782od_c_a,Ys3: list_P6327159017948738492od_c_a] :
            ( Xs
            = ( cons_P8659537020926305772od_c_a @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_271_remdups__adj_Ocases,axiom,
    ! [X3: list_P5767727266746070163od_c_a] :
      ( ( X3 != nil_Pr5380409136740620691od_c_a )
     => ( ! [X: produc2133359284458116493od_c_a] :
            ( X3
           != ( cons_P7376644893796496707od_c_a @ X @ nil_Pr5380409136740620691od_c_a ) )
       => ~ ! [X: produc2133359284458116493od_c_a,Y: produc2133359284458116493od_c_a,Xs2: list_P5767727266746070163od_c_a] :
              ( X3
             != ( cons_P7376644893796496707od_c_a @ X @ ( cons_P7376644893796496707od_c_a @ Y @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_272_remdups__adj_Ocases,axiom,
    ! [X3: list_s3765944269294497308od_c_a] :
      ( ( X3 != nil_se4123578418339231900od_c_a )
     => ( ! [X: set_Pr8357624050790843030od_c_a] :
            ( X3
           != ( cons_s8716283482848191564od_c_a @ X @ nil_se4123578418339231900od_c_a ) )
       => ~ ! [X: set_Pr8357624050790843030od_c_a,Y: set_Pr8357624050790843030od_c_a,Xs2: list_s3765944269294497308od_c_a] :
              ( X3
             != ( cons_s8716283482848191564od_c_a @ X @ ( cons_s8716283482848191564od_c_a @ Y @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_273_remdups__adj_Ocases,axiom,
    ! [X3: list_P903359562653991662od_b_c] :
      ( ( X3 != nil_Product_prod_b_c )
     => ( ! [X: product_prod_b_c] :
            ( X3
           != ( cons_P4529483553340347422od_b_c @ X @ nil_Product_prod_b_c ) )
       => ~ ! [X: product_prod_b_c,Y: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( X3
             != ( cons_P4529483553340347422od_b_c @ X @ ( cons_P4529483553340347422od_b_c @ Y @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_274_remdups__adj_Ocases,axiom,
    ! [X3: list_l2826717958560394946od_c_a] :
      ( ( X3 != nil_li2790768921375229634od_c_a )
     => ( ! [X: list_P6327159017948738492od_c_a] :
            ( X3
           != ( cons_l7906879460341455474od_c_a @ X @ nil_li2790768921375229634od_c_a ) )
       => ~ ! [X: list_P6327159017948738492od_c_a,Y: list_P6327159017948738492od_c_a,Xs2: list_l2826717958560394946od_c_a] :
              ( X3
             != ( cons_l7906879460341455474od_c_a @ X @ ( cons_l7906879460341455474od_c_a @ Y @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_275_remdups__adj_Ocases,axiom,
    ! [X3: list_c] :
      ( ( X3 != nil_c )
     => ( ! [X: c] :
            ( X3
           != ( cons_c @ X @ nil_c ) )
       => ~ ! [X: c,Y: c,Xs2: list_c] :
              ( X3
             != ( cons_c @ X @ ( cons_c @ Y @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_276_remdups__adj_Ocases,axiom,
    ! [X3: list_b] :
      ( ( X3 != nil_b )
     => ( ! [X: b] :
            ( X3
           != ( cons_b @ X @ nil_b ) )
       => ~ ! [X: b,Y: b,Xs2: list_b] :
              ( X3
             != ( cons_b @ X @ ( cons_b @ Y @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_277_remdups__adj_Ocases,axiom,
    ! [X3: list_a] :
      ( ( X3 != nil_a )
     => ( ! [X: a] :
            ( X3
           != ( cons_a @ X @ nil_a ) )
       => ~ ! [X: a,Y: a,Xs2: list_a] :
              ( X3
             != ( cons_a @ X @ ( cons_a @ Y @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_278_remdups__adj_Ocases,axiom,
    ! [X3: list_P6327159017948738492od_c_a] :
      ( ( X3 != nil_Pr1342775757158464060od_c_a )
     => ( ! [X: produc2669932201312201782od_c_a] :
            ( X3
           != ( cons_P8659537020926305772od_c_a @ X @ nil_Pr1342775757158464060od_c_a ) )
       => ~ ! [X: produc2669932201312201782od_c_a,Y: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( X3
             != ( cons_P8659537020926305772od_c_a @ X @ ( cons_P8659537020926305772od_c_a @ Y @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_279_transpose_Ocases,axiom,
    ! [X3: list_l3398063940897752857od_c_a] :
      ( ( X3 != nil_li1148627188455554457od_c_a )
     => ( ! [Xss: list_l3398063940897752857od_c_a] :
            ( X3
           != ( cons_l2389005916891889993od_c_a @ nil_Pr5380409136740620691od_c_a @ Xss ) )
       => ~ ! [X: produc2133359284458116493od_c_a,Xs2: list_P5767727266746070163od_c_a,Xss: list_l3398063940897752857od_c_a] :
              ( X3
             != ( cons_l2389005916891889993od_c_a @ ( cons_P7376644893796496707od_c_a @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_280_transpose_Ocases,axiom,
    ! [X3: list_l7289853719222193442od_c_a] :
      ( ( X3 != nil_li9161528533388831010od_c_a )
     => ( ! [Xss: list_l7289853719222193442od_c_a] :
            ( X3
           != ( cons_l4685720819657030866od_c_a @ nil_se4123578418339231900od_c_a @ Xss ) )
       => ~ ! [X: set_Pr8357624050790843030od_c_a,Xs2: list_s3765944269294497308od_c_a,Xss: list_l7289853719222193442od_c_a] :
              ( X3
             != ( cons_l4685720819657030866od_c_a @ ( cons_s8716283482848191564od_c_a @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_281_transpose_Ocases,axiom,
    ! [X3: list_l8907847357763382004od_b_c] :
      ( ( X3 != nil_li8071162985807626740od_b_c )
     => ( ! [Xss: list_l8907847357763382004od_b_c] :
            ( X3
           != ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) )
       => ~ ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Xss: list_l8907847357763382004od_b_c] :
              ( X3
             != ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_282_transpose_Ocases,axiom,
    ! [X3: list_l8402490976804658504od_c_a] :
      ( ( X3 != nil_li7065782944283248328od_c_a )
     => ( ! [Xss: list_l8402490976804658504od_c_a] :
            ( X3
           != ( cons_l6182468158186947192od_c_a @ nil_li2790768921375229634od_c_a @ Xss ) )
       => ~ ! [X: list_P6327159017948738492od_c_a,Xs2: list_l2826717958560394946od_c_a,Xss: list_l8402490976804658504od_c_a] :
              ( X3
             != ( cons_l6182468158186947192od_c_a @ ( cons_l7906879460341455474od_c_a @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_283_transpose_Ocases,axiom,
    ! [X3: list_list_c] :
      ( ( X3 != nil_list_c )
     => ( ! [Xss: list_list_c] :
            ( X3
           != ( cons_list_c @ nil_c @ Xss ) )
       => ~ ! [X: c,Xs2: list_c,Xss: list_list_c] :
              ( X3
             != ( cons_list_c @ ( cons_c @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_284_transpose_Ocases,axiom,
    ! [X3: list_list_b] :
      ( ( X3 != nil_list_b )
     => ( ! [Xss: list_list_b] :
            ( X3
           != ( cons_list_b @ nil_b @ Xss ) )
       => ~ ! [X: b,Xs2: list_b,Xss: list_list_b] :
              ( X3
             != ( cons_list_b @ ( cons_b @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_285_transpose_Ocases,axiom,
    ! [X3: list_list_a] :
      ( ( X3 != nil_list_a )
     => ( ! [Xss: list_list_a] :
            ( X3
           != ( cons_list_a @ nil_a @ Xss ) )
       => ~ ! [X: a,Xs2: list_a,Xss: list_list_a] :
              ( X3
             != ( cons_list_a @ ( cons_a @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_286_transpose_Ocases,axiom,
    ! [X3: list_l2826717958560394946od_c_a] :
      ( ( X3 != nil_li2790768921375229634od_c_a )
     => ( ! [Xss: list_l2826717958560394946od_c_a] :
            ( X3
           != ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ Xss ) )
       => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Xss: list_l2826717958560394946od_c_a] :
              ( X3
             != ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_287_min__list_Ocases,axiom,
    ! [X3: list_s3765944269294497308od_c_a] :
      ( ! [X: set_Pr8357624050790843030od_c_a,Xs2: list_s3765944269294497308od_c_a] :
          ( X3
         != ( cons_s8716283482848191564od_c_a @ X @ Xs2 ) )
     => ( X3 = nil_se4123578418339231900od_c_a ) ) ).

% min_list.cases
thf(fact_288_list_Oexhaust,axiom,
    ! [Y3: list_P5767727266746070163od_c_a] :
      ( ( Y3 != nil_Pr5380409136740620691od_c_a )
     => ~ ! [X212: produc2133359284458116493od_c_a,X222: list_P5767727266746070163od_c_a] :
            ( Y3
           != ( cons_P7376644893796496707od_c_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_289_list_Oexhaust,axiom,
    ! [Y3: list_s3765944269294497308od_c_a] :
      ( ( Y3 != nil_se4123578418339231900od_c_a )
     => ~ ! [X212: set_Pr8357624050790843030od_c_a,X222: list_s3765944269294497308od_c_a] :
            ( Y3
           != ( cons_s8716283482848191564od_c_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_290_list_Oexhaust,axiom,
    ! [Y3: list_P903359562653991662od_b_c] :
      ( ( Y3 != nil_Product_prod_b_c )
     => ~ ! [X212: product_prod_b_c,X222: list_P903359562653991662od_b_c] :
            ( Y3
           != ( cons_P4529483553340347422od_b_c @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_291_list_Oexhaust,axiom,
    ! [Y3: list_l2826717958560394946od_c_a] :
      ( ( Y3 != nil_li2790768921375229634od_c_a )
     => ~ ! [X212: list_P6327159017948738492od_c_a,X222: list_l2826717958560394946od_c_a] :
            ( Y3
           != ( cons_l7906879460341455474od_c_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_292_list_Oexhaust,axiom,
    ! [Y3: list_c] :
      ( ( Y3 != nil_c )
     => ~ ! [X212: c,X222: list_c] :
            ( Y3
           != ( cons_c @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_293_list_Oexhaust,axiom,
    ! [Y3: list_b] :
      ( ( Y3 != nil_b )
     => ~ ! [X212: b,X222: list_b] :
            ( Y3
           != ( cons_b @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_294_list_Oexhaust,axiom,
    ! [Y3: list_a] :
      ( ( Y3 != nil_a )
     => ~ ! [X212: a,X222: list_a] :
            ( Y3
           != ( cons_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_295_list_Oexhaust,axiom,
    ! [Y3: list_P6327159017948738492od_c_a] :
      ( ( Y3 != nil_Pr1342775757158464060od_c_a )
     => ~ ! [X212: produc2669932201312201782od_c_a,X222: list_P6327159017948738492od_c_a] :
            ( Y3
           != ( cons_P8659537020926305772od_c_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_296_list_OdiscI,axiom,
    ! [List: list_P6327159017948738492od_c_a,X21: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a] :
      ( ( List
        = ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) )
     => ( List != nil_Pr1342775757158464060od_c_a ) ) ).

% list.discI
thf(fact_297_distinct_Osimps_I1_J,axiom,
    distinct_a @ nil_a ).

% distinct.simps(1)
thf(fact_298_distinct_Osimps_I1_J,axiom,
    distin811446047591323055od_c_a @ nil_Pr1342775757158464060od_c_a ).

% distinct.simps(1)
thf(fact_299_list_Odistinct_I1_J,axiom,
    ! [X21: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a] :
      ( nil_Pr1342775757158464060od_c_a
     != ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_300_not__distinct__decomp,axiom,
    ! [Ws: list_a] :
      ( ~ ( distinct_a @ Ws )
     => ? [Xs2: list_a,Ys2: list_a,Zs2: list_a,Y: a] :
          ( Ws
          = ( append_a @ Xs2 @ ( append_a @ ( cons_a @ Y @ nil_a ) @ ( append_a @ Ys2 @ ( append_a @ ( cons_a @ Y @ nil_a ) @ Zs2 ) ) ) ) ) ) ).

% not_distinct_decomp
thf(fact_301_not__distinct__decomp,axiom,
    ! [Ws: list_P6327159017948738492od_c_a] :
      ( ~ ( distin811446047591323055od_c_a @ Ws )
     => ? [Xs2: list_P6327159017948738492od_c_a,Ys2: list_P6327159017948738492od_c_a,Zs2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a] :
          ( Ws
          = ( append3884164603879732689od_c_a @ Xs2 @ ( append3884164603879732689od_c_a @ ( cons_P8659537020926305772od_c_a @ Y @ nil_Pr1342775757158464060od_c_a ) @ ( append3884164603879732689od_c_a @ Ys2 @ ( append3884164603879732689od_c_a @ ( cons_P8659537020926305772od_c_a @ Y @ nil_Pr1342775757158464060od_c_a ) @ Zs2 ) ) ) ) ) ) ).

% not_distinct_decomp
thf(fact_302_acyclic__path__from__cyclic__path,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ~ ( distinct_a @ ( visited_states_a_b_c @ Q @ P5 ) )
       => ~ ! [P8: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M @ Q @ P8 )
             => ( ( ( target_a_b_c @ Q @ P5 )
                  = ( target_a_b_c @ Q @ P8 ) )
               => ~ ( distinct_a @ ( visited_states_a_b_c @ Q @ P8 ) ) ) ) ) ) ).

% acyclic_path_from_cyclic_path
thf(fact_303_rev__nonempty__induct,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: list_P6327159017948738492od_c_a > $o] :
      ( ( Xs != nil_Pr1342775757158464060od_c_a )
     => ( ! [X: produc2669932201312201782od_c_a] : ( P7 @ ( cons_P8659537020926305772od_c_a @ X @ nil_Pr1342775757158464060od_c_a ) )
       => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( ( Xs2 != nil_Pr1342775757158464060od_c_a )
             => ( ( P7 @ Xs2 )
               => ( P7 @ ( append3884164603879732689od_c_a @ Xs2 @ ( cons_P8659537020926305772od_c_a @ X @ nil_Pr1342775757158464060od_c_a ) ) ) ) )
         => ( P7 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_304_append__eq__Cons__conv,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Ys @ Zs )
        = ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( ( ( Ys = nil_Pr1342775757158464060od_c_a )
          & ( Zs
            = ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) ) )
        | ? [Ys4: list_P6327159017948738492od_c_a] :
            ( ( Ys
              = ( cons_P8659537020926305772od_c_a @ X3 @ Ys4 ) )
            & ( ( append3884164603879732689od_c_a @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_305_Cons__eq__append__conv,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( cons_P8659537020926305772od_c_a @ X3 @ Xs )
        = ( append3884164603879732689od_c_a @ Ys @ Zs ) )
      = ( ( ( Ys = nil_Pr1342775757158464060od_c_a )
          & ( ( cons_P8659537020926305772od_c_a @ X3 @ Xs )
            = Zs ) )
        | ? [Ys4: list_P6327159017948738492od_c_a] :
            ( ( ( cons_P8659537020926305772od_c_a @ X3 @ Ys4 )
              = Ys )
            & ( Xs
              = ( append3884164603879732689od_c_a @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_306_rev__exhaust,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( Xs != nil_Pr1342775757158464060od_c_a )
     => ~ ! [Ys2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a] :
            ( Xs
           != ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ Y @ nil_Pr1342775757158464060od_c_a ) ) ) ) ).

% rev_exhaust
thf(fact_307_rev__induct,axiom,
    ! [P7: list_P6327159017948738492od_c_a > $o,Xs: list_P6327159017948738492od_c_a] :
      ( ( P7 @ nil_Pr1342775757158464060od_c_a )
     => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
            ( ( P7 @ Xs2 )
           => ( P7 @ ( append3884164603879732689od_c_a @ Xs2 @ ( cons_P8659537020926305772od_c_a @ X @ nil_Pr1342775757158464060od_c_a ) ) ) )
       => ( P7 @ Xs ) ) ) ).

% rev_induct
thf(fact_308_cyclic__path__loop,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ~ ( distinct_a @ ( visited_states_a_b_c @ Q @ P5 ) )
       => ? [P12: list_P6327159017948738492od_c_a,P22: list_P6327159017948738492od_c_a,P32: list_P6327159017948738492od_c_a] :
            ( ( P5
              = ( append3884164603879732689od_c_a @ P12 @ ( append3884164603879732689od_c_a @ P22 @ P32 ) ) )
            & ( P22 != nil_Pr1342775757158464060od_c_a )
            & ( ( target_a_b_c @ Q @ P12 )
              = ( target_a_b_c @ Q @ ( append3884164603879732689od_c_a @ P12 @ P22 ) ) ) ) ) ) ).

% cyclic_path_loop
thf(fact_309_path__target__append,axiom,
    ! [Q1: a,P1: list_P6327159017948738492od_c_a,Q22: a,P2: list_P6327159017948738492od_c_a,Q3: a] :
      ( ( ( target_a_b_c @ Q1 @ P1 )
        = Q22 )
     => ( ( ( target_a_b_c @ Q22 @ P2 )
          = Q3 )
       => ( ( target_a_b_c @ Q1 @ ( append3884164603879732689od_c_a @ P1 @ P2 ) )
          = Q3 ) ) ) ).

% path_target_append
thf(fact_310_path__append__target,axiom,
    ! [Q: a,P1: list_P6327159017948738492od_c_a,P2: list_P6327159017948738492od_c_a] :
      ( ( target_a_b_c @ Q @ ( append3884164603879732689od_c_a @ P1 @ P2 ) )
      = ( target_a_b_c @ ( target_a_b_c @ Q @ P1 ) @ P2 ) ) ).

% path_append_target
thf(fact_311_path__append__transition__elim_I2_J,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,T: produc2669932201312201782od_c_a] :
      ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P5 @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) ) )
     => ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) ) ) ).

% path_append_transition_elim(2)
thf(fact_312_path__prepend__t,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q4: product_prod_c_a,P5: list_P1562850763667305982od_c_a,Q: product_prod_c_a,X3: a,Y3: b] :
      ( ( path_P1107610407223809260_a_a_b @ M @ Q4 @ P5 )
     => ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q @ ( produc5762801557034676134od_c_a @ X3 @ ( produc3440865914000881661od_c_a @ Y3 @ Q4 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) )
       => ( path_P1107610407223809260_a_a_b @ M @ Q @ ( cons_P5525787621418763054od_c_a @ ( produc7911353075612250984od_c_a @ Q @ ( produc5762801557034676134od_c_a @ X3 @ ( produc3440865914000881661od_c_a @ Y3 @ Q4 ) ) ) @ P5 ) ) ) ) ).

% path_prepend_t
thf(fact_313_path__prepend__t,axiom,
    ! [M: fsm_a_b_c,Q4: a,P5: list_P6327159017948738492od_c_a,Q: a,X3: b,Y3: c] :
      ( ( path_a_b_c @ M @ Q4 @ P5 )
     => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X3 @ ( product_Pair_c_a @ Y3 @ Q4 ) ) ) @ ( transitions_a_b_c @ M ) )
       => ( path_a_b_c @ M @ Q @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X3 @ ( product_Pair_c_a @ Y3 @ Q4 ) ) ) @ P5 ) ) ) ) ).

% path_prepend_t
thf(fact_314_Cons__eq__appendI,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs1: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( cons_P8659537020926305772od_c_a @ X3 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append3884164603879732689od_c_a @ Xs1 @ Zs ) )
       => ( ( cons_P8659537020926305772od_c_a @ X3 @ Xs )
          = ( append3884164603879732689od_c_a @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_315_append__Cons,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ Ys )
      = ( cons_P8659537020926305772od_c_a @ X3 @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_316_eq__Nil__appendI,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append3884164603879732689od_c_a @ nil_Pr1342775757158464060od_c_a @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_317_append_Oleft__neutral,axiom,
    ! [A: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ nil_Pr1342775757158464060od_c_a @ A )
      = A ) ).

% append.left_neutral
thf(fact_318_append__Nil,axiom,
    ! [Ys: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ nil_Pr1342775757158464060od_c_a @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_319_cyclic__cycle,axiom,
    ! [M: fsm_a_b_c] :
      ( ~ ( acyclic_a_b_c @ M )
     => ? [Q5: a,P: list_P6327159017948738492od_c_a] :
          ( ( path_a_b_c @ M @ Q5 @ P )
          & ( P != nil_Pr1342775757158464060od_c_a )
          & ( ( target_a_b_c @ Q5 @ P )
            = Q5 ) ) ) ).

% cyclic_cycle
thf(fact_320_acyclic__paths__from__reachable__states,axiom,
    ! [M: fsm_a_b_c,P9: list_P6327159017948738492od_c_a,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( acyclic_a_b_c @ M )
     => ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ P9 )
       => ( ( ( target_a_b_c @ ( initial_a_b_c @ M ) @ P9 )
            = Q )
         => ( ( path_a_b_c @ M @ Q @ P5 )
           => ( distinct_a @ ( visited_states_a_b_c @ Q @ P5 ) ) ) ) ) ) ).

% acyclic_paths_from_reachable_states
thf(fact_321_path__append__elim,axiom,
    ! [M: fsm_a_b_c,Q: a,P1: list_P6327159017948738492od_c_a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P1 @ P2 ) )
     => ~ ( ( path_a_b_c @ M @ Q @ P1 )
         => ~ ( path_a_b_c @ M @ ( target_a_b_c @ Q @ P1 ) @ P2 ) ) ) ).

% path_append_elim
thf(fact_322_path__suffix,axiom,
    ! [M: fsm_a_b_c,Q: a,P1: list_P6327159017948738492od_c_a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P1 @ P2 ) )
     => ( path_a_b_c @ M @ ( target_a_b_c @ Q @ P1 ) @ P2 ) ) ).

% path_suffix
thf(fact_323_cyclic__cycle__rev,axiom,
    ! [M: fsm_a_b_c,P9: list_P6327159017948738492od_c_a,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ P9 )
     => ( ( ( target_a_b_c @ ( initial_a_b_c @ M ) @ P9 )
          = Q )
       => ( ( path_a_b_c @ M @ Q @ P5 )
         => ( ( P5 != nil_Pr1342775757158464060od_c_a )
           => ( ( ( target_a_b_c @ Q @ P5 )
                = Q )
             => ~ ( acyclic_a_b_c @ M ) ) ) ) ) ) ).

% cyclic_cycle_rev
thf(fact_324_prod__induct4,axiom,
    ! [P7: produc2669932201312201782od_c_a > $o,X3: produc2669932201312201782od_c_a] :
      ( ! [A4: a,B3: b,C2: c,D: a] : ( P7 @ ( produc5762801557034676134od_c_a @ A4 @ ( produc3440865914000881661od_c_a @ B3 @ ( product_Pair_c_a @ C2 @ D ) ) ) )
     => ( P7 @ X3 ) ) ).

% prod_induct4
thf(fact_325_prod__induct3,axiom,
    ! [P7: produc2669932201312201782od_c_a > $o,X3: produc2669932201312201782od_c_a] :
      ( ! [A4: a,B3: b,C2: product_prod_c_a] : ( P7 @ ( produc5762801557034676134od_c_a @ A4 @ ( produc3440865914000881661od_c_a @ B3 @ C2 ) ) )
     => ( P7 @ X3 ) ) ).

% prod_induct3
thf(fact_326_prod__induct3,axiom,
    ! [P7: produc2133359284458116493od_c_a > $o,X3: produc2133359284458116493od_c_a] :
      ( ! [A4: b,B3: c,C2: a] : ( P7 @ ( produc3440865914000881661od_c_a @ A4 @ ( product_Pair_c_a @ B3 @ C2 ) ) )
     => ( P7 @ X3 ) ) ).

% prod_induct3
thf(fact_327_prod__cases4,axiom,
    ! [Y3: produc2669932201312201782od_c_a] :
      ~ ! [A4: a,B3: b,C2: c,D: a] :
          ( Y3
         != ( produc5762801557034676134od_c_a @ A4 @ ( produc3440865914000881661od_c_a @ B3 @ ( product_Pair_c_a @ C2 @ D ) ) ) ) ).

% prod_cases4
thf(fact_328_prod__cases3,axiom,
    ! [Y3: produc2669932201312201782od_c_a] :
      ~ ! [A4: a,B3: b,C2: product_prod_c_a] :
          ( Y3
         != ( produc5762801557034676134od_c_a @ A4 @ ( produc3440865914000881661od_c_a @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_329_prod__cases3,axiom,
    ! [Y3: produc2133359284458116493od_c_a] :
      ~ ! [A4: b,B3: c,C2: a] :
          ( Y3
         != ( produc3440865914000881661od_c_a @ A4 @ ( product_Pair_c_a @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_330_Pair__inject,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A2: a,B2: produc2133359284458116493od_c_a] :
      ( ( ( produc5762801557034676134od_c_a @ A @ B )
        = ( produc5762801557034676134od_c_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_331_Pair__inject,axiom,
    ! [A: b,B: product_prod_c_a,A2: b,B2: product_prod_c_a] :
      ( ( ( produc3440865914000881661od_c_a @ A @ B )
        = ( produc3440865914000881661od_c_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_332_Pair__inject,axiom,
    ! [A: c,B: a,A2: c,B2: a] :
      ( ( ( product_Pair_c_a @ A @ B )
        = ( product_Pair_c_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_333_prod__cases,axiom,
    ! [P7: produc2669932201312201782od_c_a > $o,P5: produc2669932201312201782od_c_a] :
      ( ! [A4: a,B3: produc2133359284458116493od_c_a] : ( P7 @ ( produc5762801557034676134od_c_a @ A4 @ B3 ) )
     => ( P7 @ P5 ) ) ).

% prod_cases
thf(fact_334_prod__cases,axiom,
    ! [P7: produc2133359284458116493od_c_a > $o,P5: produc2133359284458116493od_c_a] :
      ( ! [A4: b,B3: product_prod_c_a] : ( P7 @ ( produc3440865914000881661od_c_a @ A4 @ B3 ) )
     => ( P7 @ P5 ) ) ).

% prod_cases
thf(fact_335_prod__cases,axiom,
    ! [P7: product_prod_c_a > $o,P5: product_prod_c_a] :
      ( ! [A4: c,B3: a] : ( P7 @ ( product_Pair_c_a @ A4 @ B3 ) )
     => ( P7 @ P5 ) ) ).

% prod_cases
thf(fact_336_surj__pair,axiom,
    ! [P5: produc2669932201312201782od_c_a] :
    ? [X: a,Y: produc2133359284458116493od_c_a] :
      ( P5
      = ( produc5762801557034676134od_c_a @ X @ Y ) ) ).

% surj_pair
thf(fact_337_surj__pair,axiom,
    ! [P5: produc2133359284458116493od_c_a] :
    ? [X: b,Y: product_prod_c_a] :
      ( P5
      = ( produc3440865914000881661od_c_a @ X @ Y ) ) ).

% surj_pair
thf(fact_338_surj__pair,axiom,
    ! [P5: product_prod_c_a] :
    ? [X: c,Y: a] :
      ( P5
      = ( product_Pair_c_a @ X @ Y ) ) ).

% surj_pair
thf(fact_339_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2669932201312201782od_c_a] :
      ~ ! [A4: a,B3: produc2133359284458116493od_c_a] :
          ( Y3
         != ( produc5762801557034676134od_c_a @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_340_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2133359284458116493od_c_a] :
      ~ ! [A4: b,B3: product_prod_c_a] :
          ( Y3
         != ( produc3440865914000881661od_c_a @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_341_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_c_a] :
      ~ ! [A4: c,B3: a] :
          ( Y3
         != ( product_Pair_c_a @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_342_append__eq__append__conv2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,Ts: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Xs @ Ys )
        = ( append3884164603879732689od_c_a @ Zs @ Ts ) )
      = ( ? [Us: list_P6327159017948738492od_c_a] :
            ( ( ( Xs
                = ( append3884164603879732689od_c_a @ Zs @ Us ) )
              & ( ( append3884164603879732689od_c_a @ Us @ Ys )
                = Ts ) )
            | ( ( ( append3884164603879732689od_c_a @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append3884164603879732689od_c_a @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_343_append__eq__appendI,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Xs1: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Us2: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append3884164603879732689od_c_a @ Xs1 @ Us2 ) )
       => ( ( append3884164603879732689od_c_a @ Xs @ Ys )
          = ( append3884164603879732689od_c_a @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_344_path__append__transition__elim_I1_J,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,T: produc2669932201312201782od_c_a] :
      ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P5 @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) ) )
     => ( path_a_b_c @ M @ Q @ P5 ) ) ).

% path_append_transition_elim(1)
thf(fact_345_acyclic_Osimps,axiom,
    ( acyclic_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [P4: list_P6327159017948738492od_c_a] :
          ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P4 )
         => ( distinct_a @ ( visited_states_a_b_c @ ( initial_a_b_c @ M2 ) @ P4 ) ) ) ) ) ).

% acyclic.simps
thf(fact_346_reachable__def,axiom,
    ( reachable_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a] :
        ? [P4: list_P6327159017948738492od_c_a] :
          ( ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P4 )
          & ( ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P4 )
            = Q6 ) ) ) ) ).

% reachable_def
thf(fact_347_delete__aux_Ocases,axiom,
    ! [X3: produc3220416805844688357od_c_a] :
      ( ! [K: a] :
          ( X3
         != ( produc3580641852002036821od_c_a @ K @ nil_Pr1342775757158464060od_c_a ) )
     => ~ ! [K: a,K2: a,V: produc2133359284458116493od_c_a,Xs2: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc3580641852002036821od_c_a @ K @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K2 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_348_delete__aux_Ocases,axiom,
    ! [X3: produc7270830264722373115od_c_a] :
      ( ! [K: b] :
          ( X3
         != ( produc1443866078074754795od_c_a @ K @ nil_Pr5380409136740620691od_c_a ) )
     => ~ ! [K: b,K2: b,V: product_prod_c_a,Xs2: list_P5767727266746070163od_c_a] :
            ( X3
           != ( produc1443866078074754795od_c_a @ K @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K2 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_349_delete__aux_Ocases,axiom,
    ! [X3: produc8651263503759922642od_c_a] :
      ( ! [K: c] :
          ( X3
         != ( produc3490508561035776450od_c_a @ K @ nil_Product_prod_c_a ) )
     => ~ ! [K: c,K2: c,V: a,Xs2: list_P125642481956313003od_c_a] :
            ( X3
           != ( produc3490508561035776450od_c_a @ K @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K2 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_350_distinct__union,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( distinct_a @ ( union_a @ Xs @ Ys ) )
      = ( distinct_a @ Ys ) ) ).

% distinct_union
thf(fact_351_path__append__transition,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,T: produc2669932201312201782od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ( ( ( produc4225943634317630866od_c_a @ T )
            = ( target_a_b_c @ Q @ P5 ) )
         => ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P5 @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) ) ) ) ) ) ).

% path_append_transition
thf(fact_352_fsm__impl__from__list_H_Ocases,axiom,
    ! [X3: produc3220416805844688357od_c_a] :
      ( ! [Q5: a] :
          ( X3
         != ( produc3580641852002036821od_c_a @ Q5 @ nil_Pr1342775757158464060od_c_a ) )
     => ~ ! [Q5: a,T2: produc2669932201312201782od_c_a,Ts2: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc3580641852002036821od_c_a @ Q5 @ ( cons_P8659537020926305772od_c_a @ T2 @ Ts2 ) ) ) ) ).

% fsm_impl_from_list'.cases
thf(fact_353_deterministic__alt__def,axiom,
    ( determ4061018006534944583_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q12: product_prod_c_a,X4: a,Y5: b,Y6: b,Q13: product_prod_c_a,Q14: product_prod_c_a] :
          ( ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q12 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y5 @ Q13 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) )
            & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q12 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y6 @ Q14 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) ) )
         => ( ( Y5 = Y6 )
            & ( Q13 = Q14 ) ) ) ) ) ).

% deterministic_alt_def
thf(fact_354_deterministic__alt__def,axiom,
    ( deterministic_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q12: a,X4: b,Y5: c,Y6: c,Q13: a,Q14: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q12 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y5 @ Q13 ) ) ) @ ( transitions_a_b_c @ M2 ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q12 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y6 @ Q14 ) ) ) @ ( transitions_a_b_c @ M2 ) ) )
         => ( ( Y5 = Y6 )
            & ( Q13 = Q14 ) ) ) ) ) ).

% deterministic_alt_def
thf(fact_355_single__input__alt__def,axiom,
    ( single999786218259502647_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q12: product_prod_c_a,X4: a,X5: a,Y4: b,Y5: b,Q13: product_prod_c_a,Q14: product_prod_c_a] :
          ( ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q12 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q13 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) )
            & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q12 @ ( produc5762801557034676134od_c_a @ X5 @ ( produc3440865914000881661od_c_a @ Y5 @ Q14 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) ) )
         => ( X4 = X5 ) ) ) ) ).

% single_input_alt_def
thf(fact_356_single__input__alt__def,axiom,
    ( single_input_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q12: a,X4: b,X5: b,Y4: c,Y5: c,Q13: a,Q14: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q12 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q13 ) ) ) @ ( transitions_a_b_c @ M2 ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q12 @ ( produc3440865914000881661od_c_a @ X5 @ ( product_Pair_c_a @ Y5 @ Q14 ) ) ) @ ( transitions_a_b_c @ M2 ) ) )
         => ( X4 = X5 ) ) ) ) ).

% single_input_alt_def
thf(fact_357_fst__eqD,axiom,
    ! [X3: a,Y3: produc2133359284458116493od_c_a,A: a] :
      ( ( ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_358_fst__eqD,axiom,
    ! [X3: b,Y3: product_prod_c_a,A: b] :
      ( ( ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_359_fst__eqD,axiom,
    ! [X3: c,Y3: a,A: c] :
      ( ( ( product_fst_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_360_fst__conv,axiom,
    ! [X1: a,X2: produc2133359284458116493od_c_a] :
      ( ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_361_fst__conv,axiom,
    ! [X1: b,X2: product_prod_c_a] :
      ( ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_362_fst__conv,axiom,
    ! [X1: c,X2: a] :
      ( ( product_fst_c_a @ ( product_Pair_c_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_363_single__transition__path,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( path_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) ) ) ).

% single_transition_path
thf(fact_364_path__append__transition__elim_I3_J,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,T: produc2669932201312201782od_c_a] :
      ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P5 @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) ) )
     => ( ( produc4225943634317630866od_c_a @ T )
        = ( target_a_b_c @ Q @ P5 ) ) ) ).

% path_append_transition_elim(3)
thf(fact_365_linear__order__from__list__position_H_Ocases,axiom,
    ! [X3: list_P6327159017948738492od_c_a] :
      ( ( X3 != nil_Pr1342775757158464060od_c_a )
     => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
            ( X3
           != ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_366_find__remove_H_Ocases,axiom,
    ! [X3: produc1930072179473917589od_c_a] :
      ( ! [P6: produc2669932201312201782od_c_a > $o,Uu: list_P6327159017948738492od_c_a] :
          ( X3
         != ( produc3975879759346939847od_c_a @ P6 @ ( produc8447801329164731415od_c_a @ nil_Pr1342775757158464060od_c_a @ Uu ) ) )
     => ~ ! [P6: produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Prev: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc3975879759346939847od_c_a @ P6 @ ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_367_find__index_Ocases,axiom,
    ! [X3: produc3031831606050898730od_c_a] :
      ( ! [F: produc2669932201312201782od_c_a > $o] :
          ( X3
         != ( produc4492108061243766108od_c_a @ F @ nil_Pr1342775757158464060od_c_a ) )
     => ~ ! [F: produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc4492108061243766108od_c_a @ F @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) ) ) ).

% find_index.cases
thf(fact_368_is__prefix_Ocases,axiom,
    ! [X3: produc8143030500101326375od_c_a] :
      ( ! [Uu: list_P6327159017948738492od_c_a] :
          ( X3
         != ( produc8447801329164731415od_c_a @ nil_Pr1342775757158464060od_c_a @ Uu ) )
     => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ nil_Pr1342775757158464060od_c_a ) )
       => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a] :
              ( X3
             != ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_369_h__obs_Ocases,axiom,
    ! [X3: produc6163117450648659381od_c_a] :
      ~ ! [M3: fSM_fs6392807071807805864od_c_a,Q5: a,X: b,Y: product_prod_c_a] :
          ( X3
         != ( produc7402072633542172453od_c_a @ M3 @ ( produc5762801557034676134od_c_a @ Q5 @ ( produc3440865914000881661od_c_a @ X @ Y ) ) ) ) ).

% h_obs.cases
thf(fact_370_h__obs_Ocases,axiom,
    ! [X3: produc5988948626131613619od_c_a] :
      ~ ! [M3: fSM_fsm_impl_b_c_a,Q5: b,X: c,Y: a] :
          ( X3
         != ( produc867945525367740067od_c_a @ M3 @ ( produc3440865914000881661od_c_a @ Q5 @ ( product_Pair_c_a @ X @ Y ) ) ) ) ).

% h_obs.cases
thf(fact_371_clearjunk_Ocases,axiom,
    ! [X3: list_P6327159017948738492od_c_a] :
      ( ( X3 != nil_Pr1342775757158464060od_c_a )
     => ~ ! [P: produc2669932201312201782od_c_a,Ps: list_P6327159017948738492od_c_a] :
            ( X3
           != ( cons_P8659537020926305772od_c_a @ P @ Ps ) ) ) ).

% clearjunk.cases
thf(fact_372_map__default_Ocases,axiom,
    ! [X3: produc2329034731390255797od_c_a] :
      ( ! [K: a,V: produc2133359284458116493od_c_a,F: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a] :
          ( X3
         != ( produc4844797264803247151od_c_a @ K @ ( produc8267950104566546306od_c_a @ V @ ( produc588459825458708734od_c_a @ F @ nil_Pr1342775757158464060od_c_a ) ) ) )
     => ~ ! [K: a,V: produc2133359284458116493od_c_a,F: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,P: produc2669932201312201782od_c_a,Ps: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc4844797264803247151od_c_a @ K @ ( produc8267950104566546306od_c_a @ V @ ( produc588459825458708734od_c_a @ F @ ( cons_P8659537020926305772od_c_a @ P @ Ps ) ) ) ) ) ) ).

% map_default.cases
thf(fact_373_map__entry_Ocases,axiom,
    ! [X3: produc4958164698491676665od_c_a] :
      ( ! [K: a,F: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a] :
          ( X3
         != ( produc7215935238510903923od_c_a @ K @ ( produc588459825458708734od_c_a @ F @ nil_Pr1342775757158464060od_c_a ) ) )
     => ~ ! [K: a,F: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,P: produc2669932201312201782od_c_a,Ps: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc7215935238510903923od_c_a @ K @ ( produc588459825458708734od_c_a @ F @ ( cons_P8659537020926305772od_c_a @ P @ Ps ) ) ) ) ) ).

% map_entry.cases
thf(fact_374_longest__common__prefix_Ocases,axiom,
    ! [X3: produc8143030500101326375od_c_a] :
      ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a] :
          ( X3
         != ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) ) )
     => ( ! [Uv: list_P6327159017948738492od_c_a] :
            ( X3
           != ( produc8447801329164731415od_c_a @ nil_Pr1342775757158464060od_c_a @ Uv ) )
       => ~ ! [Uu: list_P6327159017948738492od_c_a] :
              ( X3
             != ( produc8447801329164731415od_c_a @ Uu @ nil_Pr1342775757158464060od_c_a ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_375_suffixes_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_l2826717958560394946od_c_a] :
      ( ( ( suffix1038733475819007016od_c_a @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
         => ( Y3
           != ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) )
       => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( ( X3
                = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
             => ( Y3
               != ( append2943488786620085207od_c_a @ ( suffix1038733475819007016od_c_a @ Xs2 ) @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ nil_li2790768921375229634od_c_a ) ) ) ) ) ) ).

% suffixes.elims
thf(fact_376_observable__alt__def,axiom,
    ( observ2964502264774598958_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q12: product_prod_c_a,X4: a,Y4: b,Q13: product_prod_c_a,Q14: product_prod_c_a] :
          ( ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q12 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q13 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) )
            & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q12 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q14 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) ) )
         => ( Q13 = Q14 ) ) ) ) ).

% observable_alt_def
thf(fact_377_observable__alt__def,axiom,
    ( observable_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q12: a,X4: b,Y4: c,Q13: a,Q14: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q12 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q13 ) ) ) @ ( transitions_a_b_c @ M2 ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q12 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q14 ) ) ) @ ( transitions_a_b_c @ M2 ) ) )
         => ( Q13 = Q14 ) ) ) ) ).

% observable_alt_def
thf(fact_378_fstI,axiom,
    ! [X3: produc2669932201312201782od_c_a,Y3: a,Z: produc2133359284458116493od_c_a] :
      ( ( X3
        = ( produc5762801557034676134od_c_a @ Y3 @ Z ) )
     => ( ( produc4225943634317630866od_c_a @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_379_fstI,axiom,
    ! [X3: produc2133359284458116493od_c_a,Y3: b,Z: product_prod_c_a] :
      ( ( X3
        = ( produc3440865914000881661od_c_a @ Y3 @ Z ) )
     => ( ( produc301949074746585577od_c_a @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_380_fstI,axiom,
    ! [X3: product_prod_c_a,Y3: c,Z: a] :
      ( ( X3
        = ( product_Pair_c_a @ Y3 @ Z ) )
     => ( ( product_fst_c_a @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_381_eq__fst__iff,axiom,
    ! [A: a,P5: produc2669932201312201782od_c_a] :
      ( ( A
        = ( produc4225943634317630866od_c_a @ P5 ) )
      = ( ? [B4: produc2133359284458116493od_c_a] :
            ( P5
            = ( produc5762801557034676134od_c_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_382_eq__fst__iff,axiom,
    ! [A: b,P5: produc2133359284458116493od_c_a] :
      ( ( A
        = ( produc301949074746585577od_c_a @ P5 ) )
      = ( ? [B4: product_prod_c_a] :
            ( P5
            = ( produc3440865914000881661od_c_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_383_eq__fst__iff,axiom,
    ! [A: c,P5: product_prod_c_a] :
      ( ( A
        = ( product_fst_c_a @ P5 ) )
      = ( ? [B4: a] :
            ( P5
            = ( product_Pair_c_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_384_Util_Osuffixes_Osimps_I1_J,axiom,
    ( ( suffix1038733475819007016od_c_a @ nil_Pr1342775757158464060od_c_a )
    = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) ).

% Util.suffixes.simps(1)
thf(fact_385_pow__list_Osimps_I1_J,axiom,
    ( ( pow_li2753411374138005421od_c_a @ nil_Pr1342775757158464060od_c_a )
    = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) ).

% pow_list.simps(1)
thf(fact_386_bind__simps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,F2: produc2669932201312201782od_c_a > list_P6327159017948738492od_c_a] :
      ( ( bind_P42852874897783815od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ F2 )
      = ( append3884164603879732689od_c_a @ ( F2 @ X3 ) @ ( bind_P42852874897783815od_c_a @ Xs @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_387_bind__simps_I1_J,axiom,
    ! [F2: produc2669932201312201782od_c_a > list_P6327159017948738492od_c_a] :
      ( ( bind_P42852874897783815od_c_a @ nil_Pr1342775757158464060od_c_a @ F2 )
      = nil_Pr1342775757158464060od_c_a ) ).

% bind_simps(1)
thf(fact_388_Util_Osuffixes_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( suffix1038733475819007016od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( append2943488786620085207od_c_a @ ( suffix1038733475819007016od_c_a @ Xs ) @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ nil_li2790768921375229634od_c_a ) ) ) ).

% Util.suffixes.simps(2)
thf(fact_389_prefixes__snoc,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
      ( ( prefix8878233912999784899od_c_a @ ( append3884164603879732689od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) )
      = ( append2943488786620085207od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) @ ( cons_l7906879460341455474od_c_a @ ( append3884164603879732689od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) @ nil_li2790768921375229634od_c_a ) ) ) ).

% prefixes_snoc
thf(fact_390_suffixes__eq__snoc,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a,X3: list_P6327159017948738492od_c_a] :
      ( ( ( suffix7794987338261656196od_c_a @ Ys )
        = ( append2943488786620085207od_c_a @ Xs @ ( cons_l7906879460341455474od_c_a @ X3 @ nil_li2790768921375229634od_c_a ) ) )
      = ( ( ( ( Ys = nil_Pr1342775757158464060od_c_a )
            & ( Xs = nil_li2790768921375229634od_c_a ) )
          | ? [Z2: produc2669932201312201782od_c_a,Zs3: list_P6327159017948738492od_c_a] :
              ( ( Ys
                = ( cons_P8659537020926305772od_c_a @ Z2 @ Zs3 ) )
              & ( Xs
                = ( suffix7794987338261656196od_c_a @ Zs3 ) ) ) )
        & ( X3 = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_391_prefixes__eq__snoc,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a,X3: list_P6327159017948738492od_c_a] :
      ( ( ( prefix8878233912999784899od_c_a @ Ys )
        = ( append2943488786620085207od_c_a @ Xs @ ( cons_l7906879460341455474od_c_a @ X3 @ nil_li2790768921375229634od_c_a ) ) )
      = ( ( ( ( Ys = nil_Pr1342775757158464060od_c_a )
            & ( Xs = nil_li2790768921375229634od_c_a ) )
          | ? [Z2: produc2669932201312201782od_c_a,Zs3: list_P6327159017948738492od_c_a] :
              ( ( Ys
                = ( append3884164603879732689od_c_a @ Zs3 @ ( cons_P8659537020926305772od_c_a @ Z2 @ nil_Pr1342775757158464060od_c_a ) ) )
              & ( Xs
                = ( prefix8878233912999784899od_c_a @ Zs3 ) ) ) )
        & ( X3 = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_392_sublists_Osimps_I1_J,axiom,
    ( ( sublis1354033275501720270od_c_a @ nil_Pr1342775757158464060od_c_a )
    = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) ).

% sublists.simps(1)
thf(fact_393_product__lists_Osimps_I1_J,axiom,
    ( ( produc4300910528168050469od_c_a @ nil_li2790768921375229634od_c_a )
    = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) ).

% product_lists.simps(1)
thf(fact_394_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: produc2133359284458116493od_c_a,K3: a,F2: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a] :
      ( ( update3791493425278745523_c_a_a @ V2 @ K3 @ F2 @ nil_Pr1342775757158464060od_c_a )
      = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ ( F2 @ V2 ) ) @ nil_Pr1342775757158464060od_c_a ) ) ).

% update_with_aux.simps(1)
thf(fact_395_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: product_prod_c_a,K3: b,F2: product_prod_c_a > product_prod_c_a] :
      ( ( update338640773394215068_c_a_b @ V2 @ K3 @ F2 @ nil_Pr5380409136740620691od_c_a )
      = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ ( F2 @ V2 ) ) @ nil_Pr5380409136740620691od_c_a ) ) ).

% update_with_aux.simps(1)
thf(fact_396_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: a,K3: c,F2: a > a] :
      ( ( update_with_aux_a_c @ V2 @ K3 @ F2 @ nil_Product_prod_c_a )
      = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ ( F2 @ V2 ) ) @ nil_Product_prod_c_a ) ) ).

% update_with_aux.simps(1)
thf(fact_397_concat__eq__append__conv,axiom,
    ! [Xss2: list_l2826717958560394946od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( concat7533993459791330935od_c_a @ Xss2 )
        = ( append3884164603879732689od_c_a @ Ys @ Zs ) )
      = ( ( ( Xss2 = nil_li2790768921375229634od_c_a )
         => ( ( Ys = nil_Pr1342775757158464060od_c_a )
            & ( Zs = nil_Pr1342775757158464060od_c_a ) ) )
        & ( ( Xss2 != nil_li2790768921375229634od_c_a )
         => ? [Xss1: list_l2826717958560394946od_c_a,Xs3: list_P6327159017948738492od_c_a,Xs4: list_P6327159017948738492od_c_a,Xss22: list_l2826717958560394946od_c_a] :
              ( ( Xss2
                = ( append2943488786620085207od_c_a @ Xss1 @ ( cons_l7906879460341455474od_c_a @ ( append3884164603879732689od_c_a @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys
                = ( append3884164603879732689od_c_a @ ( concat7533993459791330935od_c_a @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append3884164603879732689od_c_a @ Xs4 @ ( concat7533993459791330935od_c_a @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_398_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) @ ( append3884164603879732689od_c_a @ Ys @ ( cons_P8659537020926305772od_c_a @ Y3 @ nil_Pr1342775757158464060od_c_a ) ) ) @ ( listre7328408972406649621od_c_a @ R ) )
      = ( ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( listre7328408972406649621od_c_a @ R ) )
          & ( X3 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_399_concat__append,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a] :
      ( ( concat7533993459791330935od_c_a @ ( append2943488786620085207od_c_a @ Xs @ Ys ) )
      = ( append3884164603879732689od_c_a @ ( concat7533993459791330935od_c_a @ Xs ) @ ( concat7533993459791330935od_c_a @ Ys ) ) ) ).

% concat_append
thf(fact_400_Cons__listrel1__Cons,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) ) @ ( listre7328408972406649621od_c_a @ R ) )
      = ( ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y3 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X3 = Y3 )
          & ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( listre7328408972406649621od_c_a @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_401_concat_Osimps_I2_J,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a] :
      ( ( concat7533993459791330935od_c_a @ ( cons_l7906879460341455474od_c_a @ X3 @ Xs ) )
      = ( append3884164603879732689od_c_a @ X3 @ ( concat7533993459791330935od_c_a @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_402_concat_Osimps_I1_J,axiom,
    ( ( concat7533993459791330935od_c_a @ nil_li2790768921375229634od_c_a )
    = nil_Pr1342775757158464060od_c_a ) ).

% concat.simps(1)
thf(fact_403_listrel1I2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a,X3: produc2669932201312201782od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( listre7328408972406649621od_c_a @ R ) )
     => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ ( cons_P8659537020926305772od_c_a @ X3 @ Ys ) ) @ ( listre7328408972406649621od_c_a @ R ) ) ) ).

% listrel1I2
thf(fact_404_not__listrel1__Nil,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ~ ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ nil_Pr1342775757158464060od_c_a ) @ ( listre7328408972406649621od_c_a @ R ) ) ).

% not_listrel1_Nil
thf(fact_405_not__Nil__listrel1,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ~ ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ nil_Pr1342775757158464060od_c_a @ Xs ) @ ( listre7328408972406649621od_c_a @ R ) ) ).

% not_Nil_listrel1
thf(fact_406_append__listrel1I,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a,Us2: list_P6327159017948738492od_c_a,Vs: list_P6327159017948738492od_c_a] :
      ( ( ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( listre7328408972406649621od_c_a @ R ) )
          & ( Us2 = Vs ) )
        | ( ( Xs = Ys )
          & ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Us2 @ Vs ) @ ( listre7328408972406649621od_c_a @ R ) ) ) )
     => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Us2 ) @ ( append3884164603879732689od_c_a @ Ys @ Vs ) ) @ ( listre7328408972406649621od_c_a @ R ) ) ) ).

% append_listrel1I
thf(fact_407_prefixes_Osimps_I1_J,axiom,
    ( ( prefix8878233912999784899od_c_a @ nil_Pr1342775757158464060od_c_a )
    = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) ).

% prefixes.simps(1)
thf(fact_408_listrel1I1,axiom,
    ! [X3: produc2669932201312201782od_c_a,Y3: produc2669932201312201782od_c_a,R: set_Pr1454594181397332999od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y3 ) @ R )
     => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ ( cons_P8659537020926305772od_c_a @ Y3 @ Xs ) ) @ ( listre7328408972406649621od_c_a @ R ) ) ) ).

% listrel1I1
thf(fact_409_Cons__listrel1E1,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ Ys ) @ ( listre7328408972406649621od_c_a @ R ) )
     => ( ! [Y: produc2669932201312201782od_c_a] :
            ( ( Ys
              = ( cons_P8659537020926305772od_c_a @ Y @ Xs ) )
           => ~ ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_P6327159017948738492od_c_a] :
              ( ( Ys
                = ( cons_P8659537020926305772od_c_a @ X3 @ Zs2 ) )
             => ~ ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Zs2 ) @ ( listre7328408972406649621od_c_a @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_410_Cons__listrel1E2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) ) @ ( listre7328408972406649621od_c_a @ R ) )
     => ( ! [X: produc2669932201312201782od_c_a] :
            ( ( Xs
              = ( cons_P8659537020926305772od_c_a @ X @ Ys ) )
           => ~ ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_P6327159017948738492od_c_a] :
              ( ( Xs
                = ( cons_P8659537020926305772od_c_a @ Y3 @ Zs2 ) )
             => ~ ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Zs2 @ Ys ) @ ( listre7328408972406649621od_c_a @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_411_Sublist_Osuffixes_Osimps_I1_J,axiom,
    ( ( suffix7794987338261656196od_c_a @ nil_Pr1342775757158464060od_c_a )
    = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) ).

% Sublist.suffixes.simps(1)
thf(fact_412_listrel1E,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( listre7328408972406649621od_c_a @ R ) )
     => ~ ! [X: produc2669932201312201782od_c_a,Y: produc2669932201312201782od_c_a] :
            ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X @ Y ) @ R )
           => ! [Us3: list_P6327159017948738492od_c_a,Vs2: list_P6327159017948738492od_c_a] :
                ( ( Xs
                  = ( append3884164603879732689od_c_a @ Us3 @ ( cons_P8659537020926305772od_c_a @ X @ Vs2 ) ) )
               => ( Ys
                 != ( append3884164603879732689od_c_a @ Us3 @ ( cons_P8659537020926305772od_c_a @ Y @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_413_listrel1I,axiom,
    ! [X3: produc2669932201312201782od_c_a,Y3: produc2669932201312201782od_c_a,R: set_Pr1454594181397332999od_c_a,Xs: list_P6327159017948738492od_c_a,Us2: list_P6327159017948738492od_c_a,Vs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y3 ) @ R )
     => ( ( Xs
          = ( append3884164603879732689od_c_a @ Us2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Vs ) ) )
       => ( ( Ys
            = ( append3884164603879732689od_c_a @ Us2 @ ( cons_P8659537020926305772od_c_a @ Y3 @ Vs ) ) )
         => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( listre7328408972406649621od_c_a @ R ) ) ) ) ) ).

% listrel1I
thf(fact_414_concat__eq__appendD,axiom,
    ! [Xss2: list_l2826717958560394946od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( concat7533993459791330935od_c_a @ Xss2 )
        = ( append3884164603879732689od_c_a @ Ys @ Zs ) )
     => ( ( Xss2 != nil_li2790768921375229634od_c_a )
       => ? [Xss12: list_l2826717958560394946od_c_a,Xs2: list_P6327159017948738492od_c_a,Xs5: list_P6327159017948738492od_c_a,Xss23: list_l2826717958560394946od_c_a] :
            ( ( Xss2
              = ( append2943488786620085207od_c_a @ Xss12 @ ( cons_l7906879460341455474od_c_a @ ( append3884164603879732689od_c_a @ Xs2 @ Xs5 ) @ Xss23 ) ) )
            & ( Ys
              = ( append3884164603879732689od_c_a @ ( concat7533993459791330935od_c_a @ Xss12 ) @ Xs2 ) )
            & ( Zs
              = ( append3884164603879732689od_c_a @ Xs5 @ ( concat7533993459791330935od_c_a @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_415_Sublist_Osuffixes_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( suffix7794987338261656196od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( append2943488786620085207od_c_a @ ( suffix7794987338261656196od_c_a @ Xs ) @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ nil_li2790768921375229634od_c_a ) ) ) ).

% Sublist.suffixes.simps(2)
thf(fact_416_map__default_Osimps_I1_J,axiom,
    ! [K3: a,V2: produc2133359284458116493od_c_a,F2: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a] :
      ( ( map_de3831083019762265319od_c_a @ K3 @ V2 @ F2 @ nil_Pr1342775757158464060od_c_a )
      = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ V2 ) @ nil_Pr1342775757158464060od_c_a ) ) ).

% map_default.simps(1)
thf(fact_417_map__default_Osimps_I1_J,axiom,
    ! [K3: b,V2: product_prod_c_a,F2: product_prod_c_a > product_prod_c_a] :
      ( ( map_de913665583892520510od_c_a @ K3 @ V2 @ F2 @ nil_Pr5380409136740620691od_c_a )
      = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ V2 ) @ nil_Pr5380409136740620691od_c_a ) ) ).

% map_default.simps(1)
thf(fact_418_map__default_Osimps_I1_J,axiom,
    ! [K3: c,V2: a,F2: a > a] :
      ( ( map_default_c_a @ K3 @ V2 @ F2 @ nil_Product_prod_c_a )
      = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ V2 ) @ nil_Product_prod_c_a ) ) ).

% map_default.simps(1)
thf(fact_419_delete__aux_Oelims,axiom,
    ! [X3: a,Xa: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( delete3144518687146744757od_c_a @ X3 @ Xa )
        = Y3 )
     => ( ( ( Xa = nil_Pr1342775757158464060od_c_a )
         => ( Y3 != nil_Pr1342775757158464060od_c_a ) )
       => ~ ! [K2: a,V: produc2133359284458116493od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( ( Xa
                = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K2 @ V ) @ Xs2 ) )
             => ~ ( ( ( X3 = K2 )
                   => ( Y3 = Xs2 ) )
                  & ( ( X3 != K2 )
                   => ( Y3
                      = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K2 @ V ) @ ( delete3144518687146744757od_c_a @ X3 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_420_delete__aux_Oelims,axiom,
    ! [X3: b,Xa: list_P5767727266746070163od_c_a,Y3: list_P5767727266746070163od_c_a] :
      ( ( ( delete2063006593957858060od_c_a @ X3 @ Xa )
        = Y3 )
     => ( ( ( Xa = nil_Pr5380409136740620691od_c_a )
         => ( Y3 != nil_Pr5380409136740620691od_c_a ) )
       => ~ ! [K2: b,V: product_prod_c_a,Xs2: list_P5767727266746070163od_c_a] :
              ( ( Xa
                = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K2 @ V ) @ Xs2 ) )
             => ~ ( ( ( X3 = K2 )
                   => ( Y3 = Xs2 ) )
                  & ( ( X3 != K2 )
                   => ( Y3
                      = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K2 @ V ) @ ( delete2063006593957858060od_c_a @ X3 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_421_delete__aux_Oelims,axiom,
    ! [X3: c,Xa: list_P125642481956313003od_c_a,Y3: list_P125642481956313003od_c_a] :
      ( ( ( delete_aux_c_a @ X3 @ Xa )
        = Y3 )
     => ( ( ( Xa = nil_Product_prod_c_a )
         => ( Y3 != nil_Product_prod_c_a ) )
       => ~ ! [K2: c,V: a,Xs2: list_P125642481956313003od_c_a] :
              ( ( Xa
                = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K2 @ V ) @ Xs2 ) )
             => ~ ( ( ( X3 = K2 )
                   => ( Y3 = Xs2 ) )
                  & ( ( X3 != K2 )
                   => ( Y3
                      = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K2 @ V ) @ ( delete_aux_c_a @ X3 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_422_delete__aux__eq__Nil__conv,axiom,
    ! [K3: a,Ts: list_P6327159017948738492od_c_a] :
      ( ( ( delete3144518687146744757od_c_a @ K3 @ Ts )
        = nil_Pr1342775757158464060od_c_a )
      = ( ( Ts = nil_Pr1342775757158464060od_c_a )
        | ? [V3: produc2133359284458116493od_c_a] :
            ( Ts
            = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ V3 ) @ nil_Pr1342775757158464060od_c_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_423_delete__aux__eq__Nil__conv,axiom,
    ! [K3: b,Ts: list_P5767727266746070163od_c_a] :
      ( ( ( delete2063006593957858060od_c_a @ K3 @ Ts )
        = nil_Pr5380409136740620691od_c_a )
      = ( ( Ts = nil_Pr5380409136740620691od_c_a )
        | ? [V3: product_prod_c_a] :
            ( Ts
            = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ V3 ) @ nil_Pr5380409136740620691od_c_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_424_delete__aux__eq__Nil__conv,axiom,
    ! [K3: c,Ts: list_P125642481956313003od_c_a] :
      ( ( ( delete_aux_c_a @ K3 @ Ts )
        = nil_Product_prod_c_a )
      = ( ( Ts = nil_Product_prod_c_a )
        | ? [V3: a] :
            ( Ts
            = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ V3 ) @ nil_Product_prod_c_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_425_maps__simps_I1_J,axiom,
    ! [F2: produc2669932201312201782od_c_a > list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( maps_P5637206941461125057od_c_a @ F2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( append3884164603879732689od_c_a @ ( F2 @ X3 ) @ ( maps_P5637206941461125057od_c_a @ F2 @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_426_subseqs_Osimps_I1_J,axiom,
    ( ( subseq4269998335400966283od_c_a @ nil_Pr1342775757158464060od_c_a )
    = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) ).

% subseqs.simps(1)
thf(fact_427_suffixes_Opelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_l2826717958560394946od_c_a] :
      ( ( ( suffix1038733475819007016od_c_a @ X3 )
        = Y3 )
     => ( ( accp_l2553908896995535941od_c_a @ suffix2435388708019435401od_c_a @ X3 )
       => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
           => ( ( Y3
                = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) )
             => ~ ( accp_l2553908896995535941od_c_a @ suffix2435388708019435401od_c_a @ nil_Pr1342775757158464060od_c_a ) ) )
         => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
                ( ( X3
                  = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
               => ( ( Y3
                    = ( append2943488786620085207od_c_a @ ( suffix1038733475819007016od_c_a @ Xs2 ) @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ nil_li2790768921375229634od_c_a ) ) )
                 => ~ ( accp_l2553908896995535941od_c_a @ suffix2435388708019435401od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) ) ) ) ) ) ).

% suffixes.pelims
thf(fact_428_insert__Nil,axiom,
    ! [X3: produc2669932201312201782od_c_a] :
      ( ( insert1660371350497266002od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a )
      = ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) ).

% insert_Nil
thf(fact_429_single__input_Osimps,axiom,
    ( single_input_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M2 ) )
         => ! [Y4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ M2 ) )
             => ( ( ( produc4225943634317630866od_c_a @ X4 )
                  = ( produc4225943634317630866od_c_a @ Y4 ) )
               => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                  = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_430_distinct__insert,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( distinct_a @ ( insert_a @ X3 @ Xs ) )
      = ( distinct_a @ Xs ) ) ).

% distinct_insert
thf(fact_431_prod_Ocollapse,axiom,
    ! [Prod: produc2669932201312201782od_c_a] :
      ( ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ Prod ) @ ( produc8088130389382527188od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_432_prod_Ocollapse,axiom,
    ! [Prod: produc2133359284458116493od_c_a] :
      ( ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ Prod ) @ ( produc5221446324160585515od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_433_prod_Ocollapse,axiom,
    ! [Prod: product_prod_c_a] :
      ( ( product_Pair_c_a @ ( product_fst_c_a @ Prod ) @ ( product_snd_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_434_target__snoc,axiom,
    ! [Q: a,P5: list_P6327159017948738492od_c_a,T: produc2669932201312201782od_c_a] :
      ( ( target_a_b_c @ Q @ ( append3884164603879732689od_c_a @ P5 @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) ) )
      = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ).

% target_snoc
thf(fact_435_reachable__states__next,axiom,
    ! [Q: a,M: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ( ( ( produc4225943634317630866od_c_a @ T )
            = Q )
         => ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ) ) ) ).

% reachable_states_next
thf(fact_436_sndI,axiom,
    ! [X3: produc2669932201312201782od_c_a,Y3: a,Z: produc2133359284458116493od_c_a] :
      ( ( X3
        = ( produc5762801557034676134od_c_a @ Y3 @ Z ) )
     => ( ( produc8088130389382527188od_c_a @ X3 )
        = Z ) ) ).

% sndI
thf(fact_437_sndI,axiom,
    ! [X3: produc2133359284458116493od_c_a,Y3: b,Z: product_prod_c_a] :
      ( ( X3
        = ( produc3440865914000881661od_c_a @ Y3 @ Z ) )
     => ( ( produc5221446324160585515od_c_a @ X3 )
        = Z ) ) ).

% sndI
thf(fact_438_sndI,axiom,
    ! [X3: product_prod_c_a,Y3: c,Z: a] :
      ( ( X3
        = ( product_Pair_c_a @ Y3 @ Z ) )
     => ( ( product_snd_c_a @ X3 )
        = Z ) ) ).

% sndI
thf(fact_439_eq__snd__iff,axiom,
    ! [B: produc2133359284458116493od_c_a,P5: produc2669932201312201782od_c_a] :
      ( ( B
        = ( produc8088130389382527188od_c_a @ P5 ) )
      = ( ? [A5: a] :
            ( P5
            = ( produc5762801557034676134od_c_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_440_eq__snd__iff,axiom,
    ! [B: product_prod_c_a,P5: produc2133359284458116493od_c_a] :
      ( ( B
        = ( produc5221446324160585515od_c_a @ P5 ) )
      = ( ? [A5: b] :
            ( P5
            = ( produc3440865914000881661od_c_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_441_eq__snd__iff,axiom,
    ! [B: a,P5: product_prod_c_a] :
      ( ( B
        = ( product_snd_c_a @ P5 ) )
      = ( ? [A5: c] :
            ( P5
            = ( product_Pair_c_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_442_snd__conv,axiom,
    ! [X1: a,X2: produc2133359284458116493od_c_a] :
      ( ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_443_snd__conv,axiom,
    ! [X1: b,X2: product_prod_c_a] :
      ( ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_444_snd__conv,axiom,
    ! [X1: c,X2: a] :
      ( ( product_snd_c_a @ ( product_Pair_c_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_445_snd__eqD,axiom,
    ! [X3: a,Y3: produc2133359284458116493od_c_a,A: produc2133359284458116493od_c_a] :
      ( ( ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_446_snd__eqD,axiom,
    ! [X3: b,Y3: product_prod_c_a,A: product_prod_c_a] :
      ( ( ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_447_snd__eqD,axiom,
    ! [X3: c,Y3: a,A: a] :
      ( ( ( product_snd_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_448_cons,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c,Ts: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( ( path_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ Ts )
       => ( path_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) @ ( cons_P8659537020926305772od_c_a @ T @ Ts ) ) ) ) ).

% cons
thf(fact_449_path__cons__elim,axiom,
    ! [M: fsm_a_b_c,Q: a,T: produc2669932201312201782od_c_a,Ts: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ ( cons_P8659537020926305772od_c_a @ T @ Ts ) )
     => ~ ( ( Q
            = ( produc4225943634317630866od_c_a @ T ) )
         => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
           => ~ ( path_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ Ts ) ) ) ) ).

% path_cons_elim
thf(fact_450_reachable__states__initial__or__target,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) )
     => ( ( Q
          = ( initial_a_b_c @ M ) )
        | ? [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ M ) )
            & ( member_a @ ( produc4225943634317630866od_c_a @ X ) @ ( reacha1620305530751930115_a_b_c @ M ) )
            & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
              = Q ) ) ) ) ).

% reachable_states_initial_or_target
thf(fact_451_observable_Oelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( observable_a_b_c @ X3 )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X3 ) )
           => ! [Xa2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X3 ) )
               => ( ( ( ( produc4225943634317630866od_c_a @ X )
                      = ( produc4225943634317630866od_c_a @ Xa2 ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) )
                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                      = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) )
                 => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_452_observable_Oelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( observable_a_b_c @ X3 )
     => ! [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( transitions_a_b_c @ X3 ) )
         => ! [Xa3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X3 ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X6 )
                    = ( produc4225943634317630866od_c_a @ Xa3 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_453_observable_Oelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( observable_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X3 ) )
             => ! [Y4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X3 ) )
                 => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                        = ( produc4225943634317630866od_c_a @ Y4 ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) )
                      & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                        = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) )
                   => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                      = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_454_observable_Osimps,axiom,
    ( observable_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M2 ) )
         => ! [Y4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ M2 ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                    = ( produc4225943634317630866od_c_a @ Y4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_455_deterministic_Oelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( deterministic_a_b_c @ X3 )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X3 ) )
           => ! [Xa2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X3 ) )
               => ( ( ( ( produc4225943634317630866od_c_a @ X )
                      = ( produc4225943634317630866od_c_a @ Xa2 ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) )
                 => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                      = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) )
                    & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                      = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_456_deterministic_Oelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( deterministic_a_b_c @ X3 )
     => ! [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( transitions_a_b_c @ X3 ) )
         => ! [Xa3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X3 ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X6 )
                    = ( produc4225943634317630866od_c_a @ Xa3 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) )
               => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) )
                  & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_457_deterministic_Oelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( deterministic_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X3 ) )
             => ! [Y4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X3 ) )
                 => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                        = ( produc4225943634317630866od_c_a @ Y4 ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) )
                   => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                        = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) )
                      & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                        = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_458_deterministic_Osimps,axiom,
    ( deterministic_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M2 ) )
         => ! [Y4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ M2 ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                    = ( produc4225943634317630866od_c_a @ Y4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) )
               => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) )
                  & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_459_map__default_Osimps_I2_J,axiom,
    ! [P5: produc2669932201312201782od_c_a,K3: a,V2: produc2133359284458116493od_c_a,F2: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,Ps2: list_P6327159017948738492od_c_a] :
      ( ( ( ( produc4225943634317630866od_c_a @ P5 )
          = K3 )
       => ( ( map_de3831083019762265319od_c_a @ K3 @ V2 @ F2 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ ( F2 @ ( produc8088130389382527188od_c_a @ P5 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc4225943634317630866od_c_a @ P5 )
         != K3 )
       => ( ( map_de3831083019762265319od_c_a @ K3 @ V2 @ F2 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( cons_P8659537020926305772od_c_a @ P5 @ ( map_de3831083019762265319od_c_a @ K3 @ V2 @ F2 @ Ps2 ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_460_map__default_Osimps_I2_J,axiom,
    ! [P5: produc2133359284458116493od_c_a,K3: b,V2: product_prod_c_a,F2: product_prod_c_a > product_prod_c_a,Ps2: list_P5767727266746070163od_c_a] :
      ( ( ( ( produc301949074746585577od_c_a @ P5 )
          = K3 )
       => ( ( map_de913665583892520510od_c_a @ K3 @ V2 @ F2 @ ( cons_P7376644893796496707od_c_a @ P5 @ Ps2 ) )
          = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ ( F2 @ ( produc5221446324160585515od_c_a @ P5 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc301949074746585577od_c_a @ P5 )
         != K3 )
       => ( ( map_de913665583892520510od_c_a @ K3 @ V2 @ F2 @ ( cons_P7376644893796496707od_c_a @ P5 @ Ps2 ) )
          = ( cons_P7376644893796496707od_c_a @ P5 @ ( map_de913665583892520510od_c_a @ K3 @ V2 @ F2 @ Ps2 ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_461_map__default_Osimps_I2_J,axiom,
    ! [P5: product_prod_c_a,K3: c,V2: a,F2: a > a,Ps2: list_P125642481956313003od_c_a] :
      ( ( ( ( product_fst_c_a @ P5 )
          = K3 )
       => ( ( map_default_c_a @ K3 @ V2 @ F2 @ ( cons_P1742027962761213787od_c_a @ P5 @ Ps2 ) )
          = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ ( F2 @ ( product_snd_c_a @ P5 ) ) ) @ Ps2 ) ) )
      & ( ( ( product_fst_c_a @ P5 )
         != K3 )
       => ( ( map_default_c_a @ K3 @ V2 @ F2 @ ( cons_P1742027962761213787od_c_a @ P5 @ Ps2 ) )
          = ( cons_P1742027962761213787od_c_a @ P5 @ ( map_default_c_a @ K3 @ V2 @ F2 @ Ps2 ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_462_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2669932201312201782od_c_a] :
      ( Prod
      = ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ Prod ) @ ( produc8088130389382527188od_c_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_463_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2133359284458116493od_c_a] :
      ( Prod
      = ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ Prod ) @ ( produc5221446324160585515od_c_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_464_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_c_a] :
      ( Prod
      = ( product_Pair_c_a @ ( product_fst_c_a @ Prod ) @ ( product_snd_c_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_465_surjective__pairing,axiom,
    ! [T: produc2669932201312201782od_c_a] :
      ( T
      = ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ T ) @ ( produc8088130389382527188od_c_a @ T ) ) ) ).

% surjective_pairing
thf(fact_466_surjective__pairing,axiom,
    ! [T: produc2133359284458116493od_c_a] :
      ( T
      = ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ T ) @ ( produc5221446324160585515od_c_a @ T ) ) ) ).

% surjective_pairing
thf(fact_467_surjective__pairing,axiom,
    ! [T: product_prod_c_a] :
      ( T
      = ( product_Pair_c_a @ ( product_fst_c_a @ T ) @ ( product_snd_c_a @ T ) ) ) ).

% surjective_pairing
thf(fact_468_delete__aux_Osimps_I1_J,axiom,
    ! [K3: a] :
      ( ( delete3144518687146744757od_c_a @ K3 @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% delete_aux.simps(1)
thf(fact_469_acyclic__initial,axiom,
    ! [M: fsm_a_b_c] :
      ( ( acyclic_a_b_c @ M )
     => ~ ? [X6: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X6 @ ( transitions_a_b_c @ M ) )
            & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) )
              = ( initial_a_b_c @ M ) )
            & ? [P3: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ P3 )
                & ( ( target_a_b_c @ ( initial_a_b_c @ M ) @ P3 )
                  = ( produc4225943634317630866od_c_a @ X6 ) ) ) ) ) ).

% acyclic_initial
thf(fact_470_map__default_Oelims,axiom,
    ! [X3: a,Xa: produc2133359284458116493od_c_a,Xb: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,Xc: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( map_de3831083019762265319od_c_a @ X3 @ Xa @ Xb @ Xc )
        = Y3 )
     => ( ( ( Xc = nil_Pr1342775757158464060od_c_a )
         => ( Y3
           != ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Xa ) @ nil_Pr1342775757158464060od_c_a ) ) )
       => ~ ! [P: produc2669932201312201782od_c_a,Ps: list_P6327159017948738492od_c_a] :
              ( ( Xc
                = ( cons_P8659537020926305772od_c_a @ P @ Ps ) )
             => ~ ( ( ( ( produc4225943634317630866od_c_a @ P )
                      = X3 )
                   => ( Y3
                      = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ ( Xb @ ( produc8088130389382527188od_c_a @ P ) ) ) @ Ps ) ) )
                  & ( ( ( produc4225943634317630866od_c_a @ P )
                     != X3 )
                   => ( Y3
                      = ( cons_P8659537020926305772od_c_a @ P @ ( map_de3831083019762265319od_c_a @ X3 @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_471_map__default_Oelims,axiom,
    ! [X3: b,Xa: product_prod_c_a,Xb: product_prod_c_a > product_prod_c_a,Xc: list_P5767727266746070163od_c_a,Y3: list_P5767727266746070163od_c_a] :
      ( ( ( map_de913665583892520510od_c_a @ X3 @ Xa @ Xb @ Xc )
        = Y3 )
     => ( ( ( Xc = nil_Pr5380409136740620691od_c_a )
         => ( Y3
           != ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Xa ) @ nil_Pr5380409136740620691od_c_a ) ) )
       => ~ ! [P: produc2133359284458116493od_c_a,Ps: list_P5767727266746070163od_c_a] :
              ( ( Xc
                = ( cons_P7376644893796496707od_c_a @ P @ Ps ) )
             => ~ ( ( ( ( produc301949074746585577od_c_a @ P )
                      = X3 )
                   => ( Y3
                      = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ ( Xb @ ( produc5221446324160585515od_c_a @ P ) ) ) @ Ps ) ) )
                  & ( ( ( produc301949074746585577od_c_a @ P )
                     != X3 )
                   => ( Y3
                      = ( cons_P7376644893796496707od_c_a @ P @ ( map_de913665583892520510od_c_a @ X3 @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_472_map__default_Oelims,axiom,
    ! [X3: c,Xa: a,Xb: a > a,Xc: list_P125642481956313003od_c_a,Y3: list_P125642481956313003od_c_a] :
      ( ( ( map_default_c_a @ X3 @ Xa @ Xb @ Xc )
        = Y3 )
     => ( ( ( Xc = nil_Product_prod_c_a )
         => ( Y3
           != ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X3 @ Xa ) @ nil_Product_prod_c_a ) ) )
       => ~ ! [P: product_prod_c_a,Ps: list_P125642481956313003od_c_a] :
              ( ( Xc
                = ( cons_P1742027962761213787od_c_a @ P @ Ps ) )
             => ~ ( ( ( ( product_fst_c_a @ P )
                      = X3 )
                   => ( Y3
                      = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X3 @ ( Xb @ ( product_snd_c_a @ P ) ) ) @ Ps ) ) )
                  & ( ( ( product_fst_c_a @ P )
                     != X3 )
                   => ( Y3
                      = ( cons_P1742027962761213787od_c_a @ P @ ( map_default_c_a @ X3 @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_473_maps__simps_I2_J,axiom,
    ! [F2: produc2669932201312201782od_c_a > list_P6327159017948738492od_c_a] :
      ( ( maps_P5637206941461125057od_c_a @ F2 @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% maps_simps(2)
thf(fact_474_delete__aux_Osimps_I2_J,axiom,
    ! [K3: a,K4: a,V2: produc2133359284458116493od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( K3 = K4 )
       => ( ( delete3144518687146744757od_c_a @ K3 @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K4 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete3144518687146744757od_c_a @ K3 @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K4 @ V2 ) @ Xs ) )
          = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K4 @ V2 ) @ ( delete3144518687146744757od_c_a @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_475_delete__aux_Osimps_I2_J,axiom,
    ! [K3: b,K4: b,V2: product_prod_c_a,Xs: list_P5767727266746070163od_c_a] :
      ( ( ( K3 = K4 )
       => ( ( delete2063006593957858060od_c_a @ K3 @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K4 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete2063006593957858060od_c_a @ K3 @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K4 @ V2 ) @ Xs ) )
          = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K4 @ V2 ) @ ( delete2063006593957858060od_c_a @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_476_delete__aux_Osimps_I2_J,axiom,
    ! [K3: c,K4: c,V2: a,Xs: list_P125642481956313003od_c_a] :
      ( ( ( K3 = K4 )
       => ( ( delete_aux_c_a @ K3 @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K4 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete_aux_c_a @ K3 @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K4 @ V2 ) @ Xs ) )
          = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K4 @ V2 ) @ ( delete_aux_c_a @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_477_update__with__aux_Osimps_I2_J,axiom,
    ! [P5: produc2669932201312201782od_c_a,K3: a,V2: produc2133359284458116493od_c_a,F2: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,Ps2: list_P6327159017948738492od_c_a] :
      ( ( ( ( produc4225943634317630866od_c_a @ P5 )
          = K3 )
       => ( ( update3791493425278745523_c_a_a @ V2 @ K3 @ F2 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ ( F2 @ ( produc8088130389382527188od_c_a @ P5 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc4225943634317630866od_c_a @ P5 )
         != K3 )
       => ( ( update3791493425278745523_c_a_a @ V2 @ K3 @ F2 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( cons_P8659537020926305772od_c_a @ P5 @ ( update3791493425278745523_c_a_a @ V2 @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_478_update__with__aux_Osimps_I2_J,axiom,
    ! [P5: produc2133359284458116493od_c_a,K3: b,V2: product_prod_c_a,F2: product_prod_c_a > product_prod_c_a,Ps2: list_P5767727266746070163od_c_a] :
      ( ( ( ( produc301949074746585577od_c_a @ P5 )
          = K3 )
       => ( ( update338640773394215068_c_a_b @ V2 @ K3 @ F2 @ ( cons_P7376644893796496707od_c_a @ P5 @ Ps2 ) )
          = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ ( F2 @ ( produc5221446324160585515od_c_a @ P5 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc301949074746585577od_c_a @ P5 )
         != K3 )
       => ( ( update338640773394215068_c_a_b @ V2 @ K3 @ F2 @ ( cons_P7376644893796496707od_c_a @ P5 @ Ps2 ) )
          = ( cons_P7376644893796496707od_c_a @ P5 @ ( update338640773394215068_c_a_b @ V2 @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_479_update__with__aux_Osimps_I2_J,axiom,
    ! [P5: product_prod_c_a,K3: c,V2: a,F2: a > a,Ps2: list_P125642481956313003od_c_a] :
      ( ( ( ( product_fst_c_a @ P5 )
          = K3 )
       => ( ( update_with_aux_a_c @ V2 @ K3 @ F2 @ ( cons_P1742027962761213787od_c_a @ P5 @ Ps2 ) )
          = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ ( F2 @ ( product_snd_c_a @ P5 ) ) ) @ Ps2 ) ) )
      & ( ( ( product_fst_c_a @ P5 )
         != K3 )
       => ( ( update_with_aux_a_c @ V2 @ K3 @ F2 @ ( cons_P1742027962761213787od_c_a @ P5 @ Ps2 ) )
          = ( cons_P1742027962761213787od_c_a @ P5 @ ( update_with_aux_a_c @ V2 @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_480_single__input_Oelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( single_input_a_b_c @ X3 )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X3 ) )
           => ! [Xa2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X3 ) )
               => ( ( ( produc4225943634317630866od_c_a @ X )
                    = ( produc4225943634317630866od_c_a @ Xa2 ) )
                 => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_481_single__input_Oelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( single_input_a_b_c @ X3 )
     => ! [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( transitions_a_b_c @ X3 ) )
         => ! [Xa3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X3 ) )
             => ( ( ( produc4225943634317630866od_c_a @ X6 )
                  = ( produc4225943634317630866od_c_a @ Xa3 ) )
               => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) )
                  = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_482_single__input_Oelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( single_input_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X3 ) )
             => ! [Y4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X3 ) )
                 => ( ( ( produc4225943634317630866od_c_a @ X4 )
                      = ( produc4225943634317630866od_c_a @ Y4 ) )
                   => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_483_LS__prepend__transition,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) )
       => ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ Io ) @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_484_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P7: a > produc2133359284458116493od_c_a > $o,X3: a,Y3: produc2133359284458116493od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( P7 @ X3 @ Y3 )
     => ( ( A
          = ( produc5762801557034676134od_c_a @ X3 @ Y3 ) )
       => ( P7 @ ( produc4225943634317630866od_c_a @ A ) @ ( produc8088130389382527188od_c_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_485_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P7: b > product_prod_c_a > $o,X3: b,Y3: product_prod_c_a,A: produc2133359284458116493od_c_a] :
      ( ( P7 @ X3 @ Y3 )
     => ( ( A
          = ( produc3440865914000881661od_c_a @ X3 @ Y3 ) )
       => ( P7 @ ( produc301949074746585577od_c_a @ A ) @ ( produc5221446324160585515od_c_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_486_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P7: c > a > $o,X3: c,Y3: a,A: product_prod_c_a] :
      ( ( P7 @ X3 @ Y3 )
     => ( ( A
          = ( product_Pair_c_a @ X3 @ Y3 ) )
       => ( P7 @ ( product_fst_c_a @ A ) @ ( product_snd_c_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_487_conjI__realizer,axiom,
    ! [P7: a > $o,P5: a,Q2: produc2133359284458116493od_c_a > $o,Q: produc2133359284458116493od_c_a] :
      ( ( P7 @ P5 )
     => ( ( Q2 @ Q )
       => ( ( P7 @ ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ P5 @ Q ) ) )
          & ( Q2 @ ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ P5 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_488_conjI__realizer,axiom,
    ! [P7: b > $o,P5: b,Q2: product_prod_c_a > $o,Q: product_prod_c_a] :
      ( ( P7 @ P5 )
     => ( ( Q2 @ Q )
       => ( ( P7 @ ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ P5 @ Q ) ) )
          & ( Q2 @ ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ P5 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_489_conjI__realizer,axiom,
    ! [P7: c > $o,P5: c,Q2: a > $o,Q: a] :
      ( ( P7 @ P5 )
     => ( ( Q2 @ Q )
       => ( ( P7 @ ( product_fst_c_a @ ( product_Pair_c_a @ P5 @ Q ) ) )
          & ( Q2 @ ( product_snd_c_a @ ( product_Pair_c_a @ P5 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_490_exI__realizer,axiom,
    ! [P7: produc2133359284458116493od_c_a > a > $o,Y3: produc2133359284458116493od_c_a,X3: a] :
      ( ( P7 @ Y3 @ X3 )
     => ( P7 @ ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) ) @ ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_491_exI__realizer,axiom,
    ! [P7: product_prod_c_a > b > $o,Y3: product_prod_c_a,X3: b] :
      ( ( P7 @ Y3 @ X3 )
     => ( P7 @ ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) ) @ ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_492_exI__realizer,axiom,
    ! [P7: a > c > $o,Y3: a,X3: c] :
      ( ( P7 @ Y3 @ X3 )
     => ( P7 @ ( product_snd_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) ) @ ( product_fst_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_493_path_Ocases,axiom,
    ! [A1: fsm_a_b_c,A22: a,A32: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ A1 @ A22 @ A32 )
     => ( ( ( A32 = nil_Pr1342775757158464060od_c_a )
         => ~ ( member_a @ A22 @ ( states_a_b_c @ A1 ) ) )
       => ~ ! [T2: produc2669932201312201782od_c_a] :
              ( ( A22
                = ( produc4225943634317630866od_c_a @ T2 ) )
             => ! [Ts2: list_P6327159017948738492od_c_a] :
                  ( ( A32
                    = ( cons_P8659537020926305772od_c_a @ T2 @ Ts2 ) )
                 => ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ A1 ) )
                   => ~ ( path_a_b_c @ A1 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ Ts2 ) ) ) ) ) ) ).

% path.cases
thf(fact_494_path_Osimps,axiom,
    ( path_a_b_c
    = ( ^ [A12: fsm_a_b_c,A23: a,A33: list_P6327159017948738492od_c_a] :
          ( ? [Q6: a,M2: fsm_a_b_c] :
              ( ( A12 = M2 )
              & ( A23 = Q6 )
              & ( A33 = nil_Pr1342775757158464060od_c_a )
              & ( member_a @ Q6 @ ( states_a_b_c @ M2 ) ) )
          | ? [T3: produc2669932201312201782od_c_a,M2: fsm_a_b_c,Ts3: list_P6327159017948738492od_c_a] :
              ( ( A12 = M2 )
              & ( A23
                = ( produc4225943634317630866od_c_a @ T3 ) )
              & ( A33
                = ( cons_P8659537020926305772od_c_a @ T3 @ Ts3 ) )
              & ( member5502040268208861279od_c_a @ T3 @ ( transitions_a_b_c @ M2 ) )
              & ( path_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ Ts3 ) ) ) ) ) ).

% path.simps
thf(fact_495_map__entry_Oelims,axiom,
    ! [X3: a,Xa: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,Xb: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( map_en8679759376116354648od_c_a @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xb = nil_Pr1342775757158464060od_c_a )
         => ( Y3 != nil_Pr1342775757158464060od_c_a ) )
       => ~ ! [P: produc2669932201312201782od_c_a,Ps: list_P6327159017948738492od_c_a] :
              ( ( Xb
                = ( cons_P8659537020926305772od_c_a @ P @ Ps ) )
             => ~ ( ( ( ( produc4225943634317630866od_c_a @ P )
                      = X3 )
                   => ( Y3
                      = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ ( Xa @ ( produc8088130389382527188od_c_a @ P ) ) ) @ Ps ) ) )
                  & ( ( ( produc4225943634317630866od_c_a @ P )
                     != X3 )
                   => ( Y3
                      = ( cons_P8659537020926305772od_c_a @ P @ ( map_en8679759376116354648od_c_a @ X3 @ Xa @ Ps ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_496_map__entry_Oelims,axiom,
    ! [X3: b,Xa: product_prod_c_a > product_prod_c_a,Xb: list_P5767727266746070163od_c_a,Y3: list_P5767727266746070163od_c_a] :
      ( ( ( map_en316799446415087791od_c_a @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xb = nil_Pr5380409136740620691od_c_a )
         => ( Y3 != nil_Pr5380409136740620691od_c_a ) )
       => ~ ! [P: produc2133359284458116493od_c_a,Ps: list_P5767727266746070163od_c_a] :
              ( ( Xb
                = ( cons_P7376644893796496707od_c_a @ P @ Ps ) )
             => ~ ( ( ( ( produc301949074746585577od_c_a @ P )
                      = X3 )
                   => ( Y3
                      = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ ( Xa @ ( produc5221446324160585515od_c_a @ P ) ) ) @ Ps ) ) )
                  & ( ( ( produc301949074746585577od_c_a @ P )
                     != X3 )
                   => ( Y3
                      = ( cons_P7376644893796496707od_c_a @ P @ ( map_en316799446415087791od_c_a @ X3 @ Xa @ Ps ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_497_map__entry_Oelims,axiom,
    ! [X3: c,Xa: a > a,Xb: list_P125642481956313003od_c_a,Y3: list_P125642481956313003od_c_a] :
      ( ( ( map_entry_c_a @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xb = nil_Product_prod_c_a )
         => ( Y3 != nil_Product_prod_c_a ) )
       => ~ ! [P: product_prod_c_a,Ps: list_P125642481956313003od_c_a] :
              ( ( Xb
                = ( cons_P1742027962761213787od_c_a @ P @ Ps ) )
             => ~ ( ( ( ( product_fst_c_a @ P )
                      = X3 )
                   => ( Y3
                      = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X3 @ ( Xa @ ( product_snd_c_a @ P ) ) ) @ Ps ) ) )
                  & ( ( ( product_fst_c_a @ P )
                     != X3 )
                   => ( Y3
                      = ( cons_P1742027962761213787od_c_a @ P @ ( map_entry_c_a @ X3 @ Xa @ Ps ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_498_fsm__initial,axiom,
    ! [M: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M ) @ ( states_a_b_c @ M ) ) ).

% fsm_initial
thf(fact_499_fsm__transition__source,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( member_a @ ( produc4225943634317630866od_c_a @ T ) @ ( states_a_b_c @ M ) ) ) ).

% fsm_transition_source
thf(fact_500_fsm__transition__target,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( states_a_b_c @ M ) ) ) ).

% fsm_transition_target
thf(fact_501_path__begin__state,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( member_a @ Q @ ( states_a_b_c @ M ) ) ) ).

% path_begin_state
thf(fact_502_reachable__state__is__state,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) )
     => ( member_a @ Q @ ( states_a_b_c @ M ) ) ) ).

% reachable_state_is_state
thf(fact_503_map__entry_Osimps_I1_J,axiom,
    ! [K3: a,F2: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a] :
      ( ( map_en8679759376116354648od_c_a @ K3 @ F2 @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% map_entry.simps(1)
thf(fact_504_path__target__is__state,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( member_a @ ( target_a_b_c @ Q @ P5 ) @ ( states_a_b_c @ M ) ) ) ).

% path_target_is_state
thf(fact_505_nil,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( path_a_b_c @ M @ Q @ nil_Pr1342775757158464060od_c_a ) ) ).

% nil
thf(fact_506_path__nil__elim,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( path_a_b_c @ M @ Q @ nil_Pr1342775757158464060od_c_a )
     => ( member_a @ Q @ ( states_a_b_c @ M ) ) ) ).

% path_nil_elim
thf(fact_507_language__contains__empty__sequence,axiom,
    ! [M: fsm_a_b_c] : ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ).

% language_contains_empty_sequence
thf(fact_508_map__entry_Osimps_I2_J,axiom,
    ! [P5: produc2669932201312201782od_c_a,K3: a,F2: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,Ps2: list_P6327159017948738492od_c_a] :
      ( ( ( ( produc4225943634317630866od_c_a @ P5 )
          = K3 )
       => ( ( map_en8679759376116354648od_c_a @ K3 @ F2 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ ( F2 @ ( produc8088130389382527188od_c_a @ P5 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc4225943634317630866od_c_a @ P5 )
         != K3 )
       => ( ( map_en8679759376116354648od_c_a @ K3 @ F2 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( cons_P8659537020926305772od_c_a @ P5 @ ( map_en8679759376116354648od_c_a @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_509_map__entry_Osimps_I2_J,axiom,
    ! [P5: produc2133359284458116493od_c_a,K3: b,F2: product_prod_c_a > product_prod_c_a,Ps2: list_P5767727266746070163od_c_a] :
      ( ( ( ( produc301949074746585577od_c_a @ P5 )
          = K3 )
       => ( ( map_en316799446415087791od_c_a @ K3 @ F2 @ ( cons_P7376644893796496707od_c_a @ P5 @ Ps2 ) )
          = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ ( F2 @ ( produc5221446324160585515od_c_a @ P5 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc301949074746585577od_c_a @ P5 )
         != K3 )
       => ( ( map_en316799446415087791od_c_a @ K3 @ F2 @ ( cons_P7376644893796496707od_c_a @ P5 @ Ps2 ) )
          = ( cons_P7376644893796496707od_c_a @ P5 @ ( map_en316799446415087791od_c_a @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_510_map__entry_Osimps_I2_J,axiom,
    ! [P5: product_prod_c_a,K3: c,F2: a > a,Ps2: list_P125642481956313003od_c_a] :
      ( ( ( ( product_fst_c_a @ P5 )
          = K3 )
       => ( ( map_entry_c_a @ K3 @ F2 @ ( cons_P1742027962761213787od_c_a @ P5 @ Ps2 ) )
          = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ ( F2 @ ( product_snd_c_a @ P5 ) ) ) @ Ps2 ) ) )
      & ( ( ( product_fst_c_a @ P5 )
         != K3 )
       => ( ( map_entry_c_a @ K3 @ F2 @ ( cons_P1742027962761213787od_c_a @ P5 @ Ps2 ) )
          = ( cons_P1742027962761213787od_c_a @ P5 @ ( map_entry_c_a @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_511_LS__from__LS__acyclic,axiom,
    ! [M: fsm_a_b_c] :
      ( ( acyclic_a_b_c @ M )
     => ( ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) )
        = ( lS_acyclic_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ) ).

% LS_from_LS_acyclic
thf(fact_512_map__default_Opelims,axiom,
    ! [X3: a,Xa: produc2133359284458116493od_c_a,Xb: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,Xc: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( map_de3831083019762265319od_c_a @ X3 @ Xa @ Xb @ Xc )
        = Y3 )
     => ( ( accp_P750033058094773484od_c_a @ map_de7659607532393035568od_c_a @ ( produc4844797264803247151od_c_a @ X3 @ ( produc8267950104566546306od_c_a @ Xa @ ( produc588459825458708734od_c_a @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Pr1342775757158464060od_c_a )
           => ( ( Y3
                = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Xa ) @ nil_Pr1342775757158464060od_c_a ) )
             => ~ ( accp_P750033058094773484od_c_a @ map_de7659607532393035568od_c_a @ ( produc4844797264803247151od_c_a @ X3 @ ( produc8267950104566546306od_c_a @ Xa @ ( produc588459825458708734od_c_a @ Xb @ nil_Pr1342775757158464060od_c_a ) ) ) ) ) )
         => ~ ! [P: produc2669932201312201782od_c_a,Ps: list_P6327159017948738492od_c_a] :
                ( ( Xc
                  = ( cons_P8659537020926305772od_c_a @ P @ Ps ) )
               => ( ( ( ( ( produc4225943634317630866od_c_a @ P )
                        = X3 )
                     => ( Y3
                        = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ ( Xb @ ( produc8088130389382527188od_c_a @ P ) ) ) @ Ps ) ) )
                    & ( ( ( produc4225943634317630866od_c_a @ P )
                       != X3 )
                     => ( Y3
                        = ( cons_P8659537020926305772od_c_a @ P @ ( map_de3831083019762265319od_c_a @ X3 @ Xa @ Xb @ Ps ) ) ) ) )
                 => ~ ( accp_P750033058094773484od_c_a @ map_de7659607532393035568od_c_a @ ( produc4844797264803247151od_c_a @ X3 @ ( produc8267950104566546306od_c_a @ Xa @ ( produc588459825458708734od_c_a @ Xb @ ( cons_P8659537020926305772od_c_a @ P @ Ps ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_513_map__default_Opelims,axiom,
    ! [X3: b,Xa: product_prod_c_a,Xb: product_prod_c_a > product_prod_c_a,Xc: list_P5767727266746070163od_c_a,Y3: list_P5767727266746070163od_c_a] :
      ( ( ( map_de913665583892520510od_c_a @ X3 @ Xa @ Xb @ Xc )
        = Y3 )
     => ( ( accp_P5976026805589096668od_c_a @ map_de7189691545484040583od_c_a @ ( produc8849119538608437151od_c_a @ X3 @ ( produc3208542616115268593od_c_a @ Xa @ ( produc2491487165461192149od_c_a @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Pr5380409136740620691od_c_a )
           => ( ( Y3
                = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Xa ) @ nil_Pr5380409136740620691od_c_a ) )
             => ~ ( accp_P5976026805589096668od_c_a @ map_de7189691545484040583od_c_a @ ( produc8849119538608437151od_c_a @ X3 @ ( produc3208542616115268593od_c_a @ Xa @ ( produc2491487165461192149od_c_a @ Xb @ nil_Pr5380409136740620691od_c_a ) ) ) ) ) )
         => ~ ! [P: produc2133359284458116493od_c_a,Ps: list_P5767727266746070163od_c_a] :
                ( ( Xc
                  = ( cons_P7376644893796496707od_c_a @ P @ Ps ) )
               => ( ( ( ( ( produc301949074746585577od_c_a @ P )
                        = X3 )
                     => ( Y3
                        = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ ( Xb @ ( produc5221446324160585515od_c_a @ P ) ) ) @ Ps ) ) )
                    & ( ( ( produc301949074746585577od_c_a @ P )
                       != X3 )
                     => ( Y3
                        = ( cons_P7376644893796496707od_c_a @ P @ ( map_de913665583892520510od_c_a @ X3 @ Xa @ Xb @ Ps ) ) ) ) )
                 => ~ ( accp_P5976026805589096668od_c_a @ map_de7189691545484040583od_c_a @ ( produc8849119538608437151od_c_a @ X3 @ ( produc3208542616115268593od_c_a @ Xa @ ( produc2491487165461192149od_c_a @ Xb @ ( cons_P7376644893796496707od_c_a @ P @ Ps ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_514_map__default_Opelims,axiom,
    ! [X3: c,Xa: a,Xb: a > a,Xc: list_P125642481956313003od_c_a,Y3: list_P125642481956313003od_c_a] :
      ( ( ( map_default_c_a @ X3 @ Xa @ Xb @ Xc )
        = Y3 )
     => ( ( accp_P3947048988216089614od_c_a @ map_default_rel_c_a @ ( produc5401631760763263825od_c_a @ X3 @ ( produc8592089209173205282od_c_a @ Xa @ ( produc5907466840628013od_c_a @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Product_prod_c_a )
           => ( ( Y3
                = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X3 @ Xa ) @ nil_Product_prod_c_a ) )
             => ~ ( accp_P3947048988216089614od_c_a @ map_default_rel_c_a @ ( produc5401631760763263825od_c_a @ X3 @ ( produc8592089209173205282od_c_a @ Xa @ ( produc5907466840628013od_c_a @ Xb @ nil_Product_prod_c_a ) ) ) ) ) )
         => ~ ! [P: product_prod_c_a,Ps: list_P125642481956313003od_c_a] :
                ( ( Xc
                  = ( cons_P1742027962761213787od_c_a @ P @ Ps ) )
               => ( ( ( ( ( product_fst_c_a @ P )
                        = X3 )
                     => ( Y3
                        = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X3 @ ( Xb @ ( product_snd_c_a @ P ) ) ) @ Ps ) ) )
                    & ( ( ( product_fst_c_a @ P )
                       != X3 )
                     => ( Y3
                        = ( cons_P1742027962761213787od_c_a @ P @ ( map_default_c_a @ X3 @ Xa @ Xb @ Ps ) ) ) ) )
                 => ~ ( accp_P3947048988216089614od_c_a @ map_default_rel_c_a @ ( produc5401631760763263825od_c_a @ X3 @ ( produc8592089209173205282od_c_a @ Xa @ ( produc5907466840628013od_c_a @ Xb @ ( cons_P1742027962761213787od_c_a @ P @ Ps ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_515_fsm__transition__output,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M ) ) ) ).

% fsm_transition_output
thf(fact_516_cycle__incoming__transition__ex,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,T: produc2669932201312201782od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ( P5 != nil_Pr1342775757158464060od_c_a )
       => ( ( ( target_a_b_c @ Q @ P5 )
            = Q )
         => ( ( member5502040268208861279od_c_a @ T @ ( set_Pr5664641634098845387od_c_a @ P5 ) )
           => ? [X: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ P5 ) )
                & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                  = ( produc4225943634317630866od_c_a @ T ) ) ) ) ) ) ) ).

% cycle_incoming_transition_ex
thf(fact_517_map__entry_Opelims,axiom,
    ! [X3: a,Xa: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,Xb: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( map_en8679759376116354648od_c_a @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P7130708734473209648od_c_a @ map_en5663065482706518783od_c_a @ ( produc7215935238510903923od_c_a @ X3 @ ( produc588459825458708734od_c_a @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Pr1342775757158464060od_c_a )
           => ( ( Y3 = nil_Pr1342775757158464060od_c_a )
             => ~ ( accp_P7130708734473209648od_c_a @ map_en5663065482706518783od_c_a @ ( produc7215935238510903923od_c_a @ X3 @ ( produc588459825458708734od_c_a @ Xa @ nil_Pr1342775757158464060od_c_a ) ) ) ) )
         => ~ ! [P: produc2669932201312201782od_c_a,Ps: list_P6327159017948738492od_c_a] :
                ( ( Xb
                  = ( cons_P8659537020926305772od_c_a @ P @ Ps ) )
               => ( ( ( ( ( produc4225943634317630866od_c_a @ P )
                        = X3 )
                     => ( Y3
                        = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ ( Xa @ ( produc8088130389382527188od_c_a @ P ) ) ) @ Ps ) ) )
                    & ( ( ( produc4225943634317630866od_c_a @ P )
                       != X3 )
                     => ( Y3
                        = ( cons_P8659537020926305772od_c_a @ P @ ( map_en8679759376116354648od_c_a @ X3 @ Xa @ Ps ) ) ) ) )
                 => ~ ( accp_P7130708734473209648od_c_a @ map_en5663065482706518783od_c_a @ ( produc7215935238510903923od_c_a @ X3 @ ( produc588459825458708734od_c_a @ Xa @ ( cons_P8659537020926305772od_c_a @ P @ Ps ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_518_map__entry_Opelims,axiom,
    ! [X3: b,Xa: product_prod_c_a > product_prod_c_a,Xb: list_P5767727266746070163od_c_a,Y3: list_P5767727266746070163od_c_a] :
      ( ( ( map_en316799446415087791od_c_a @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P6087421567936909704od_c_a @ map_en3384752097407880278od_c_a @ ( produc2763197380088095307od_c_a @ X3 @ ( produc2491487165461192149od_c_a @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Pr5380409136740620691od_c_a )
           => ( ( Y3 = nil_Pr5380409136740620691od_c_a )
             => ~ ( accp_P6087421567936909704od_c_a @ map_en3384752097407880278od_c_a @ ( produc2763197380088095307od_c_a @ X3 @ ( produc2491487165461192149od_c_a @ Xa @ nil_Pr5380409136740620691od_c_a ) ) ) ) )
         => ~ ! [P: produc2133359284458116493od_c_a,Ps: list_P5767727266746070163od_c_a] :
                ( ( Xb
                  = ( cons_P7376644893796496707od_c_a @ P @ Ps ) )
               => ( ( ( ( ( produc301949074746585577od_c_a @ P )
                        = X3 )
                     => ( Y3
                        = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ ( Xa @ ( produc5221446324160585515od_c_a @ P ) ) ) @ Ps ) ) )
                    & ( ( ( produc301949074746585577od_c_a @ P )
                       != X3 )
                     => ( Y3
                        = ( cons_P7376644893796496707od_c_a @ P @ ( map_en316799446415087791od_c_a @ X3 @ Xa @ Ps ) ) ) ) )
                 => ~ ( accp_P6087421567936909704od_c_a @ map_en3384752097407880278od_c_a @ ( produc2763197380088095307od_c_a @ X3 @ ( produc2491487165461192149od_c_a @ Xa @ ( cons_P7376644893796496707od_c_a @ P @ Ps ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_519_map__entry_Opelims,axiom,
    ! [X3: c,Xa: a > a,Xb: list_P125642481956313003od_c_a,Y3: list_P125642481956313003od_c_a] :
      ( ( ( map_entry_c_a @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P4197195109920256353od_c_a @ map_entry_rel_c_a @ ( produc1741992747163535396od_c_a @ X3 @ ( produc5907466840628013od_c_a @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Product_prod_c_a )
           => ( ( Y3 = nil_Product_prod_c_a )
             => ~ ( accp_P4197195109920256353od_c_a @ map_entry_rel_c_a @ ( produc1741992747163535396od_c_a @ X3 @ ( produc5907466840628013od_c_a @ Xa @ nil_Product_prod_c_a ) ) ) ) )
         => ~ ! [P: product_prod_c_a,Ps: list_P125642481956313003od_c_a] :
                ( ( Xb
                  = ( cons_P1742027962761213787od_c_a @ P @ Ps ) )
               => ( ( ( ( ( product_fst_c_a @ P )
                        = X3 )
                     => ( Y3
                        = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X3 @ ( Xa @ ( product_snd_c_a @ P ) ) ) @ Ps ) ) )
                    & ( ( ( product_fst_c_a @ P )
                       != X3 )
                     => ( Y3
                        = ( cons_P1742027962761213787od_c_a @ P @ ( map_entry_c_a @ X3 @ Xa @ Ps ) ) ) ) )
                 => ~ ( accp_P4197195109920256353od_c_a @ map_entry_rel_c_a @ ( produc1741992747163535396od_c_a @ X3 @ ( produc5907466840628013od_c_a @ Xa @ ( cons_P1742027962761213787od_c_a @ P @ Ps ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_520_SuccD,axiom,
    ! [K3: a,Kl: set_list_a,Kl2: list_a] :
      ( ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) )
     => ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K3 @ nil_a ) ) @ Kl ) ) ).

% SuccD
thf(fact_521_SuccD,axiom,
    ! [K3: produc2669932201312201782od_c_a,Kl: set_li1159382662694783132od_c_a,Kl2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ K3 @ ( bNF_Gr5367812047628651857od_c_a @ Kl @ Kl2 ) )
     => ( member7410604586820865893od_c_a @ ( append3884164603879732689od_c_a @ Kl2 @ ( cons_P8659537020926305772od_c_a @ K3 @ nil_Pr1342775757158464060od_c_a ) ) @ Kl ) ) ).

% SuccD
thf(fact_522_in__set__insert,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ( insert_a @ X3 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_523_in__set__insert,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( insert1660371350497266002od_c_a @ X3 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_524_not__in__set__insert,axiom,
    ! [X3: a,Xs: list_a] :
      ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ( insert_a @ X3 @ Xs )
        = ( cons_a @ X3 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_525_not__in__set__insert,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( insert1660371350497266002od_c_a @ X3 @ Xs )
        = ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_526_set__concat__elem,axiom,
    ! [X3: a,Xss2: list_list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ ( concat_a @ Xss2 ) ) )
     => ~ ! [Xs2: list_a] :
            ( ( member_list_a @ Xs2 @ ( set_list_a2 @ Xss2 ) )
           => ~ ( member_a @ X3 @ ( set_a2 @ Xs2 ) ) ) ) ).

% set_concat_elem
thf(fact_527_set__concat__elem,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xss2: list_l2826717958560394946od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( concat7533993459791330935od_c_a @ Xss2 ) ) )
     => ~ ! [Xs2: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ Xs2 @ ( set_li8663184538779258833od_c_a @ Xss2 ) )
           => ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) ) ) ) ).

% set_concat_elem
thf(fact_528_set__ConsD,axiom,
    ! [Y3: a,X3: a,Xs: list_a] :
      ( ( member_a @ Y3 @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member_a @ Y3 @ ( set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_529_set__ConsD,axiom,
    ! [Y3: produc2669932201312201782od_c_a,X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_530_list_Oset__cases,axiom,
    ! [E: a,A: list_a] :
      ( ( member_a @ E @ ( set_a2 @ A ) )
     => ( ! [Z22: list_a] :
            ( A
           != ( cons_a @ E @ Z22 ) )
       => ~ ! [Z1: a,Z22: list_a] :
              ( ( A
                = ( cons_a @ Z1 @ Z22 ) )
             => ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_531_list_Oset__cases,axiom,
    ! [E: produc2669932201312201782od_c_a,A: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ E @ ( set_Pr5664641634098845387od_c_a @ A ) )
     => ( ! [Z22: list_P6327159017948738492od_c_a] :
            ( A
           != ( cons_P8659537020926305772od_c_a @ E @ Z22 ) )
       => ~ ! [Z1: produc2669932201312201782od_c_a,Z22: list_P6327159017948738492od_c_a] :
              ( ( A
                = ( cons_P8659537020926305772od_c_a @ Z1 @ Z22 ) )
             => ~ ( member5502040268208861279od_c_a @ E @ ( set_Pr5664641634098845387od_c_a @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_532_list_Oset__intros_I1_J,axiom,
    ! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_533_list_Oset__intros_I1_J,axiom,
    ! [X21: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a] : ( member5502040268208861279od_c_a @ X21 @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_534_list_Oset__intros_I2_J,axiom,
    ! [Y3: a,X22: list_a,X21: a] :
      ( ( member_a @ Y3 @ ( set_a2 @ X22 ) )
     => ( member_a @ Y3 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_535_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a,X21: produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ X22 ) )
     => ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_536_list__concat__non__elem,axiom,
    ! [X3: a,Xs: list_a,Ys: list_a] :
      ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
       => ~ ( member_a @ X3 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_537_list__concat__non__elem,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_538_list__prefix__elem,axiom,
    ! [X3: a,Xs: list_a,Ys: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) )
     => ( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
       => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_539_list__prefix__elem,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) )
     => ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_540_list__set__sym,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ X3 @ Y3 ) )
      = ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Y3 @ X3 ) ) ) ).

% list_set_sym
thf(fact_541_split__list__first__prop__iff,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: produc2669932201312201782od_c_a > $o] :
      ( ( ? [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
            & ( P7 @ X4 ) ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,X4: produc2669932201312201782od_c_a] :
            ( ? [Zs3: list_P6327159017948738492od_c_a] :
                ( Xs
                = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X4 @ Zs3 ) ) )
            & ( P7 @ X4 )
            & ! [Y4: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y4 @ ( set_Pr5664641634098845387od_c_a @ Ys3 ) )
               => ~ ( P7 @ Y4 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_542_split__list__last__prop__iff,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: produc2669932201312201782od_c_a > $o] :
      ( ( ? [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
            & ( P7 @ X4 ) ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,X4: produc2669932201312201782od_c_a,Zs3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X4 @ Zs3 ) ) )
            & ( P7 @ X4 )
            & ! [Y4: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y4 @ ( set_Pr5664641634098845387od_c_a @ Zs3 ) )
               => ~ ( P7 @ Y4 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_543_in__set__conv__decomp__first,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
      = ( ? [Ys3: list_a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
            & ~ ( member_a @ X3 @ ( set_a2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_544_in__set__conv__decomp__first,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,Zs3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs3 ) ) )
            & ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_545_in__set__conv__decomp__last,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
      = ( ? [Ys3: list_a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
            & ~ ( member_a @ X3 @ ( set_a2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_546_in__set__conv__decomp__last,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,Zs3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs3 ) ) )
            & ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_547_split__list__first__propE,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: produc2669932201312201782od_c_a > $o] :
      ( ? [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
          & ( P7 @ X6 ) )
     => ~ ! [Ys2: list_P6327159017948738492od_c_a,X: produc2669932201312201782od_c_a] :
            ( ? [Zs2: list_P6327159017948738492od_c_a] :
                ( Xs
                = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X @ Zs2 ) ) )
           => ( ( P7 @ X )
             => ~ ! [Xa3: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa3 @ ( set_Pr5664641634098845387od_c_a @ Ys2 ) )
                   => ~ ( P7 @ Xa3 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_548_split__list__last__propE,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: produc2669932201312201782od_c_a > $o] :
      ( ? [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
          & ( P7 @ X6 ) )
     => ~ ! [Ys2: list_P6327159017948738492od_c_a,X: produc2669932201312201782od_c_a,Zs2: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X @ Zs2 ) ) )
           => ( ( P7 @ X )
             => ~ ! [Xa3: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa3 @ ( set_Pr5664641634098845387od_c_a @ Zs2 ) )
                   => ~ ( P7 @ Xa3 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_549_split__list__first__prop,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: produc2669932201312201782od_c_a > $o] :
      ( ? [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
          & ( P7 @ X6 ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,X: produc2669932201312201782od_c_a] :
          ( ? [Zs2: list_P6327159017948738492od_c_a] :
              ( Xs
              = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X @ Zs2 ) ) )
          & ( P7 @ X )
          & ! [Xa3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa3 @ ( set_Pr5664641634098845387od_c_a @ Ys2 ) )
             => ~ ( P7 @ Xa3 ) ) ) ) ).

% split_list_first_prop
thf(fact_550_split__list__last__prop,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: produc2669932201312201782od_c_a > $o] :
      ( ? [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
          & ( P7 @ X6 ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,X: produc2669932201312201782od_c_a,Zs2: list_P6327159017948738492od_c_a] :
          ( ( Xs
            = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X @ Zs2 ) ) )
          & ( P7 @ X )
          & ! [Xa3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa3 @ ( set_Pr5664641634098845387od_c_a @ Zs2 ) )
             => ~ ( P7 @ Xa3 ) ) ) ) ).

% split_list_last_prop
thf(fact_551_in__set__conv__decomp,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
      = ( ? [Ys3: list_a,Zs3: list_a] :
            ( Xs
            = ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_552_in__set__conv__decomp,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,Zs3: list_P6327159017948738492od_c_a] :
            ( Xs
            = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_553_append__Cons__eq__iff,axiom,
    ! [X3: a,Xs: list_a,Ys: list_a,Xs6: list_a,Ys5: list_a] :
      ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
       => ( ( ( append_a @ Xs @ ( cons_a @ X3 @ Ys ) )
            = ( append_a @ Xs6 @ ( cons_a @ X3 @ Ys5 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_554_append__Cons__eq__iff,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Xs6: list_P6327159017948738492od_c_a,Ys5: list_P6327159017948738492od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ( ( ( append3884164603879732689od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ X3 @ Ys ) )
            = ( append3884164603879732689od_c_a @ Xs6 @ ( cons_P8659537020926305772od_c_a @ X3 @ Ys5 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_555_split__list__propE,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: produc2669932201312201782od_c_a > $o] :
      ( ? [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
          & ( P7 @ X6 ) )
     => ~ ! [Ys2: list_P6327159017948738492od_c_a,X: produc2669932201312201782od_c_a] :
            ( ? [Zs2: list_P6327159017948738492od_c_a] :
                ( Xs
                = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X @ Zs2 ) ) )
           => ~ ( P7 @ X ) ) ) ).

% split_list_propE
thf(fact_556_split__list__first,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
          & ~ ( member_a @ X3 @ ( set_a2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_557_split__list__first,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,Zs2: list_P6327159017948738492od_c_a] :
          ( ( Xs
            = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs2 ) ) )
          & ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_558_split__list__prop,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,P7: produc2669932201312201782od_c_a > $o] :
      ( ? [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
          & ( P7 @ X6 ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,X: produc2669932201312201782od_c_a] :
          ( ? [Zs2: list_P6327159017948738492od_c_a] :
              ( Xs
              = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X @ Zs2 ) ) )
          & ( P7 @ X ) ) ) ).

% split_list_prop
thf(fact_559_split__list__last,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
          & ~ ( member_a @ X3 @ ( set_a2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_560_split__list__last,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,Zs2: list_P6327159017948738492od_c_a] :
          ( ( Xs
            = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs2 ) ) )
          & ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_561_split__list,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs2: list_a] :
          ( Xs
          = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_562_split__list,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,Zs2: list_P6327159017948738492od_c_a] :
          ( Xs
          = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_563_distinct_Osimps_I2_J,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( distinct_a @ ( cons_a @ X3 @ Xs ) )
      = ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
        & ( distinct_a @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_564_distinct_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( distin811446047591323055od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
        & ( distin811446047591323055od_c_a @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_565_List_Oinsert__def,axiom,
    ( insert_a
    = ( ^ [X4: a,Xs3: list_a] : ( if_list_a @ ( member_a @ X4 @ ( set_a2 @ Xs3 ) ) @ Xs3 @ ( cons_a @ X4 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_566_List_Oinsert__def,axiom,
    ( insert1660371350497266002od_c_a
    = ( ^ [X4: produc2669932201312201782od_c_a,Xs3: list_P6327159017948738492od_c_a] : ( if_lis1795051829816687490od_c_a @ ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ Xs3 ) ) @ Xs3 @ ( cons_P8659537020926305772od_c_a @ X4 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_567_not__distinct__conv__prefix,axiom,
    ! [As: list_a] :
      ( ( ~ ( distinct_a @ As ) )
      = ( ? [Xs3: list_a,Y4: a,Ys3: list_a] :
            ( ( member_a @ Y4 @ ( set_a2 @ Xs3 ) )
            & ( distinct_a @ Xs3 )
            & ( As
              = ( append_a @ Xs3 @ ( cons_a @ Y4 @ Ys3 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_568_not__distinct__conv__prefix,axiom,
    ! [As: list_P6327159017948738492od_c_a] :
      ( ( ~ ( distin811446047591323055od_c_a @ As ) )
      = ( ? [Xs3: list_P6327159017948738492od_c_a,Y4: produc2669932201312201782od_c_a,Ys3: list_P6327159017948738492od_c_a] :
            ( ( member5502040268208861279od_c_a @ Y4 @ ( set_Pr5664641634098845387od_c_a @ Xs3 ) )
            & ( distin811446047591323055od_c_a @ Xs3 )
            & ( As
              = ( append3884164603879732689od_c_a @ Xs3 @ ( cons_P8659537020926305772od_c_a @ Y4 @ Ys3 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_569_reachable__states__path,axiom,
    ! [Q: a,M: fsm_a_b_c,P5: list_P6327159017948738492od_c_a,T: produc2669932201312201782od_c_a] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) )
     => ( ( path_a_b_c @ M @ Q @ P5 )
       => ( ( member5502040268208861279od_c_a @ T @ ( set_Pr5664641634098845387od_c_a @ P5 ) )
         => ( member_a @ ( produc4225943634317630866od_c_a @ T ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ) ) ) ).

% reachable_states_path
thf(fact_570_visited__states__prefix,axiom,
    ! [Q4: a,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q4 @ ( set_a2 @ ( visited_states_a_b_c @ Q @ P5 ) ) )
     => ? [P12: list_P6327159017948738492od_c_a,P22: list_P6327159017948738492od_c_a] :
          ( ( P5
            = ( append3884164603879732689od_c_a @ P12 @ P22 ) )
          & ( ( target_a_b_c @ Q @ P12 )
            = Q4 ) ) ) ).

% visited_states_prefix
thf(fact_571_cyclic__path__transition__states__property_I1_J,axiom,
    ! [P5: list_P6327159017948738492od_c_a,P7: a > $o,M: fsm_a_b_c,Q: a] :
      ( ? [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ P5 ) )
          & ( P7 @ ( produc4225943634317630866od_c_a @ X6 ) ) )
     => ( ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ P5 ) )
           => ( ( P7 @ ( produc4225943634317630866od_c_a @ X ) )
             => ( P7 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) ) ) ) )
       => ( ( path_a_b_c @ M @ Q @ P5 )
         => ( ( ( target_a_b_c @ Q @ P5 )
              = Q )
           => ! [X6: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ P5 ) )
               => ( P7 @ ( produc4225943634317630866od_c_a @ X6 ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(1)
thf(fact_572_cyclic__path__transition__states__property_I2_J,axiom,
    ! [P5: list_P6327159017948738492od_c_a,P7: a > $o,M: fsm_a_b_c,Q: a] :
      ( ? [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ P5 ) )
          & ( P7 @ ( produc4225943634317630866od_c_a @ X6 ) ) )
     => ( ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ P5 ) )
           => ( ( P7 @ ( produc4225943634317630866od_c_a @ X ) )
             => ( P7 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) ) ) ) )
       => ( ( path_a_b_c @ M @ Q @ P5 )
         => ( ( ( target_a_b_c @ Q @ P5 )
              = Q )
           => ! [X6: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ P5 ) )
               => ( P7 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(2)
thf(fact_573_SuccI,axiom,
    ! [Kl2: list_a,K3: a,Kl: set_list_a] :
      ( ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K3 @ nil_a ) ) @ Kl )
     => ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_574_SuccI,axiom,
    ! [Kl2: list_P6327159017948738492od_c_a,K3: produc2669932201312201782od_c_a,Kl: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ ( append3884164603879732689od_c_a @ Kl2 @ ( cons_P8659537020926305772od_c_a @ K3 @ nil_Pr1342775757158464060od_c_a ) ) @ Kl )
     => ( member5502040268208861279od_c_a @ K3 @ ( bNF_Gr5367812047628651857od_c_a @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_575_output__complete_Oelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( output3834324022317553264_a_b_c @ X3 )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X3 ) )
           => ! [Xa2: c] :
                ( ( member_c @ Xa2 @ ( outputs_a_b_c @ X3 ) )
               => ? [Xb2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xb2 @ ( transitions_a_b_c @ X3 ) )
                    & ( ( produc4225943634317630866od_c_a @ X )
                      = ( produc4225943634317630866od_c_a @ Xb2 ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                      = Xa2 ) ) ) ) ) ).

% output_complete.elims(3)
thf(fact_576_output__complete_Oelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( output3834324022317553264_a_b_c @ X3 )
     => ! [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ ( transitions_a_b_c @ X3 ) )
         => ! [Xa3: c] :
              ( ( member_c @ Xa3 @ ( outputs_a_b_c @ X3 ) )
             => ? [Xb3: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Xb3 @ ( transitions_a_b_c @ X3 ) )
                  & ( ( produc4225943634317630866od_c_a @ X6 )
                    = ( produc4225943634317630866od_c_a @ Xb3 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb3 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb3 ) ) )
                    = Xa3 ) ) ) ) ) ).

% output_complete.elims(2)
thf(fact_577_output__complete_Oelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( output3834324022317553264_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X3 ) )
             => ! [Y4: c] :
                  ( ( member_c @ Y4 @ ( outputs_a_b_c @ X3 ) )
                 => ? [Z2: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Z2 @ ( transitions_a_b_c @ X3 ) )
                      & ( ( produc4225943634317630866od_c_a @ X4 )
                        = ( produc4225943634317630866od_c_a @ Z2 ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z2 ) ) )
                      & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Z2 ) ) )
                        = Y4 ) ) ) ) ) ) ) ).

% output_complete.elims(1)
thf(fact_578_output__complete_Osimps,axiom,
    ( output3834324022317553264_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M2 ) )
         => ! [Y4: c] :
              ( ( member_c @ Y4 @ ( outputs_a_b_c @ M2 ) )
             => ? [Z2: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Z2 @ ( transitions_a_b_c @ M2 ) )
                  & ( ( produc4225943634317630866od_c_a @ X4 )
                    = ( produc4225943634317630866od_c_a @ Z2 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z2 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Z2 ) ) )
                    = Y4 ) ) ) ) ) ) ).

% output_complete.simps
thf(fact_579_output__complete__alt__def,axiom,
    ( output1640370057126496855_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q6: product_prod_c_a,X4: a] :
          ( ? [Y4: b,Q7: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q6 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( output2371653660386375239_a_a_b @ M2 ) )
             => ? [Q7: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q6 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ) ) ) ).

% output_complete_alt_def
thf(fact_580_output__complete__alt__def,axiom,
    ( output3834324022317553264_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q6: a,X4: b] :
          ( ? [Y4: c,Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q6 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transitions_a_b_c @ M2 ) )
         => ! [Y4: c] :
              ( ( member_c @ Y4 @ ( outputs_a_b_c @ M2 ) )
             => ? [Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q6 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transitions_a_b_c @ M2 ) ) ) ) ) ) ).

% output_complete_alt_def
thf(fact_581_empty__Shift,axiom,
    ! [Kl: set_list_a,K3: a] :
      ( ( member_list_a @ nil_a @ Kl )
     => ( ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl @ nil_a ) )
       => ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_582_empty__Shift,axiom,
    ! [Kl: set_li1159382662694783132od_c_a,K3: produc2669932201312201782od_c_a] :
      ( ( member7410604586820865893od_c_a @ nil_Pr1342775757158464060od_c_a @ Kl )
     => ( ( member5502040268208861279od_c_a @ K3 @ ( bNF_Gr5367812047628651857od_c_a @ Kl @ nil_Pr1342775757158464060od_c_a ) )
       => ( member7410604586820865893od_c_a @ nil_Pr1342775757158464060od_c_a @ ( bNF_Gr6081535474448196565od_c_a @ Kl @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_583_Succ__Shift,axiom,
    ! [Kl: set_li1159382662694783132od_c_a,K3: produc2669932201312201782od_c_a,Kl2: list_P6327159017948738492od_c_a] :
      ( ( bNF_Gr5367812047628651857od_c_a @ ( bNF_Gr6081535474448196565od_c_a @ Kl @ K3 ) @ Kl2 )
      = ( bNF_Gr5367812047628651857od_c_a @ Kl @ ( cons_P8659537020926305772od_c_a @ K3 @ Kl2 ) ) ) ).

% Succ_Shift
thf(fact_584_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l2826717958560394946od_c_a] :
      ( ( ( concat7533993459791330935od_c_a @ Xss2 )
        = nil_Pr1342775757158464060od_c_a )
      = ( ! [X4: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X4 @ ( set_li8663184538779258833od_c_a @ Xss2 ) )
           => ( X4 = nil_Pr1342775757158464060od_c_a ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_585_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l2826717958560394946od_c_a] :
      ( ( nil_Pr1342775757158464060od_c_a
        = ( concat7533993459791330935od_c_a @ Xss2 ) )
      = ( ! [X4: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X4 @ ( set_li8663184538779258833od_c_a @ Xss2 ) )
           => ( X4 = nil_Pr1342775757158464060od_c_a ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_586_prefixes__Cons_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) ) ) )
     => ( member7410604586820865893od_c_a @ Xs @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Ys ) ) ) ) ).

% prefixes_Cons(2)
thf(fact_587_prefixes__Cons_I1_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) ) ) )
     => ( X3 = Y3 ) ) ).

% prefixes_Cons(1)
thf(fact_588_prefixes__set__Nil,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ nil_Pr1342775757158464060od_c_a @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) ) ) ).

% prefixes_set_Nil
thf(fact_589_prefixes__set__ob,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Xss2: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ Xs @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xss2 ) ) )
     => ~ ! [Xs5: list_P6327159017948738492od_c_a] :
            ( Xss2
           != ( append3884164603879732689od_c_a @ Xs @ Xs5 ) ) ) ).

% prefixes_set_ob
thf(fact_590_prefixes__prepend,axiom,
    ! [Xs6: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ Xs6 @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) ) )
     => ( member7410604586820865893od_c_a @ ( append3884164603879732689od_c_a @ Ys @ Xs6 ) @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ ( append3884164603879732689od_c_a @ Ys @ Xs ) ) ) ) ) ).

% prefixes_prepend
thf(fact_591_Cons__in__subseqsD,axiom,
    ! [Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) @ ( set_li8663184538779258833od_c_a @ ( subseq4269998335400966283od_c_a @ Xs ) ) )
     => ( member7410604586820865893od_c_a @ Ys @ ( set_li8663184538779258833od_c_a @ ( subseq4269998335400966283od_c_a @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_592_subseqs__distinctD,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( member_list_a @ Ys @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
     => ( ( distinct_a @ Xs )
       => ( distinct_a @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_593_prefixes__prefix__suffix__ob,axiom,
    ! [A: list_P6327159017948738492od_c_a,B: list_P6327159017948738492od_c_a,C: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ A @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ ( append3884164603879732689od_c_a @ B @ C ) ) ) )
     => ( ~ ( member7410604586820865893od_c_a @ A @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ B ) ) )
       => ~ ! [C3: list_P6327159017948738492od_c_a] :
              ( ? [C4: list_P6327159017948738492od_c_a] :
                  ( C
                  = ( append3884164603879732689od_c_a @ C3 @ C4 ) )
             => ( ( A
                  = ( append3884164603879732689od_c_a @ B @ C3 ) )
               => ( C3 = nil_Pr1342775757158464060od_c_a ) ) ) ) ) ).

% prefixes_prefix_suffix_ob
thf(fact_594_ShiftD,axiom,
    ! [Kl2: list_P6327159017948738492od_c_a,Kl: set_li1159382662694783132od_c_a,K3: produc2669932201312201782od_c_a] :
      ( ( member7410604586820865893od_c_a @ Kl2 @ ( bNF_Gr6081535474448196565od_c_a @ Kl @ K3 ) )
     => ( member7410604586820865893od_c_a @ ( cons_P8659537020926305772od_c_a @ K3 @ Kl2 ) @ Kl ) ) ).

% ShiftD
thf(fact_595_distinct__product__lists,axiom,
    ! [Xss2: list_list_a] :
      ( ! [X: list_a] :
          ( ( member_list_a @ X @ ( set_list_a2 @ Xss2 ) )
         => ( distinct_a @ X ) )
     => ( distinct_list_a @ ( product_lists_a @ Xss2 ) ) ) ).

% distinct_product_lists
thf(fact_596_language__equivalence__from__isomorphism__reachable,axiom,
    ! [F2: a > product_prod_c_a,M1: fsm_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b] :
      ( ( bij_be3575907416252707741od_c_a @ F2 @ ( reacha4407761112724606147_a_a_b @ M1 ) @ ( reacha5035603111973943274_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initial_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: a,X: a,Y: b,Q8: a] :
              ( ( member_a @ Q5 @ ( reacha4407761112724606147_a_a_b @ M1 ) )
             => ( ( member_a @ Q8 @ ( reacha4407761112724606147_a_a_b @ M1 ) )
               => ( ( member4187420457662959969od_b_a @ ( produc7037890857971478952od_b_a @ Q5 @ ( produc6867761796305373823od_b_a @ X @ ( product_Pair_b_a @ Y @ Q8 ) ) ) @ ( transitions_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ( lS_a_a_b @ M1 @ ( initial_a_a_b @ M1 ) )
            = ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( initia681974817753967189_a_a_b @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_reachable
thf(fact_597_language__equivalence__from__isomorphism__reachable,axiom,
    ! [F2: produc2669932201312201782od_c_a > product_prod_c_a,M1: fsm_Pr9212659883570776987_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b] :
      ( ( bij_be2014444163610170213od_c_a @ F2 @ ( reacha4177095180937997819_a_a_b @ M1 ) @ ( reacha5035603111973943274_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initia6128492417919064678_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: produc2669932201312201782od_c_a,X: a,Y: b,Q8: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Q5 @ ( reacha4177095180937997819_a_a_b @ M1 ) )
             => ( ( member5502040268208861279od_c_a @ Q8 @ ( reacha4177095180937997819_a_a_b @ M1 ) )
               => ( ( member425229614235723617od_c_a @ ( produc1944339458867481512od_c_a @ Q5 @ ( produc1283362787453724087od_c_a @ X @ ( produc6426307015734985358od_c_a @ Y @ Q8 ) ) ) @ ( transi8346546345654214412_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ( lS_Pro6519348473384642011_a_a_b @ M1 @ ( initia6128492417919064678_a_a_b @ M1 ) )
            = ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( initia681974817753967189_a_a_b @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_reachable
thf(fact_598_language__equivalence__from__isomorphism__reachable,axiom,
    ! [F2: product_prod_c_a > product_prod_c_a,M1: fsm_Pr1886043760914421514_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b] :
      ( ( bij_be4905085173284547766od_c_a @ F2 @ ( reacha5035603111973943274_a_a_b @ M1 ) @ ( reacha5035603111973943274_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initia681974817753967189_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: product_prod_c_a,X: a,Y: b,Q8: product_prod_c_a] :
              ( ( member5074992350434858958od_c_a @ Q5 @ ( reacha5035603111973943274_a_a_b @ M1 ) )
             => ( ( member5074992350434858958od_c_a @ Q8 @ ( reacha5035603111973943274_a_a_b @ M1 ) )
               => ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q5 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ Q8 ) ) ) @ ( transi2941759273464634107_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ( lS_Pro5114676609533226954_a_a_b @ M1 @ ( initia681974817753967189_a_a_b @ M1 ) )
            = ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( initia681974817753967189_a_a_b @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_reachable
thf(fact_599_language__equivalence__from__isomorphism__reachable,axiom,
    ! [F2: produc2669932201312201782od_c_a > a,M1: fsm_Pr6425204305901329755_a_b_c,M22: fsm_a_b_c] :
      ( ( bij_be5632368444736963774_c_a_a @ F2 @ ( reacha1389639598965321787_a_b_c @ M1 ) @ ( reacha1620305530751930115_a_b_c @ M22 ) )
     => ( ( ( F2 @ ( initia3341036835946388646_a_b_c @ M1 ) )
          = ( initial_a_b_c @ M22 ) )
       => ( ! [Q5: produc2669932201312201782od_c_a,X: b,Y: c,Q8: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Q5 @ ( reacha1389639598965321787_a_b_c @ M1 ) )
             => ( ( member5502040268208861279od_c_a @ Q8 @ ( reacha1389639598965321787_a_b_c @ M1 ) )
               => ( ( member6671621370117864543od_c_a @ ( produc6668960782114422694od_c_a @ Q5 @ ( produc727174699211671861od_c_a @ X @ ( produc8442212246646489037od_c_a @ Y @ Q8 ) ) ) @ ( transi5559090763681538380_a_b_c @ M1 ) )
                  = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ ( F2 @ Q5 ) @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transitions_a_b_c @ M22 ) ) ) ) )
         => ( ( lS_Pro3731892891411965979_a_b_c @ M1 @ ( initia3341036835946388646_a_b_c @ M1 ) )
            = ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_reachable
thf(fact_600_language__equivalence__from__isomorphism__reachable,axiom,
    ! [F2: a > a,M1: fsm_a_b_c,M22: fsm_a_b_c] :
      ( ( bij_betw_a_a @ F2 @ ( reacha1620305530751930115_a_b_c @ M1 ) @ ( reacha1620305530751930115_a_b_c @ M22 ) )
     => ( ( ( F2 @ ( initial_a_b_c @ M1 ) )
          = ( initial_a_b_c @ M22 ) )
       => ( ! [Q5: a,X: b,Y: c,Q8: a] :
              ( ( member_a @ Q5 @ ( reacha1620305530751930115_a_b_c @ M1 ) )
             => ( ( member_a @ Q8 @ ( reacha1620305530751930115_a_b_c @ M1 ) )
               => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q5 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ Q8 ) ) ) @ ( transitions_a_b_c @ M1 ) )
                  = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ ( F2 @ Q5 ) @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transitions_a_b_c @ M22 ) ) ) ) )
         => ( ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) )
            = ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_reachable
thf(fact_601_language__equivalence__from__isomorphism,axiom,
    ! [F2: a > product_prod_c_a,M1: fsm_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b,Q: a] :
      ( ( bij_be3575907416252707741od_c_a @ F2 @ ( states_a_a_b @ M1 ) @ ( states367398909958982703_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initial_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: a,X: a,Y: b,Q8: a] :
              ( ( member_a @ Q5 @ ( states_a_a_b @ M1 ) )
             => ( ( member_a @ Q8 @ ( states_a_a_b @ M1 ) )
               => ( ( member4187420457662959969od_b_a @ ( produc7037890857971478952od_b_a @ Q5 @ ( produc6867761796305373823od_b_a @ X @ ( product_Pair_b_a @ Y @ Q8 ) ) ) @ ( transitions_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ( member_a @ Q @ ( states_a_a_b @ M1 ) )
           => ( ( lS_a_a_b @ M1 @ Q )
              = ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism
thf(fact_602_language__equivalence__from__isomorphism,axiom,
    ! [F2: produc2669932201312201782od_c_a > product_prod_c_a,M1: fsm_Pr9212659883570776987_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b,Q: produc2669932201312201782od_c_a] :
      ( ( bij_be2014444163610170213od_c_a @ F2 @ ( states3683092704449898560_a_a_b @ M1 ) @ ( states367398909958982703_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initia6128492417919064678_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: produc2669932201312201782od_c_a,X: a,Y: b,Q8: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Q5 @ ( states3683092704449898560_a_a_b @ M1 ) )
             => ( ( member5502040268208861279od_c_a @ Q8 @ ( states3683092704449898560_a_a_b @ M1 ) )
               => ( ( member425229614235723617od_c_a @ ( produc1944339458867481512od_c_a @ Q5 @ ( produc1283362787453724087od_c_a @ X @ ( produc6426307015734985358od_c_a @ Y @ Q8 ) ) ) @ ( transi8346546345654214412_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ( member5502040268208861279od_c_a @ Q @ ( states3683092704449898560_a_a_b @ M1 ) )
           => ( ( lS_Pro6519348473384642011_a_a_b @ M1 @ Q )
              = ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism
thf(fact_603_language__equivalence__from__isomorphism,axiom,
    ! [F2: produc2669932201312201782od_c_a > a,M1: fsm_Pr6425204305901329755_a_b_c,M22: fsm_a_b_c,Q: produc2669932201312201782od_c_a] :
      ( ( bij_be5632368444736963774_c_a_a @ F2 @ ( states895637122477222528_a_b_c @ M1 ) @ ( states_a_b_c @ M22 ) )
     => ( ( ( F2 @ ( initia3341036835946388646_a_b_c @ M1 ) )
          = ( initial_a_b_c @ M22 ) )
       => ( ! [Q5: produc2669932201312201782od_c_a,X: b,Y: c,Q8: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Q5 @ ( states895637122477222528_a_b_c @ M1 ) )
             => ( ( member5502040268208861279od_c_a @ Q8 @ ( states895637122477222528_a_b_c @ M1 ) )
               => ( ( member6671621370117864543od_c_a @ ( produc6668960782114422694od_c_a @ Q5 @ ( produc727174699211671861od_c_a @ X @ ( produc8442212246646489037od_c_a @ Y @ Q8 ) ) ) @ ( transi5559090763681538380_a_b_c @ M1 ) )
                  = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ ( F2 @ Q5 ) @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transitions_a_b_c @ M22 ) ) ) ) )
         => ( ( member5502040268208861279od_c_a @ Q @ ( states895637122477222528_a_b_c @ M1 ) )
           => ( ( lS_Pro3731892891411965979_a_b_c @ M1 @ Q )
              = ( lS_a_b_c @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism
thf(fact_604_language__equivalence__from__isomorphism,axiom,
    ! [F2: product_prod_c_a > product_prod_c_a,M1: fsm_Pr1886043760914421514_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a] :
      ( ( bij_be4905085173284547766od_c_a @ F2 @ ( states367398909958982703_a_a_b @ M1 ) @ ( states367398909958982703_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initia681974817753967189_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: product_prod_c_a,X: a,Y: b,Q8: product_prod_c_a] :
              ( ( member5074992350434858958od_c_a @ Q5 @ ( states367398909958982703_a_a_b @ M1 ) )
             => ( ( member5074992350434858958od_c_a @ Q8 @ ( states367398909958982703_a_a_b @ M1 ) )
               => ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q5 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ Q8 ) ) ) @ ( transi2941759273464634107_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ( member5074992350434858958od_c_a @ Q @ ( states367398909958982703_a_a_b @ M1 ) )
           => ( ( lS_Pro5114676609533226954_a_a_b @ M1 @ Q )
              = ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism
thf(fact_605_language__equivalence__from__isomorphism,axiom,
    ! [F2: a > a,M1: fsm_a_b_c,M22: fsm_a_b_c,Q: a] :
      ( ( bij_betw_a_a @ F2 @ ( states_a_b_c @ M1 ) @ ( states_a_b_c @ M22 ) )
     => ( ( ( F2 @ ( initial_a_b_c @ M1 ) )
          = ( initial_a_b_c @ M22 ) )
       => ( ! [Q5: a,X: b,Y: c,Q8: a] :
              ( ( member_a @ Q5 @ ( states_a_b_c @ M1 ) )
             => ( ( member_a @ Q8 @ ( states_a_b_c @ M1 ) )
               => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q5 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ Q8 ) ) ) @ ( transitions_a_b_c @ M1 ) )
                  = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ ( F2 @ Q5 ) @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transitions_a_b_c @ M22 ) ) ) ) )
         => ( ( member_a @ Q @ ( states_a_b_c @ M1 ) )
           => ( ( lS_a_b_c @ M1 @ Q )
              = ( lS_a_b_c @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism
thf(fact_606_non__sym__dist__pairs_H__elems__distinct_I2_J,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a,Y3: produc2669932201312201782od_c_a] :
      ( ( distin811446047591323055od_c_a @ Xs )
     => ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y3 ) @ ( set_Pr3422107008073957564od_c_a @ ( non_sy1373097905802202817od_c_a @ Xs ) ) )
       => ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(2)
thf(fact_607_non__sym__dist__pairs_H__elems__distinct_I2_J,axiom,
    ! [Xs: list_a,X3: a,Y3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
       => ( member_a @ Y3 @ ( set_a2 @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(2)
thf(fact_608_non__sym__dist__pairs_H__elems__distinct_I1_J,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a,Y3: produc2669932201312201782od_c_a] :
      ( ( distin811446047591323055od_c_a @ Xs )
     => ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y3 ) @ ( set_Pr3422107008073957564od_c_a @ ( non_sy1373097905802202817od_c_a @ Xs ) ) )
       => ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(1)
thf(fact_609_non__sym__dist__pairs_H__elems__distinct_I1_J,axiom,
    ! [Xs: list_a,X3: a,Y3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
       => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(1)
thf(fact_610_map__ran__Cons__sel,axiom,
    ! [F2: a > produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,P5: produc2669932201312201782od_c_a,Ps2: list_P6327159017948738492od_c_a] :
      ( ( map_ra3149561528272431559od_c_a @ F2 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
      = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ P5 ) @ ( F2 @ ( produc4225943634317630866od_c_a @ P5 ) @ ( produc8088130389382527188od_c_a @ P5 ) ) ) @ ( map_ra3149561528272431559od_c_a @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_611_list__ordered__pairs__set__containment,axiom,
    ! [X3: a,Xs: list_a,Y3: a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ( member_a @ Y3 @ ( set_a2 @ Xs ) )
       => ( ( X3 != Y3 )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( list_ordered_pairs_a @ Xs ) ) )
            | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ ( set_Product_prod_a_a2 @ ( list_ordered_pairs_a @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_612_list__ordered__pairs__set__containment,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
       => ( ( X3 != Y3 )
         => ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y3 ) @ ( set_Pr3422107008073957564od_c_a @ ( list_o2952673153366482557od_c_a @ Xs ) ) )
            | ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ Y3 @ X3 ) @ ( set_Pr3422107008073957564od_c_a @ ( list_o2952673153366482557od_c_a @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_613_map__ran__simps_I1_J,axiom,
    ! [F2: a > produc2133359284458116493od_c_a > produc2133359284458116493od_c_a] :
      ( ( map_ra3149561528272431559od_c_a @ F2 @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% map_ran_simps(1)
thf(fact_614_map__ran__simps_I2_J,axiom,
    ! [F2: a > produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,K3: a,V2: produc2133359284458116493od_c_a,Ps2: list_P6327159017948738492od_c_a] :
      ( ( map_ra3149561528272431559od_c_a @ F2 @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra3149561528272431559od_c_a @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_615_map__ran__simps_I2_J,axiom,
    ! [F2: b > product_prod_c_a > product_prod_c_a,K3: b,V2: product_prod_c_a,Ps2: list_P5767727266746070163od_c_a] :
      ( ( map_ra7041774465467605832od_c_a @ F2 @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra7041774465467605832od_c_a @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_616_map__ran__simps_I2_J,axiom,
    ! [F2: c > a > a,K3: c,V2: a,Ps2: list_P125642481956313003od_c_a] :
      ( ( map_ran_c_a_a @ F2 @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ran_c_a_a @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_617_non__sym__dist__pairs_H_Osimps_I1_J,axiom,
    ( ( non_sy1373097905802202817od_c_a @ nil_Pr1342775757158464060od_c_a )
    = nil_Pr98046886160973869od_c_a ) ).

% non_sym_dist_pairs'.simps(1)
thf(fact_618_list__ordered__pairs_Osimps_I1_J,axiom,
    ( ( list_o2952673153366482557od_c_a @ nil_Pr1342775757158464060od_c_a )
    = nil_Pr98046886160973869od_c_a ) ).

% list_ordered_pairs.simps(1)
thf(fact_619_non__sym__dist__pairs_H__elems__distinct_I3_J,axiom,
    ! [Xs: list_a,X3: a,Y3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
       => ( X3 != Y3 ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(3)
thf(fact_620_non__sym__dist__pairs_H__elems__non__refl,axiom,
    ! [Xs: list_a,X3: a,Y3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs'_elems_non_refl
thf(fact_621_language__equivalence__from__isomorphism__helper__reachable,axiom,
    ! [F2: a > product_prod_c_a,M1: fsm_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b] :
      ( ( bij_be3575907416252707741od_c_a @ F2 @ ( reacha4407761112724606147_a_a_b @ M1 ) @ ( reacha5035603111973943274_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initial_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: a,X: a,Y: b,Q8: a] :
              ( ( member_a @ Q5 @ ( reacha4407761112724606147_a_a_b @ M1 ) )
             => ( ( member_a @ Q8 @ ( reacha4407761112724606147_a_a_b @ M1 ) )
               => ( ( member4187420457662959969od_b_a @ ( produc7037890857971478952od_b_a @ Q5 @ ( produc6867761796305373823od_b_a @ X @ ( product_Pair_b_a @ Y @ Q8 ) ) ) @ ( transitions_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ord_le153002609085230766od_a_b @ ( lS_a_a_b @ M1 @ ( initial_a_a_b @ M1 ) ) @ ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( initia681974817753967189_a_a_b @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper_reachable
thf(fact_622_language__equivalence__from__isomorphism__helper__reachable,axiom,
    ! [F2: produc2669932201312201782od_c_a > product_prod_c_a,M1: fsm_Pr9212659883570776987_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b] :
      ( ( bij_be2014444163610170213od_c_a @ F2 @ ( reacha4177095180937997819_a_a_b @ M1 ) @ ( reacha5035603111973943274_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initia6128492417919064678_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: produc2669932201312201782od_c_a,X: a,Y: b,Q8: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Q5 @ ( reacha4177095180937997819_a_a_b @ M1 ) )
             => ( ( member5502040268208861279od_c_a @ Q8 @ ( reacha4177095180937997819_a_a_b @ M1 ) )
               => ( ( member425229614235723617od_c_a @ ( produc1944339458867481512od_c_a @ Q5 @ ( produc1283362787453724087od_c_a @ X @ ( produc6426307015734985358od_c_a @ Y @ Q8 ) ) ) @ ( transi8346546345654214412_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ord_le153002609085230766od_a_b @ ( lS_Pro6519348473384642011_a_a_b @ M1 @ ( initia6128492417919064678_a_a_b @ M1 ) ) @ ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( initia681974817753967189_a_a_b @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper_reachable
thf(fact_623_language__equivalence__from__isomorphism__helper__reachable,axiom,
    ! [F2: product_prod_c_a > product_prod_c_a,M1: fsm_Pr1886043760914421514_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b] :
      ( ( bij_be4905085173284547766od_c_a @ F2 @ ( reacha5035603111973943274_a_a_b @ M1 ) @ ( reacha5035603111973943274_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initia681974817753967189_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: product_prod_c_a,X: a,Y: b,Q8: product_prod_c_a] :
              ( ( member5074992350434858958od_c_a @ Q5 @ ( reacha5035603111973943274_a_a_b @ M1 ) )
             => ( ( member5074992350434858958od_c_a @ Q8 @ ( reacha5035603111973943274_a_a_b @ M1 ) )
               => ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q5 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ Q8 ) ) ) @ ( transi2941759273464634107_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ord_le153002609085230766od_a_b @ ( lS_Pro5114676609533226954_a_a_b @ M1 @ ( initia681974817753967189_a_a_b @ M1 ) ) @ ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( initia681974817753967189_a_a_b @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper_reachable
thf(fact_624_language__equivalence__from__isomorphism__helper__reachable,axiom,
    ! [F2: produc2669932201312201782od_c_a > a,M1: fsm_Pr6425204305901329755_a_b_c,M22: fsm_a_b_c] :
      ( ( bij_be5632368444736963774_c_a_a @ F2 @ ( reacha1389639598965321787_a_b_c @ M1 ) @ ( reacha1620305530751930115_a_b_c @ M22 ) )
     => ( ( ( F2 @ ( initia3341036835946388646_a_b_c @ M1 ) )
          = ( initial_a_b_c @ M22 ) )
       => ( ! [Q5: produc2669932201312201782od_c_a,X: b,Y: c,Q8: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Q5 @ ( reacha1389639598965321787_a_b_c @ M1 ) )
             => ( ( member5502040268208861279od_c_a @ Q8 @ ( reacha1389639598965321787_a_b_c @ M1 ) )
               => ( ( member6671621370117864543od_c_a @ ( produc6668960782114422694od_c_a @ Q5 @ ( produc727174699211671861od_c_a @ X @ ( produc8442212246646489037od_c_a @ Y @ Q8 ) ) ) @ ( transi5559090763681538380_a_b_c @ M1 ) )
                  = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ ( F2 @ Q5 ) @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transitions_a_b_c @ M22 ) ) ) ) )
         => ( ord_le282488521294790766od_b_c @ ( lS_Pro3731892891411965979_a_b_c @ M1 @ ( initia3341036835946388646_a_b_c @ M1 ) ) @ ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper_reachable
thf(fact_625_language__equivalence__from__isomorphism__helper__reachable,axiom,
    ! [F2: a > a,M1: fsm_a_b_c,M22: fsm_a_b_c] :
      ( ( bij_betw_a_a @ F2 @ ( reacha1620305530751930115_a_b_c @ M1 ) @ ( reacha1620305530751930115_a_b_c @ M22 ) )
     => ( ( ( F2 @ ( initial_a_b_c @ M1 ) )
          = ( initial_a_b_c @ M22 ) )
       => ( ! [Q5: a,X: b,Y: c,Q8: a] :
              ( ( member_a @ Q5 @ ( reacha1620305530751930115_a_b_c @ M1 ) )
             => ( ( member_a @ Q8 @ ( reacha1620305530751930115_a_b_c @ M1 ) )
               => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q5 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ Q8 ) ) ) @ ( transitions_a_b_c @ M1 ) )
                  = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ ( F2 @ Q5 ) @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transitions_a_b_c @ M22 ) ) ) ) )
         => ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) ) @ ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper_reachable
thf(fact_626_language__equivalence__from__isomorphism__helper,axiom,
    ! [F2: a > product_prod_c_a,M1: fsm_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b,Q: a] :
      ( ( bij_be3575907416252707741od_c_a @ F2 @ ( states_a_a_b @ M1 ) @ ( states367398909958982703_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initial_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: a,X: a,Y: b,Q8: a] :
              ( ( member_a @ Q5 @ ( states_a_a_b @ M1 ) )
             => ( ( member_a @ Q8 @ ( states_a_a_b @ M1 ) )
               => ( ( member4187420457662959969od_b_a @ ( produc7037890857971478952od_b_a @ Q5 @ ( produc6867761796305373823od_b_a @ X @ ( product_Pair_b_a @ Y @ Q8 ) ) ) @ ( transitions_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ( member_a @ Q @ ( states_a_a_b @ M1 ) )
           => ( ord_le153002609085230766od_a_b @ ( lS_a_a_b @ M1 @ Q ) @ ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper
thf(fact_627_language__equivalence__from__isomorphism__helper,axiom,
    ! [F2: produc2669932201312201782od_c_a > product_prod_c_a,M1: fsm_Pr9212659883570776987_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b,Q: produc2669932201312201782od_c_a] :
      ( ( bij_be2014444163610170213od_c_a @ F2 @ ( states3683092704449898560_a_a_b @ M1 ) @ ( states367398909958982703_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initia6128492417919064678_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: produc2669932201312201782od_c_a,X: a,Y: b,Q8: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Q5 @ ( states3683092704449898560_a_a_b @ M1 ) )
             => ( ( member5502040268208861279od_c_a @ Q8 @ ( states3683092704449898560_a_a_b @ M1 ) )
               => ( ( member425229614235723617od_c_a @ ( produc1944339458867481512od_c_a @ Q5 @ ( produc1283362787453724087od_c_a @ X @ ( produc6426307015734985358od_c_a @ Y @ Q8 ) ) ) @ ( transi8346546345654214412_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ( member5502040268208861279od_c_a @ Q @ ( states3683092704449898560_a_a_b @ M1 ) )
           => ( ord_le153002609085230766od_a_b @ ( lS_Pro6519348473384642011_a_a_b @ M1 @ Q ) @ ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper
thf(fact_628_language__equivalence__from__isomorphism__helper,axiom,
    ! [F2: produc2669932201312201782od_c_a > a,M1: fsm_Pr6425204305901329755_a_b_c,M22: fsm_a_b_c,Q: produc2669932201312201782od_c_a] :
      ( ( bij_be5632368444736963774_c_a_a @ F2 @ ( states895637122477222528_a_b_c @ M1 ) @ ( states_a_b_c @ M22 ) )
     => ( ( ( F2 @ ( initia3341036835946388646_a_b_c @ M1 ) )
          = ( initial_a_b_c @ M22 ) )
       => ( ! [Q5: produc2669932201312201782od_c_a,X: b,Y: c,Q8: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Q5 @ ( states895637122477222528_a_b_c @ M1 ) )
             => ( ( member5502040268208861279od_c_a @ Q8 @ ( states895637122477222528_a_b_c @ M1 ) )
               => ( ( member6671621370117864543od_c_a @ ( produc6668960782114422694od_c_a @ Q5 @ ( produc727174699211671861od_c_a @ X @ ( produc8442212246646489037od_c_a @ Y @ Q8 ) ) ) @ ( transi5559090763681538380_a_b_c @ M1 ) )
                  = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ ( F2 @ Q5 ) @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transitions_a_b_c @ M22 ) ) ) ) )
         => ( ( member5502040268208861279od_c_a @ Q @ ( states895637122477222528_a_b_c @ M1 ) )
           => ( ord_le282488521294790766od_b_c @ ( lS_Pro3731892891411965979_a_b_c @ M1 @ Q ) @ ( lS_a_b_c @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper
thf(fact_629_language__equivalence__from__isomorphism__helper,axiom,
    ! [F2: product_prod_c_a > product_prod_c_a,M1: fsm_Pr1886043760914421514_a_a_b,M22: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a] :
      ( ( bij_be4905085173284547766od_c_a @ F2 @ ( states367398909958982703_a_a_b @ M1 ) @ ( states367398909958982703_a_a_b @ M22 ) )
     => ( ( ( F2 @ ( initia681974817753967189_a_a_b @ M1 ) )
          = ( initia681974817753967189_a_a_b @ M22 ) )
       => ( ! [Q5: product_prod_c_a,X: a,Y: b,Q8: product_prod_c_a] :
              ( ( member5074992350434858958od_c_a @ Q5 @ ( states367398909958982703_a_a_b @ M1 ) )
             => ( ( member5074992350434858958od_c_a @ Q8 @ ( states367398909958982703_a_a_b @ M1 ) )
               => ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q5 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ Q8 ) ) ) @ ( transi2941759273464634107_a_a_b @ M1 ) )
                  = ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ ( F2 @ Q5 ) @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transi2941759273464634107_a_a_b @ M22 ) ) ) ) )
         => ( ( member5074992350434858958od_c_a @ Q @ ( states367398909958982703_a_a_b @ M1 ) )
           => ( ord_le153002609085230766od_a_b @ ( lS_Pro5114676609533226954_a_a_b @ M1 @ Q ) @ ( lS_Pro5114676609533226954_a_a_b @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper
thf(fact_630_language__equivalence__from__isomorphism__helper,axiom,
    ! [F2: a > a,M1: fsm_a_b_c,M22: fsm_a_b_c,Q: a] :
      ( ( bij_betw_a_a @ F2 @ ( states_a_b_c @ M1 ) @ ( states_a_b_c @ M22 ) )
     => ( ( ( F2 @ ( initial_a_b_c @ M1 ) )
          = ( initial_a_b_c @ M22 ) )
       => ( ! [Q5: a,X: b,Y: c,Q8: a] :
              ( ( member_a @ Q5 @ ( states_a_b_c @ M1 ) )
             => ( ( member_a @ Q8 @ ( states_a_b_c @ M1 ) )
               => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q5 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ Q8 ) ) ) @ ( transitions_a_b_c @ M1 ) )
                  = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ ( F2 @ Q5 ) @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ ( F2 @ Q8 ) ) ) ) @ ( transitions_a_b_c @ M22 ) ) ) ) )
         => ( ( member_a @ Q @ ( states_a_b_c @ M1 ) )
           => ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ M1 @ Q ) @ ( lS_a_b_c @ M22 @ ( F2 @ Q ) ) ) ) ) ) ) ).

% language_equivalence_from_isomorphism_helper
thf(fact_631_list__ordered__pairs_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P4800044111251741997od_c_a] :
      ( ( ( list_o2952673153366482557od_c_a @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
         => ( Y3 != nil_Pr98046886160973869od_c_a ) )
       => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( ( X3
                = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
             => ( Y3
               != ( append3794360244814202818od_c_a @ ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X ) @ Xs2 ) @ ( list_o2952673153366482557od_c_a @ Xs2 ) ) ) ) ) ) ).

% list_ordered_pairs.elims
thf(fact_632_non__sym__dist__pairs_H_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P4800044111251741997od_c_a] :
      ( ( ( non_sy1373097905802202817od_c_a @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
         => ( Y3 != nil_Pr98046886160973869od_c_a ) )
       => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( ( X3
                = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
             => ( Y3
               != ( append3794360244814202818od_c_a @ ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X ) @ Xs2 ) @ ( non_sy1373097905802202817od_c_a @ Xs2 ) ) ) ) ) ) ).

% non_sym_dist_pairs'.elims
thf(fact_633_output__complete_Opelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( output3834324022317553264_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X3 )
           => ! [X: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X3 ) )
               => ! [Xa2: c] :
                    ( ( member_c @ Xa2 @ ( outputs_a_b_c @ X3 ) )
                   => ? [Xb2: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Xb2 @ ( transitions_a_b_c @ X3 ) )
                        & ( ( produc4225943634317630866od_c_a @ X )
                          = ( produc4225943634317630866od_c_a @ Xb2 ) )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                          = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                        & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                          = Xa2 ) ) ) ) ) ) ) ).

% output_complete.pelims(3)
thf(fact_634_output__complete_Opelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( output3834324022317553264_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X3 )
           => ~ ! [X6: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ X6 @ ( transitions_a_b_c @ X3 ) )
                 => ! [Xa3: c] :
                      ( ( member_c @ Xa3 @ ( outputs_a_b_c @ X3 ) )
                     => ? [Xb3: produc2669932201312201782od_c_a] :
                          ( ( member5502040268208861279od_c_a @ Xb3 @ ( transitions_a_b_c @ X3 ) )
                          & ( ( produc4225943634317630866od_c_a @ X6 )
                            = ( produc4225943634317630866od_c_a @ Xb3 ) )
                          & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) )
                            = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb3 ) ) )
                          & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb3 ) ) )
                            = Xa3 ) ) ) ) ) ) ) ).

% output_complete.pelims(2)
thf(fact_635_list_Omap__disc__iff,axiom,
    ! [F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr6705208461955284164od_c_a @ F2 @ A )
        = nil_Pr1342775757158464060od_c_a )
      = ( A = nil_Pr1342775757158464060od_c_a ) ) ).

% list.map_disc_iff
thf(fact_636_Nil__is__map__conv,axiom,
    ! [F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( nil_Pr1342775757158464060od_c_a
        = ( map_Pr6705208461955284164od_c_a @ F2 @ Xs ) )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% Nil_is_map_conv
thf(fact_637_map__is__Nil__conv,axiom,
    ! [F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr6705208461955284164od_c_a @ F2 @ Xs )
        = nil_Pr1342775757158464060od_c_a )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% map_is_Nil_conv
thf(fact_638_map__append,axiom,
    ! [F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( map_Pr6705208461955284164od_c_a @ F2 @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
      = ( append3884164603879732689od_c_a @ ( map_Pr6705208461955284164od_c_a @ F2 @ Xs ) @ ( map_Pr6705208461955284164od_c_a @ F2 @ Ys ) ) ) ).

% map_append
thf(fact_639_list_Osimps_I9_J,axiom,
    ! [F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,X21: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a] :
      ( ( map_Pr6705208461955284164od_c_a @ F2 @ ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) )
      = ( cons_P8659537020926305772od_c_a @ ( F2 @ X21 ) @ ( map_Pr6705208461955284164od_c_a @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_640_Cons__eq__map__D,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( cons_P8659537020926305772od_c_a @ X3 @ Xs )
        = ( map_Pr6705208461955284164od_c_a @ F2 @ Ys ) )
     => ? [Z3: produc2669932201312201782od_c_a,Zs2: list_P6327159017948738492od_c_a] :
          ( ( Ys
            = ( cons_P8659537020926305772od_c_a @ Z3 @ Zs2 ) )
          & ( X3
            = ( F2 @ Z3 ) )
          & ( Xs
            = ( map_Pr6705208461955284164od_c_a @ F2 @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_641_map__eq__Cons__D,axiom,
    ! [F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr6705208461955284164od_c_a @ F2 @ Xs )
        = ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) )
     => ? [Z3: produc2669932201312201782od_c_a,Zs2: list_P6327159017948738492od_c_a] :
          ( ( Xs
            = ( cons_P8659537020926305772od_c_a @ Z3 @ Zs2 ) )
          & ( ( F2 @ Z3 )
            = Y3 )
          & ( ( map_Pr6705208461955284164od_c_a @ F2 @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_642_Cons__eq__map__conv,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( cons_P8659537020926305772od_c_a @ X3 @ Xs )
        = ( map_Pr6705208461955284164od_c_a @ F2 @ Ys ) )
      = ( ? [Z2: produc2669932201312201782od_c_a,Zs3: list_P6327159017948738492od_c_a] :
            ( ( Ys
              = ( cons_P8659537020926305772od_c_a @ Z2 @ Zs3 ) )
            & ( X3
              = ( F2 @ Z2 ) )
            & ( Xs
              = ( map_Pr6705208461955284164od_c_a @ F2 @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_643_map__eq__Cons__conv,axiom,
    ! [F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr6705208461955284164od_c_a @ F2 @ Xs )
        = ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) )
      = ( ? [Z2: produc2669932201312201782od_c_a,Zs3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( cons_P8659537020926305772od_c_a @ Z2 @ Zs3 ) )
            & ( ( F2 @ Z2 )
              = Y3 )
            & ( ( map_Pr6705208461955284164od_c_a @ F2 @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_644_list_Osimps_I8_J,axiom,
    ! [F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ( map_Pr6705208461955284164od_c_a @ F2 @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% list.simps(8)
thf(fact_645_map__idI,axiom,
    ! [Xs: list_a,F2: a > a] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ( ( F2 @ X )
            = X ) )
     => ( ( map_a_a @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_646_map__idI,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ! [X: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
         => ( ( F2 @ X )
            = X ) )
     => ( ( map_Pr6705208461955284164od_c_a @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_647_list_Omap__ident__strong,axiom,
    ! [T: list_a,F2: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_a_a @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_648_list_Omap__ident__strong,axiom,
    ! [T: list_P6327159017948738492od_c_a,F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ! [Z3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ Z3 @ ( set_Pr5664641634098845387od_c_a @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_Pr6705208461955284164od_c_a @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_649_set__map__elem,axiom,
    ! [Y3: a,F2: a > a,Xs: list_a] :
      ( ( member_a @ Y3 @ ( set_a2 @ ( map_a_a @ F2 @ Xs ) ) )
     => ~ ! [X: a] :
            ( ( Y3
              = ( F2 @ X ) )
           => ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_650_set__map__elem,axiom,
    ! [Y3: a,F2: produc2669932201312201782od_c_a > a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Y3 @ ( set_a2 @ ( map_Pr5179659585931931020_c_a_a @ F2 @ Xs ) ) )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( Y3
              = ( F2 @ X ) )
           => ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_651_set__map__elem,axiom,
    ! [Y3: produc2669932201312201782od_c_a,F2: a > produc2669932201312201782od_c_a,Xs: list_a] :
      ( ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ ( map_a_192154525783254780od_c_a @ F2 @ Xs ) ) )
     => ~ ! [X: a] :
            ( ( Y3
              = ( F2 @ X ) )
           => ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_652_set__map__elem,axiom,
    ! [Y3: produc2669932201312201782od_c_a,F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ ( map_Pr6705208461955284164od_c_a @ F2 @ Xs ) ) )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( Y3
              = ( F2 @ X ) )
           => ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_653_map__set,axiom,
    ! [X3: a,Xs: list_a,F2: a > a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( member_a @ ( F2 @ X3 ) @ ( set_a2 @ ( map_a_a @ F2 @ Xs ) ) ) ) ).

% map_set
thf(fact_654_map__set,axiom,
    ! [X3: a,Xs: list_a,F2: a > produc2669932201312201782od_c_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( member5502040268208861279od_c_a @ ( F2 @ X3 ) @ ( set_Pr5664641634098845387od_c_a @ ( map_a_192154525783254780od_c_a @ F2 @ Xs ) ) ) ) ).

% map_set
thf(fact_655_map__set,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,F2: produc2669932201312201782od_c_a > a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( member_a @ ( F2 @ X3 ) @ ( set_a2 @ ( map_Pr5179659585931931020_c_a_a @ F2 @ Xs ) ) ) ) ).

% map_set
thf(fact_656_map__set,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( member5502040268208861279od_c_a @ ( F2 @ X3 ) @ ( set_Pr5664641634098845387od_c_a @ ( map_Pr6705208461955284164od_c_a @ F2 @ Xs ) ) ) ) ).

% map_set
thf(fact_657_append__eq__map__conv,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Ys @ Zs )
        = ( map_Pr6705208461955284164od_c_a @ F2 @ Xs ) )
      = ( ? [Us: list_P6327159017948738492od_c_a,Vs3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Us @ Vs3 ) )
            & ( Ys
              = ( map_Pr6705208461955284164od_c_a @ F2 @ Us ) )
            & ( Zs
              = ( map_Pr6705208461955284164od_c_a @ F2 @ Vs3 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_658_map__eq__append__conv,axiom,
    ! [F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr6705208461955284164od_c_a @ F2 @ Xs )
        = ( append3884164603879732689od_c_a @ Ys @ Zs ) )
      = ( ? [Us: list_P6327159017948738492od_c_a,Vs3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Us @ Vs3 ) )
            & ( Ys
              = ( map_Pr6705208461955284164od_c_a @ F2 @ Us ) )
            & ( Zs
              = ( map_Pr6705208461955284164od_c_a @ F2 @ Vs3 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_659_subset__code_I1_J,axiom,
    ! [Xs: list_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B5 )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( member_a @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_660_subset__code_I1_J,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,B5: set_Pr8357624050790843030od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs ) @ B5 )
      = ( ! [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
           => ( member5502040268208861279od_c_a @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_661_set__subset__Cons,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] : ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs ) @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_662_list__prefix__subset,axiom,
    ! [Ts: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ? [Ys6: list_P6327159017948738492od_c_a] :
          ( Ts
          = ( append3884164603879732689od_c_a @ Xs @ Ys6 ) )
     => ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs ) @ ( set_Pr5664641634098845387od_c_a @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_663_list__append__subset3,axiom,
    ! [Xs1: list_P6327159017948738492od_c_a,Ys1: list_P6327159017948738492od_c_a,Xs22: list_P6327159017948738492od_c_a,Ys22: list_P6327159017948738492od_c_a,Xs32: list_P6327159017948738492od_c_a,Ys32: list_P6327159017948738492od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs1 ) @ ( set_Pr5664641634098845387od_c_a @ Ys1 ) )
     => ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs22 ) @ ( set_Pr5664641634098845387od_c_a @ Ys22 ) )
       => ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs32 ) @ ( set_Pr5664641634098845387od_c_a @ Ys32 ) )
         => ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs1 @ ( append3884164603879732689od_c_a @ Xs22 @ Xs32 ) ) ) @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Ys1 @ ( append3884164603879732689od_c_a @ Ys22 @ Ys32 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_664_set__map__subset,axiom,
    ! [X3: a,Xs: list_a,T: a,F2: a > a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ( member_a @ T @ ( set_a2 @ ( map_a_a @ F2 @ ( cons_a @ X3 @ nil_a ) ) ) )
       => ( member_a @ T @ ( set_a2 @ ( map_a_a @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_665_set__map__subset,axiom,
    ! [X3: a,Xs: list_a,T: produc2669932201312201782od_c_a,F2: a > produc2669932201312201782od_c_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ( member5502040268208861279od_c_a @ T @ ( set_Pr5664641634098845387od_c_a @ ( map_a_192154525783254780od_c_a @ F2 @ ( cons_a @ X3 @ nil_a ) ) ) )
       => ( member5502040268208861279od_c_a @ T @ ( set_Pr5664641634098845387od_c_a @ ( map_a_192154525783254780od_c_a @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_666_set__map__subset,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,T: a,F2: produc2669932201312201782od_c_a > a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( member_a @ T @ ( set_a2 @ ( map_Pr5179659585931931020_c_a_a @ F2 @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) ) )
       => ( member_a @ T @ ( set_a2 @ ( map_Pr5179659585931931020_c_a_a @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_667_set__map__subset,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,T: produc2669932201312201782od_c_a,F2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( member5502040268208861279od_c_a @ T @ ( set_Pr5664641634098845387od_c_a @ ( map_Pr6705208461955284164od_c_a @ F2 @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) ) )
       => ( member5502040268208861279od_c_a @ T @ ( set_Pr5664641634098845387od_c_a @ ( map_Pr6705208461955284164od_c_a @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_668_prefixes__set__subset,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] : ( ord_le2998388488506175548od_c_a @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) ) @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) ) ) ).

% prefixes_set_subset
thf(fact_669_non__sym__dist__pairs_H_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( non_sy1373097905802202817od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( append3794360244814202818od_c_a @ ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X3 ) @ Xs ) @ ( non_sy1373097905802202817od_c_a @ Xs ) ) ) ).

% non_sym_dist_pairs'.simps(2)
thf(fact_670_list__ordered__pairs_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( list_o2952673153366482557od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( append3794360244814202818od_c_a @ ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X3 ) @ Xs ) @ ( list_o2952673153366482557od_c_a @ Xs ) ) ) ).

% list_ordered_pairs.simps(2)
thf(fact_671_visited__states__are__states,axiom,
    ! [M: fsm_a_b_c,Q1: a,P5: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q1 @ P5 )
     => ( ord_less_eq_set_a @ ( set_a2 @ ( visited_states_a_b_c @ Q1 @ P5 ) ) @ ( states_a_b_c @ M ) ) ) ).

% visited_states_are_states
thf(fact_672_output__complete_Opelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( output3834324022317553264_a_b_c @ X3 )
        = Y3 )
     => ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X3 )
       => ~ ( ( Y3
              = ( ! [X4: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X3 ) )
                   => ! [Y4: c] :
                        ( ( member_c @ Y4 @ ( outputs_a_b_c @ X3 ) )
                       => ? [Z2: produc2669932201312201782od_c_a] :
                            ( ( member5502040268208861279od_c_a @ Z2 @ ( transitions_a_b_c @ X3 ) )
                            & ( ( produc4225943634317630866od_c_a @ X4 )
                              = ( produc4225943634317630866od_c_a @ Z2 ) )
                            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z2 ) ) )
                            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Z2 ) ) )
                              = Y4 ) ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X3 ) ) ) ) ).

% output_complete.pelims(1)
thf(fact_673_list__ordered__pairs_Opelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P4800044111251741997od_c_a] :
      ( ( ( list_o2952673153366482557od_c_a @ X3 )
        = Y3 )
     => ( ( accp_l2553908896995535941od_c_a @ list_o869247370875604852od_c_a @ X3 )
       => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
           => ( ( Y3 = nil_Pr98046886160973869od_c_a )
             => ~ ( accp_l2553908896995535941od_c_a @ list_o869247370875604852od_c_a @ nil_Pr1342775757158464060od_c_a ) ) )
         => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
                ( ( X3
                  = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
               => ( ( Y3
                    = ( append3794360244814202818od_c_a @ ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X ) @ Xs2 ) @ ( list_o2952673153366482557od_c_a @ Xs2 ) ) )
                 => ~ ( accp_l2553908896995535941od_c_a @ list_o869247370875604852od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) ) ) ) ) ) ).

% list_ordered_pairs.pelims
thf(fact_674_non__sym__dist__pairs_H_Opelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P4800044111251741997od_c_a] :
      ( ( ( non_sy1373097905802202817od_c_a @ X3 )
        = Y3 )
     => ( ( accp_l2553908896995535941od_c_a @ non_sy2652613377995764784od_c_a @ X3 )
       => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
           => ( ( Y3 = nil_Pr98046886160973869od_c_a )
             => ~ ( accp_l2553908896995535941od_c_a @ non_sy2652613377995764784od_c_a @ nil_Pr1342775757158464060od_c_a ) ) )
         => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
                ( ( X3
                  = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
               => ( ( Y3
                    = ( append3794360244814202818od_c_a @ ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X ) @ Xs2 ) @ ( non_sy1373097905802202817od_c_a @ Xs2 ) ) )
                 => ~ ( accp_l2553908896995535941od_c_a @ non_sy2652613377995764784od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) ) ) ) ) ) ).

% non_sym_dist_pairs'.pelims
thf(fact_675_deterministic_Opelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( deterministic_a_b_c @ X3 )
        = Y3 )
     => ( ( accp_fsm_a_b_c @ determ4397191569268572707_a_b_c @ X3 )
       => ~ ( ( Y3
              = ( ! [X4: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X3 ) )
                   => ! [Y4: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X3 ) )
                       => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                              = ( produc4225943634317630866od_c_a @ Y4 ) )
                            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) )
                         => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) )
                            & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                              = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ determ4397191569268572707_a_b_c @ X3 ) ) ) ) ).

% deterministic.pelims(1)
thf(fact_676_deterministic_Opelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( deterministic_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ determ4397191569268572707_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ determ4397191569268572707_a_b_c @ X3 )
           => ~ ! [X6: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ X6 @ ( transitions_a_b_c @ X3 ) )
                 => ! [Xa3: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X3 ) )
                     => ( ( ( ( produc4225943634317630866od_c_a @ X6 )
                            = ( produc4225943634317630866od_c_a @ Xa3 ) )
                          & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) )
                            = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) )
                       => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) )
                            = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) )
                          & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) )
                            = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.pelims(2)
thf(fact_677_deterministic_Opelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( deterministic_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ determ4397191569268572707_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ determ4397191569268572707_a_b_c @ X3 )
           => ! [X: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X3 ) )
               => ! [Xa2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X3 ) )
                   => ( ( ( ( produc4225943634317630866od_c_a @ X )
                          = ( produc4225943634317630866od_c_a @ Xa2 ) )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                          = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) )
                     => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                          = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) )
                        & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                          = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.pelims(3)
thf(fact_678_observable_Opelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( observable_a_b_c @ X3 )
        = Y3 )
     => ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X3 )
       => ~ ( ( Y3
              = ( ! [X4: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X3 ) )
                   => ! [Y4: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X3 ) )
                       => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                              = ( produc4225943634317630866od_c_a @ Y4 ) )
                            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) )
                            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) )
                         => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                            = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X3 ) ) ) ) ).

% observable.pelims(1)
thf(fact_679_concat__map__elem,axiom,
    ! [Y3: a,F2: a > list_a,Xs: list_a] :
      ( ( member_a @ Y3 @ ( set_a2 @ ( concat_a @ ( map_a_list_a @ F2 @ Xs ) ) ) )
     => ~ ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ~ ( member_a @ Y3 @ ( set_a2 @ ( F2 @ X ) ) ) ) ) ).

% concat_map_elem
thf(fact_680_concat__map__elem,axiom,
    ! [Y3: a,F2: produc2669932201312201782od_c_a > list_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Y3 @ ( set_a2 @ ( concat_a @ ( map_Pr7036266158781360914list_a @ F2 @ Xs ) ) ) )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
           => ~ ( member_a @ Y3 @ ( set_a2 @ ( F2 @ X ) ) ) ) ) ).

% concat_map_elem
thf(fact_681_concat__map__elem,axiom,
    ! [Y3: produc2669932201312201782od_c_a,F2: a > list_P6327159017948738492od_c_a,Xs: list_a] :
      ( ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ ( concat7533993459791330935od_c_a @ ( map_a_889415987282285442od_c_a @ F2 @ Xs ) ) ) )
     => ~ ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ~ ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ ( F2 @ X ) ) ) ) ) ).

% concat_map_elem
thf(fact_682_concat__map__elem,axiom,
    ! [Y3: produc2669932201312201782od_c_a,F2: produc2669932201312201782od_c_a > list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ ( concat7533993459791330935od_c_a @ ( map_Pr5537105415683728714od_c_a @ F2 @ Xs ) ) ) )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
           => ~ ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ ( F2 @ X ) ) ) ) ) ).

% concat_map_elem
thf(fact_683_distinct__set__subseqs,axiom,
    ! [Xs: list_a] :
      ( ( distinct_a @ Xs )
     => ( distinct_set_a @ ( map_list_a_set_a @ set_a2 @ ( subseqs_a @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_684_transition__subset__path,axiom,
    ! [A3: fsm_a_b_c,B5: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ A3 ) @ ( transitions_a_b_c @ B5 ) )
     => ( ( path_a_b_c @ A3 @ Q @ P5 )
       => ( ( member_a @ Q @ ( states_a_b_c @ B5 ) )
         => ( path_a_b_c @ B5 @ Q @ P5 ) ) ) ) ).

% transition_subset_path
thf(fact_685_path__transitions,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ P5 ) @ ( transitions_a_b_c @ M ) ) ) ).

% path_transitions
thf(fact_686_prefixes_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( prefix8878233912999784899od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ ( map_li335866320336015812od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 ) @ ( prefix8878233912999784899od_c_a @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_687_sublists_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( sublis1354033275501720270od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( append2943488786620085207od_c_a @ ( sublis1354033275501720270od_c_a @ Xs ) @ ( map_li335866320336015812od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 ) @ ( prefix8878233912999784899od_c_a @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_688_observable_Opelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( observable_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X3 )
           => ! [X: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X3 ) )
               => ! [Xa2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X3 ) )
                   => ( ( ( ( produc4225943634317630866od_c_a @ X )
                          = ( produc4225943634317630866od_c_a @ Xa2 ) )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                          = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) )
                        & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                          = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) )
                     => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                        = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(3)
thf(fact_689_observable_Opelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( observable_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X3 )
           => ~ ! [X6: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ X6 @ ( transitions_a_b_c @ X3 ) )
                 => ! [Xa3: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X3 ) )
                     => ( ( ( ( produc4225943634317630866od_c_a @ X6 )
                            = ( produc4225943634317630866od_c_a @ Xa3 ) )
                          & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) )
                            = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) )
                          & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) )
                            = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) )
                       => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) ) )
                          = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(2)
thf(fact_690_acyclic_Opelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( acyclic_a_b_c @ X3 )
        = Y3 )
     => ( ( accp_fsm_a_b_c @ acyclic_rel_a_b_c @ X3 )
       => ~ ( ( Y3
              = ( ! [P4: list_P6327159017948738492od_c_a] :
                    ( ( path_a_b_c @ X3 @ ( initial_a_b_c @ X3 ) @ P4 )
                   => ( distinct_a @ ( visited_states_a_b_c @ ( initial_a_b_c @ X3 ) @ P4 ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ acyclic_rel_a_b_c @ X3 ) ) ) ) ).

% acyclic.pelims(1)
thf(fact_691_acyclic_Opelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( acyclic_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ acyclic_rel_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ acyclic_rel_a_b_c @ X3 )
           => ~ ! [P3: list_P6327159017948738492od_c_a] :
                  ( ( path_a_b_c @ X3 @ ( initial_a_b_c @ X3 ) @ P3 )
                 => ( distinct_a @ ( visited_states_a_b_c @ ( initial_a_b_c @ X3 ) @ P3 ) ) ) ) ) ) ).

% acyclic.pelims(2)
thf(fact_692_acyclic_Opelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( acyclic_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ acyclic_rel_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ acyclic_rel_a_b_c @ X3 )
           => ! [P: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ X3 @ ( initial_a_b_c @ X3 ) @ P )
               => ( distinct_a @ ( visited_states_a_b_c @ ( initial_a_b_c @ X3 ) @ P ) ) ) ) ) ) ).

% acyclic.pelims(3)
thf(fact_693_single__input_Opelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( single_input_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ single6965823701822693171_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ single6965823701822693171_a_b_c @ X3 )
           => ! [X: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X3 ) )
               => ! [Xa2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X3 ) )
                   => ( ( ( produc4225943634317630866od_c_a @ X )
                        = ( produc4225943634317630866od_c_a @ Xa2 ) )
                     => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% single_input.pelims(3)
thf(fact_694_single__input_Opelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( single_input_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ single6965823701822693171_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ single6965823701822693171_a_b_c @ X3 )
           => ~ ! [X6: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ X6 @ ( transitions_a_b_c @ X3 ) )
                 => ! [Xa3: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X3 ) )
                     => ( ( ( produc4225943634317630866od_c_a @ X6 )
                          = ( produc4225943634317630866od_c_a @ Xa3 ) )
                       => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X6 ) )
                          = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ) ) ).

% single_input.pelims(2)
thf(fact_695_single__input_Opelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( single_input_a_b_c @ X3 )
        = Y3 )
     => ( ( accp_fsm_a_b_c @ single6965823701822693171_a_b_c @ X3 )
       => ~ ( ( Y3
              = ( ! [X4: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X3 ) )
                   => ! [Y4: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X3 ) )
                       => ( ( ( produc4225943634317630866od_c_a @ X4 )
                            = ( produc4225943634317630866od_c_a @ Y4 ) )
                         => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                            = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ single6965823701822693171_a_b_c @ X3 ) ) ) ) ).

% single_input.pelims(1)
thf(fact_696_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,K3: a,V1: produc2133359284458116493od_c_a,V22: produc2133359284458116493od_c_a] :
      ( ( distinct_a @ ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Xs ) )
     => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ V1 ) @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
       => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ V22 ) @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_697_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,K3: b,V1: product_prod_c_a,V22: product_prod_c_a] :
      ( ( distinct_b @ ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Xs ) )
     => ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ V1 ) @ ( set_Pr5098181003905080610od_c_a @ Xs ) )
       => ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ V22 ) @ ( set_Pr5098181003905080610od_c_a @ Xs ) )
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_698_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P125642481956313003od_c_a,K3: c,V1: a,V22: a] :
      ( ( distinct_c @ ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Xs ) )
     => ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ K3 @ V1 ) @ ( set_Product_prod_c_a2 @ Xs ) )
       => ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ K3 @ V22 ) @ ( set_Product_prod_c_a2 @ Xs ) )
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_699_linear__order__from__list__position_H_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P4800044111251741997od_c_a] :
      ( ( ( linear2334534536944050656od_c_a @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
         => ( Y3 != nil_Pr98046886160973869od_c_a ) )
       => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( ( X3
                = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
             => ( Y3
               != ( cons_P3602142545535101917od_c_a @ ( produc4004402368211057175od_c_a @ X @ X ) @ ( append3794360244814202818od_c_a @ ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X ) @ Xs2 ) @ ( linear2334534536944050656od_c_a @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.elims
thf(fact_700_linear__order__from__list__position_H_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( linear2334534536944050656od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( cons_P3602142545535101917od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ X3 ) @ ( append3794360244814202818od_c_a @ ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X3 ) @ Xs ) @ ( linear2334534536944050656od_c_a @ Xs ) ) ) ) ).

% linear_order_from_list_position'.simps(2)
thf(fact_701_linear__order__from__list__position_H_Opelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P4800044111251741997od_c_a] :
      ( ( ( linear2334534536944050656od_c_a @ X3 )
        = Y3 )
     => ( ( accp_l2553908896995535941od_c_a @ linear80608147231238097od_c_a @ X3 )
       => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
           => ( ( Y3 = nil_Pr98046886160973869od_c_a )
             => ~ ( accp_l2553908896995535941od_c_a @ linear80608147231238097od_c_a @ nil_Pr1342775757158464060od_c_a ) ) )
         => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
                ( ( X3
                  = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
               => ( ( Y3
                    = ( cons_P3602142545535101917od_c_a @ ( produc4004402368211057175od_c_a @ X @ X ) @ ( append3794360244814202818od_c_a @ ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X ) @ Xs2 ) @ ( linear2334534536944050656od_c_a @ Xs2 ) ) ) )
                 => ~ ( accp_l2553908896995535941od_c_a @ linear80608147231238097od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.pelims
thf(fact_702_linear__order__from__list__position_H_Osimps_I1_J,axiom,
    ( ( linear2334534536944050656od_c_a @ nil_Pr1342775757158464060od_c_a )
    = nil_Pr98046886160973869od_c_a ) ).

% linear_order_from_list_position'.simps(1)
thf(fact_703_List_Oproduct_Osimps_I2_J,axiom,
    ! [X3: a,Xs: list_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( produc7027247015253682516od_c_a @ ( cons_a @ X3 @ Xs ) @ Ys )
      = ( append3884164603879732689od_c_a @ ( map_Pr1968850252878565677od_c_a @ ( produc5762801557034676134od_c_a @ X3 ) @ Ys ) @ ( produc7027247015253682516od_c_a @ Xs @ Ys ) ) ) ).

% List.product.simps(2)
thf(fact_704_List_Oproduct_Osimps_I2_J,axiom,
    ! [X3: b,Xs: list_b,Ys: list_P125642481956313003od_c_a] :
      ( ( produc460432258840367019od_c_a @ ( cons_b @ X3 @ Xs ) @ Ys )
      = ( append2814445079200870952od_c_a @ ( map_Pr1026158493455843436od_c_a @ ( produc3440865914000881661od_c_a @ X3 ) @ Ys ) @ ( produc460432258840367019od_c_a @ Xs @ Ys ) ) ) ).

% List.product.simps(2)
thf(fact_705_List_Oproduct_Osimps_I2_J,axiom,
    ! [X3: c,Xs: list_c,Ys: list_a] :
      ( ( product_c_a @ ( cons_c @ X3 @ Xs ) @ Ys )
      = ( append8983669691956257088od_c_a @ ( map_a_2285140998955227243od_c_a @ ( product_Pair_c_a @ X3 ) @ Ys ) @ ( product_c_a @ Xs @ Ys ) ) ) ).

% List.product.simps(2)
thf(fact_706_subrelI,axiom,
    ! [R: set_Pr8357624050790843030od_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ! [X: a,Y: produc2133359284458116493od_c_a] :
          ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X @ Y ) @ R )
         => ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X @ Y ) @ S ) )
     => ( ord_le3226338165915859510od_c_a @ R @ S ) ) ).

% subrelI
thf(fact_707_subrelI,axiom,
    ! [R: set_Pr272002107461315949od_c_a,S: set_Pr272002107461315949od_c_a] :
      ( ! [X: b,Y: product_prod_c_a] :
          ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X @ Y ) @ R )
         => ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X @ Y ) @ S ) )
     => ( ord_le4080174072498570509od_c_a @ R @ S ) ) ).

% subrelI
thf(fact_708_subrelI,axiom,
    ! [R: set_Product_prod_c_a,S: set_Product_prod_c_a] :
      ( ! [X: c,Y: a] :
          ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ R )
         => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ S ) )
     => ( ord_le8698776994054418981od_c_a @ R @ S ) ) ).

% subrelI
thf(fact_709_language__initial__path__append__transition,axiom,
    ! [Ios: list_P903359562653991662od_b_c,Io: product_prod_b_c,M: fsm_a_b_c] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Ios @ ( cons_P4529483553340347422od_b_c @ Io @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
     => ~ ! [P: list_P6327159017948738492od_c_a,T2: produc2669932201312201782od_c_a] :
            ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append3884164603879732689od_c_a @ P @ ( cons_P8659537020926305772od_c_a @ T2 @ nil_Pr1342775757158464060od_c_a ) ) )
           => ( ( map_Pr776894537227705078od_b_c
                @ ^ [U: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) ) )
                @ ( append3884164603879732689od_c_a @ P @ ( cons_P8659537020926305772od_c_a @ T2 @ nil_Pr1342775757158464060od_c_a ) ) )
             != ( append2547753245680614915od_b_c @ Ios @ ( cons_P4529483553340347422od_b_c @ Io @ nil_Product_prod_b_c ) ) ) ) ) ).

% language_initial_path_append_transition
thf(fact_710_language__append__path__ob,axiom,
    ! [Io: list_P903359562653991662od_b_c,X3: b,Y3: c,M: fsm_a_b_c] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
     => ~ ! [P: list_P6327159017948738492od_c_a,T2: produc2669932201312201782od_c_a] :
            ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append3884164603879732689od_c_a @ P @ ( cons_P8659537020926305772od_c_a @ T2 @ nil_Pr1342775757158464060od_c_a ) ) )
           => ( ( ( map_Pr776894537227705078od_b_c
                  @ ^ [U: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) ) )
                  @ P )
                = Io )
             => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) )
                  = X3 )
               => ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) )
                 != Y3 ) ) ) ) ) ).

% language_append_path_ob
thf(fact_711_suffixes__snoc,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
      ( ( suffix7794987338261656196od_c_a @ ( append3884164603879732689od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) )
      = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a
        @ ( map_li335866320336015812od_c_a
          @ ^ [Ys3: list_P6327159017948738492od_c_a] : ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) )
          @ ( suffix7794987338261656196od_c_a @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_712_pred__equals__eq2,axiom,
    ! [R2: set_Pr8357624050790843030od_c_a,S2: set_Pr8357624050790843030od_c_a] :
      ( ( ( ^ [X4: a,Y4: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y4 ) @ R2 ) )
        = ( ^ [X4: a,Y4: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y4 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_713_pred__equals__eq2,axiom,
    ! [R2: set_Pr272002107461315949od_c_a,S2: set_Pr272002107461315949od_c_a] :
      ( ( ( ^ [X4: b,Y4: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y4 ) @ R2 ) )
        = ( ^ [X4: b,Y4: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y4 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_714_pred__equals__eq2,axiom,
    ! [R2: set_Product_prod_c_a,S2: set_Product_prod_c_a] :
      ( ( ( ^ [X4: c,Y4: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y4 ) @ R2 ) )
        = ( ^ [X4: c,Y4: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y4 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_715_Shift__def,axiom,
    ( bNF_Gr6081535474448196565od_c_a
    = ( ^ [Kl3: set_li1159382662694783132od_c_a,K5: produc2669932201312201782od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [Kl4: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ ( cons_P8659537020926305772od_c_a @ K5 @ Kl4 ) @ Kl3 ) ) ) ) ).

% Shift_def
thf(fact_716_product__concat__map,axiom,
    ( produc7027247015253682516od_c_a
    = ( ^ [Xs3: list_a,Ys3: list_P5767727266746070163od_c_a] :
          ( concat7533993459791330935od_c_a
          @ ( map_a_889415987282285442od_c_a
            @ ^ [X4: a] : ( map_Pr1968850252878565677od_c_a @ ( produc5762801557034676134od_c_a @ X4 ) @ Ys3 )
            @ Xs3 ) ) ) ) ).

% product_concat_map
thf(fact_717_product__concat__map,axiom,
    ( produc460432258840367019od_c_a
    = ( ^ [Xs3: list_b,Ys3: list_P125642481956313003od_c_a] :
          ( concat3978415732298191054od_c_a
          @ ( map_b_696456053270819608od_c_a
            @ ^ [X4: b] : ( map_Pr1026158493455843436od_c_a @ ( produc3440865914000881661od_c_a @ X4 ) @ Ys3 )
            @ Xs3 ) ) ) ) ).

% product_concat_map
thf(fact_718_product__concat__map,axiom,
    ( product_c_a
    = ( ^ [Xs3: list_c,Ys3: list_a] :
          ( concat497087365762831846od_c_a
          @ ( map_c_3537221148099011311od_c_a
            @ ^ [X4: c] : ( map_a_2285140998955227243od_c_a @ ( product_Pair_c_a @ X4 ) @ Ys3 )
            @ Xs3 ) ) ) ) ).

% product_concat_map
thf(fact_719_update__nonempty,axiom,
    ! [K3: a,V2: produc2133359284458116493od_c_a,Al: list_P6327159017948738492od_c_a] :
      ( ( update8541657393971699723od_c_a @ K3 @ V2 @ Al )
     != nil_Pr1342775757158464060od_c_a ) ).

% update_nonempty
thf(fact_720_map__pair__fst,axiom,
    ! [F2: a > produc2133359284458116493od_c_a,Xs: list_a] :
      ( ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a
        @ ( map_a_192154525783254780od_c_a
          @ ^ [X4: a] : ( produc5762801557034676134od_c_a @ X4 @ ( F2 @ X4 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_721_map__pair__fst,axiom,
    ! [F2: b > product_prod_c_a,Xs: list_b] :
      ( ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a
        @ ( map_b_1729217027404323474od_c_a
          @ ^ [X4: b] : ( produc3440865914000881661od_c_a @ X4 @ ( F2 @ X4 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_722_map__pair__fst,axiom,
    ! [F2: c > a,Xs: list_c] :
      ( ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a
        @ ( map_c_1006260398291594985od_c_a
          @ ^ [X4: c] : ( product_Pair_c_a @ X4 @ ( F2 @ X4 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_723_Succ__def,axiom,
    ( bNF_Gr5367812047628651857od_c_a
    = ( ^ [Kl3: set_li1159382662694783132od_c_a,Kl4: list_P6327159017948738492od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [K5: produc2669932201312201782od_c_a] : ( member7410604586820865893od_c_a @ ( append3884164603879732689od_c_a @ Kl4 @ ( cons_P8659537020926305772od_c_a @ K5 @ nil_Pr1342775757158464060od_c_a ) ) @ Kl3 ) ) ) ) ).

% Succ_def
thf(fact_724_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Xss2: list_l2826717958560394946od_c_a] :
      ( ( produc4300910528168050469od_c_a @ ( cons_l7906879460341455474od_c_a @ Xs @ Xss2 ) )
      = ( concat1386605678430990717od_c_a
        @ ( map_Pr5369532675236462928od_c_a
          @ ^ [X4: produc2669932201312201782od_c_a] : ( map_li335866320336015812od_c_a @ ( cons_P8659537020926305772od_c_a @ X4 ) @ ( produc4300910528168050469od_c_a @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_725_subseqs_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( subseq4269998335400966283od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( append2943488786620085207od_c_a @ ( map_li335866320336015812od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 ) @ ( subseq4269998335400966283od_c_a @ Xs ) ) @ ( subseq4269998335400966283od_c_a @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_726_pow__list_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( pow_li2753411374138005421od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( append2943488786620085207od_c_a @ ( pow_li2753411374138005421od_c_a @ Xs ) @ ( map_li335866320336015812od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 ) @ ( pow_li2753411374138005421od_c_a @ Xs ) ) ) ) ).

% pow_list.simps(2)
thf(fact_727_visited__states_Oelims,axiom,
    ! [X3: a,Xa: list_P6327159017948738492od_c_a,Y3: list_a] :
      ( ( ( visited_states_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( Y3
        = ( cons_a @ X3
          @ ( map_Pr5179659585931931020_c_a_a
            @ ^ [A5: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A5 ) ) )
            @ Xa ) ) ) ) ).

% visited_states.elims
thf(fact_728_visited__states_Osimps,axiom,
    ( visited_states_a_b_c
    = ( ^ [Q6: a,P4: list_P6327159017948738492od_c_a] :
          ( cons_a @ Q6
          @ ( map_Pr5179659585931931020_c_a_a
            @ ^ [A5: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A5 ) ) )
            @ P4 ) ) ) ) ).

% visited_states.simps
thf(fact_729_concat__pair__set,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( set_Product_prod_a_a2
        @ ( concat6071998529708183912od_a_a
          @ ( map_a_6618942900564114931od_a_a
            @ ^ [X4: a] : ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec3336397797384452498od_a_a
        @ ^ [Xy: product_prod_a_a] :
            ( ( member_a @ ( product_fst_a_a @ Xy ) @ ( set_a2 @ Xs ) )
            & ( member_a @ ( product_snd_a_a @ Xy ) @ ( set_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_730_concat__pair__set,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Xs: list_a] :
      ( ( set_Pr4509209463540227572od_c_a
        @ ( concat331025720195512736od_c_a
          @ ( map_a_3289375268142426667od_c_a
            @ ^ [X4: a] : ( map_Pr3414642998751257325od_c_a @ ( produc4410401784823481679od_c_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec8871988621716747210od_c_a
        @ ^ [Xy: produc871828282075189343od_c_a] :
            ( ( member_a @ ( produc3141450672998377275od_c_a @ Xy ) @ ( set_a2 @ Xs ) )
            & ( member5502040268208861279od_c_a @ ( produc367105772464697469od_c_a @ Xy ) @ ( set_Pr5664641634098845387od_c_a @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_731_concat__pair__set,axiom,
    ! [Ys: list_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( set_Pr2373150112579439748_c_a_a
        @ ( concat7418338406089500720_c_a_a
          @ ( map_Pr7338260324097972867_c_a_a
            @ ^ [X4: produc2669932201312201782od_c_a] : ( map_a_6906649024970834869_c_a_a @ ( produc174534808117382111_c_a_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec6735929270755959386_c_a_a
        @ ^ [Xy: produc7959140967969177327_c_a_a] :
            ( ( member5502040268208861279od_c_a @ ( produc8128955733147053515_c_a_a @ Xy ) @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
            & ( member_a @ ( produc5354610832613373709_c_a_a @ Xy ) @ ( set_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_732_concat__pair__set,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( set_Pr3422107008073957564od_c_a
        @ ( concat3706020456378546280od_c_a
          @ ( map_Pr7648788594696427195od_c_a
            @ ^ [X4: produc2669932201312201782od_c_a] : ( map_Pr6248875406537007797od_c_a @ ( produc4004402368211057175od_c_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec1700201398098035858od_c_a
        @ ^ [Xy: produc1647090505208597031od_c_a] :
            ( ( member5502040268208861279od_c_a @ ( produc2189173959066220547od_c_a @ Xy ) @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
            & ( member5502040268208861279od_c_a @ ( produc3204662071173827397od_c_a @ Xy ) @ ( set_Pr5664641634098845387od_c_a @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_733_concat__pair__set,axiom,
    ! [Ys: list_P5767727266746070163od_c_a,Xs: list_a] :
      ( ( set_Pr5664641634098845387od_c_a
        @ ( concat7533993459791330935od_c_a
          @ ( map_a_889415987282285442od_c_a
            @ ^ [X4: a] : ( map_Pr1968850252878565677od_c_a @ ( produc5762801557034676134od_c_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec8405073525749625505od_c_a
        @ ^ [Xy: produc2669932201312201782od_c_a] :
            ( ( member_a @ ( produc4225943634317630866od_c_a @ Xy ) @ ( set_a2 @ Xs ) )
            & ( member1160383752307813046od_c_a @ ( produc8088130389382527188od_c_a @ Xy ) @ ( set_Pr5098181003905080610od_c_a @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_734_concat__pair__set,axiom,
    ! [Ys: list_P125642481956313003od_c_a,Xs: list_b] :
      ( ( set_Pr5098181003905080610od_c_a
        @ ( concat3978415732298191054od_c_a
          @ ( map_b_696456053270819608od_c_a
            @ ^ [X4: b] : ( map_Pr1026158493455843436od_c_a @ ( produc3440865914000881661od_c_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec3135042693873630968od_c_a
        @ ^ [Xy: produc2133359284458116493od_c_a] :
            ( ( member_b @ ( produc301949074746585577od_c_a @ Xy ) @ ( set_b2 @ Xs ) )
            & ( member5074992350434858958od_c_a @ ( produc5221446324160585515od_c_a @ Xy ) @ ( set_Product_prod_c_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_735_concat__pair__set,axiom,
    ! [Ys: list_a,Xs: list_c] :
      ( ( set_Product_prod_c_a2
        @ ( concat497087365762831846od_c_a
          @ ( map_c_3537221148099011311od_c_a
            @ ^ [X4: c] : ( map_a_2285140998955227243od_c_a @ ( product_Pair_c_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec6984858670293876240od_c_a
        @ ^ [Xy: product_prod_c_a] :
            ( ( member_c @ ( product_fst_c_a @ Xy ) @ ( set_c2 @ Xs ) )
            & ( member_a @ ( product_snd_c_a @ Xy ) @ ( set_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_736_language__state__containment,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,Io: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ( ( map_Pr776894537227705078od_b_c
            @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
            @ P5 )
          = Io )
       => ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) ) ) ) ).

% language_state_containment
thf(fact_737_language__intro,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( member6330420149250801815od_b_c
        @ ( map_Pr776894537227705078od_b_c
          @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
          @ P5 )
        @ ( lS_a_b_c @ M @ Q ) ) ) ).

% language_intro
thf(fact_738_update_Osimps_I2_J,axiom,
    ! [P5: produc2669932201312201782od_c_a,K3: a,V2: produc2133359284458116493od_c_a,Ps2: list_P6327159017948738492od_c_a] :
      ( ( ( ( produc4225943634317630866od_c_a @ P5 )
          = K3 )
       => ( ( update8541657393971699723od_c_a @ K3 @ V2 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ V2 ) @ Ps2 ) ) )
      & ( ( ( produc4225943634317630866od_c_a @ P5 )
         != K3 )
       => ( ( update8541657393971699723od_c_a @ K3 @ V2 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( cons_P8659537020926305772od_c_a @ P5 @ ( update8541657393971699723od_c_a @ K3 @ V2 @ Ps2 ) ) ) ) ) ).

% update.simps(2)
thf(fact_739_update_Osimps_I2_J,axiom,
    ! [P5: produc2133359284458116493od_c_a,K3: b,V2: product_prod_c_a,Ps2: list_P5767727266746070163od_c_a] :
      ( ( ( ( produc301949074746585577od_c_a @ P5 )
          = K3 )
       => ( ( update960876589708498274od_c_a @ K3 @ V2 @ ( cons_P7376644893796496707od_c_a @ P5 @ Ps2 ) )
          = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ V2 ) @ Ps2 ) ) )
      & ( ( ( produc301949074746585577od_c_a @ P5 )
         != K3 )
       => ( ( update960876589708498274od_c_a @ K3 @ V2 @ ( cons_P7376644893796496707od_c_a @ P5 @ Ps2 ) )
          = ( cons_P7376644893796496707od_c_a @ P5 @ ( update960876589708498274od_c_a @ K3 @ V2 @ Ps2 ) ) ) ) ) ).

% update.simps(2)
thf(fact_740_update_Osimps_I2_J,axiom,
    ! [P5: product_prod_c_a,K3: c,V2: a,Ps2: list_P125642481956313003od_c_a] :
      ( ( ( ( product_fst_c_a @ P5 )
          = K3 )
       => ( ( update_c_a @ K3 @ V2 @ ( cons_P1742027962761213787od_c_a @ P5 @ Ps2 ) )
          = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ V2 ) @ Ps2 ) ) )
      & ( ( ( product_fst_c_a @ P5 )
         != K3 )
       => ( ( update_c_a @ K3 @ V2 @ ( cons_P1742027962761213787od_c_a @ P5 @ Ps2 ) )
          = ( cons_P1742027962761213787od_c_a @ P5 @ ( update_c_a @ K3 @ V2 @ Ps2 ) ) ) ) ) ).

% update.simps(2)
thf(fact_741_update_Osimps_I1_J,axiom,
    ! [K3: a,V2: produc2133359284458116493od_c_a] :
      ( ( update8541657393971699723od_c_a @ K3 @ V2 @ nil_Pr1342775757158464060od_c_a )
      = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ V2 ) @ nil_Pr1342775757158464060od_c_a ) ) ).

% update.simps(1)
thf(fact_742_update_Osimps_I1_J,axiom,
    ! [K3: b,V2: product_prod_c_a] :
      ( ( update960876589708498274od_c_a @ K3 @ V2 @ nil_Pr5380409136740620691od_c_a )
      = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ V2 ) @ nil_Pr5380409136740620691od_c_a ) ) ).

% update.simps(1)
thf(fact_743_update_Osimps_I1_J,axiom,
    ! [K3: c,V2: a] :
      ( ( update_c_a @ K3 @ V2 @ nil_Product_prod_c_a )
      = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K3 @ V2 ) @ nil_Product_prod_c_a ) ) ).

% update.simps(1)
thf(fact_744_pow__list_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_l2826717958560394946od_c_a] :
      ( ( ( pow_li2753411374138005421od_c_a @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
         => ( Y3
           != ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) )
       => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
              ( ( X3
                = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
             => ( Y3
               != ( append2943488786620085207od_c_a @ ( pow_li2753411374138005421od_c_a @ Xs2 ) @ ( map_li335866320336015812od_c_a @ ( cons_P8659537020926305772od_c_a @ X ) @ ( pow_li2753411374138005421od_c_a @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.elims
thf(fact_745_List_Oproduct_Osimps_I1_J,axiom,
    ! [Uu2: list_P5767727266746070163od_c_a] :
      ( ( produc7027247015253682516od_c_a @ nil_a @ Uu2 )
      = nil_Pr1342775757158464060od_c_a ) ).

% List.product.simps(1)
thf(fact_746_language__state__split,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q1: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M @ Q1 ) )
     => ~ ! [P12: list_P6327159017948738492od_c_a] :
            ( ( path_a_b_c @ M @ Q1 @ P12 )
           => ! [P22: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M @ ( target_a_b_c @ Q1 @ P12 ) @ P22 )
               => ( ( ( map_Pr776894537227705078od_b_c
                      @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                      @ P12 )
                    = Io1 )
                 => ( ( map_Pr776894537227705078od_b_c
                      @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                      @ P22 )
                   != Io2 ) ) ) ) ) ).

% language_state_split
thf(fact_747_path__map__target,axiom,
    ! [F4: a > product_prod_c_a,Q: a,F1: a > product_prod_c_a,F22: b > a,F3: c > b,P5: list_P6327159017948738492od_c_a] :
      ( ( target6303326113586172192_a_a_b @ ( F4 @ Q )
        @ ( map_Pr8547749104283478534od_c_a
          @ ^ [T3: produc2669932201312201782od_c_a] : ( produc7911353075612250984od_c_a @ ( F1 @ ( produc4225943634317630866od_c_a @ T3 ) ) @ ( produc5762801557034676134od_c_a @ ( F22 @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ ( produc3440865914000881661od_c_a @ ( F3 @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) ) @ ( F4 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) ) ) ) )
          @ P5 ) )
      = ( F4 @ ( target_a_b_c @ Q @ P5 ) ) ) ).

% path_map_target
thf(fact_748_path__map__target,axiom,
    ! [F4: a > a,Q: a,F1: a > a,F22: b > b,F3: c > c,P5: list_P6327159017948738492od_c_a] :
      ( ( target_a_b_c @ ( F4 @ Q )
        @ ( map_Pr6705208461955284164od_c_a
          @ ^ [T3: produc2669932201312201782od_c_a] : ( produc5762801557034676134od_c_a @ ( F1 @ ( produc4225943634317630866od_c_a @ T3 ) ) @ ( produc3440865914000881661od_c_a @ ( F22 @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ ( product_Pair_c_a @ ( F3 @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) ) @ ( F4 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) ) ) ) )
          @ P5 ) )
      = ( F4 @ ( target_a_b_c @ Q @ P5 ) ) ) ).

% path_map_target
thf(fact_749_language__split,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M: fsm_a_b_c] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
     => ~ ! [P12: list_P6327159017948738492od_c_a,P22: list_P6327159017948738492od_c_a] :
            ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append3884164603879732689od_c_a @ P12 @ P22 ) )
           => ( ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                  @ P12 )
                = Io1 )
             => ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                  @ P22 )
               != Io2 ) ) ) ) ).

% language_split
thf(fact_750_distinct__product,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( distinct_a @ Xs )
     => ( ( distinct_a @ Ys )
       => ( distin132333870042060960od_a_a @ ( product_a_a @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_751_language__path__append__transition,axiom,
    ! [Ios: list_P903359562653991662od_b_c,Io: product_prod_b_c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Ios @ ( cons_P4529483553340347422od_b_c @ Io @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ Q ) )
     => ~ ! [P: list_P6327159017948738492od_c_a,T2: produc2669932201312201782od_c_a] :
            ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P @ ( cons_P8659537020926305772od_c_a @ T2 @ nil_Pr1342775757158464060od_c_a ) ) )
           => ( ( map_Pr776894537227705078od_b_c
                @ ^ [U: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ U ) ) ) )
                @ ( append3884164603879732689od_c_a @ P @ ( cons_P8659537020926305772od_c_a @ T2 @ nil_Pr1342775757158464060od_c_a ) ) )
             != ( append2547753245680614915od_b_c @ Ios @ ( cons_P4529483553340347422od_b_c @ Io @ nil_Product_prod_b_c ) ) ) ) ) ).

% language_path_append_transition
thf(fact_752_pow__list_Opelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_l2826717958560394946od_c_a] :
      ( ( ( pow_li2753411374138005421od_c_a @ X3 )
        = Y3 )
     => ( ( accp_l2553908896995535941od_c_a @ pow_li4689656146799022660od_c_a @ X3 )
       => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
           => ( ( Y3
                = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) )
             => ~ ( accp_l2553908896995535941od_c_a @ pow_li4689656146799022660od_c_a @ nil_Pr1342775757158464060od_c_a ) ) )
         => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
                ( ( X3
                  = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
               => ( ( Y3
                    = ( append2943488786620085207od_c_a @ ( pow_li2753411374138005421od_c_a @ Xs2 ) @ ( map_li335866320336015812od_c_a @ ( cons_P8659537020926305772od_c_a @ X ) @ ( pow_li2753411374138005421od_c_a @ Xs2 ) ) ) )
                 => ~ ( accp_l2553908896995535941od_c_a @ pow_li4689656146799022660od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.pelims
thf(fact_753_product__code,axiom,
    ! [Xs: list_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( produc4596061662451530337od_c_a @ ( set_a2 @ Xs ) @ ( set_Pr5098181003905080610od_c_a @ Ys ) )
      = ( set_Pr5664641634098845387od_c_a
        @ ( concat7533993459791330935od_c_a
          @ ( map_a_889415987282285442od_c_a
            @ ^ [X4: a] : ( map_Pr1968850252878565677od_c_a @ ( produc5762801557034676134od_c_a @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_754_product__code,axiom,
    ! [Xs: list_b,Ys: list_P125642481956313003od_c_a] :
      ( ( produc5646432941235140024od_c_a @ ( set_b2 @ Xs ) @ ( set_Product_prod_c_a2 @ Ys ) )
      = ( set_Pr5098181003905080610od_c_a
        @ ( concat3978415732298191054od_c_a
          @ ( map_b_696456053270819608od_c_a
            @ ^ [X4: b] : ( map_Pr1026158493455843436od_c_a @ ( produc3440865914000881661od_c_a @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_755_product__code,axiom,
    ! [Xs: list_c,Ys: list_a] :
      ( ( product_product_c_a @ ( set_c2 @ Xs ) @ ( set_a2 @ Ys ) )
      = ( set_Product_prod_c_a2
        @ ( concat497087365762831846od_c_a
          @ ( map_c_3537221148099011311od_c_a
            @ ^ [X4: c] : ( map_a_2285140998955227243od_c_a @ ( product_Pair_c_a @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_756_LS__acyclic__def,axiom,
    ( lS_acy3021907329840865005od_c_a
    = ( ^ [M2: fsm_a_1173558889237794545od_c_a,Q6: a] :
          ( collec6273869032445462695od_c_a
          @ ^ [Uu3: list_P6327159017948738492od_c_a] :
            ? [P4: list_P8924682930347365808_c_a_a] :
              ( ( Uu3
                = ( map_Pr6172405204570161616od_c_a
                  @ ^ [T3: produc6003456256095645226_c_a_a] : ( produc5762801557034676134od_c_a @ ( produc2803981006637266269_c_a_a @ ( produc8671850722014827208_c_a_a @ T3 ) ) @ ( produc1462292479987911732_c_a_a @ ( produc29636106103586463_c_a_a @ ( produc8671850722014827208_c_a_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a4575340523007214163od_c_a @ M2 @ Q6 @ P4 )
              & ( distinct_a @ ( visite4103323964629638516od_c_a @ Q6 @ P4 ) ) ) ) ) ) ).

% LS_acyclic_def
thf(fact_757_LS__acyclic__def,axiom,
    ( lS_acy4968821731943269956od_c_a
    = ( ^ [M2: fsm_a_4970532336856918856od_c_a,Q6: a] :
          ( collec1779435015300967550od_c_a
          @ ^ [Uu3: list_P5767727266746070163od_c_a] :
            ? [P4: list_P3747170188271196375_c_a_a] :
              ( ( Uu3
                = ( map_Pr1630241477723153472od_c_a
                  @ ^ [T3: produc2473480641396933073_c_a_a] : ( produc3440865914000881661od_c_a @ ( produc3813224185952080260_c_a_a @ ( produc8586270406267960047_c_a_a @ T3 ) ) @ ( produc4522685202854701596_c_a_a @ ( produc7675410941016976582_c_a_a @ ( produc8586270406267960047_c_a_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a4972803899755839402od_c_a @ M2 @ Q6 @ P4 )
              & ( distinct_a @ ( visite1724772098679956427od_c_a @ Q6 @ P4 ) ) ) ) ) ) ).

% LS_acyclic_def
thf(fact_758_LS__acyclic__def,axiom,
    ( lS_acyclic_a_c_a
    = ( ^ [M2: fsm_a_c_a,Q6: a] :
          ( collec1503280309375431318od_c_a
          @ ^ [Uu3: list_P125642481956313003od_c_a] :
            ? [P4: list_P6647267739079350397od_a_a] :
              ( ( Uu3
                = ( map_Pr5708375262016554034od_c_a
                  @ ^ [T3: produc5255912757217091959od_a_a] : ( product_Pair_c_a @ ( produc5237419938360121514od_a_a @ ( produc8719988090260684437od_a_a @ T3 ) ) @ ( product_fst_a_a @ ( produc933545150919345644od_a_a @ ( produc8719988090260684437od_a_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a_c_a @ M2 @ Q6 @ P4 )
              & ( distinct_a @ ( visited_states_a_c_a @ Q6 @ P4 ) ) ) ) ) ) ).

% LS_acyclic_def
thf(fact_759_LS__acyclic__def,axiom,
    ( lS_acyclic_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu3: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ M2 @ Q6 @ P4 )
              & ( distinct_a @ ( visited_states_a_b_c @ Q6 @ P4 ) ) ) ) ) ) ).

% LS_acyclic_def
thf(fact_760_transitions__from__set,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( transi6113640857693861514_a_b_c @ M @ Q )
        = ( collec8405073525749625505od_c_a
          @ ^ [T3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ T3 @ ( transitions_a_b_c @ M ) )
              & ( ( produc4225943634317630866od_c_a @ T3 )
                = Q ) ) ) ) ) ).

% transitions_from_set
thf(fact_761_path__prefix__take,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,I: nat] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( path_a_b_c @ M @ Q @ ( take_P3357827036823063076od_c_a @ I @ P5 ) ) ) ).

% path_prefix_take
thf(fact_762_in__set__takeD,axiom,
    ! [X3: a,N: nat,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ ( take_a @ N @ Xs ) ) )
     => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_763_in__set__takeD,axiom,
    ! [X3: produc2669932201312201782od_c_a,N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( take_P3357827036823063076od_c_a @ N @ Xs ) ) )
     => ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ).

% in_set_takeD
thf(fact_764_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_P3357827036823063076od_c_a @ N @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% take_Nil
thf(fact_765_distinct__take,axiom,
    ! [Xs: list_a,I: nat] :
      ( ( distinct_a @ Xs )
     => ( distinct_a @ ( take_a @ I @ Xs ) ) ) ).

% distinct_take
thf(fact_766_prefixes__set,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) )
      = ( collec6273869032445462695od_c_a
        @ ^ [Xs4: list_P6327159017948738492od_c_a] :
          ? [Xs7: list_P6327159017948738492od_c_a] :
            ( ( append3884164603879732689od_c_a @ Xs4 @ Xs7 )
            = Xs ) ) ) ).

% prefixes_set
thf(fact_767_suffixes__set,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( set_li8663184538779258833od_c_a @ ( suffix1038733475819007016od_c_a @ Xs ) )
      = ( collec6273869032445462695od_c_a
        @ ^ [Zs3: list_P6327159017948738492od_c_a] :
          ? [Ys3: list_P6327159017948738492od_c_a] :
            ( ( append3884164603879732689od_c_a @ Ys3 @ Zs3 )
            = Xs ) ) ) ).

% suffixes_set
thf(fact_768_reachable__states__def,axiom,
    ( reacha1620305530751930115_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
          ( collect_a
          @ ^ [Uu3: a] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( target_a_b_c @ ( initial_a_b_c @ M2 ) @ P4 ) )
              & ( path_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) @ P4 ) ) ) ) ) ).

% reachable_states_def
thf(fact_769_LS_Oelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Y3: set_li6436108459499378894od_b_c] :
      ( ( ( lS_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( Y3
        = ( collec2280997390073109977od_b_c
          @ ^ [Uu3: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ X3 @ Xa @ P4 ) ) ) ) ) ).

% LS.elims
thf(fact_770_LS_Osimps,axiom,
    ( lS_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu3: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ M2 @ Q6 @ P4 ) ) ) ) ) ).

% LS.simps
thf(fact_771_language__state__for__input_Osimps,axiom,
    ( langua3305995968491465752_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a,Xs3: list_b] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu3: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ M2 @ Q6 @ P4 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                    @ P4 ) )
                = Xs3 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_772_language__state__for__input_Oelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: list_b,Y3: set_li6436108459499378894od_b_c] :
      ( ( ( langua3305995968491465752_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( collec2280997390073109977od_b_c
          @ ^ [Uu3: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ X3 @ Xa @ P4 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                    @ P4 ) )
                = Xb ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_773_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a,Xss3: set_list_b] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu3: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ M2 @ Q6 @ P4 )
              & ( member_list_b
                @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                    @ P4 ) )
                @ Xss3 ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_774_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: set_list_b,Y3: set_li6436108459499378894od_b_c] :
      ( ( ( lS_i_n_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( collec2280997390073109977od_b_c
          @ ^ [Uu3: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ X3 @ Xa @ P4 )
              & ( member_list_b
                @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_775_remove__proper__prefixes__def,axiom,
    ( remove8467471134740590387od_c_a
    = ( ^ [Xs3: set_li1159382662694783132od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [X4: list_P6327159017948738492od_c_a] :
              ( ( member7410604586820865893od_c_a @ X4 @ Xs3 )
              & ~ ? [X5: list_P6327159017948738492od_c_a] :
                    ( ( X5 != nil_Pr1342775757158464060od_c_a )
                    & ( member7410604586820865893od_c_a @ ( append3884164603879732689od_c_a @ X4 @ X5 ) @ Xs3 ) ) ) ) ) ) ).

% remove_proper_prefixes_def
thf(fact_776_h__out_Oelims,axiom,
    ! [X3: fsm_Pr1886043760914421514_a_a_b,Xa: produc5771957243474376000_c_a_a,Y3: set_b] :
      ( ( ( h_out_8774345398997575435_a_a_b @ X3 @ Xa )
        = Y3 )
     => ~ ! [Q5: product_prod_c_a,X: a] :
            ( ( Xa
              = ( produc7661602042108997680_c_a_a @ Q5 @ X ) )
           => ( Y3
             != ( collect_b
                @ ^ [Y4: b] :
                  ? [Q7: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q5 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ X3 ) ) ) ) ) ) ).

% h_out.elims
thf(fact_777_h__out_Oelims,axiom,
    ! [X3: fsm_a_b_c,Xa: product_prod_a_b,Y3: set_c] :
      ( ( ( h_out_a_b_c @ X3 @ Xa )
        = Y3 )
     => ~ ! [Q5: a,X: b] :
            ( ( Xa
              = ( product_Pair_a_b @ Q5 @ X ) )
           => ( Y3
             != ( collect_c
                @ ^ [Y4: c] :
                  ? [Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q5 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transitions_a_b_c @ X3 ) ) ) ) ) ) ).

% h_out.elims
thf(fact_778_h__out_Oelims,axiom,
    ! [X3: fsm_a_836089231483141141_c_a_c,Xa: produc2669932201312201782od_c_a,Y3: set_c] :
      ( ( ( h_out_1068671350592293910_c_a_c @ X3 @ Xa )
        = Y3 )
     => ~ ! [Q5: a,X: produc2133359284458116493od_c_a] :
            ( ( Xa
              = ( produc5762801557034676134od_c_a @ Q5 @ X ) )
           => ( Y3
             != ( collect_c
                @ ^ [Y4: c] :
                  ? [Q7: a] : ( member8587115632268026449od_c_a @ ( produc8441414538239260312od_c_a @ Q5 @ ( produc1679040934828577519od_c_a @ X @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transi4754753175853207558_c_a_c @ X3 ) ) ) ) ) ) ).

% h_out.elims
thf(fact_779_h__out_Osimps,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,X3: a] :
      ( ( h_out_8774345398997575435_a_a_b @ M @ ( produc7661602042108997680_c_a_a @ Q @ X3 ) )
      = ( collect_b
        @ ^ [Y4: b] :
          ? [Q7: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q @ ( produc5762801557034676134od_c_a @ X3 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) ) ) ).

% h_out.simps
thf(fact_780_h__out_Osimps,axiom,
    ! [M: fsm_a_b_c,Q: a,X3: b] :
      ( ( h_out_a_b_c @ M @ ( product_Pair_a_b @ Q @ X3 ) )
      = ( collect_c
        @ ^ [Y4: c] :
          ? [Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X3 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transitions_a_b_c @ M ) ) ) ) ).

% h_out.simps
thf(fact_781_h__out_Osimps,axiom,
    ! [M: fsm_a_836089231483141141_c_a_c,Q: a,X3: produc2133359284458116493od_c_a] :
      ( ( h_out_1068671350592293910_c_a_c @ M @ ( produc5762801557034676134od_c_a @ Q @ X3 ) )
      = ( collect_c
        @ ^ [Y4: c] :
          ? [Q7: a] : ( member8587115632268026449od_c_a @ ( produc8441414538239260312od_c_a @ Q @ ( produc1679040934828577519od_c_a @ X3 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transi4754753175853207558_c_a_c @ M ) ) ) ) ).

% h_out.simps
thf(fact_782_language__state__for__input_Opelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: list_b,Y3: set_li6436108459499378894od_b_c] :
      ( ( ( langua3305995968491465752_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P4587740637805451515list_b @ langua4063633228699856747_a_b_c @ ( produc5717594475321894882list_b @ X3 @ ( produc6670463076781050526list_b @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( collec2280997390073109977od_b_c
                @ ^ [Uu3: list_P903359562653991662od_b_c] :
                  ? [P4: list_P6327159017948738492od_c_a] :
                    ( ( Uu3
                      = ( map_Pr776894537227705078od_b_c
                        @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                        @ P4 ) )
                    & ( path_a_b_c @ X3 @ Xa @ P4 )
                    & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                        @ ( map_Pr776894537227705078od_b_c
                          @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                          @ P4 ) )
                      = Xb ) ) ) )
           => ~ ( accp_P4587740637805451515list_b @ langua4063633228699856747_a_b_c @ ( produc5717594475321894882list_b @ X3 @ ( produc6670463076781050526list_b @ Xa @ Xb ) ) ) ) ) ) ).

% language_state_for_input.pelims
thf(fact_783_h__out__alt__def,axiom,
    ! [M: fsm_a_b_c,Q: a,X3: b] :
      ( ( h_out_a_b_c @ M @ ( product_Pair_a_b @ Q @ X3 ) )
      = ( collect_c
        @ ^ [Uu3: c] :
          ? [T3: produc2669932201312201782od_c_a] :
            ( ( Uu3
              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
            & ( member5502040268208861279od_c_a @ T3 @ ( transitions_a_b_c @ M ) )
            & ( ( produc4225943634317630866od_c_a @ T3 )
              = Q )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) )
              = X3 ) ) ) ) ).

% h_out_alt_def
thf(fact_784_LS_Opelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Y3: set_li6436108459499378894od_b_c] :
      ( ( ( lS_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P8775296850406592587_b_c_a @ lS_rel_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) )
       => ~ ( ( Y3
              = ( collec2280997390073109977od_b_c
                @ ^ [Uu3: list_P903359562653991662od_b_c] :
                  ? [P4: list_P6327159017948738492od_c_a] :
                    ( ( Uu3
                      = ( map_Pr776894537227705078od_b_c
                        @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                        @ P4 ) )
                    & ( path_a_b_c @ X3 @ Xa @ P4 ) ) ) )
           => ~ ( accp_P8775296850406592587_b_c_a @ lS_rel_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) ) ) ) ) ).

% LS.pelims
thf(fact_785_h__out_Opelims,axiom,
    ! [X3: fsm_Pr1886043760914421514_a_a_b,Xa: produc5771957243474376000_c_a_a,Y3: set_b] :
      ( ( ( h_out_8774345398997575435_a_a_b @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P6856346757249708390_c_a_a @ h_out_8051809192792576070_a_a_b @ ( produc9181388106632008269_c_a_a @ X3 @ Xa ) )
       => ~ ! [Q5: product_prod_c_a,X: a] :
              ( ( Xa
                = ( produc7661602042108997680_c_a_a @ Q5 @ X ) )
             => ( ( Y3
                  = ( collect_b
                    @ ^ [Y4: b] :
                      ? [Q7: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q5 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ X3 ) ) ) )
               => ~ ( accp_P6856346757249708390_c_a_a @ h_out_8051809192792576070_a_a_b @ ( produc9181388106632008269_c_a_a @ X3 @ ( produc7661602042108997680_c_a_a @ Q5 @ X ) ) ) ) ) ) ) ).

% h_out.pelims
thf(fact_786_h__out_Opelims,axiom,
    ! [X3: fsm_a_b_c,Xa: product_prod_a_b,Y3: set_c] :
      ( ( ( h_out_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P7742720022723642101od_a_b @ h_out_rel_a_b_c @ ( produc8315626074455449564od_a_b @ X3 @ Xa ) )
       => ~ ! [Q5: a,X: b] :
              ( ( Xa
                = ( product_Pair_a_b @ Q5 @ X ) )
             => ( ( Y3
                  = ( collect_c
                    @ ^ [Y4: c] :
                      ? [Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q5 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transitions_a_b_c @ X3 ) ) ) )
               => ~ ( accp_P7742720022723642101od_a_b @ h_out_rel_a_b_c @ ( produc8315626074455449564od_a_b @ X3 @ ( product_Pair_a_b @ Q5 @ X ) ) ) ) ) ) ) ).

% h_out.pelims
thf(fact_787_h__out_Opelims,axiom,
    ! [X3: fsm_a_836089231483141141_c_a_c,Xa: produc2669932201312201782od_c_a,Y3: set_c] :
      ( ( ( h_out_1068671350592293910_c_a_c @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P6123706172504556817od_c_a @ h_out_5416972558647935569_c_a_c @ ( produc201330100546959608od_c_a @ X3 @ Xa ) )
       => ~ ! [Q5: a,X: produc2133359284458116493od_c_a] :
              ( ( Xa
                = ( produc5762801557034676134od_c_a @ Q5 @ X ) )
             => ( ( Y3
                  = ( collect_c
                    @ ^ [Y4: c] :
                      ? [Q7: a] : ( member8587115632268026449od_c_a @ ( produc8441414538239260312od_c_a @ Q5 @ ( produc1679040934828577519od_c_a @ X @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transi4754753175853207558_c_a_c @ X3 ) ) ) )
               => ~ ( accp_P6123706172504556817od_c_a @ h_out_5416972558647935569_c_a_c @ ( produc201330100546959608od_c_a @ X3 @ ( produc5762801557034676134od_c_a @ Q5 @ X ) ) ) ) ) ) ) ).

% h_out.pelims
thf(fact_788_path__io__split_I2_J,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ( ( map_Pr776894537227705078od_b_c
            @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
            @ P5 )
          = ( append2547753245680614915od_b_c @ Io1 @ Io2 ) )
       => ( ( map_Pr776894537227705078od_b_c
            @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
            @ ( take_P3357827036823063076od_c_a @ ( size_s3392097710323735898od_b_c @ Io1 ) @ P5 ) )
          = Io1 ) ) ) ).

% path_io_split(2)
thf(fact_789_path__io__split_I1_J,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ( ( map_Pr776894537227705078od_b_c
            @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
            @ P5 )
          = ( append2547753245680614915od_b_c @ Io1 @ Io2 ) )
       => ( path_a_b_c @ M @ Q @ ( take_P3357827036823063076od_c_a @ ( size_s3392097710323735898od_b_c @ Io1 ) @ P5 ) ) ) ) ).

% path_io_split(1)
thf(fact_790_append__eq__append__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Us2: list_P6327159017948738492od_c_a,Vs: list_P6327159017948738492od_c_a] :
      ( ( ( ( size_s3386368156187063848od_c_a @ Xs )
          = ( size_s3386368156187063848od_c_a @ Ys ) )
        | ( ( size_s3386368156187063848od_c_a @ Us2 )
          = ( size_s3386368156187063848od_c_a @ Vs ) ) )
     => ( ( ( append3884164603879732689od_c_a @ Xs @ Us2 )
          = ( append3884164603879732689od_c_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_791_list__induct2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,P7: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o] :
      ( ( ( size_s3386368156187063848od_c_a @ Xs )
        = ( size_s3386368156187063848od_c_a @ Ys ) )
     => ( ( P7 @ nil_Pr1342775757158464060od_c_a @ nil_Pr1342775757158464060od_c_a )
       => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a] :
              ( ( ( size_s3386368156187063848od_c_a @ Xs2 )
                = ( size_s3386368156187063848od_c_a @ Ys2 ) )
             => ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) ) ) )
         => ( P7 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_792_list__induct3,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,P7: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o] :
      ( ( ( size_s3386368156187063848od_c_a @ Xs )
        = ( size_s3386368156187063848od_c_a @ Ys ) )
     => ( ( ( size_s3386368156187063848od_c_a @ Ys )
          = ( size_s3386368156187063848od_c_a @ Zs ) )
       => ( ( P7 @ nil_Pr1342775757158464060od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_Pr1342775757158464060od_c_a )
         => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a,Z3: produc2669932201312201782od_c_a,Zs2: list_P6327159017948738492od_c_a] :
                ( ( ( size_s3386368156187063848od_c_a @ Xs2 )
                  = ( size_s3386368156187063848od_c_a @ Ys2 ) )
               => ( ( ( size_s3386368156187063848od_c_a @ Ys2 )
                    = ( size_s3386368156187063848od_c_a @ Zs2 ) )
                 => ( ( P7 @ Xs2 @ Ys2 @ Zs2 )
                   => ( P7 @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) @ ( cons_P8659537020926305772od_c_a @ Z3 @ Zs2 ) ) ) ) )
           => ( P7 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_793_list__induct4,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,Ws: list_P6327159017948738492od_c_a,P7: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o] :
      ( ( ( size_s3386368156187063848od_c_a @ Xs )
        = ( size_s3386368156187063848od_c_a @ Ys ) )
     => ( ( ( size_s3386368156187063848od_c_a @ Ys )
          = ( size_s3386368156187063848od_c_a @ Zs ) )
       => ( ( ( size_s3386368156187063848od_c_a @ Zs )
            = ( size_s3386368156187063848od_c_a @ Ws ) )
         => ( ( P7 @ nil_Pr1342775757158464060od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_Pr1342775757158464060od_c_a )
           => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a,Z3: produc2669932201312201782od_c_a,Zs2: list_P6327159017948738492od_c_a,W: produc2669932201312201782od_c_a,Ws2: list_P6327159017948738492od_c_a] :
                  ( ( ( size_s3386368156187063848od_c_a @ Xs2 )
                    = ( size_s3386368156187063848od_c_a @ Ys2 ) )
                 => ( ( ( size_s3386368156187063848od_c_a @ Ys2 )
                      = ( size_s3386368156187063848od_c_a @ Zs2 ) )
                   => ( ( ( size_s3386368156187063848od_c_a @ Zs2 )
                        = ( size_s3386368156187063848od_c_a @ Ws2 ) )
                     => ( ( P7 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
                       => ( P7 @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) @ ( cons_P8659537020926305772od_c_a @ Z3 @ Zs2 ) @ ( cons_P8659537020926305772od_c_a @ W @ Ws2 ) ) ) ) ) )
             => ( P7 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_794_rev__induct2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,P7: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a > $o] :
      ( ( ( size_s3386368156187063848od_c_a @ Xs )
        = ( size_s3386368156187063848od_c_a @ Ys ) )
     => ( ( P7 @ nil_Pr1342775757158464060od_c_a @ nil_Pr1342775757158464060od_c_a )
       => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a] :
              ( ( ( size_s3386368156187063848od_c_a @ Xs2 )
                = ( size_s3386368156187063848od_c_a @ Ys2 ) )
             => ( ( P7 @ Xs2 @ Ys2 )
               => ( P7 @ ( append3884164603879732689od_c_a @ Xs2 @ ( cons_P8659537020926305772od_c_a @ X @ nil_Pr1342775757158464060od_c_a ) ) @ ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ Y @ nil_Pr1342775757158464060od_c_a ) ) ) ) )
         => ( P7 @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_795_same__length__different,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( Xs != Ys )
     => ( ( ( size_s3386368156187063848od_c_a @ Xs )
          = ( size_s3386368156187063848od_c_a @ Ys ) )
       => ? [Pre: list_P6327159017948738492od_c_a,X: produc2669932201312201782od_c_a,Xs5: list_P6327159017948738492od_c_a,Y: produc2669932201312201782od_c_a,Ys7: list_P6327159017948738492od_c_a] :
            ( ( X != Y )
            & ( Xs
              = ( append3884164603879732689od_c_a @ Pre @ ( append3884164603879732689od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ nil_Pr1342775757158464060od_c_a ) @ Xs5 ) ) )
            & ( Ys
              = ( append3884164603879732689od_c_a @ Pre @ ( append3884164603879732689od_c_a @ ( cons_P8659537020926305772od_c_a @ Y @ nil_Pr1342775757158464060od_c_a ) @ Ys7 ) ) ) ) ) ) ).

% same_length_different
thf(fact_796_LS_092_060_094sub_062i_092_060_094sub_062n_Opelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: set_list_b,Y3: set_li6436108459499378894od_b_c] :
      ( ( ( lS_i_n_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P1375664208687034459list_b @ lS_i_n_rel_a_b_c @ ( produc6733253750833833794list_b @ X3 @ ( produc9038752348217253374list_b @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( collec2280997390073109977od_b_c
                @ ^ [Uu3: list_P903359562653991662od_b_c] :
                  ? [P4: list_P6327159017948738492od_c_a] :
                    ( ( Uu3
                      = ( map_Pr776894537227705078od_b_c
                        @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                        @ P4 ) )
                    & ( path_a_b_c @ X3 @ Xa @ P4 )
                    & ( member_list_b
                      @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                        @ ( map_Pr776894537227705078od_b_c
                          @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
                          @ P4 ) )
                      @ Xb ) ) ) )
           => ~ ( accp_P1375664208687034459list_b @ lS_i_n_rel_a_b_c @ ( produc6733253750833833794list_b @ X3 @ ( produc9038752348217253374list_b @ Xa @ Xb ) ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.pelims
thf(fact_797_path__io__split_I3_J,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ( ( map_Pr776894537227705078od_b_c
            @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
            @ P5 )
          = ( append2547753245680614915od_b_c @ Io1 @ Io2 ) )
       => ( path_a_b_c @ M @ ( target_a_b_c @ Q @ ( take_P3357827036823063076od_c_a @ ( size_s3392097710323735898od_b_c @ Io1 ) @ P5 ) ) @ ( drop_P8434746179008708476od_c_a @ ( size_s3392097710323735898od_b_c @ Io1 ) @ P5 ) ) ) ) ).

% path_io_split(3)
thf(fact_798_add__prefixes__set,axiom,
    ! [Xs: list_l2826717958560394946od_c_a] :
      ( ( set_li8663184538779258833od_c_a @ ( add_pr535045906478010658od_c_a @ Xs ) )
      = ( collec6273869032445462695od_c_a
        @ ^ [Xs4: list_P6327159017948738492od_c_a] :
          ? [Xs7: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ ( append3884164603879732689od_c_a @ Xs4 @ Xs7 ) @ ( set_li8663184538779258833od_c_a @ Xs ) ) ) ) ).

% add_prefixes_set
thf(fact_799_path__io__split_I4_J,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ( ( map_Pr776894537227705078od_b_c
            @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
            @ P5 )
          = ( append2547753245680614915od_b_c @ Io1 @ Io2 ) )
       => ( ( map_Pr776894537227705078od_b_c
            @ ^ [T3: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) )
            @ ( drop_P8434746179008708476od_c_a @ ( size_s3392097710323735898od_b_c @ Io1 ) @ P5 ) )
          = Io2 ) ) ) ).

% path_io_split(4)
thf(fact_800_append__take__drop__id,axiom,
    ! [N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ ( take_P3357827036823063076od_c_a @ N @ Xs ) @ ( drop_P8434746179008708476od_c_a @ N @ Xs ) )
      = Xs ) ).

% append_take_drop_id
thf(fact_801_in__set__dropD,axiom,
    ! [X3: a,N: nat,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ ( drop_a @ N @ Xs ) ) )
     => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_802_in__set__dropD,axiom,
    ! [X3: produc2669932201312201782od_c_a,N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( drop_P8434746179008708476od_c_a @ N @ Xs ) ) )
     => ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ).

% in_set_dropD
thf(fact_803_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_P8434746179008708476od_c_a @ N @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% drop_Nil
thf(fact_804_distinct__drop,axiom,
    ! [Xs: list_a,I: nat] :
      ( ( distinct_a @ Xs )
     => ( distinct_a @ ( drop_a @ I @ Xs ) ) ) ).

% distinct_drop
thf(fact_805_append__eq__conv__conj,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Xs @ Ys )
        = Zs )
      = ( ( Xs
          = ( take_P3357827036823063076od_c_a @ ( size_s3386368156187063848od_c_a @ Xs ) @ Zs ) )
        & ( Ys
          = ( drop_P8434746179008708476od_c_a @ ( size_s3386368156187063848od_c_a @ Xs ) @ Zs ) ) ) ) ).

% append_eq_conv_conj
thf(fact_806_cyclic__path__shift_I1_J,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,I: nat] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ( ( target_a_b_c @ Q @ P5 )
          = Q )
       => ( path_a_b_c @ M @ ( target_a_b_c @ Q @ ( take_P3357827036823063076od_c_a @ I @ P5 ) ) @ ( append3884164603879732689od_c_a @ ( drop_P8434746179008708476od_c_a @ I @ P5 ) @ ( take_P3357827036823063076od_c_a @ I @ P5 ) ) ) ) ) ).

% cyclic_path_shift(1)
thf(fact_807_cyclic__path__shift_I2_J,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,I: nat] :
      ( ( path_a_b_c @ M @ Q @ P5 )
     => ( ( ( target_a_b_c @ Q @ P5 )
          = Q )
       => ( ( target_a_b_c @ ( target_a_b_c @ Q @ ( take_P3357827036823063076od_c_a @ I @ P5 ) ) @ ( append3884164603879732689od_c_a @ ( drop_P8434746179008708476od_c_a @ I @ P5 ) @ ( take_P3357827036823063076od_c_a @ I @ P5 ) ) )
          = ( target_a_b_c @ Q @ ( take_P3357827036823063076od_c_a @ I @ P5 ) ) ) ) ) ).

% cyclic_path_shift(2)
thf(fact_808_prefix__pairs__set,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( set_Pr1402144133550743484od_c_a @ ( prefix1345720000202332220od_c_a @ Xs ) )
      = ( collec6600936072614665618od_c_a
        @ ^ [Uu3: produc8143030500101326375od_c_a] :
          ? [Zs3: list_P6327159017948738492od_c_a,Ys3: list_P6327159017948738492od_c_a] :
            ( ( Uu3
              = ( produc8447801329164731415od_c_a @ Zs3 @ Ys3 ) )
            & ? [Xs12: list_P6327159017948738492od_c_a,Xs23: list_P6327159017948738492od_c_a] :
                ( ( ( append3884164603879732689od_c_a @ Zs3 @ Xs12 )
                  = Ys3 )
                & ( ( append3884164603879732689od_c_a @ Ys3 @ Xs23 )
                  = Xs )
                & ( Xs12 != nil_Pr1342775757158464060od_c_a ) ) ) ) ) ).

% prefix_pairs_set
thf(fact_809_prefix__pairs__set__alt,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( set_Pr1402144133550743484od_c_a @ ( prefix1345720000202332220od_c_a @ Xs ) )
      = ( collec6600936072614665618od_c_a
        @ ^ [Uu3: produc8143030500101326375od_c_a] :
          ? [Xs12: list_P6327159017948738492od_c_a,Xs23: list_P6327159017948738492od_c_a] :
            ( ( Uu3
              = ( produc8447801329164731415od_c_a @ Xs12 @ ( append3884164603879732689od_c_a @ Xs12 @ Xs23 ) ) )
            & ( Xs23 != nil_Pr1342775757158464060od_c_a )
            & ? [Xs33: list_P6327159017948738492od_c_a] :
                ( ( append3884164603879732689od_c_a @ Xs12 @ ( append3884164603879732689od_c_a @ Xs23 @ Xs33 ) )
                = Xs ) ) ) ) ).

% prefix_pairs_set_alt
thf(fact_810_prefix__pairs_Osimps_I1_J,axiom,
    ( ( prefix1345720000202332220od_c_a @ nil_Pr1342775757158464060od_c_a )
    = nil_Pr7268063790293894701od_c_a ) ).

% prefix_pairs.simps(1)
thf(fact_811_distinct__n__lists,axiom,
    ! [Xs: list_a,N: nat] :
      ( ( distinct_a @ Xs )
     => ( distinct_list_a @ ( n_lists_a @ N @ Xs ) ) ) ).

% distinct_n_lists
thf(fact_812_set__Cons__def,axiom,
    ( set_Cons_a
    = ( ^ [A6: set_a,XS: set_list_a] :
          ( collect_list_a
          @ ^ [Z2: list_a] :
            ? [X4: a,Xs3: list_a] :
              ( ( Z2
                = ( cons_a @ X4 @ Xs3 ) )
              & ( member_a @ X4 @ A6 )
              & ( member_list_a @ Xs3 @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_813_set__Cons__def,axiom,
    ( set_Co7906722538203907820od_c_a
    = ( ^ [A6: set_Pr8357624050790843030od_c_a,XS: set_li1159382662694783132od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [Z2: list_P6327159017948738492od_c_a] :
            ? [X4: produc2669932201312201782od_c_a,Xs3: list_P6327159017948738492od_c_a] :
              ( ( Z2
                = ( cons_P8659537020926305772od_c_a @ X4 @ Xs3 ) )
              & ( member5502040268208861279od_c_a @ X4 @ A6 )
              & ( member7410604586820865893od_c_a @ Xs3 @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_814_transpose_Opinduct,axiom,
    ! [A0: list_l2826717958560394946od_c_a,P7: list_l2826717958560394946od_c_a > $o] :
      ( ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ A0 )
     => ( ( ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ nil_li2790768921375229634od_c_a )
         => ( P7 @ nil_li2790768921375229634od_c_a ) )
       => ( ! [Xss: list_l2826717958560394946od_c_a] :
              ( ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ Xss ) )
             => ( ( P7 @ Xss )
               => ( P7 @ ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ Xss ) ) ) )
         => ( ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Xss: list_l2826717958560394946od_c_a] :
                ( ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ Xss ) )
               => ( ( P7
                    @ ( cons_l7906879460341455474od_c_a @ Xs2
                      @ ( concat1386605678430990717od_c_a
                        @ ( map_li3895047706134046922od_c_a
                          @ ( case_l5048770793687561178od_c_a @ nil_li2790768921375229634od_c_a
                            @ ^ [H: produc2669932201312201782od_c_a,T3: list_P6327159017948738492od_c_a] : ( cons_l7906879460341455474od_c_a @ T3 @ nil_li2790768921375229634od_c_a ) )
                          @ Xss ) ) ) )
                 => ( P7 @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ Xss ) ) ) )
           => ( P7 @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_815_transpose_Oelims,axiom,
    ! [X3: list_l2826717958560394946od_c_a,Y3: list_l2826717958560394946od_c_a] :
      ( ( ( transp4308190897923546054od_c_a @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_li2790768921375229634od_c_a )
         => ( Y3 != nil_li2790768921375229634od_c_a ) )
       => ( ! [Xss: list_l2826717958560394946od_c_a] :
              ( ( X3
                = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ Xss ) )
             => ( Y3
               != ( transp4308190897923546054od_c_a @ Xss ) ) )
         => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Xss: list_l2826717958560394946od_c_a] :
                ( ( X3
                  = ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ Xss ) )
               => ( Y3
                 != ( cons_l7906879460341455474od_c_a
                    @ ( cons_P8659537020926305772od_c_a @ X
                      @ ( concat7533993459791330935od_c_a
                        @ ( map_li335866320336015812od_c_a
                          @ ( case_l4158997801718927072od_c_a @ nil_Pr1342775757158464060od_c_a
                            @ ^ [H: produc2669932201312201782od_c_a,T3: list_P6327159017948738492od_c_a] : ( cons_P8659537020926305772od_c_a @ H @ nil_Pr1342775757158464060od_c_a ) )
                          @ Xss ) ) )
                    @ ( transp4308190897923546054od_c_a
                      @ ( cons_l7906879460341455474od_c_a @ Xs2
                        @ ( concat1386605678430990717od_c_a
                          @ ( map_li3895047706134046922od_c_a
                            @ ( case_l5048770793687561178od_c_a @ nil_li2790768921375229634od_c_a
                              @ ^ [H: produc2669932201312201782od_c_a,T3: list_P6327159017948738492od_c_a] : ( cons_l7906879460341455474od_c_a @ T3 @ nil_li2790768921375229634od_c_a ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_816_transpose_Osimps_I3_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Xss2: list_l2826717958560394946od_c_a] :
      ( ( transp4308190897923546054od_c_a @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ Xss2 ) )
      = ( cons_l7906879460341455474od_c_a
        @ ( cons_P8659537020926305772od_c_a @ X3
          @ ( concat7533993459791330935od_c_a
            @ ( map_li335866320336015812od_c_a
              @ ( case_l4158997801718927072od_c_a @ nil_Pr1342775757158464060od_c_a
                @ ^ [H: produc2669932201312201782od_c_a,T3: list_P6327159017948738492od_c_a] : ( cons_P8659537020926305772od_c_a @ H @ nil_Pr1342775757158464060od_c_a ) )
              @ Xss2 ) ) )
        @ ( transp4308190897923546054od_c_a
          @ ( cons_l7906879460341455474od_c_a @ Xs
            @ ( concat1386605678430990717od_c_a
              @ ( map_li3895047706134046922od_c_a
                @ ( case_l5048770793687561178od_c_a @ nil_li2790768921375229634od_c_a
                  @ ^ [H: produc2669932201312201782od_c_a,T3: list_P6327159017948738492od_c_a] : ( cons_l7906879460341455474od_c_a @ T3 @ nil_li2790768921375229634od_c_a ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_817_transpose_Opelims,axiom,
    ! [X3: list_l2826717958560394946od_c_a,Y3: list_l2826717958560394946od_c_a] :
      ( ( ( transp4308190897923546054od_c_a @ X3 )
        = Y3 )
     => ( ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ X3 )
       => ( ( ( X3 = nil_li2790768921375229634od_c_a )
           => ( ( Y3 = nil_li2790768921375229634od_c_a )
             => ~ ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ nil_li2790768921375229634od_c_a ) ) )
         => ( ! [Xss: list_l2826717958560394946od_c_a] :
                ( ( X3
                  = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ Xss ) )
               => ( ( Y3
                    = ( transp4308190897923546054od_c_a @ Xss ) )
                 => ~ ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ Xss ) ) ) )
           => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Xss: list_l2826717958560394946od_c_a] :
                  ( ( X3
                    = ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ Xss ) )
                 => ( ( Y3
                      = ( cons_l7906879460341455474od_c_a
                        @ ( cons_P8659537020926305772od_c_a @ X
                          @ ( concat7533993459791330935od_c_a
                            @ ( map_li335866320336015812od_c_a
                              @ ( case_l4158997801718927072od_c_a @ nil_Pr1342775757158464060od_c_a
                                @ ^ [H: produc2669932201312201782od_c_a,T3: list_P6327159017948738492od_c_a] : ( cons_P8659537020926305772od_c_a @ H @ nil_Pr1342775757158464060od_c_a ) )
                              @ Xss ) ) )
                        @ ( transp4308190897923546054od_c_a
                          @ ( cons_l7906879460341455474od_c_a @ Xs2
                            @ ( concat1386605678430990717od_c_a
                              @ ( map_li3895047706134046922od_c_a
                                @ ( case_l5048770793687561178od_c_a @ nil_li2790768921375229634od_c_a
                                  @ ^ [H: produc2669932201312201782od_c_a,T3: list_P6327159017948738492od_c_a] : ( cons_l7906879460341455474od_c_a @ T3 @ nil_li2790768921375229634od_c_a ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_818_list_Odisc__eq__case_I2_J,axiom,
    ! [List: list_P6327159017948738492od_c_a] :
      ( ( List != nil_Pr1342775757158464060od_c_a )
      = ( case_l1173855409784678916od_c_a @ $false
        @ ^ [Uu3: produc2669932201312201782od_c_a,Uv2: list_P6327159017948738492od_c_a] : $true
        @ List ) ) ).

% list.disc_eq_case(2)
thf(fact_819_list_Odisc__eq__case_I1_J,axiom,
    ! [List: list_P6327159017948738492od_c_a] :
      ( ( List = nil_Pr1342775757158464060od_c_a )
      = ( case_l1173855409784678916od_c_a @ $true
        @ ^ [Uu3: produc2669932201312201782od_c_a,Uv2: list_P6327159017948738492od_c_a] : $false
        @ List ) ) ).

% list.disc_eq_case(1)
thf(fact_820_transpose_Opsimps_I2_J,axiom,
    ! [Xss2: list_l2826717958560394946od_c_a] :
      ( ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ Xss2 ) )
     => ( ( transp4308190897923546054od_c_a @ ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ Xss2 ) )
        = ( transp4308190897923546054od_c_a @ Xss2 ) ) ) ).

% transpose.psimps(2)
thf(fact_821_transpose_Osimps_I2_J,axiom,
    ! [Xss2: list_l2826717958560394946od_c_a] :
      ( ( transp4308190897923546054od_c_a @ ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ Xss2 ) )
      = ( transp4308190897923546054od_c_a @ Xss2 ) ) ).

% transpose.simps(2)
thf(fact_822_transpose__empty,axiom,
    ! [Xs: list_l2826717958560394946od_c_a] :
      ( ( ( transp4308190897923546054od_c_a @ Xs )
        = nil_li2790768921375229634od_c_a )
      = ( ! [X4: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X4 @ ( set_li8663184538779258833od_c_a @ Xs ) )
           => ( X4 = nil_Pr1342775757158464060od_c_a ) ) ) ) ).

% transpose_empty
thf(fact_823_transpose_Opsimps_I3_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Xss2: list_l2826717958560394946od_c_a] :
      ( ( accp_l6922217013464387275od_c_a @ transp1675003631036763499od_c_a @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ Xss2 ) )
     => ( ( transp4308190897923546054od_c_a @ ( cons_l7906879460341455474od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ Xss2 ) )
        = ( cons_l7906879460341455474od_c_a
          @ ( cons_P8659537020926305772od_c_a @ X3
            @ ( concat7533993459791330935od_c_a
              @ ( map_li335866320336015812od_c_a
                @ ( case_l4158997801718927072od_c_a @ nil_Pr1342775757158464060od_c_a
                  @ ^ [H: produc2669932201312201782od_c_a,T3: list_P6327159017948738492od_c_a] : ( cons_P8659537020926305772od_c_a @ H @ nil_Pr1342775757158464060od_c_a ) )
                @ Xss2 ) ) )
          @ ( transp4308190897923546054od_c_a
            @ ( cons_l7906879460341455474od_c_a @ Xs
              @ ( concat1386605678430990717od_c_a
                @ ( map_li3895047706134046922od_c_a
                  @ ( case_l5048770793687561178od_c_a @ nil_li2790768921375229634od_c_a
                    @ ^ [H: produc2669932201312201782od_c_a,T3: list_P6327159017948738492od_c_a] : ( cons_l7906879460341455474od_c_a @ T3 @ nil_li2790768921375229634od_c_a ) )
                  @ Xss2 ) ) ) ) ) ) ) ).

% transpose.psimps(3)
thf(fact_824_image2__def,axiom,
    ( bNF_Gr4071498237745607475od_c_a
    = ( ^ [A6: set_a,F5: a > a,G: a > produc2133359284458116493od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [Uu3: produc2669932201312201782od_c_a] :
            ? [A5: a] :
              ( ( Uu3
                = ( produc5762801557034676134od_c_a @ ( F5 @ A5 ) @ ( G @ A5 ) ) )
              & ( member_a @ A5 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_825_image2__def,axiom,
    ( bNF_Gr3962133264466529899od_c_a
    = ( ^ [A6: set_Pr8357624050790843030od_c_a,F5: produc2669932201312201782od_c_a > a,G: produc2669932201312201782od_c_a > produc2133359284458116493od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [Uu3: produc2669932201312201782od_c_a] :
            ? [A5: produc2669932201312201782od_c_a] :
              ( ( Uu3
                = ( produc5762801557034676134od_c_a @ ( F5 @ A5 ) @ ( G @ A5 ) ) )
              & ( member5502040268208861279od_c_a @ A5 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_826_image2__def,axiom,
    ( bNF_Gr7990187263158702218od_c_a
    = ( ^ [A6: set_a,F5: a > b,G: a > product_prod_c_a] :
          ( collec3135042693873630968od_c_a
          @ ^ [Uu3: produc2133359284458116493od_c_a] :
            ? [A5: a] :
              ( ( Uu3
                = ( produc3440865914000881661od_c_a @ ( F5 @ A5 ) @ ( G @ A5 ) ) )
              & ( member_a @ A5 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_827_image2__def,axiom,
    ( bNF_Gr924734927316018114od_c_a
    = ( ^ [A6: set_Pr8357624050790843030od_c_a,F5: produc2669932201312201782od_c_a > b,G: produc2669932201312201782od_c_a > product_prod_c_a] :
          ( collec3135042693873630968od_c_a
          @ ^ [Uu3: produc2133359284458116493od_c_a] :
            ? [A5: produc2669932201312201782od_c_a] :
              ( ( Uu3
                = ( produc3440865914000881661od_c_a @ ( F5 @ A5 ) @ ( G @ A5 ) ) )
              & ( member5502040268208861279od_c_a @ A5 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_828_image2__def,axiom,
    ( bNF_Gr5415220321506865442_a_c_a
    = ( ^ [A6: set_a,F5: a > c,G: a > a] :
          ( collec6984858670293876240od_c_a
          @ ^ [Uu3: product_prod_c_a] :
            ? [A5: a] :
              ( ( Uu3
                = ( product_Pair_c_a @ ( F5 @ A5 ) @ ( G @ A5 ) ) )
              & ( member_a @ A5 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_829_image2__def,axiom,
    ( bNF_Gr8390375702759816282_a_c_a
    = ( ^ [A6: set_Pr8357624050790843030od_c_a,F5: produc2669932201312201782od_c_a > c,G: produc2669932201312201782od_c_a > a] :
          ( collec6984858670293876240od_c_a
          @ ^ [Uu3: product_prod_c_a] :
            ? [A5: produc2669932201312201782od_c_a] :
              ( ( Uu3
                = ( product_Pair_c_a @ ( F5 @ A5 ) @ ( G @ A5 ) ) )
              & ( member5502040268208861279od_c_a @ A5 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_830_merge__simps_I2_J,axiom,
    ! [Qs: list_P6327159017948738492od_c_a,P5: produc2669932201312201782od_c_a,Ps2: list_P6327159017948738492od_c_a] :
      ( ( merge_433830206917135860od_c_a @ Qs @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
      = ( update8541657393971699723od_c_a @ ( produc4225943634317630866od_c_a @ P5 ) @ ( produc8088130389382527188od_c_a @ P5 ) @ ( merge_433830206917135860od_c_a @ Qs @ Ps2 ) ) ) ).

% merge_simps(2)
thf(fact_831_Cons__in__lex,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) ) @ ( lex_Pr7986905161953557664od_c_a @ R ) )
      = ( ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y3 ) @ R )
          & ( ( size_s3386368156187063848od_c_a @ Xs )
            = ( size_s3386368156187063848od_c_a @ Ys ) ) )
        | ( ( X3 = Y3 )
          & ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( lex_Pr7986905161953557664od_c_a @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_832_merge__simps_I1_J,axiom,
    ! [Qs: list_P6327159017948738492od_c_a] :
      ( ( merge_433830206917135860od_c_a @ Qs @ nil_Pr1342775757158464060od_c_a )
      = Qs ) ).

% merge_simps(1)
thf(fact_833_relcomp_Ocases,axiom,
    ! [A1: c,A22: produc2133359284458116493od_c_a,R: set_Product_prod_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member6622777490300016861od_c_a @ ( produc6883538779125831716od_c_a @ A1 @ A22 ) @ ( relcom6295975468129122254od_c_a @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A1 @ B3 ) @ R )
           => ~ ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_834_relcomp_Ocases,axiom,
    ! [A1: a,A22: produc2133359284458116493od_c_a,R: set_Product_prod_a_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A1 @ A22 ) @ ( relcom430949252934960332od_c_a @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ R )
           => ~ ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_835_relcomp_Ocases,axiom,
    ! [A1: a,A22: produc2133359284458116493od_c_a,R: set_Pr8357624050790843030od_c_a,S: set_Pr3173710681352674119od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A1 @ A22 ) @ ( relcom3801338480298919421od_c_a @ R @ S ) )
     => ~ ! [B3: produc2133359284458116493od_c_a] :
            ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A1 @ B3 ) @ R )
           => ~ ( member7469191060503964048od_c_a @ ( produc9198192719383092439od_c_a @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_836_relcomp_Ocases,axiom,
    ! [A1: b,A22: product_prod_c_a,R: set_Product_prod_b_b,S: set_Pr272002107461315949od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A1 @ A22 ) @ ( relcom6637550213880284580od_c_a @ R @ S ) )
     => ~ ! [B3: b] :
            ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A1 @ B3 ) @ R )
           => ~ ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_837_relcomp_Ocases,axiom,
    ! [A1: b,A22: product_prod_c_a,R: set_Pr272002107461315949od_c_a,S: set_Pr5414900262478774087od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A1 @ A22 ) @ ( relcom7792157020509014910od_c_a @ R @ S ) )
     => ~ ! [B3: product_prod_c_a] :
            ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A1 @ B3 ) @ R )
           => ~ ( member3715682548391523728od_c_a @ ( produc968124180156873943od_c_a @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_838_relcomp_Ocases,axiom,
    ! [A1: c,A22: a,R: set_Product_prod_c_c,S: set_Product_prod_c_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A1 @ A22 ) @ ( relcomp_c_c_a @ R @ S ) )
     => ~ ! [B3: c] :
            ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A1 @ B3 ) @ R )
           => ~ ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_839_relcomp_Ocases,axiom,
    ! [A1: c,A22: a,R: set_Product_prod_c_a,S: set_Product_prod_a_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A1 @ A22 ) @ ( relcomp_c_a_a @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A1 @ B3 ) @ R )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A22 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_840_relcomp_Osimps,axiom,
    ! [A1: c,A22: produc2133359284458116493od_c_a,R: set_Product_prod_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member6622777490300016861od_c_a @ ( produc6883538779125831716od_c_a @ A1 @ A22 ) @ ( relcom6295975468129122254od_c_a @ R @ S ) )
      = ( ? [A5: c,B4: a,C5: produc2133359284458116493od_c_a] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A5 @ B4 ) @ R )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_841_relcomp_Osimps,axiom,
    ! [A1: a,A22: produc2133359284458116493od_c_a,R: set_Product_prod_a_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A1 @ A22 ) @ ( relcom430949252934960332od_c_a @ R @ S ) )
      = ( ? [A5: a,B4: a,C5: produc2133359284458116493od_c_a] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ B4 ) @ R )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_842_relcomp_Osimps,axiom,
    ! [A1: a,A22: produc2133359284458116493od_c_a,R: set_Pr8357624050790843030od_c_a,S: set_Pr3173710681352674119od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A1 @ A22 ) @ ( relcom3801338480298919421od_c_a @ R @ S ) )
      = ( ? [A5: a,B4: produc2133359284458116493od_c_a,C5: produc2133359284458116493od_c_a] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A5 @ B4 ) @ R )
            & ( member7469191060503964048od_c_a @ ( produc9198192719383092439od_c_a @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_843_relcomp_Osimps,axiom,
    ! [A1: b,A22: product_prod_c_a,R: set_Product_prod_b_b,S: set_Pr272002107461315949od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A1 @ A22 ) @ ( relcom6637550213880284580od_c_a @ R @ S ) )
      = ( ? [A5: b,B4: b,C5: product_prod_c_a] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A5 @ B4 ) @ R )
            & ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_844_relcomp_Osimps,axiom,
    ! [A1: b,A22: product_prod_c_a,R: set_Pr272002107461315949od_c_a,S: set_Pr5414900262478774087od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A1 @ A22 ) @ ( relcom7792157020509014910od_c_a @ R @ S ) )
      = ( ? [A5: b,B4: product_prod_c_a,C5: product_prod_c_a] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A5 @ B4 ) @ R )
            & ( member3715682548391523728od_c_a @ ( produc968124180156873943od_c_a @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_845_relcomp_Osimps,axiom,
    ! [A1: c,A22: a,R: set_Product_prod_c_c,S: set_Product_prod_c_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A1 @ A22 ) @ ( relcomp_c_c_a @ R @ S ) )
      = ( ? [A5: c,B4: c,C5: a] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A5 @ B4 ) @ R )
            & ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_846_relcomp_Osimps,axiom,
    ! [A1: c,A22: a,R: set_Product_prod_c_a,S: set_Product_prod_a_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A1 @ A22 ) @ ( relcomp_c_a_a @ R @ S ) )
      = ( ? [A5: c,B4: a,C5: a] :
            ( ( A1 = A5 )
            & ( A22 = C5 )
            & ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A5 @ B4 ) @ R )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ C5 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_847_relcomp_OrelcompI,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,C: produc2133359284458116493od_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B @ C ) @ S )
       => ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ C ) @ ( relcom430949252934960332od_c_a @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_848_relcomp_OrelcompI,axiom,
    ! [A: b,B: b,R: set_Product_prod_b_b,C: product_prod_c_a,S: set_Pr272002107461315949od_c_a] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R )
     => ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ B @ C ) @ S )
       => ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ C ) @ ( relcom6637550213880284580od_c_a @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_849_relcomp_OrelcompI,axiom,
    ! [A: c,B: c,R: set_Product_prod_c_c,C: a,S: set_Product_prod_c_a] :
      ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A @ B ) @ R )
     => ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ B @ C ) @ S )
       => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ C ) @ ( relcomp_c_c_a @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_850_relcomp_OrelcompI,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,R: set_Pr8357624050790843030od_c_a,C: produc2133359284458116493od_c_a,S: set_Pr3173710681352674119od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ R )
     => ( ( member7469191060503964048od_c_a @ ( produc9198192719383092439od_c_a @ B @ C ) @ S )
       => ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ C ) @ ( relcom3801338480298919421od_c_a @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_851_relcomp_OrelcompI,axiom,
    ! [A: b,B: product_prod_c_a,R: set_Pr272002107461315949od_c_a,C: product_prod_c_a,S: set_Pr5414900262478774087od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ R )
     => ( ( member3715682548391523728od_c_a @ ( produc968124180156873943od_c_a @ B @ C ) @ S )
       => ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ C ) @ ( relcom7792157020509014910od_c_a @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_852_relcomp_OrelcompI,axiom,
    ! [A: c,B: a,R: set_Product_prod_c_a,C: a,S: set_Product_prod_a_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ S )
       => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ C ) @ ( relcomp_c_a_a @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_853_relcomp_OrelcompI,axiom,
    ! [A: c,B: a,R: set_Product_prod_c_a,C: produc2133359284458116493od_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ R )
     => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B @ C ) @ S )
       => ( member6622777490300016861od_c_a @ ( produc6883538779125831716od_c_a @ A @ C ) @ ( relcom6295975468129122254od_c_a @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_854_relcompE,axiom,
    ! [Xz: produc3790669423403357364od_c_a,R: set_Product_prod_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member6622777490300016861od_c_a @ Xz @ ( relcom6295975468129122254od_c_a @ R @ S ) )
     => ~ ! [X: c,Y: a,Z3: produc2133359284458116493od_c_a] :
            ( ( Xz
              = ( produc6883538779125831716od_c_a @ X @ Z3 ) )
           => ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ R )
             => ~ ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Y @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_855_relcompE,axiom,
    ! [Xz: produc2669932201312201782od_c_a,R: set_Product_prod_a_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ Xz @ ( relcom430949252934960332od_c_a @ R @ S ) )
     => ~ ! [X: a,Y: a,Z3: produc2133359284458116493od_c_a] :
            ( ( Xz
              = ( produc5762801557034676134od_c_a @ X @ Z3 ) )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
             => ~ ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Y @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_856_relcompE,axiom,
    ! [Xz: produc2669932201312201782od_c_a,R: set_Pr8357624050790843030od_c_a,S: set_Pr3173710681352674119od_c_a] :
      ( ( member5502040268208861279od_c_a @ Xz @ ( relcom3801338480298919421od_c_a @ R @ S ) )
     => ~ ! [X: a,Y: produc2133359284458116493od_c_a,Z3: produc2133359284458116493od_c_a] :
            ( ( Xz
              = ( produc5762801557034676134od_c_a @ X @ Z3 ) )
           => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X @ Y ) @ R )
             => ~ ( member7469191060503964048od_c_a @ ( produc9198192719383092439od_c_a @ Y @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_857_relcompE,axiom,
    ! [Xz: produc2133359284458116493od_c_a,R: set_Product_prod_b_b,S: set_Pr272002107461315949od_c_a] :
      ( ( member1160383752307813046od_c_a @ Xz @ ( relcom6637550213880284580od_c_a @ R @ S ) )
     => ~ ! [X: b,Y: b,Z3: product_prod_c_a] :
            ( ( Xz
              = ( produc3440865914000881661od_c_a @ X @ Z3 ) )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R )
             => ~ ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ Y @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_858_relcompE,axiom,
    ! [Xz: produc2133359284458116493od_c_a,R: set_Pr272002107461315949od_c_a,S: set_Pr5414900262478774087od_c_a] :
      ( ( member1160383752307813046od_c_a @ Xz @ ( relcom7792157020509014910od_c_a @ R @ S ) )
     => ~ ! [X: b,Y: product_prod_c_a,Z3: product_prod_c_a] :
            ( ( Xz
              = ( produc3440865914000881661od_c_a @ X @ Z3 ) )
           => ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X @ Y ) @ R )
             => ~ ( member3715682548391523728od_c_a @ ( produc968124180156873943od_c_a @ Y @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_859_relcompE,axiom,
    ! [Xz: product_prod_c_a,R: set_Product_prod_c_c,S: set_Product_prod_c_a] :
      ( ( member5074992350434858958od_c_a @ Xz @ ( relcomp_c_c_a @ R @ S ) )
     => ~ ! [X: c,Y: c,Z3: a] :
            ( ( Xz
              = ( product_Pair_c_a @ X @ Z3 ) )
           => ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ X @ Y ) @ R )
             => ~ ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ Y @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_860_relcompE,axiom,
    ! [Xz: product_prod_c_a,R: set_Product_prod_c_a,S: set_Product_prod_a_a] :
      ( ( member5074992350434858958od_c_a @ Xz @ ( relcomp_c_a_a @ R @ S ) )
     => ~ ! [X: c,Y: a,Z3: a] :
            ( ( Xz
              = ( product_Pair_c_a @ X @ Z3 ) )
           => ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X @ Y ) @ R )
             => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z3 ) @ S ) ) ) ) ).

% relcompE
thf(fact_861_relcompEpair,axiom,
    ! [A: c,C: produc2133359284458116493od_c_a,R: set_Product_prod_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member6622777490300016861od_c_a @ ( produc6883538779125831716od_c_a @ A @ C ) @ ( relcom6295975468129122254od_c_a @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B3 ) @ R )
           => ~ ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_862_relcompEpair,axiom,
    ! [A: a,C: produc2133359284458116493od_c_a,R: set_Product_prod_a_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ C ) @ ( relcom430949252934960332od_c_a @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B3 ) @ R )
           => ~ ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_863_relcompEpair,axiom,
    ! [A: a,C: produc2133359284458116493od_c_a,R: set_Pr8357624050790843030od_c_a,S: set_Pr3173710681352674119od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ C ) @ ( relcom3801338480298919421od_c_a @ R @ S ) )
     => ~ ! [B3: produc2133359284458116493od_c_a] :
            ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B3 ) @ R )
           => ~ ( member7469191060503964048od_c_a @ ( produc9198192719383092439od_c_a @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_864_relcompEpair,axiom,
    ! [A: b,C: product_prod_c_a,R: set_Product_prod_b_b,S: set_Pr272002107461315949od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ C ) @ ( relcom6637550213880284580od_c_a @ R @ S ) )
     => ~ ! [B3: b] :
            ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ R )
           => ~ ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_865_relcompEpair,axiom,
    ! [A: b,C: product_prod_c_a,R: set_Pr272002107461315949od_c_a,S: set_Pr5414900262478774087od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ C ) @ ( relcom7792157020509014910od_c_a @ R @ S ) )
     => ~ ! [B3: product_prod_c_a] :
            ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B3 ) @ R )
           => ~ ( member3715682548391523728od_c_a @ ( produc968124180156873943od_c_a @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_866_relcompEpair,axiom,
    ! [A: c,C: a,R: set_Product_prod_c_c,S: set_Product_prod_c_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ C ) @ ( relcomp_c_c_a @ R @ S ) )
     => ~ ! [B3: c] :
            ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A @ B3 ) @ R )
           => ~ ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_867_relcompEpair,axiom,
    ! [A: c,C: a,R: set_Product_prod_c_a,S: set_Product_prod_a_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ C ) @ ( relcomp_c_a_a @ R @ S ) )
     => ~ ! [B3: a] :
            ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B3 ) @ R )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ C ) @ S ) ) ) ).

% relcompEpair
thf(fact_868_Nil__notin__lex,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ~ ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ nil_Pr1342775757158464060od_c_a @ Ys ) @ ( lex_Pr7986905161953557664od_c_a @ R ) ) ).

% Nil_notin_lex
thf(fact_869_Nil2__notin__lex,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ~ ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ nil_Pr1342775757158464060od_c_a ) @ ( lex_Pr7986905161953557664od_c_a @ R ) ) ).

% Nil2_notin_lex
thf(fact_870_lex__append__leftI,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Ys @ Zs ) @ ( lex_Pr7986905161953557664od_c_a @ R ) )
     => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) @ ( append3884164603879732689od_c_a @ Xs @ Zs ) ) @ ( lex_Pr7986905161953557664od_c_a @ R ) ) ) ).

% lex_append_leftI
thf(fact_871_image2__eqI,axiom,
    ! [B: a,F2: a > a,X3: a,C: produc2133359284458116493od_c_a,G2: a > produc2133359284458116493od_c_a,A3: set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member_a @ X3 @ A3 )
         => ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B @ C ) @ ( bNF_Gr4071498237745607475od_c_a @ A3 @ F2 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_872_image2__eqI,axiom,
    ! [B: a,F2: produc2669932201312201782od_c_a > a,X3: produc2669932201312201782od_c_a,C: produc2133359284458116493od_c_a,G2: produc2669932201312201782od_c_a > produc2133359284458116493od_c_a,A3: set_Pr8357624050790843030od_c_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member5502040268208861279od_c_a @ X3 @ A3 )
         => ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ B @ C ) @ ( bNF_Gr3962133264466529899od_c_a @ A3 @ F2 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_873_image2__eqI,axiom,
    ! [B: b,F2: a > b,X3: a,C: product_prod_c_a,G2: a > product_prod_c_a,A3: set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member_a @ X3 @ A3 )
         => ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ B @ C ) @ ( bNF_Gr7990187263158702218od_c_a @ A3 @ F2 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_874_image2__eqI,axiom,
    ! [B: b,F2: produc2669932201312201782od_c_a > b,X3: produc2669932201312201782od_c_a,C: product_prod_c_a,G2: produc2669932201312201782od_c_a > product_prod_c_a,A3: set_Pr8357624050790843030od_c_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member5502040268208861279od_c_a @ X3 @ A3 )
         => ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ B @ C ) @ ( bNF_Gr924734927316018114od_c_a @ A3 @ F2 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_875_image2__eqI,axiom,
    ! [B: c,F2: a > c,X3: a,C: a,G2: a > a,A3: set_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member_a @ X3 @ A3 )
         => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ B @ C ) @ ( bNF_Gr5415220321506865442_a_c_a @ A3 @ F2 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_876_image2__eqI,axiom,
    ! [B: c,F2: produc2669932201312201782od_c_a > c,X3: produc2669932201312201782od_c_a,C: a,G2: produc2669932201312201782od_c_a > a,A3: set_Pr8357624050790843030od_c_a] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( C
          = ( G2 @ X3 ) )
       => ( ( member5502040268208861279od_c_a @ X3 @ A3 )
         => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ B @ C ) @ ( bNF_Gr8390375702759816282_a_c_a @ A3 @ F2 @ G2 ) ) ) ) ) ).

% image2_eqI
thf(fact_877_lex__append__left__iff,axiom,
    ! [R: set_Pr1454594181397332999od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ! [X: produc2669932201312201782od_c_a] :
          ~ ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X @ X ) @ R )
     => ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) @ ( append3884164603879732689od_c_a @ Xs @ Zs ) ) @ ( lex_Pr7986905161953557664od_c_a @ R ) )
        = ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Ys @ Zs ) @ ( lex_Pr7986905161953557664od_c_a @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_878_lex__append__leftD,axiom,
    ! [R: set_Pr1454594181397332999od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ! [X: produc2669932201312201782od_c_a] :
          ~ ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X @ X ) @ R )
     => ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) @ ( append3884164603879732689od_c_a @ Xs @ Zs ) ) @ ( lex_Pr7986905161953557664od_c_a @ R ) )
       => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Ys @ Zs ) @ ( lex_Pr7986905161953557664od_c_a @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_879_lex__append__rightI,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a,Vs: list_P6327159017948738492od_c_a,Us2: list_P6327159017948738492od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( lex_Pr7986905161953557664od_c_a @ R ) )
     => ( ( ( size_s3386368156187063848od_c_a @ Vs )
          = ( size_s3386368156187063848od_c_a @ Us2 ) )
       => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Us2 ) @ ( append3884164603879732689od_c_a @ Ys @ Vs ) ) @ ( lex_Pr7986905161953557664od_c_a @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_880_lex__conv,axiom,
    ( lex_Pr7986905161953557664od_c_a
    = ( ^ [R3: set_Pr1454594181397332999od_c_a] :
          ( collec6600936072614665618od_c_a
          @ ( produc3830619377415620716_c_a_o
            @ ^ [Xs3: list_P6327159017948738492od_c_a,Ys3: list_P6327159017948738492od_c_a] :
                ( ( ( size_s3386368156187063848od_c_a @ Xs3 )
                  = ( size_s3386368156187063848od_c_a @ Ys3 ) )
                & ? [Xys: list_P6327159017948738492od_c_a,X4: produc2669932201312201782od_c_a,Y4: produc2669932201312201782od_c_a,Xs4: list_P6327159017948738492od_c_a,Ys4: list_P6327159017948738492od_c_a] :
                    ( ( Xs3
                      = ( append3884164603879732689od_c_a @ Xys @ ( cons_P8659537020926305772od_c_a @ X4 @ Xs4 ) ) )
                    & ( Ys3
                      = ( append3884164603879732689od_c_a @ Xys @ ( cons_P8659537020926305772od_c_a @ Y4 @ Ys4 ) ) )
                    & ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ).

% lex_conv
thf(fact_881_n__lists_Osimps_I2_J,axiom,
    ! [N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( n_list6750465182534647364od_c_a @ ( suc @ N ) @ Xs )
      = ( concat1386605678430990717od_c_a
        @ ( map_li3895047706134046922od_c_a
          @ ^ [Ys3: list_P6327159017948738492od_c_a] :
              ( map_Pr5537105415683728714od_c_a
              @ ^ [Y4: produc2669932201312201782od_c_a] : ( cons_P8659537020926305772od_c_a @ Y4 @ Ys3 )
              @ Xs )
          @ ( n_list6750465182534647364od_c_a @ N @ Xs ) ) ) ) ).

% n_lists.simps(2)
thf(fact_882_lists__of__length_Osimps_I2_J,axiom,
    ! [T4: list_P6327159017948738492od_c_a,N: nat] :
      ( ( lists_3695925642795093265od_c_a @ T4 @ ( suc @ N ) )
      = ( concat1386605678430990717od_c_a
        @ ( map_li3895047706134046922od_c_a
          @ ^ [Xs3: list_P6327159017948738492od_c_a] :
              ( map_Pr5537105415683728714od_c_a
              @ ^ [X4: produc2669932201312201782od_c_a] : ( cons_P8659537020926305772od_c_a @ X4 @ Xs3 )
              @ T4 )
          @ ( lists_3695925642795093265od_c_a @ T4 @ N ) ) ) ) ).

% lists_of_length.simps(2)
thf(fact_883_visited__states_Opelims,axiom,
    ! [X3: a,Xa: list_P6327159017948738492od_c_a,Y3: list_a] :
      ( ( ( visited_states_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P5710213560906687854od_c_a @ visite3107294483184927197_a_b_c @ ( produc3580641852002036821od_c_a @ X3 @ Xa ) )
       => ~ ( ( Y3
              = ( cons_a @ X3
                @ ( map_Pr5179659585931931020_c_a_a
                  @ ^ [A5: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A5 ) ) )
                  @ Xa ) ) )
           => ~ ( accp_P5710213560906687854od_c_a @ visite3107294483184927197_a_b_c @ ( produc3580641852002036821od_c_a @ X3 @ Xa ) ) ) ) ) ).

% visited_states.pelims
thf(fact_884_case__prodI2,axiom,
    ! [P5: produc2669932201312201782od_c_a,C: a > produc2133359284458116493od_c_a > $o] :
      ( ! [A4: a,B3: produc2133359284458116493od_c_a] :
          ( ( P5
            = ( produc5762801557034676134od_c_a @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc1940289858450342685_c_a_o @ C @ P5 ) ) ).

% case_prodI2
thf(fact_885_case__prodI2,axiom,
    ! [P5: produc2133359284458116493od_c_a,C: b > product_prod_c_a > $o] :
      ( ! [A4: b,B3: product_prod_c_a] :
          ( ( P5
            = ( produc3440865914000881661od_c_a @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc6211283585785106182_c_a_o @ C @ P5 ) ) ).

% case_prodI2
thf(fact_886_case__prodI2,axiom,
    ! [P5: product_prod_c_a,C: c > a > $o] :
      ( ! [A4: c,B3: a] :
          ( ( P5
            = ( product_Pair_c_a @ A4 @ B3 ) )
         => ( C @ A4 @ B3 ) )
     => ( produc8907516716866730350_c_a_o @ C @ P5 ) ) ).

% case_prodI2
thf(fact_887_case__prodI,axiom,
    ! [F2: a > produc2133359284458116493od_c_a > $o,A: a,B: produc2133359284458116493od_c_a] :
      ( ( F2 @ A @ B )
     => ( produc1940289858450342685_c_a_o @ F2 @ ( produc5762801557034676134od_c_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_888_case__prodI,axiom,
    ! [F2: b > product_prod_c_a > $o,A: b,B: product_prod_c_a] :
      ( ( F2 @ A @ B )
     => ( produc6211283585785106182_c_a_o @ F2 @ ( produc3440865914000881661od_c_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_889_case__prodI,axiom,
    ! [F2: c > a > $o,A: c,B: a] :
      ( ( F2 @ A @ B )
     => ( produc8907516716866730350_c_a_o @ F2 @ ( product_Pair_c_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_890_take__Suc__Cons,axiom,
    ! [N: nat,X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( take_P3357827036823063076od_c_a @ ( suc @ N ) @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( cons_P8659537020926305772od_c_a @ X3 @ ( take_P3357827036823063076od_c_a @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_891_drop__Suc__Cons,axiom,
    ! [N: nat,X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( drop_P8434746179008708476od_c_a @ ( suc @ N ) @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( drop_P8434746179008708476od_c_a @ N @ Xs ) ) ).

% drop_Suc_Cons
thf(fact_892_case__prodE,axiom,
    ! [C: a > produc2133359284458116493od_c_a > $o,P5: produc2669932201312201782od_c_a] :
      ( ( produc1940289858450342685_c_a_o @ C @ P5 )
     => ~ ! [X: a,Y: produc2133359284458116493od_c_a] :
            ( ( P5
              = ( produc5762801557034676134od_c_a @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_893_case__prodE,axiom,
    ! [C: b > product_prod_c_a > $o,P5: produc2133359284458116493od_c_a] :
      ( ( produc6211283585785106182_c_a_o @ C @ P5 )
     => ~ ! [X: b,Y: product_prod_c_a] :
            ( ( P5
              = ( produc3440865914000881661od_c_a @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_894_case__prodE,axiom,
    ! [C: c > a > $o,P5: product_prod_c_a] :
      ( ( produc8907516716866730350_c_a_o @ C @ P5 )
     => ~ ! [X: c,Y: a] :
            ( ( P5
              = ( product_Pair_c_a @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_895_case__prodD,axiom,
    ! [F2: a > produc2133359284458116493od_c_a > $o,A: a,B: produc2133359284458116493od_c_a] :
      ( ( produc1940289858450342685_c_a_o @ F2 @ ( produc5762801557034676134od_c_a @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_896_case__prodD,axiom,
    ! [F2: b > product_prod_c_a > $o,A: b,B: product_prod_c_a] :
      ( ( produc6211283585785106182_c_a_o @ F2 @ ( produc3440865914000881661od_c_a @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_897_case__prodD,axiom,
    ! [F2: c > a > $o,A: c,B: a] :
      ( ( produc8907516716866730350_c_a_o @ F2 @ ( product_Pair_c_a @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_898_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc2669932201312201782od_c_a,A3: a > produc2133359284458116493od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( collec8405073525749625505od_c_a @ ( produc1940289858450342685_c_a_o @ A3 ) ) )
     => ( A3 @ ( produc4225943634317630866od_c_a @ X3 ) @ ( produc8088130389382527188od_c_a @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_899_length__Suc__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,N: nat] :
      ( ( ( size_s3386368156187063848od_c_a @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: produc2669932201312201782od_c_a,Ys3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( cons_P8659537020926305772od_c_a @ Y4 @ Ys3 ) )
            & ( ( size_s3386368156187063848od_c_a @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_900_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( suc @ N )
        = ( size_s3386368156187063848od_c_a @ Xs ) )
      = ( ? [Y4: produc2669932201312201782od_c_a,Ys3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( cons_P8659537020926305772od_c_a @ Y4 @ Ys3 ) )
            & ( ( size_s3386368156187063848od_c_a @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_901_length__Cons,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( size_s3386368156187063848od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
      = ( suc @ ( size_s3386368156187063848od_c_a @ Xs ) ) ) ).

% length_Cons
thf(fact_902_maximal__distinct__prefix,axiom,
    ! [Xs: list_a] :
      ( ~ ( distinct_a @ Xs )
     => ~ ! [N2: nat] :
            ( ( distinct_a @ ( take_a @ ( suc @ N2 ) @ Xs ) )
           => ( distinct_a @ ( take_a @ ( suc @ ( suc @ N2 ) ) @ Xs ) ) ) ) ).

% maximal_distinct_prefix
thf(fact_903_visited__states__length,axiom,
    ! [Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( size_size_list_a @ ( visited_states_a_b_c @ Q @ P5 ) )
      = ( suc @ ( size_s3386368156187063848od_c_a @ P5 ) ) ) ).

% visited_states_length
thf(fact_904_relcomp__unfold,axiom,
    ( relcom6295975468129122254od_c_a
    = ( ^ [R3: set_Product_prod_c_a,S3: set_Pr8357624050790843030od_c_a] :
          ( collec302438710986005279od_c_a
          @ ( produc6787939801517004575_c_a_o
            @ ^ [X4: c,Z2: produc2133359284458116493od_c_a] :
              ? [Y4: a] :
                ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y4 ) @ R3 )
                & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Y4 @ Z2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_905_visited__states__take,axiom,
    ! [N: nat,Q: a,P5: list_P6327159017948738492od_c_a] :
      ( ( take_a @ ( suc @ N ) @ ( visited_states_a_b_c @ Q @ P5 ) )
      = ( visited_states_a_b_c @ Q @ ( take_P3357827036823063076od_c_a @ N @ P5 ) ) ) ).

% visited_states_take
thf(fact_906_length__append__singleton,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
      ( ( size_s3386368156187063848od_c_a @ ( append3884164603879732689od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) )
      = ( suc @ ( size_s3386368156187063848od_c_a @ Xs ) ) ) ).

% length_append_singleton
thf(fact_907_length__Suc__conv__rev,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,N: nat] :
      ( ( ( size_s3386368156187063848od_c_a @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: produc2669932201312201782od_c_a,Ys3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ Y4 @ nil_Pr1342775757158464060od_c_a ) ) )
            & ( ( size_s3386368156187063848od_c_a @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_908_listrel1__def,axiom,
    ( listre7328408972406649621od_c_a
    = ( ^ [R3: set_Pr1454594181397332999od_c_a] :
          ( collec6600936072614665618od_c_a
          @ ( produc3830619377415620716_c_a_o
            @ ^ [Xs3: list_P6327159017948738492od_c_a,Ys3: list_P6327159017948738492od_c_a] :
              ? [Us: list_P6327159017948738492od_c_a,Z2: produc2669932201312201782od_c_a,Z4: produc2669932201312201782od_c_a,Vs3: list_P6327159017948738492od_c_a] :
                ( ( Xs3
                  = ( append3884164603879732689od_c_a @ Us @ ( cons_P8659537020926305772od_c_a @ Z2 @ Vs3 ) ) )
                & ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ Z2 @ Z4 ) @ R3 )
                & ( Ys3
                  = ( append3884164603879732689od_c_a @ Us @ ( cons_P8659537020926305772od_c_a @ Z4 @ Vs3 ) ) ) ) ) ) ) ) ).

% listrel1_def
thf(fact_909_delete__aux_Opelims,axiom,
    ! [X3: a,Xa: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( ( delete3144518687146744757od_c_a @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P5710213560906687854od_c_a @ delete6751563828040927714od_c_a @ ( produc3580641852002036821od_c_a @ X3 @ Xa ) )
       => ( ( ( Xa = nil_Pr1342775757158464060od_c_a )
           => ( ( Y3 = nil_Pr1342775757158464060od_c_a )
             => ~ ( accp_P5710213560906687854od_c_a @ delete6751563828040927714od_c_a @ ( produc3580641852002036821od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) ) )
         => ~ ! [K2: a,V: produc2133359284458116493od_c_a,Xs2: list_P6327159017948738492od_c_a] :
                ( ( Xa
                  = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K2 @ V ) @ Xs2 ) )
               => ( ( ( ( X3 = K2 )
                     => ( Y3 = Xs2 ) )
                    & ( ( X3 != K2 )
                     => ( Y3
                        = ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K2 @ V ) @ ( delete3144518687146744757od_c_a @ X3 @ Xs2 ) ) ) ) )
                 => ~ ( accp_P5710213560906687854od_c_a @ delete6751563828040927714od_c_a @ ( produc3580641852002036821od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K2 @ V ) @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_910_delete__aux_Opelims,axiom,
    ! [X3: b,Xa: list_P5767727266746070163od_c_a,Y3: list_P5767727266746070163od_c_a] :
      ( ( ( delete2063006593957858060od_c_a @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P3497580143769170564od_c_a @ delete601219079491431993od_c_a @ ( produc1443866078074754795od_c_a @ X3 @ Xa ) )
       => ( ( ( Xa = nil_Pr5380409136740620691od_c_a )
           => ( ( Y3 = nil_Pr5380409136740620691od_c_a )
             => ~ ( accp_P3497580143769170564od_c_a @ delete601219079491431993od_c_a @ ( produc1443866078074754795od_c_a @ X3 @ nil_Pr5380409136740620691od_c_a ) ) ) )
         => ~ ! [K2: b,V: product_prod_c_a,Xs2: list_P5767727266746070163od_c_a] :
                ( ( Xa
                  = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K2 @ V ) @ Xs2 ) )
               => ( ( ( ( X3 = K2 )
                     => ( Y3 = Xs2 ) )
                    & ( ( X3 != K2 )
                     => ( Y3
                        = ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K2 @ V ) @ ( delete2063006593957858060od_c_a @ X3 @ Xs2 ) ) ) ) )
                 => ~ ( accp_P3497580143769170564od_c_a @ delete601219079491431993od_c_a @ ( produc1443866078074754795od_c_a @ X3 @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K2 @ V ) @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_911_delete__aux_Opelims,axiom,
    ! [X3: c,Xa: list_P125642481956313003od_c_a,Y3: list_P125642481956313003od_c_a] :
      ( ( ( delete_aux_c_a @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P3633840959627935579od_c_a @ delete_aux_rel_c_a @ ( produc3490508561035776450od_c_a @ X3 @ Xa ) )
       => ( ( ( Xa = nil_Product_prod_c_a )
           => ( ( Y3 = nil_Product_prod_c_a )
             => ~ ( accp_P3633840959627935579od_c_a @ delete_aux_rel_c_a @ ( produc3490508561035776450od_c_a @ X3 @ nil_Product_prod_c_a ) ) ) )
         => ~ ! [K2: c,V: a,Xs2: list_P125642481956313003od_c_a] :
                ( ( Xa
                  = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K2 @ V ) @ Xs2 ) )
               => ( ( ( ( X3 = K2 )
                     => ( Y3 = Xs2 ) )
                    & ( ( X3 != K2 )
                     => ( Y3
                        = ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K2 @ V ) @ ( delete_aux_c_a @ X3 @ Xs2 ) ) ) ) )
                 => ~ ( accp_P3633840959627935579od_c_a @ delete_aux_rel_c_a @ ( produc3490508561035776450od_c_a @ X3 @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K2 @ V ) @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_912_lexn__conv,axiom,
    ( lexn_P7929102822784682204od_c_a
    = ( ^ [R3: set_Pr1454594181397332999od_c_a,N3: nat] :
          ( collec6600936072614665618od_c_a
          @ ( produc3830619377415620716_c_a_o
            @ ^ [Xs3: list_P6327159017948738492od_c_a,Ys3: list_P6327159017948738492od_c_a] :
                ( ( ( size_s3386368156187063848od_c_a @ Xs3 )
                  = N3 )
                & ( ( size_s3386368156187063848od_c_a @ Ys3 )
                  = N3 )
                & ? [Xys: list_P6327159017948738492od_c_a,X4: produc2669932201312201782od_c_a,Y4: produc2669932201312201782od_c_a,Xs4: list_P6327159017948738492od_c_a,Ys4: list_P6327159017948738492od_c_a] :
                    ( ( Xs3
                      = ( append3884164603879732689od_c_a @ Xys @ ( cons_P8659537020926305772od_c_a @ X4 @ Xs4 ) ) )
                    & ( Ys3
                      = ( append3884164603879732689od_c_a @ Xys @ ( cons_P8659537020926305772od_c_a @ Y4 @ Ys4 ) ) )
                    & ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ).

% lexn_conv
thf(fact_913_h__from_Oelims,axiom,
    ! [X3: fsm_Pr1886043760914421514_a_a_b,Xa: product_prod_c_a,Y3: set_Pr8357624050790843030od_c_a] :
      ( ( ( h_from4453792899594357127_a_a_b @ X3 @ Xa )
        = Y3 )
     => ( Y3
        = ( collec8405073525749625505od_c_a
          @ ( produc1940289858450342685_c_a_o
            @ ^ [X4: a] :
                ( produc6211283585785106182_c_a_o
                @ ^ [Y4: b,Q7: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Xa @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ X3 ) ) ) ) ) ) ) ).

% h_from.elims
thf(fact_914_h__from_Oelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Y3: set_Pr272002107461315949od_c_a] :
      ( ( ( h_from_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( Y3
        = ( collec3135042693873630968od_c_a
          @ ( produc6211283585785106182_c_a_o
            @ ^ [X4: b] :
                ( produc8907516716866730350_c_a_o
                @ ^ [Y4: c,Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Xa @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transitions_a_b_c @ X3 ) ) ) ) ) ) ) ).

% h_from.elims
thf(fact_915_mem__case__prodI,axiom,
    ! [Z: a,C: a > produc2133359284458116493od_c_a > set_a,A: a,B: produc2133359284458116493od_c_a] :
      ( ( member_a @ Z @ ( C @ A @ B ) )
     => ( member_a @ Z @ ( produc2035080022814879075_set_a @ C @ ( produc5762801557034676134od_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_916_mem__case__prodI,axiom,
    ! [Z: produc2669932201312201782od_c_a,C: a > produc2133359284458116493od_c_a > set_Pr8357624050790843030od_c_a,A: a,B: produc2133359284458116493od_c_a] :
      ( ( member5502040268208861279od_c_a @ Z @ ( C @ A @ B ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc6802637503518801051od_c_a @ C @ ( produc5762801557034676134od_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_917_mem__case__prodI,axiom,
    ! [Z: a,C: b > product_prod_c_a > set_a,A: b,B: product_prod_c_a] :
      ( ( member_a @ Z @ ( C @ A @ B ) )
     => ( member_a @ Z @ ( produc711231852148568268_set_a @ C @ ( produc3440865914000881661od_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_918_mem__case__prodI,axiom,
    ! [Z: produc2669932201312201782od_c_a,C: b > product_prod_c_a > set_Pr8357624050790843030od_c_a,A: b,B: product_prod_c_a] :
      ( ( member5502040268208861279od_c_a @ Z @ ( C @ A @ B ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc3761857390937722116od_c_a @ C @ ( produc3440865914000881661od_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_919_mem__case__prodI,axiom,
    ! [Z: a,C: c > a > set_a,A: c,B: a] :
      ( ( member_a @ Z @ ( C @ A @ B ) )
     => ( member_a @ Z @ ( produc2545871480639997748_set_a @ C @ ( product_Pair_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_920_mem__case__prodI,axiom,
    ! [Z: produc2669932201312201782od_c_a,C: c > a > set_Pr8357624050790843030od_c_a,A: c,B: a] :
      ( ( member5502040268208861279od_c_a @ Z @ ( C @ A @ B ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc8162871031289310060od_c_a @ C @ ( product_Pair_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_921_mem__case__prodI2,axiom,
    ! [P5: produc2669932201312201782od_c_a,Z: a,C: a > produc2133359284458116493od_c_a > set_a] :
      ( ! [A4: a,B3: produc2133359284458116493od_c_a] :
          ( ( P5
            = ( produc5762801557034676134od_c_a @ A4 @ B3 ) )
         => ( member_a @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_a @ Z @ ( produc2035080022814879075_set_a @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_922_mem__case__prodI2,axiom,
    ! [P5: produc2669932201312201782od_c_a,Z: produc2669932201312201782od_c_a,C: a > produc2133359284458116493od_c_a > set_Pr8357624050790843030od_c_a] :
      ( ! [A4: a,B3: produc2133359284458116493od_c_a] :
          ( ( P5
            = ( produc5762801557034676134od_c_a @ A4 @ B3 ) )
         => ( member5502040268208861279od_c_a @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc6802637503518801051od_c_a @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_923_mem__case__prodI2,axiom,
    ! [P5: produc2133359284458116493od_c_a,Z: a,C: b > product_prod_c_a > set_a] :
      ( ! [A4: b,B3: product_prod_c_a] :
          ( ( P5
            = ( produc3440865914000881661od_c_a @ A4 @ B3 ) )
         => ( member_a @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_a @ Z @ ( produc711231852148568268_set_a @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_924_mem__case__prodI2,axiom,
    ! [P5: produc2133359284458116493od_c_a,Z: produc2669932201312201782od_c_a,C: b > product_prod_c_a > set_Pr8357624050790843030od_c_a] :
      ( ! [A4: b,B3: product_prod_c_a] :
          ( ( P5
            = ( produc3440865914000881661od_c_a @ A4 @ B3 ) )
         => ( member5502040268208861279od_c_a @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc3761857390937722116od_c_a @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_925_mem__case__prodI2,axiom,
    ! [P5: product_prod_c_a,Z: a,C: c > a > set_a] :
      ( ! [A4: c,B3: a] :
          ( ( P5
            = ( product_Pair_c_a @ A4 @ B3 ) )
         => ( member_a @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member_a @ Z @ ( produc2545871480639997748_set_a @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_926_mem__case__prodI2,axiom,
    ! [P5: product_prod_c_a,Z: produc2669932201312201782od_c_a,C: c > a > set_Pr8357624050790843030od_c_a] :
      ( ! [A4: c,B3: a] :
          ( ( P5
            = ( product_Pair_c_a @ A4 @ B3 ) )
         => ( member5502040268208861279od_c_a @ Z @ ( C @ A4 @ B3 ) ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc8162871031289310060od_c_a @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_927_mem__case__prodE,axiom,
    ! [Z: a,C: a > produc2133359284458116493od_c_a > set_a,P5: produc2669932201312201782od_c_a] :
      ( ( member_a @ Z @ ( produc2035080022814879075_set_a @ C @ P5 ) )
     => ~ ! [X: a,Y: produc2133359284458116493od_c_a] :
            ( ( P5
              = ( produc5762801557034676134od_c_a @ X @ Y ) )
           => ~ ( member_a @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_928_mem__case__prodE,axiom,
    ! [Z: produc2669932201312201782od_c_a,C: a > produc2133359284458116493od_c_a > set_Pr8357624050790843030od_c_a,P5: produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ Z @ ( produc6802637503518801051od_c_a @ C @ P5 ) )
     => ~ ! [X: a,Y: produc2133359284458116493od_c_a] :
            ( ( P5
              = ( produc5762801557034676134od_c_a @ X @ Y ) )
           => ~ ( member5502040268208861279od_c_a @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_929_mem__case__prodE,axiom,
    ! [Z: a,C: b > product_prod_c_a > set_a,P5: produc2133359284458116493od_c_a] :
      ( ( member_a @ Z @ ( produc711231852148568268_set_a @ C @ P5 ) )
     => ~ ! [X: b,Y: product_prod_c_a] :
            ( ( P5
              = ( produc3440865914000881661od_c_a @ X @ Y ) )
           => ~ ( member_a @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_930_mem__case__prodE,axiom,
    ! [Z: produc2669932201312201782od_c_a,C: b > product_prod_c_a > set_Pr8357624050790843030od_c_a,P5: produc2133359284458116493od_c_a] :
      ( ( member5502040268208861279od_c_a @ Z @ ( produc3761857390937722116od_c_a @ C @ P5 ) )
     => ~ ! [X: b,Y: product_prod_c_a] :
            ( ( P5
              = ( produc3440865914000881661od_c_a @ X @ Y ) )
           => ~ ( member5502040268208861279od_c_a @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_931_mem__case__prodE,axiom,
    ! [Z: a,C: c > a > set_a,P5: product_prod_c_a] :
      ( ( member_a @ Z @ ( produc2545871480639997748_set_a @ C @ P5 ) )
     => ~ ! [X: c,Y: a] :
            ( ( P5
              = ( product_Pair_c_a @ X @ Y ) )
           => ~ ( member_a @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_932_mem__case__prodE,axiom,
    ! [Z: produc2669932201312201782od_c_a,C: c > a > set_Pr8357624050790843030od_c_a,P5: product_prod_c_a] :
      ( ( member5502040268208861279od_c_a @ Z @ ( produc8162871031289310060od_c_a @ C @ P5 ) )
     => ~ ! [X: c,Y: a] :
            ( ( P5
              = ( product_Pair_c_a @ X @ Y ) )
           => ~ ( member5502040268208861279od_c_a @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_933_h__from_Osimps,axiom,
    ( h_from4453792899594357127_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b,Q6: product_prod_c_a] :
          ( collec8405073525749625505od_c_a
          @ ( produc1940289858450342685_c_a_o
            @ ^ [X4: a] :
                ( produc6211283585785106182_c_a_o
                @ ^ [Y4: b,Q7: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q6 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ) ) ) ) ).

% h_from.simps
thf(fact_934_h__from_Osimps,axiom,
    ( h_from_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a] :
          ( collec3135042693873630968od_c_a
          @ ( produc6211283585785106182_c_a_o
            @ ^ [X4: b] :
                ( produc8907516716866730350_c_a_o
                @ ^ [Y4: c,Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q6 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transitions_a_b_c @ M2 ) ) ) ) ) ) ) ).

% h_from.simps
thf(fact_935_h__from_Opelims,axiom,
    ! [X3: fsm_Pr1886043760914421514_a_a_b,Xa: product_prod_c_a,Y3: set_Pr8357624050790843030od_c_a] :
      ( ( ( h_from4453792899594357127_a_a_b @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P268257991765507659od_c_a @ h_from7835141602277640394_a_a_b @ ( produc816620231618388914od_c_a @ X3 @ Xa ) )
       => ~ ( ( Y3
              = ( collec8405073525749625505od_c_a
                @ ( produc1940289858450342685_c_a_o
                  @ ^ [X4: a] :
                      ( produc6211283585785106182_c_a_o
                      @ ^ [Y4: b,Q7: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Xa @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ X3 ) ) ) ) ) )
           => ~ ( accp_P268257991765507659od_c_a @ h_from7835141602277640394_a_a_b @ ( produc816620231618388914od_c_a @ X3 @ Xa ) ) ) ) ) ).

% h_from.pelims
thf(fact_936_h__from_Opelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Y3: set_Pr272002107461315949od_c_a] :
      ( ( ( h_from_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P8775296850406592587_b_c_a @ h_from_rel_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) )
       => ~ ( ( Y3
              = ( collec3135042693873630968od_c_a
                @ ( produc6211283585785106182_c_a_o
                  @ ^ [X4: b] :
                      ( produc8907516716866730350_c_a_o
                      @ ^ [Y4: c,Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Xa @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transitions_a_b_c @ X3 ) ) ) ) ) )
           => ~ ( accp_P8775296850406592587_b_c_a @ h_from_rel_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) ) ) ) ) ).

% h_from.pelims
thf(fact_937_lists__of__length_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xa: nat,Y3: list_l2826717958560394946od_c_a] :
      ( ( ( lists_3695925642795093265od_c_a @ X3 @ Xa )
        = Y3 )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y3
           != ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) )
       => ~ ! [N2: nat] :
              ( ( Xa
                = ( suc @ N2 ) )
             => ( Y3
               != ( concat1386605678430990717od_c_a
                  @ ( map_li3895047706134046922od_c_a
                    @ ^ [Xs3: list_P6327159017948738492od_c_a] :
                        ( map_Pr5537105415683728714od_c_a
                        @ ^ [X4: produc2669932201312201782od_c_a] : ( cons_P8659537020926305772od_c_a @ X4 @ Xs3 )
                        @ X3 )
                    @ ( lists_3695925642795093265od_c_a @ X3 @ N2 ) ) ) ) ) ) ) ).

% lists_of_length.elims
thf(fact_938_h__simps,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,X3: a] :
      ( ( h_Prod4541284455510304017_a_a_b @ M @ ( produc7661602042108997680_c_a_a @ Q @ X3 ) )
      = ( collec3135042693873630968od_c_a
        @ ( produc6211283585785106182_c_a_o
          @ ^ [Y4: b,Q7: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q @ ( produc5762801557034676134od_c_a @ X3 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) ) ) ) ).

% h_simps
thf(fact_939_h__simps,axiom,
    ! [M: fsm_a_b_c,Q: a,X3: b] :
      ( ( h_a_b_c @ M @ ( product_Pair_a_b @ Q @ X3 ) )
      = ( collec6984858670293876240od_c_a
        @ ( produc8907516716866730350_c_a_o
          @ ^ [Y4: c,Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X3 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transitions_a_b_c @ M ) ) ) ) ) ).

% h_simps
thf(fact_940_h__simps,axiom,
    ! [M: fsm_a_836089231483141141_c_a_c,Q: a,X3: produc2133359284458116493od_c_a] :
      ( ( h_a_Pr5118833440477893660_c_a_c @ M @ ( produc5762801557034676134od_c_a @ Q @ X3 ) )
      = ( collec6984858670293876240od_c_a
        @ ( produc8907516716866730350_c_a_o
          @ ^ [Y4: c,Q7: a] : ( member8587115632268026449od_c_a @ ( produc8441414538239260312od_c_a @ Q @ ( produc1679040934828577519od_c_a @ X3 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ ( transi4754753175853207558_c_a_c @ M ) ) ) ) ) ).

% h_simps
thf(fact_941_length__0__conv,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( ( size_s3386368156187063848od_c_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% length_0_conv
thf(fact_942_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( nil_Pr1342775757158464060od_c_a
        = ( take_P3357827036823063076od_c_a @ N @ Xs ) )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_Pr1342775757158464060od_c_a ) ) ) ).

% take_eq_Nil2
thf(fact_943_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( take_P3357827036823063076od_c_a @ N @ Xs )
        = nil_Pr1342775757158464060od_c_a )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_Pr1342775757158464060od_c_a ) ) ) ).

% take_eq_Nil
thf(fact_944_take0,axiom,
    ( ( take_P3357827036823063076od_c_a @ zero_zero_nat )
    = ( ^ [Xs3: list_P6327159017948738492od_c_a] : nil_Pr1342775757158464060od_c_a ) ) ).

% take0
thf(fact_945_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list6750465182534647364od_c_a @ N @ nil_Pr1342775757158464060od_c_a )
          = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list6750465182534647364od_c_a @ N @ nil_Pr1342775757158464060od_c_a )
          = nil_li2790768921375229634od_c_a ) ) ) ).

% n_lists_Nil
thf(fact_946_list_Osize_I3_J,axiom,
    ( ( size_s3386368156187063848od_c_a @ nil_Pr1342775757158464060od_c_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_947_take__0,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( take_P3357827036823063076od_c_a @ zero_zero_nat @ Xs )
      = nil_Pr1342775757158464060od_c_a ) ).

% take_0
thf(fact_948_lists__of__length_Osimps_I1_J,axiom,
    ! [T4: list_P6327159017948738492od_c_a] :
      ( ( lists_3695925642795093265od_c_a @ T4 @ zero_zero_nat )
      = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) ).

% lists_of_length.simps(1)
thf(fact_949_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( n_list6750465182534647364od_c_a @ zero_zero_nat @ Xs )
      = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) ) ).

% n_lists.simps(1)
thf(fact_950_lists__of__length_Opelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xa: nat,Y3: list_l2826717958560394946od_c_a] :
      ( ( ( lists_3695925642795093265od_c_a @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P9180951279490314994_a_nat @ lists_1085471087770999264od_c_a @ ( produc2049479133603754037_a_nat @ X3 @ Xa ) )
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y3
                = ( cons_l7906879460341455474od_c_a @ nil_Pr1342775757158464060od_c_a @ nil_li2790768921375229634od_c_a ) )
             => ~ ( accp_P9180951279490314994_a_nat @ lists_1085471087770999264od_c_a @ ( produc2049479133603754037_a_nat @ X3 @ zero_zero_nat ) ) ) )
         => ~ ! [N2: nat] :
                ( ( Xa
                  = ( suc @ N2 ) )
               => ( ( Y3
                    = ( concat1386605678430990717od_c_a
                      @ ( map_li3895047706134046922od_c_a
                        @ ^ [Xs3: list_P6327159017948738492od_c_a] :
                            ( map_Pr5537105415683728714od_c_a
                            @ ^ [X4: produc2669932201312201782od_c_a] : ( cons_P8659537020926305772od_c_a @ X4 @ Xs3 )
                            @ X3 )
                        @ ( lists_3695925642795093265od_c_a @ X3 @ N2 ) ) ) )
                 => ~ ( accp_P9180951279490314994_a_nat @ lists_1085471087770999264od_c_a @ ( produc2049479133603754037_a_nat @ X3 @ ( suc @ N2 ) ) ) ) ) ) ) ) ).

% lists_of_length.pelims
thf(fact_951_case__prod__Pair__iden,axiom,
    ! [P5: produc2669932201312201782od_c_a] :
      ( ( produc4268383296163969595od_c_a @ produc5762801557034676134od_c_a @ P5 )
      = P5 ) ).

% case_prod_Pair_iden
thf(fact_952_case__prod__Pair__iden,axiom,
    ! [P5: produc2133359284458116493od_c_a] :
      ( ( produc6855526503766603259od_c_a @ produc3440865914000881661od_c_a @ P5 )
      = P5 ) ).

% case_prod_Pair_iden
thf(fact_953_case__prod__Pair__iden,axiom,
    ! [P5: product_prod_c_a] :
      ( ( produc2713253650426281595od_c_a @ product_Pair_c_a @ P5 )
      = P5 ) ).

% case_prod_Pair_iden
thf(fact_954_lexord__def,axiom,
    ( lexord8026273315949980969od_c_a
    = ( ^ [R3: set_Pr1454594181397332999od_c_a] :
          ( collec6600936072614665618od_c_a
          @ ( produc3830619377415620716_c_a_o
            @ ^ [X4: list_P6327159017948738492od_c_a,Y4: list_P6327159017948738492od_c_a] :
              ? [A5: produc2669932201312201782od_c_a,V3: list_P6327159017948738492od_c_a] :
                ( ( Y4
                  = ( append3884164603879732689od_c_a @ X4 @ ( cons_P8659537020926305772od_c_a @ A5 @ V3 ) ) )
                | ? [U: list_P6327159017948738492od_c_a,B4: produc2669932201312201782od_c_a,C5: produc2669932201312201782od_c_a,W2: list_P6327159017948738492od_c_a,Z2: list_P6327159017948738492od_c_a] :
                    ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ B4 @ C5 ) @ R3 )
                    & ( X4
                      = ( append3884164603879732689od_c_a @ U @ ( cons_P8659537020926305772od_c_a @ B4 @ W2 ) ) )
                    & ( Y4
                      = ( append3884164603879732689od_c_a @ U @ ( cons_P8659537020926305772od_c_a @ C5 @ Z2 ) ) ) ) ) ) ) ) ) ).

% lexord_def
thf(fact_955_acyclic__paths__up__to__length_H__set,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,Prev2: list_P1562850763667305982od_c_a,HF: product_prod_c_a > set_Pr8357624050790843030od_c_a,VisitedStates: set_Product_prod_c_a,K3: nat] :
      ( ( path_P1107610407223809260_a_a_b @ M @ ( p_sour1959829948604423246_a_a_b @ Q @ Prev2 ) @ Prev2 )
     => ( ! [Q8: product_prod_c_a] :
            ( ( HF @ Q8 )
            = ( collec8405073525749625505od_c_a
              @ ^ [Uu3: produc2669932201312201782od_c_a] :
                ? [X4: a,Y4: b,Q9: product_prod_c_a] :
                  ( ( Uu3
                    = ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q9 ) ) )
                  & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q8 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q9 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) ) ) )
       => ( ( distin3780794742951484702od_c_a @ ( visite7082950643002702093_a_a_b @ ( p_sour1959829948604423246_a_a_b @ Q @ Prev2 ) @ Prev2 ) )
         => ( ( VisitedStates
              = ( set_Product_prod_c_a2 @ ( visite7082950643002702093_a_a_b @ ( p_sour1959829948604423246_a_a_b @ Q @ Prev2 ) @ Prev2 ) ) )
           => ( ( acycli2757849055487800696_a_a_b @ Prev2 @ ( target6303326113586172192_a_a_b @ ( p_sour1959829948604423246_a_a_b @ Q @ Prev2 ) @ Prev2 ) @ HF @ VisitedStates @ K3 )
              = ( collec453890259647268073od_c_a
                @ ^ [Uu3: list_P1562850763667305982od_c_a] :
                  ? [P4: list_P1562850763667305982od_c_a] :
                    ( ( Uu3
                      = ( append311360319448987923od_c_a @ Prev2 @ P4 ) )
                    & ( path_P1107610407223809260_a_a_b @ M @ ( p_sour1959829948604423246_a_a_b @ Q @ Prev2 ) @ ( append311360319448987923od_c_a @ Prev2 @ P4 ) )
                    & ( ord_less_eq_nat @ ( size_s1349739684588871786od_c_a @ P4 ) @ K3 )
                    & ( distin3780794742951484702od_c_a @ ( visite7082950643002702093_a_a_b @ ( p_sour1959829948604423246_a_a_b @ Q @ Prev2 ) @ ( append311360319448987923od_c_a @ Prev2 @ P4 ) ) ) ) ) ) ) ) ) ) ).

% acyclic_paths_up_to_length'_set
thf(fact_956_acyclic__paths__up__to__length_H__set,axiom,
    ! [M: fsm_a_b_c,Q: a,Prev2: list_P6327159017948738492od_c_a,HF: a > set_Pr272002107461315949od_c_a,VisitedStates: set_a,K3: nat] :
      ( ( path_a_b_c @ M @ ( p_source_a_b_c @ Q @ Prev2 ) @ Prev2 )
     => ( ! [Q8: a] :
            ( ( HF @ Q8 )
            = ( collec3135042693873630968od_c_a
              @ ^ [Uu3: produc2133359284458116493od_c_a] :
                ? [X4: b,Y4: c,Q9: a] :
                  ( ( Uu3
                    = ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q9 ) ) )
                  & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q8 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q9 ) ) ) @ ( transitions_a_b_c @ M ) ) ) ) )
       => ( ( distinct_a @ ( visited_states_a_b_c @ ( p_source_a_b_c @ Q @ Prev2 ) @ Prev2 ) )
         => ( ( VisitedStates
              = ( set_a2 @ ( visited_states_a_b_c @ ( p_source_a_b_c @ Q @ Prev2 ) @ Prev2 ) ) )
           => ( ( acycli8838441729103940241_a_b_c @ Prev2 @ ( target_a_b_c @ ( p_source_a_b_c @ Q @ Prev2 ) @ Prev2 ) @ HF @ VisitedStates @ K3 )
              = ( collec6273869032445462695od_c_a
                @ ^ [Uu3: list_P6327159017948738492od_c_a] :
                  ? [P4: list_P6327159017948738492od_c_a] :
                    ( ( Uu3
                      = ( append3884164603879732689od_c_a @ Prev2 @ P4 ) )
                    & ( path_a_b_c @ M @ ( p_source_a_b_c @ Q @ Prev2 ) @ ( append3884164603879732689od_c_a @ Prev2 @ P4 ) )
                    & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P4 ) @ K3 )
                    & ( distinct_a @ ( visited_states_a_b_c @ ( p_source_a_b_c @ Q @ Prev2 ) @ ( append3884164603879732689od_c_a @ Prev2 @ P4 ) ) ) ) ) ) ) ) ) ) ).

% acyclic_paths_up_to_length'_set
thf(fact_957_drop__all,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ Xs ) @ N )
     => ( ( drop_P8434746179008708476od_c_a @ N @ Xs )
        = nil_Pr1342775757158464060od_c_a ) ) ).

% drop_all
thf(fact_958_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( drop_P8434746179008708476od_c_a @ N @ Xs )
        = nil_Pr1342775757158464060od_c_a )
      = ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_959_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( nil_Pr1342775757158464060od_c_a
        = ( drop_P8434746179008708476od_c_a @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_960_lexord__cons__cons,axiom,
    ! [A: produc2669932201312201782od_c_a,X3: list_P6327159017948738492od_c_a,B: produc2669932201312201782od_c_a,Y3: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( cons_P8659537020926305772od_c_a @ A @ X3 ) @ ( cons_P8659537020926305772od_c_a @ B @ Y3 ) ) @ ( lexord8026273315949980969od_c_a @ R ) )
      = ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ B ) @ R )
        | ( ( A = B )
          & ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ X3 @ Y3 ) @ ( lexord8026273315949980969od_c_a @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_961_lexord__Nil__left,axiom,
    ! [Y3: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ nil_Pr1342775757158464060od_c_a @ Y3 ) @ ( lexord8026273315949980969od_c_a @ R ) )
      = ( ? [A5: produc2669932201312201782od_c_a,X4: list_P6327159017948738492od_c_a] :
            ( Y3
            = ( cons_P8659537020926305772od_c_a @ A5 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_962_lexord__sufI,axiom,
    ! [U2: list_P6327159017948738492od_c_a,W3: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a,V2: list_P6327159017948738492od_c_a,Z: list_P6327159017948738492od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ U2 @ W3 ) @ ( lexord8026273315949980969od_c_a @ R ) )
     => ( ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ W3 ) @ ( size_s3386368156187063848od_c_a @ U2 ) )
       => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ U2 @ V2 ) @ ( append3884164603879732689od_c_a @ W3 @ Z ) ) @ ( lexord8026273315949980969od_c_a @ R ) ) ) ) ).

% lexord_sufI
thf(fact_963_recursion__renaming__helper,axiom,
    ! [F1: nat > nat,P7: nat > $o,F22: nat > nat,K3: nat] :
      ( ( F1
        = ( ^ [X4: nat] : ( if_nat @ ( P7 @ X4 ) @ X4 @ ( F1 @ ( suc @ X4 ) ) ) ) )
     => ( ( F22
          = ( ^ [X4: nat] : ( if_nat @ ( P7 @ X4 ) @ X4 @ ( F22 @ ( suc @ X4 ) ) ) ) )
       => ( ! [X: nat] :
              ( ( ord_less_eq_nat @ K3 @ X )
             => ( P7 @ X ) )
         => ( F1 = F22 ) ) ) ) ).

% recursion_renaming_helper
thf(fact_964_minimal__fixpoint__helper_I1_J,axiom,
    ! [F2: nat > nat,P7: nat > $o,K3: nat,X3: nat] :
      ( ( F2
        = ( ^ [X4: nat] : ( if_nat @ ( P7 @ X4 ) @ X4 @ ( F2 @ ( suc @ X4 ) ) ) ) )
     => ( ! [X: nat] :
            ( ( ord_less_eq_nat @ K3 @ X )
           => ( P7 @ X ) )
       => ( P7 @ ( F2 @ X3 ) ) ) ) ).

% minimal_fixpoint_helper(1)
thf(fact_965_impossible__Cons,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
      ( ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ Xs ) @ ( size_s3386368156187063848od_c_a @ Ys ) )
     => ( Xs
       != ( cons_P8659537020926305772od_c_a @ X3 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_966_acyclic__paths__up__to__length_H__prev,axiom,
    ! [P9: list_P6327159017948738492od_c_a,Prev2: list_P6327159017948738492od_c_a,Prev3: list_P6327159017948738492od_c_a,Q: a,HF: a > set_Pr272002107461315949od_c_a,VisitedStates: set_a,K3: nat] :
      ( ( member7410604586820865893od_c_a @ P9 @ ( acycli8838441729103940241_a_b_c @ ( append3884164603879732689od_c_a @ Prev2 @ Prev3 ) @ Q @ HF @ VisitedStates @ K3 ) )
     => ? [P10: list_P6327159017948738492od_c_a] :
          ( P9
          = ( append3884164603879732689od_c_a @ Prev2 @ P10 ) ) ) ).

% acyclic_paths_up_to_length'_prev
thf(fact_967_lexord__Nil__right,axiom,
    ! [X3: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ~ ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) @ ( lexord8026273315949980969od_c_a @ R ) ) ).

% lexord_Nil_right
thf(fact_968_lexord__append__leftI,axiom,
    ! [U2: list_P6327159017948738492od_c_a,V2: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a,X3: list_P6327159017948738492od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ U2 @ V2 ) @ ( lexord8026273315949980969od_c_a @ R ) )
     => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ X3 @ U2 ) @ ( append3884164603879732689od_c_a @ X3 @ V2 ) ) @ ( lexord8026273315949980969od_c_a @ R ) ) ) ).

% lexord_append_leftI
thf(fact_969_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_P6327159017948738492od_c_a] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s3386368156187063848od_c_a @ Xs ) )
      = ( ? [X4: produc2669932201312201782od_c_a,Ys3: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( cons_P8659537020926305772od_c_a @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s3386368156187063848od_c_a @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_970_take__le,axiom,
    ! [I: nat,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ord_less_eq_nat @ I @ ( size_s3386368156187063848od_c_a @ Xs ) )
     => ( ( take_P3357827036823063076od_c_a @ I @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
        = ( take_P3357827036823063076od_c_a @ I @ Xs ) ) ) ).

% take_le
thf(fact_971_lexord__partial__trans,axiom,
    ! [Xs: list_a,R: set_Product_prod_a_a,Ys: list_a,Zs: list_a] :
      ( ! [X: a,Y: a,Z3: a] :
          ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z3 ) @ R )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z3 ) @ R ) ) ) )
     => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lexord_a @ R ) )
       => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lexord_a @ R ) )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs ) @ ( lexord_a @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_972_lexord__partial__trans,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ! [X: produc2669932201312201782od_c_a,Y: produc2669932201312201782od_c_a,Z3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
         => ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X @ Y ) @ R )
           => ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ Y @ Z3 ) @ R )
             => ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X @ Z3 ) @ R ) ) ) )
     => ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( lexord8026273315949980969od_c_a @ R ) )
       => ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Ys @ Zs ) @ ( lexord8026273315949980969od_c_a @ R ) )
         => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Zs ) @ ( lexord8026273315949980969od_c_a @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_973_lexord__append__leftD,axiom,
    ! [X3: list_P6327159017948738492od_c_a,U2: list_P6327159017948738492od_c_a,V2: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ X3 @ U2 ) @ ( append3884164603879732689od_c_a @ X3 @ V2 ) ) @ ( lexord8026273315949980969od_c_a @ R ) )
     => ( ! [A4: produc2669932201312201782od_c_a] :
            ~ ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A4 @ A4 ) @ R )
       => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ U2 @ V2 ) @ ( lexord8026273315949980969od_c_a @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_974_lexord__append__rightI,axiom,
    ! [Y3: list_P6327159017948738492od_c_a,X3: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ? [B6: produc2669932201312201782od_c_a,Z5: list_P6327159017948738492od_c_a] :
          ( Y3
          = ( cons_P8659537020926305772od_c_a @ B6 @ Z5 ) )
     => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ X3 @ ( append3884164603879732689od_c_a @ X3 @ Y3 ) ) @ ( lexord8026273315949980969od_c_a @ R ) ) ) ).

% lexord_append_rightI
thf(fact_975_lexord__sufE,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Qs: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Zs ) @ ( append3884164603879732689od_c_a @ Ys @ Qs ) ) @ ( lexord8026273315949980969od_c_a @ R ) )
     => ( ( Xs != Ys )
       => ( ( ( size_s3386368156187063848od_c_a @ Xs )
            = ( size_s3386368156187063848od_c_a @ Ys ) )
         => ( ( ( size_s3386368156187063848od_c_a @ Zs )
              = ( size_s3386368156187063848od_c_a @ Qs ) )
           => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Xs @ Ys ) @ ( lexord8026273315949980969od_c_a @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_976_append__eq__append__conv__if,axiom,
    ! [Xs_1: list_P6327159017948738492od_c_a,Xs_2: list_P6327159017948738492od_c_a,Ys_1: list_P6327159017948738492od_c_a,Ys_2: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Xs_1 @ Xs_2 )
        = ( append3884164603879732689od_c_a @ Ys_1 @ Ys_2 ) )
      = ( ( ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ Xs_1 ) @ ( size_s3386368156187063848od_c_a @ Ys_1 ) )
         => ( ( Xs_1
              = ( take_P3357827036823063076od_c_a @ ( size_s3386368156187063848od_c_a @ Xs_1 ) @ Ys_1 ) )
            & ( Xs_2
              = ( append3884164603879732689od_c_a @ ( drop_P8434746179008708476od_c_a @ ( size_s3386368156187063848od_c_a @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ Xs_1 ) @ ( size_s3386368156187063848od_c_a @ Ys_1 ) )
         => ( ( ( take_P3357827036823063076od_c_a @ ( size_s3386368156187063848od_c_a @ Ys_1 ) @ Xs_1 )
              = Ys_1 )
            & ( ( append3884164603879732689od_c_a @ ( drop_P8434746179008708476od_c_a @ ( size_s3386368156187063848od_c_a @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
              = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
thf(fact_977_lexord__append__left__rightI,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a,R: set_Pr1454594181397332999od_c_a,U2: list_P6327159017948738492od_c_a,X3: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ B ) @ R )
     => ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ U2 @ ( cons_P8659537020926305772od_c_a @ A @ X3 ) ) @ ( append3884164603879732689od_c_a @ U2 @ ( cons_P8659537020926305772od_c_a @ B @ Y3 ) ) ) @ ( lexord8026273315949980969od_c_a @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_978_lexord__same__pref__iff,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) @ ( append3884164603879732689od_c_a @ Xs @ Zs ) ) @ ( lexord8026273315949980969od_c_a @ R ) )
      = ( ? [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
            & ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X4 @ X4 ) @ R ) )
        | ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Ys @ Zs ) @ ( lexord8026273315949980969od_c_a @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_979_cyclic__path__pumping,axiom,
    ! [M: fsm_a_b_c,P5: list_P6327159017948738492od_c_a,N: nat] :
      ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ P5 )
     => ( ~ ( distinct_a @ ( visited_states_a_b_c @ ( initial_a_b_c @ M ) @ P5 ) )
       => ? [P: list_P6327159017948738492od_c_a] :
            ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ P )
            & ( ord_less_eq_nat @ N @ ( size_s3386368156187063848od_c_a @ P ) ) ) ) ) ).

% cyclic_path_pumping
thf(fact_980_paths__up__to__length__set,axiom,
    ! [Q: a,M: fsm_a_b_c,K3: nat] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( paths_5655032219139660648_a_b_c @ M @ Q @ K3 )
        = ( collec6273869032445462695od_c_a
          @ ^ [P4: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M @ Q @ P4 )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P4 ) @ K3 ) ) ) ) ) ).

% paths_up_to_length_set
thf(fact_981_acyclic__paths__up__to__length_Osimps,axiom,
    ( acycli3978232057192500090_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a,K5: nat] :
          ( collec6273869032445462695od_c_a
          @ ^ [P4: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M2 @ Q6 @ P4 )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P4 ) @ K5 )
              & ( distinct_a @ ( visited_states_a_b_c @ Q6 @ P4 ) ) ) ) ) ) ).

% acyclic_paths_up_to_length.simps
thf(fact_982_acyclic__paths__up__to__length_Oelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( acycli3978232057192500090_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( collec6273869032445462695od_c_a
          @ ^ [P4: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ X3 @ Xa @ P4 )
              & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P4 ) @ Xb )
              & ( distinct_a @ ( visited_states_a_b_c @ Xa @ P4 ) ) ) ) ) ) ).

% acyclic_paths_up_to_length.elims
thf(fact_983_acyclic__paths__up__to__length_Opelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( acycli3978232057192500090_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P540219132740423814_a_nat @ acycli6498804396026743433_a_b_c @ ( produc6635463149712959049_a_nat @ X3 @ ( product_Pair_a_nat @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( collec6273869032445462695od_c_a
                @ ^ [P4: list_P6327159017948738492od_c_a] :
                    ( ( path_a_b_c @ X3 @ Xa @ P4 )
                    & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P4 ) @ Xb )
                    & ( distinct_a @ ( visited_states_a_b_c @ Xa @ P4 ) ) ) ) )
           => ~ ( accp_P540219132740423814_a_nat @ acycli6498804396026743433_a_b_c @ ( produc6635463149712959049_a_nat @ X3 @ ( product_Pair_a_nat @ Xa @ Xb ) ) ) ) ) ) ).

% acyclic_paths_up_to_length.pelims
thf(fact_984_nths__Cons,axiom,
    ! [X3: produc2669932201312201782od_c_a,L: list_P6327159017948738492od_c_a,A3: set_nat] :
      ( ( nths_P7808834103366214970od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ L ) @ A3 )
      = ( append3884164603879732689od_c_a @ ( if_lis1795051829816687490od_c_a @ ( member_nat @ zero_zero_nat @ A3 ) @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) @ nil_Pr1342775757158464060od_c_a )
        @ ( nths_P7808834103366214970od_c_a @ L
          @ ( collect_nat
            @ ^ [J: nat] : ( member_nat @ ( suc @ J ) @ A3 ) ) ) ) ) ).

% nths_Cons
thf(fact_985_paths__up__to__length_H__set,axiom,
    ! [Q: a,M: fsm_a_b_c,Prev2: list_P6327159017948738492od_c_a,K3: nat] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( path_a_b_c @ M @ Q @ Prev2 )
       => ( ( paths_5428161815106503551_a_b_c @ Prev2 @ ( target_a_b_c @ Q @ Prev2 ) @ ( h_a_b_c @ M ) @ ( inputs_a_b_c @ M ) @ K3 )
          = ( collec6273869032445462695od_c_a
            @ ^ [Uu3: list_P6327159017948738492od_c_a] :
              ? [P4: list_P6327159017948738492od_c_a] :
                ( ( Uu3
                  = ( append3884164603879732689od_c_a @ Prev2 @ P4 ) )
                & ( path_a_b_c @ M @ ( target_a_b_c @ Q @ Prev2 ) @ P4 )
                & ( ord_less_eq_nat @ ( size_s3386368156187063848od_c_a @ P4 ) @ K3 ) ) ) ) ) ) ).

% paths_up_to_length'_set
thf(fact_986_nths__nil,axiom,
    ! [A3: set_nat] :
      ( ( nths_P7808834103366214970od_c_a @ nil_Pr1342775757158464060od_c_a @ A3 )
      = nil_Pr1342775757158464060od_c_a ) ).

% nths_nil
thf(fact_987_nths__singleton,axiom,
    ! [A3: set_nat,X3: produc2669932201312201782od_c_a] :
      ( ( ( member_nat @ zero_zero_nat @ A3 )
       => ( ( nths_P7808834103366214970od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) @ A3 )
          = ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) )
      & ( ~ ( member_nat @ zero_zero_nat @ A3 )
       => ( ( nths_P7808834103366214970od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) @ A3 )
          = nil_Pr1342775757158464060od_c_a ) ) ) ).

% nths_singleton
thf(fact_988_fsm__transition__input,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M ) ) ) ).

% fsm_transition_input
thf(fact_989_distinct__nthsI,axiom,
    ! [Xs: list_a,I2: set_nat] :
      ( ( distinct_a @ Xs )
     => ( distinct_a @ ( nths_a @ Xs @ I2 ) ) ) ).

% distinct_nthsI
thf(fact_990_notin__set__nthsI,axiom,
    ! [X3: a,Xs: list_a,I2: set_nat] :
      ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ~ ( member_a @ X3 @ ( set_a2 @ ( nths_a @ Xs @ I2 ) ) ) ) ).

% notin_set_nthsI
thf(fact_991_notin__set__nthsI,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,I2: set_nat] :
      ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( nths_P7808834103366214970od_c_a @ Xs @ I2 ) ) ) ) ).

% notin_set_nthsI
thf(fact_992_in__set__nthsD,axiom,
    ! [X3: a,Xs: list_a,I2: set_nat] :
      ( ( member_a @ X3 @ ( set_a2 @ ( nths_a @ Xs @ I2 ) ) )
     => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_993_in__set__nthsD,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,I2: set_nat] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( nths_P7808834103366214970od_c_a @ Xs @ I2 ) ) )
     => ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_994_paths__up__to__length_Oelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( paths_5655032219139660648_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( paths_5428161815106503551_a_b_c @ nil_Pr1342775757158464060od_c_a @ Xa @ ( h_a_b_c @ X3 ) @ ( inputs_a_b_c @ X3 ) @ Xb ) ) ) ).

% paths_up_to_length.elims
thf(fact_995_paths__up__to__length_Osimps,axiom,
    ( paths_5655032219139660648_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a] : ( paths_5428161815106503551_a_b_c @ nil_Pr1342775757158464060od_c_a @ Q6 @ ( h_a_b_c @ M2 ) @ ( inputs_a_b_c @ M2 ) ) ) ) ).

% paths_up_to_length.simps
thf(fact_996_paths__up__to__length_Opelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( paths_5655032219139660648_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P540219132740423814_a_nat @ paths_134489500739134491_a_b_c @ ( produc6635463149712959049_a_nat @ X3 @ ( product_Pair_a_nat @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( paths_5428161815106503551_a_b_c @ nil_Pr1342775757158464060od_c_a @ Xa @ ( h_a_b_c @ X3 ) @ ( inputs_a_b_c @ X3 ) @ Xb ) )
           => ~ ( accp_P540219132740423814_a_nat @ paths_134489500739134491_a_b_c @ ( produc6635463149712959049_a_nat @ X3 @ ( product_Pair_a_nat @ Xa @ Xb ) ) ) ) ) ) ).

% paths_up_to_length.pelims
thf(fact_997_completely__specified__path__extension,axiom,
    ! [M: fsm_a_b_c,Q: a,P5: list_P6327159017948738492od_c_a,X3: b] :
      ( ( comple6900478119722646120_a_b_c @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( ( path_a_b_c @ M @ Q @ P5 )
         => ( ( member_b @ X3 @ ( inputs_a_b_c @ M ) )
           => ~ ! [T2: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ M ) )
                 => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) )
                      = X3 )
                   => ( ( produc4225943634317630866od_c_a @ T2 )
                     != ( target_a_b_c @ Q @ P5 ) ) ) ) ) ) ) ) ).

% completely_specified_path_extension
thf(fact_998_completely__specified__alt__def,axiom,
    ( comple3114736741132006991_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b] :
        ! [X4: product_prod_c_a] :
          ( ( member5074992350434858958od_c_a @ X4 @ ( states367398909958982703_a_a_b @ M2 ) )
         => ! [Y4: a] :
              ( ( member_a @ Y4 @ ( inputs8636541066849307272_a_a_b @ M2 ) )
             => ? [Q7: product_prod_c_a,Z2: b] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ X4 @ ( produc5762801557034676134od_c_a @ Y4 @ ( produc3440865914000881661od_c_a @ Z2 @ Q7 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ) ) ) ).

% completely_specified_alt_def
thf(fact_999_completely__specified__alt__def,axiom,
    ( comple6900478119722646120_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X4: a] :
          ( ( member_a @ X4 @ ( states_a_b_c @ M2 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs_a_b_c @ M2 ) )
             => ? [Q7: a,Z2: c] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ ( product_Pair_c_a @ Z2 @ Q7 ) ) ) @ ( transitions_a_b_c @ M2 ) ) ) ) ) ) ).

% completely_specified_alt_def
thf(fact_1000_completely__specified_Oelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( comple6900478119722646120_a_b_c @ X3 )
     => ~ ! [X: a] :
            ( ( member_a @ X @ ( states_a_b_c @ X3 ) )
           => ! [Xa2: b] :
                ( ( member_b @ Xa2 @ ( inputs_a_b_c @ X3 ) )
               => ? [Xb2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xb2 @ ( transitions_a_b_c @ X3 ) )
                    & ( ( produc4225943634317630866od_c_a @ Xb2 )
                      = X )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) )
                      = Xa2 ) ) ) ) ) ).

% completely_specified.elims(3)
thf(fact_1001_completely__specified_Oelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( comple6900478119722646120_a_b_c @ X3 )
     => ! [X6: a] :
          ( ( member_a @ X6 @ ( states_a_b_c @ X3 ) )
         => ! [Xa3: b] :
              ( ( member_b @ Xa3 @ ( inputs_a_b_c @ X3 ) )
             => ? [Xb3: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Xb3 @ ( transitions_a_b_c @ X3 ) )
                  & ( ( produc4225943634317630866od_c_a @ Xb3 )
                    = X6 )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb3 ) )
                    = Xa3 ) ) ) ) ) ).

% completely_specified.elims(2)
thf(fact_1002_completely__specified_Oelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( comple6900478119722646120_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X4: a] :
              ( ( member_a @ X4 @ ( states_a_b_c @ X3 ) )
             => ! [Y4: b] :
                  ( ( member_b @ Y4 @ ( inputs_a_b_c @ X3 ) )
                 => ? [Z2: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Z2 @ ( transitions_a_b_c @ X3 ) )
                      & ( ( produc4225943634317630866od_c_a @ Z2 )
                        = X4 )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z2 ) )
                        = Y4 ) ) ) ) ) ) ) ).

% completely_specified.elims(1)
thf(fact_1003_completely__specified_Osimps,axiom,
    ( comple6900478119722646120_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X4: a] :
          ( ( member_a @ X4 @ ( states_a_b_c @ M2 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs_a_b_c @ M2 ) )
             => ? [Z2: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Z2 @ ( transitions_a_b_c @ M2 ) )
                  & ( ( produc4225943634317630866od_c_a @ Z2 )
                    = X4 )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z2 ) )
                    = Y4 ) ) ) ) ) ) ).

% completely_specified.simps
thf(fact_1004_completely__specified__language__extension,axiom,
    ! [M: fsm_a_1173558889237794545od_c_a,Q: a,Io: list_P6327159017948738492od_c_a,X3: a] :
      ( ( comple4016279332040951990od_c_a @ M )
     => ( ( member_a @ Q @ ( states6037215131715960470od_c_a @ M ) )
       => ( ( member7410604586820865893od_c_a @ Io @ ( lS_a_a3505603215192706865od_c_a @ M @ Q ) )
         => ( ( member_a @ X3 @ ( inputs1662344551406107631od_c_a @ M ) )
           => ~ ! [Y: produc2133359284458116493od_c_a] :
                  ~ ( member7410604586820865893od_c_a @ ( append3884164603879732689od_c_a @ Io @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y ) @ nil_Pr1342775757158464060od_c_a ) ) @ ( lS_a_a3505603215192706865od_c_a @ M @ Q ) ) ) ) ) ) ).

% completely_specified_language_extension
thf(fact_1005_completely__specified__language__extension,axiom,
    ! [M: fsm_Pr4777319640450098473od_c_a,Q: produc2669932201312201782od_c_a,Io: list_P6327159017948738492od_c_a,X3: a] :
      ( ( comple6579661845503832814od_c_a @ M )
     => ( ( member5502040268208861279od_c_a @ Q @ ( states4150737990171199694od_c_a @ M ) )
       => ( ( member7410604586820865893od_c_a @ Io @ ( lS_Pro6861894141136986217od_c_a @ M @ Q ) )
         => ( ( member_a @ X3 @ ( inputs4712173150264771367od_c_a @ M ) )
           => ~ ! [Y: produc2133359284458116493od_c_a] :
                  ~ ( member7410604586820865893od_c_a @ ( append3884164603879732689od_c_a @ Io @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y ) @ nil_Pr1342775757158464060od_c_a ) ) @ ( lS_Pro6861894141136986217od_c_a @ M @ Q ) ) ) ) ) ) ).

% completely_specified_language_extension
thf(fact_1006_completely__specified__language__extension,axiom,
    ! [M: fsm_a_4970532336856918856od_c_a,Q: a,Io: list_P5767727266746070163od_c_a,X3: b] :
      ( ( comple6979930233664037133od_c_a @ M )
     => ( ( member_a @ Q @ ( states4232592402491012845od_c_a @ M ) )
       => ( ( member2650335117833904444od_c_a @ Io @ ( lS_a_b8979870102065257096od_c_a @ M @ Q ) )
         => ( ( member_b @ X3 @ ( inputs3278362522526561606od_c_a @ M ) )
           => ~ ! [Y: product_prod_c_a] :
                  ~ ( member2650335117833904444od_c_a @ ( append2814445079200870952od_c_a @ Io @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y ) @ nil_Pr5380409136740620691od_c_a ) ) @ ( lS_a_b8979870102065257096od_c_a @ M @ Q ) ) ) ) ) ) ).

% completely_specified_language_extension
thf(fact_1007_completely__specified__language__extension,axiom,
    ! [M: fsm_Pr7194325094602570112od_c_a,Q: produc2669932201312201782od_c_a,Io: list_P5767727266746070163od_c_a,X3: b] :
      ( ( comple146904887516201797od_c_a @ M )
     => ( ( member5502040268208861279od_c_a @ Q @ ( states3181181425523751205od_c_a @ M ) )
       => ( ( member2650335117833904444od_c_a @ Io @ ( lS_Pro6466918861714006976od_c_a @ M @ Q ) )
         => ( ( member_b @ X3 @ ( inputs4481568158855885950od_c_a @ M ) )
           => ~ ! [Y: product_prod_c_a] :
                  ~ ( member2650335117833904444od_c_a @ ( append2814445079200870952od_c_a @ Io @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y ) @ nil_Pr5380409136740620691od_c_a ) ) @ ( lS_Pro6466918861714006976od_c_a @ M @ Q ) ) ) ) ) ) ).

% completely_specified_language_extension
thf(fact_1008_completely__specified__language__extension,axiom,
    ! [M: fsm_a_c_a,Q: a,Io: list_P125642481956313003od_c_a,X3: c] :
      ( ( comple4113022537749970085_a_c_a @ M )
     => ( ( member_a @ Q @ ( states_a_c_a @ M ) )
       => ( ( member5552703068553123156od_c_a @ Io @ ( lS_a_c_a @ M @ Q ) )
         => ( ( member_c @ X3 @ ( inputs_a_c_a @ M ) )
           => ~ ! [Y: a] :
                  ~ ( member5552703068553123156od_c_a @ ( append8983669691956257088od_c_a @ Io @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X3 @ Y ) @ nil_Product_prod_c_a ) ) @ ( lS_a_c_a @ M @ Q ) ) ) ) ) ) ).

% completely_specified_language_extension
thf(fact_1009_completely__specified__language__extension,axiom,
    ! [M: fsm_Pr3637748715322196120_a_c_a,Q: produc2669932201312201782od_c_a,Io: list_P125642481956313003od_c_a,X3: c] :
      ( ( comple2527714743810021085_a_c_a @ M )
     => ( ( member5502040268208861279od_c_a @ Q @ ( states7331553577359322301_a_c_a @ M ) )
       => ( ( member5552703068553123156od_c_a @ Io @ ( lS_Pro944437309439289944_a_c_a @ M @ Q ) )
         => ( ( member_c @ X3 @ ( inputs3975702456018823446_a_c_a @ M ) )
           => ~ ! [Y: a] :
                  ~ ( member5552703068553123156od_c_a @ ( append8983669691956257088od_c_a @ Io @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X3 @ Y ) @ nil_Product_prod_c_a ) ) @ ( lS_Pro944437309439289944_a_c_a @ M @ Q ) ) ) ) ) ) ).

% completely_specified_language_extension
thf(fact_1010_completely__specified_Opelims_I3_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ~ ( comple6900478119722646120_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X3 )
           => ! [X: a] :
                ( ( member_a @ X @ ( states_a_b_c @ X3 ) )
               => ! [Xa2: b] :
                    ( ( member_b @ Xa2 @ ( inputs_a_b_c @ X3 ) )
                   => ? [Xb2: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Xb2 @ ( transitions_a_b_c @ X3 ) )
                        & ( ( produc4225943634317630866od_c_a @ Xb2 )
                          = X )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) )
                          = Xa2 ) ) ) ) ) ) ) ).

% completely_specified.pelims(3)
thf(fact_1011_completely__specified_Opelims_I2_J,axiom,
    ! [X3: fsm_a_b_c] :
      ( ( comple6900478119722646120_a_b_c @ X3 )
     => ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X3 )
       => ~ ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X3 )
           => ~ ! [X6: a] :
                  ( ( member_a @ X6 @ ( states_a_b_c @ X3 ) )
                 => ! [Xa3: b] :
                      ( ( member_b @ Xa3 @ ( inputs_a_b_c @ X3 ) )
                     => ? [Xb3: produc2669932201312201782od_c_a] :
                          ( ( member5502040268208861279od_c_a @ Xb3 @ ( transitions_a_b_c @ X3 ) )
                          & ( ( produc4225943634317630866od_c_a @ Xb3 )
                            = X6 )
                          & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb3 ) )
                            = Xa3 ) ) ) ) ) ) ) ).

% completely_specified.pelims(2)
thf(fact_1012_completely__specified_Opelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Y3: $o] :
      ( ( ( comple6900478119722646120_a_b_c @ X3 )
        = Y3 )
     => ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X3 )
       => ~ ( ( Y3
              = ( ! [X4: a] :
                    ( ( member_a @ X4 @ ( states_a_b_c @ X3 ) )
                   => ! [Y4: b] :
                        ( ( member_b @ Y4 @ ( inputs_a_b_c @ X3 ) )
                       => ? [Z2: produc2669932201312201782od_c_a] :
                            ( ( member5502040268208861279od_c_a @ Z2 @ ( transitions_a_b_c @ X3 ) )
                            & ( ( produc4225943634317630866od_c_a @ Z2 )
                              = X4 )
                            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z2 ) )
                              = Y4 ) ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X3 ) ) ) ) ).

% completely_specified.pelims(1)
thf(fact_1013_completely__specified__state_Osimps,axiom,
    ( comple2039557650379672579_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a] :
        ! [X4: b] :
          ( ( member_b @ X4 @ ( inputs_a_b_c @ M2 ) )
         => ? [Y4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ M2 ) )
              & ( ( produc4225943634317630866od_c_a @ Y4 )
                = Q6 )
              & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) )
                = X4 ) ) ) ) ) ).

% completely_specified_state.simps
thf(fact_1014_completely__specified__state_Oelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Y3: $o] :
      ( ( ( comple2039557650379672579_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( Y3
        = ( ! [X4: b] :
              ( ( member_b @ X4 @ ( inputs_a_b_c @ X3 ) )
             => ? [Y4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X3 ) )
                  & ( ( produc4225943634317630866od_c_a @ Y4 )
                    = Xa )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) )
                    = X4 ) ) ) ) ) ) ).

% completely_specified_state.elims(1)
thf(fact_1015_completely__specified__state_Oelims_I2_J,axiom,
    ! [X3: fsm_a_b_c,Xa: a] :
      ( ( comple2039557650379672579_a_b_c @ X3 @ Xa )
     => ! [X6: b] :
          ( ( member_b @ X6 @ ( inputs_a_b_c @ X3 ) )
         => ? [Xa2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X3 ) )
              & ( ( produc4225943634317630866od_c_a @ Xa2 )
                = Xa )
              & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) )
                = X6 ) ) ) ) ).

% completely_specified_state.elims(2)
thf(fact_1016_completely__specified__state_Oelims_I3_J,axiom,
    ! [X3: fsm_a_b_c,Xa: a] :
      ( ~ ( comple2039557650379672579_a_b_c @ X3 @ Xa )
     => ~ ! [X: b] :
            ( ( member_b @ X @ ( inputs_a_b_c @ X3 ) )
           => ? [Xa3: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X3 ) )
                & ( ( produc4225943634317630866od_c_a @ Xa3 )
                  = Xa )
                & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) )
                  = X ) ) ) ) ).

% completely_specified_state.elims(3)
thf(fact_1017_completely__specified__state_Opelims_I1_J,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Y3: $o] :
      ( ( ( comple2039557650379672579_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) )
       => ~ ( ( Y3
              = ( ! [X4: b] :
                    ( ( member_b @ X4 @ ( inputs_a_b_c @ X3 ) )
                   => ? [Y4: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X3 ) )
                        & ( ( produc4225943634317630866od_c_a @ Y4 )
                          = Xa )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) )
                          = X4 ) ) ) ) )
           => ~ ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) ) ) ) ) ).

% completely_specified_state.pelims(1)
thf(fact_1018_completely__specified__state_Opelims_I2_J,axiom,
    ! [X3: fsm_a_b_c,Xa: a] :
      ( ( comple2039557650379672579_a_b_c @ X3 @ Xa )
     => ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) )
       => ~ ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) )
           => ~ ! [X6: b] :
                  ( ( member_b @ X6 @ ( inputs_a_b_c @ X3 ) )
                 => ? [Xa2: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X3 ) )
                      & ( ( produc4225943634317630866od_c_a @ Xa2 )
                        = Xa )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) )
                        = X6 ) ) ) ) ) ) ).

% completely_specified_state.pelims(2)
thf(fact_1019_completely__specified__state_Opelims_I3_J,axiom,
    ! [X3: fsm_a_b_c,Xa: a] :
      ( ~ ( comple2039557650379672579_a_b_c @ X3 @ Xa )
     => ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) )
       => ~ ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X3 @ Xa ) )
           => ! [X: b] :
                ( ( member_b @ X @ ( inputs_a_b_c @ X3 ) )
               => ? [Xa3: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X3 ) )
                    & ( ( produc4225943634317630866od_c_a @ Xa3 )
                      = Xa )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) )
                      = X ) ) ) ) ) ) ).

% completely_specified_state.pelims(3)
thf(fact_1020_path__of__length__ex,axiom,
    ! [M: fsm_a_b_c,Q: a,K3: nat] :
      ( ( comple6900478119722646120_a_b_c @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( ( ( inputs_a_b_c @ M )
           != bot_bot_set_b )
         => ? [P: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M @ Q @ P )
              & ( ( size_s3386368156187063848od_c_a @ P )
                = K3 ) ) ) ) ) ).

% path_of_length_ex
thf(fact_1021_paths__of__length_Opelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( paths_2034914492968346983_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P540219132740423814_a_nat @ paths_6980869764649105628_a_b_c @ ( produc6635463149712959049_a_nat @ X3 @ ( product_Pair_a_nat @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( paths_2590340026247609854_a_b_c @ nil_Pr1342775757158464060od_c_a @ Xa @ ( h_a_b_c @ X3 ) @ ( inputs_a_b_c @ X3 ) @ Xb ) )
           => ~ ( accp_P540219132740423814_a_nat @ paths_6980869764649105628_a_b_c @ ( produc6635463149712959049_a_nat @ X3 @ ( product_Pair_a_nat @ Xa @ Xb ) ) ) ) ) ) ).

% paths_of_length.pelims
thf(fact_1022_nths__empty,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( nths_P7808834103366214970od_c_a @ Xs @ bot_bot_set_nat )
      = nil_Pr1342775757158464060od_c_a ) ).

% nths_empty
thf(fact_1023_set__empty,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( ( set_Pr5664641634098845387od_c_a @ Xs )
        = bot_bo267491471446728962od_c_a )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% set_empty
thf(fact_1024_set__empty2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( bot_bo267491471446728962od_c_a
        = ( set_Pr5664641634098845387od_c_a @ Xs ) )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% set_empty2
thf(fact_1025_empty__set,axiom,
    ( bot_bo267491471446728962od_c_a
    = ( set_Pr5664641634098845387od_c_a @ nil_Pr1342775757158464060od_c_a ) ) ).

% empty_set
thf(fact_1026_paths__of__length_Osimps,axiom,
    ( paths_2034914492968346983_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a] : ( paths_2590340026247609854_a_b_c @ nil_Pr1342775757158464060od_c_a @ Q6 @ ( h_a_b_c @ M2 ) @ ( inputs_a_b_c @ M2 ) ) ) ) ).

% paths_of_length.simps
thf(fact_1027_paths__of__length_Oelims,axiom,
    ! [X3: fsm_a_b_c,Xa: a,Xb: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( paths_2034914492968346983_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( paths_2590340026247609854_a_b_c @ nil_Pr1342775757158464060od_c_a @ Xa @ ( h_a_b_c @ X3 ) @ ( inputs_a_b_c @ X3 ) @ Xb ) ) ) ).

% paths_of_length.elims
thf(fact_1028_output__complete__alt__def__h,axiom,
    ( output5131981854805885794_c_a_c
    = ( ^ [M2: fsm_a_836089231483141141_c_a_c] :
        ! [Q6: a,X4: produc2133359284458116493od_c_a] :
          ( ( ( h_a_Pr5118833440477893660_c_a_c @ M2 @ ( produc5762801557034676134od_c_a @ Q6 @ X4 ) )
           != bot_bo2086078286244720881od_c_a )
         => ! [Y4: c] :
              ( ( member_c @ Y4 @ ( output7059084347266203474_c_a_c @ M2 ) )
             => ? [Q7: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ Y4 @ Q7 ) @ ( h_a_Pr5118833440477893660_c_a_c @ M2 @ ( produc5762801557034676134od_c_a @ Q6 @ X4 ) ) ) ) ) ) ) ).

% output_complete_alt_def_h
thf(fact_1029_language__empty__io,axiom,
    ! [M: fsm_a_b_c] :
      ( ( ( ( inputs_a_b_c @ M )
          = bot_bot_set_b )
        | ( ( outputs_a_b_c @ M )
          = bot_bot_set_c ) )
     => ( ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) )
        = ( insert6227932334100060350od_b_c @ nil_Product_prod_b_c @ bot_bo4166481423041325370od_b_c ) ) ) ).

% language_empty_io
thf(fact_1030_distinct__concat,axiom,
    ! [Xs: list_list_a] :
      ( ( distinct_list_a @ Xs )
     => ( ! [Ys2: list_a] :
            ( ( member_list_a @ Ys2 @ ( set_list_a2 @ Xs ) )
           => ( distinct_a @ Ys2 ) )
       => ( ! [Ys2: list_a,Zs2: list_a] :
              ( ( member_list_a @ Ys2 @ ( set_list_a2 @ Xs ) )
             => ( ( member_list_a @ Zs2 @ ( set_list_a2 @ Xs ) )
               => ( ( Ys2 != Zs2 )
                 => ( ( inf_inf_set_a @ ( set_a2 @ Ys2 ) @ ( set_a2 @ Zs2 ) )
                    = bot_bot_set_a ) ) ) )
         => ( distinct_a @ ( concat_a @ Xs ) ) ) ) ) ).

% distinct_concat
thf(fact_1031_list_Osimps_I15_J,axiom,
    ! [X21: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a] :
      ( ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) )
      = ( insert8105206408023001606od_c_a @ X21 @ ( set_Pr5664641634098845387od_c_a @ X22 ) ) ) ).

% list.simps(15)
thf(fact_1032_distinct__append,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( distin811446047591323055od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
      = ( ( distin811446047591323055od_c_a @ Xs )
        & ( distin811446047591323055od_c_a @ Ys )
        & ( ( inf_in3815988127369930372od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs ) @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
          = bot_bo267491471446728962od_c_a ) ) ) ).

% distinct_append
thf(fact_1033_distinct__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( distinct_a @ ( append_a @ Xs @ Ys ) )
      = ( ( distinct_a @ Xs )
        & ( distinct_a @ Ys )
        & ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
          = bot_bot_set_a ) ) ) ).

% distinct_append
thf(fact_1034_prefixes__set__Cons__insert,axiom,
    ! [W4: list_P6327159017948738492od_c_a,Xy2: produc2669932201312201782od_c_a] :
      ( ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ ( append3884164603879732689od_c_a @ W4 @ ( cons_P8659537020926305772od_c_a @ Xy2 @ nil_Pr1342775757158464060od_c_a ) ) ) )
      = ( insert4789241225314331020od_c_a @ ( append3884164603879732689od_c_a @ W4 @ ( cons_P8659537020926305772od_c_a @ Xy2 @ nil_Pr1342775757158464060od_c_a ) ) @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ W4 ) ) ) ) ).

% prefixes_set_Cons_insert
thf(fact_1035_set__take__disj__set__drop__if__distinct,axiom,
    ! [Vs: list_a,I: nat,J2: nat] :
      ( ( distinct_a @ Vs )
     => ( ( ord_less_eq_nat @ I @ J2 )
       => ( ( inf_inf_set_a @ ( set_a2 @ ( take_a @ I @ Vs ) ) @ ( set_a2 @ ( drop_a @ J2 @ Vs ) ) )
          = bot_bot_set_a ) ) ) ).

% set_take_disj_set_drop_if_distinct
thf(fact_1036_distinct__concat__iff,axiom,
    ! [Xs: list_list_a] :
      ( ( distinct_a @ ( concat_a @ Xs ) )
      = ( ( distinct_list_a @ ( removeAll_list_a @ nil_a @ Xs ) )
        & ! [Ys3: list_a] :
            ( ( member_list_a @ Ys3 @ ( set_list_a2 @ Xs ) )
           => ( distinct_a @ Ys3 ) )
        & ! [Ys3: list_a,Zs3: list_a] :
            ( ( ( member_list_a @ Ys3 @ ( set_list_a2 @ Xs ) )
              & ( member_list_a @ Zs3 @ ( set_list_a2 @ Xs ) )
              & ( Ys3 != Zs3 ) )
           => ( ( inf_inf_set_a @ ( set_a2 @ Ys3 ) @ ( set_a2 @ Zs3 ) )
              = bot_bot_set_a ) ) ) ) ).

% distinct_concat_iff
thf(fact_1037_distinct__concat__iff,axiom,
    ! [Xs: list_l2826717958560394946od_c_a] :
      ( ( distin811446047591323055od_c_a @ ( concat7533993459791330935od_c_a @ Xs ) )
      = ( ( distin5812642747228818613od_c_a @ ( remove7772166321875213256od_c_a @ nil_Pr1342775757158464060od_c_a @ Xs ) )
        & ! [Ys3: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ Ys3 @ ( set_li8663184538779258833od_c_a @ Xs ) )
           => ( distin811446047591323055od_c_a @ Ys3 ) )
        & ! [Ys3: list_P6327159017948738492od_c_a,Zs3: list_P6327159017948738492od_c_a] :
            ( ( ( member7410604586820865893od_c_a @ Ys3 @ ( set_li8663184538779258833od_c_a @ Xs ) )
              & ( member7410604586820865893od_c_a @ Zs3 @ ( set_li8663184538779258833od_c_a @ Xs ) )
              & ( Ys3 != Zs3 ) )
           => ( ( inf_in3815988127369930372od_c_a @ ( set_Pr5664641634098845387od_c_a @ Ys3 ) @ ( set_Pr5664641634098845387od_c_a @ Zs3 ) )
              = bot_bo267491471446728962od_c_a ) ) ) ) ).

% distinct_concat_iff
thf(fact_1038_removeAll__id,axiom,
    ! [X3: a,Xs: list_a] :
      ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ( removeAll_a @ X3 @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1039_removeAll__id,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( remove4874121529251957314od_c_a @ X3 @ Xs )
        = Xs ) ) ).

% removeAll_id
thf(fact_1040_removeAll__append,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( remove4874121529251957314od_c_a @ X3 @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
      = ( append3884164603879732689od_c_a @ ( remove4874121529251957314od_c_a @ X3 @ Xs ) @ ( remove4874121529251957314od_c_a @ X3 @ Ys ) ) ) ).

% removeAll_append
thf(fact_1041_distinct__removeAll,axiom,
    ! [Xs: list_a,X3: a] :
      ( ( distinct_a @ Xs )
     => ( distinct_a @ ( removeAll_a @ X3 @ Xs ) ) ) ).

% distinct_removeAll
thf(fact_1042_removeAll_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Y3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( X3 = Y3 )
       => ( ( remove4874121529251957314od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ Y3 @ Xs ) )
          = ( remove4874121529251957314od_c_a @ X3 @ Xs ) ) )
      & ( ( X3 != Y3 )
       => ( ( remove4874121529251957314od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ Y3 @ Xs ) )
          = ( cons_P8659537020926305772od_c_a @ Y3 @ ( remove4874121529251957314od_c_a @ X3 @ Xs ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_1043_removeAll_Osimps_I1_J,axiom,
    ! [X3: produc2669932201312201782od_c_a] :
      ( ( remove4874121529251957314od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% removeAll.simps(1)
thf(fact_1044_acyclic__paths__up__to__length_H_Oelims,axiom,
    ! [X3: list_P1562850763667305982od_c_a,Xa: product_prod_c_a,Xb: product_prod_c_a > set_Pr8357624050790843030od_c_a,Xc: set_Product_prod_c_a,Xd: nat,Y3: set_li4189853136481112030od_c_a] :
      ( ( ( acycli2757849055487800696_a_a_b @ X3 @ Xa @ Xb @ Xc @ Xd )
        = Y3 )
     => ( ( ( Xd = zero_zero_nat )
         => ( Y3
           != ( insert471206331724843982od_c_a @ X3 @ bot_bo4204560030348456010od_c_a ) ) )
       => ~ ! [K: nat] :
              ( ( Xd
                = ( suc @ K ) )
             => ( Y3
               != ( insert471206331724843982od_c_a @ X3
                  @ ( comple4314218821256468939od_c_a
                    @ ( image_5093262280946859181od_c_a
                      @ ( produc7762011161211578723od_c_a
                        @ ^ [X4: a] :
                            ( produc4720993599074471756od_c_a
                            @ ^ [Y4: b,Q7: product_prod_c_a] : ( acycli2757849055487800696_a_a_b @ ( append311360319448987923od_c_a @ X3 @ ( cons_P5525787621418763054od_c_a @ ( produc7911353075612250984od_c_a @ Xa @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ nil_Pr7913020553331505022od_c_a ) ) @ Q7 @ Xb @ ( insert8183397254950580085od_c_a @ Q7 @ Xc ) @ K ) ) )
                      @ ( filter617373377718276967od_c_a
                        @ ( produc1940289858450342685_c_a_o
                          @ ^ [X4: a] :
                              ( produc6211283585785106182_c_a_o
                              @ ^ [Y4: b,Q7: product_prod_c_a] :
                                  ~ ( member5074992350434858958od_c_a @ Q7 @ Xc ) ) )
                        @ ( Xb @ Xa ) ) ) ) ) ) ) ) ) ).

% acyclic_paths_up_to_length'.elims
thf(fact_1045_acyclic__paths__up__to__length_H_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xa: a,Xb: a > set_Pr272002107461315949od_c_a,Xc: set_a,Xd: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( acycli8838441729103940241_a_b_c @ X3 @ Xa @ Xb @ Xc @ Xd )
        = Y3 )
     => ( ( ( Xd = zero_zero_nat )
         => ( Y3
           != ( insert4789241225314331020od_c_a @ X3 @ bot_bo6236370880139903240od_c_a ) ) )
       => ~ ! [K: nat] :
              ( ( Xd
                = ( suc @ K ) )
             => ( Y3
               != ( insert4789241225314331020od_c_a @ X3
                  @ ( comple2269964827201748105od_c_a
                    @ ( image_259459118490653780od_c_a
                      @ ( produc4758774801380361226od_c_a
                        @ ^ [X4: b] :
                            ( produc4676147736864820850od_c_a
                            @ ^ [Y4: c,Q7: a] : ( acycli8838441729103940241_a_b_c @ ( append3884164603879732689od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ Xa @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ nil_Pr1342775757158464060od_c_a ) ) @ Q7 @ Xb @ ( insert_a2 @ Q7 @ Xc ) @ K ) ) )
                      @ ( filter5629755252890391998od_c_a
                        @ ( produc6211283585785106182_c_a_o
                          @ ^ [X4: b] :
                              ( produc8907516716866730350_c_a_o
                              @ ^ [Y4: c,Q7: a] :
                                  ~ ( member_a @ Q7 @ Xc ) ) )
                        @ ( Xb @ Xa ) ) ) ) ) ) ) ) ) ).

% acyclic_paths_up_to_length'.elims
thf(fact_1046_listset_Osimps_I1_J,axiom,
    ( ( listse6082093074526806811od_c_a @ nil_se4123578418339231900od_c_a )
    = ( insert4789241225314331020od_c_a @ nil_Pr1342775757158464060od_c_a @ bot_bo6236370880139903240od_c_a ) ) ).

% listset.simps(1)
thf(fact_1047_acyclic__paths__up__to__length_H_Osimps_I2_J,axiom,
    ! [Prev2: list_P1562850763667305982od_c_a,Q: product_prod_c_a,HF: product_prod_c_a > set_Pr8357624050790843030od_c_a,VisitedStates: set_Product_prod_c_a,K3: nat] :
      ( ( acycli2757849055487800696_a_a_b @ Prev2 @ Q @ HF @ VisitedStates @ ( suc @ K3 ) )
      = ( insert471206331724843982od_c_a @ Prev2
        @ ( comple4314218821256468939od_c_a
          @ ( image_5093262280946859181od_c_a
            @ ( produc7762011161211578723od_c_a
              @ ^ [X4: a] :
                  ( produc4720993599074471756od_c_a
                  @ ^ [Y4: b,Q7: product_prod_c_a] : ( acycli2757849055487800696_a_a_b @ ( append311360319448987923od_c_a @ Prev2 @ ( cons_P5525787621418763054od_c_a @ ( produc7911353075612250984od_c_a @ Q @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ nil_Pr7913020553331505022od_c_a ) ) @ Q7 @ HF @ ( insert8183397254950580085od_c_a @ Q7 @ VisitedStates ) @ K3 ) ) )
            @ ( filter617373377718276967od_c_a
              @ ( produc1940289858450342685_c_a_o
                @ ^ [X4: a] :
                    ( produc6211283585785106182_c_a_o
                    @ ^ [Y4: b,Q7: product_prod_c_a] :
                        ~ ( member5074992350434858958od_c_a @ Q7 @ VisitedStates ) ) )
              @ ( HF @ Q ) ) ) ) ) ) ).

% acyclic_paths_up_to_length'.simps(2)
thf(fact_1048_acyclic__paths__up__to__length_H_Osimps_I2_J,axiom,
    ! [Prev2: list_P6327159017948738492od_c_a,Q: a,HF: a > set_Pr272002107461315949od_c_a,VisitedStates: set_a,K3: nat] :
      ( ( acycli8838441729103940241_a_b_c @ Prev2 @ Q @ HF @ VisitedStates @ ( suc @ K3 ) )
      = ( insert4789241225314331020od_c_a @ Prev2
        @ ( comple2269964827201748105od_c_a
          @ ( image_259459118490653780od_c_a
            @ ( produc4758774801380361226od_c_a
              @ ^ [X4: b] :
                  ( produc4676147736864820850od_c_a
                  @ ^ [Y4: c,Q7: a] : ( acycli8838441729103940241_a_b_c @ ( append3884164603879732689od_c_a @ Prev2 @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ nil_Pr1342775757158464060od_c_a ) ) @ Q7 @ HF @ ( insert_a2 @ Q7 @ VisitedStates ) @ K3 ) ) )
            @ ( filter5629755252890391998od_c_a
              @ ( produc6211283585785106182_c_a_o
                @ ^ [X4: b] :
                    ( produc8907516716866730350_c_a_o
                    @ ^ [Y4: c,Q7: a] :
                        ~ ( member_a @ Q7 @ VisitedStates ) ) )
              @ ( HF @ Q ) ) ) ) ) ) ).

% acyclic_paths_up_to_length'.simps(2)
thf(fact_1049_pair__imageI,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A3: set_Pr8357624050790843030od_c_a,F2: a > produc2133359284458116493od_c_a > a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ A3 )
     => ( member_a @ ( F2 @ A @ B ) @ ( image_9171994905208062541_c_a_a @ ( produc4731461908963710979_c_a_a @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1050_pair__imageI,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A3: set_Pr8357624050790843030od_c_a,F2: a > produc2133359284458116493od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ A3 )
     => ( member5502040268208861279od_c_a @ ( F2 @ A @ B ) @ ( image_2169818278710580869od_c_a @ ( produc4268383296163969595od_c_a @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1051_pair__imageI,axiom,
    ! [A: b,B: product_prod_c_a,A3: set_Pr272002107461315949od_c_a,F2: b > product_prod_c_a > a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ A3 )
     => ( member_a @ ( F2 @ A @ B ) @ ( image_8972666397057282742_c_a_a @ ( produc2354600891577745260_c_a_a @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1052_pair__imageI,axiom,
    ! [A: b,B: product_prod_c_a,A3: set_Pr272002107461315949od_c_a,F2: b > product_prod_c_a > produc2669932201312201782od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ A3 )
     => ( member5502040268208861279od_c_a @ ( F2 @ A @ B ) @ ( image_8151808240737915886od_c_a @ ( produc606978492514889380od_c_a @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1053_pair__imageI,axiom,
    ! [A: c,B: a,A3: set_Product_prod_c_a,F2: c > a > a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ A3 )
     => ( member_a @ ( F2 @ A @ B ) @ ( image_5407010446420943518_c_a_a @ ( produc1561580084227405268_c_a_a @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1054_pair__imageI,axiom,
    ! [A: c,B: a,A3: set_Product_prod_c_a,F2: c > a > produc2669932201312201782od_c_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ A3 )
     => ( member5502040268208861279od_c_a @ ( F2 @ A @ B ) @ ( image_3444923613702048726od_c_a @ ( produc3316798848510373132od_c_a @ F2 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1055_paths__of__length_H_Osimps_I2_J,axiom,
    ! [Prev2: list_P6327159017948738492od_c_a,Q: a,HM: product_prod_a_b > set_Product_prod_c_a,IM: set_b,K3: nat] :
      ( ( paths_2590340026247609854_a_b_c @ Prev2 @ Q @ HM @ IM @ ( suc @ K3 ) )
      = ( comple2269964827201748105od_c_a
        @ ( image_8047758555021968235od_c_a
          @ ^ [T3: produc2669932201312201782od_c_a] : ( paths_2590340026247609854_a_b_c @ ( append3884164603879732689od_c_a @ Prev2 @ ( cons_P8659537020926305772od_c_a @ T3 @ nil_Pr1342775757158464060od_c_a ) ) @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ HM @ IM @ K3 )
          @ ( transi7772877549217426849_a_b_c @ HM @ IM @ Q ) ) ) ) ).

% paths_of_length'.simps(2)
thf(fact_1056_paths__up__to__length_H_Osimps_I2_J,axiom,
    ! [Prev2: list_P6327159017948738492od_c_a,Q: a,HM: product_prod_a_b > set_Product_prod_c_a,IM: set_b,K3: nat] :
      ( ( paths_5428161815106503551_a_b_c @ Prev2 @ Q @ HM @ IM @ ( suc @ K3 ) )
      = ( insert4789241225314331020od_c_a @ Prev2
        @ ( comple2269964827201748105od_c_a
          @ ( image_8047758555021968235od_c_a
            @ ^ [T3: produc2669932201312201782od_c_a] : ( paths_5428161815106503551_a_b_c @ ( append3884164603879732689od_c_a @ Prev2 @ ( cons_P8659537020926305772od_c_a @ T3 @ nil_Pr1342775757158464060od_c_a ) ) @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ HM @ IM @ K3 )
            @ ( transi7772877549217426849_a_b_c @ HM @ IM @ Q ) ) ) ) ) ).

% paths_up_to_length'.simps(2)
thf(fact_1057_paths__up__to__length_H_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xa: a,Xb: product_prod_a_b > set_Product_prod_c_a,Xc: set_b,Xd: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( paths_5428161815106503551_a_b_c @ X3 @ Xa @ Xb @ Xc @ Xd )
        = Y3 )
     => ( ( ( Xd = zero_zero_nat )
         => ( Y3
           != ( insert4789241225314331020od_c_a @ X3 @ bot_bo6236370880139903240od_c_a ) ) )
       => ~ ! [K: nat] :
              ( ( Xd
                = ( suc @ K ) )
             => ( Y3
               != ( insert4789241225314331020od_c_a @ X3
                  @ ( comple2269964827201748105od_c_a
                    @ ( image_8047758555021968235od_c_a
                      @ ^ [T3: produc2669932201312201782od_c_a] : ( paths_5428161815106503551_a_b_c @ ( append3884164603879732689od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ T3 @ nil_Pr1342775757158464060od_c_a ) ) @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ Xb @ Xc @ K )
                      @ ( transi7772877549217426849_a_b_c @ Xb @ Xc @ Xa ) ) ) ) ) ) ) ) ).

% paths_up_to_length'.elims
thf(fact_1058_paths__of__length_H_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xa: a,Xb: product_prod_a_b > set_Product_prod_c_a,Xc: set_b,Xd: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( paths_2590340026247609854_a_b_c @ X3 @ Xa @ Xb @ Xc @ Xd )
        = Y3 )
     => ( ( ( Xd = zero_zero_nat )
         => ( Y3
           != ( insert4789241225314331020od_c_a @ X3 @ bot_bo6236370880139903240od_c_a ) ) )
       => ~ ! [K: nat] :
              ( ( Xd
                = ( suc @ K ) )
             => ( Y3
               != ( comple2269964827201748105od_c_a
                  @ ( image_8047758555021968235od_c_a
                    @ ^ [T3: produc2669932201312201782od_c_a] : ( paths_2590340026247609854_a_b_c @ ( append3884164603879732689od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ T3 @ nil_Pr1342775757158464060od_c_a ) ) @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ Xb @ Xc @ K )
                    @ ( transi7772877549217426849_a_b_c @ Xb @ Xc @ Xa ) ) ) ) ) ) ) ).

% paths_of_length'.elims
thf(fact_1059_paths__of__length_H_Opelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xa: a,Xb: product_prod_a_b > set_Product_prod_c_a,Xc: set_b,Xd: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( paths_2590340026247609854_a_b_c @ X3 @ Xa @ Xb @ Xc @ Xd )
        = Y3 )
     => ( ( accp_P9111333681250203678_b_nat @ paths_169346709130657029_a_b_c @ ( produc2836127534114196229_b_nat @ X3 @ ( produc4539511198857703066_b_nat @ Xa @ ( produc6126712305228101945_b_nat @ Xb @ ( produc4973803328240822360_b_nat @ Xc @ Xd ) ) ) ) )
       => ( ( ( Xd = zero_zero_nat )
           => ( ( Y3
                = ( insert4789241225314331020od_c_a @ X3 @ bot_bo6236370880139903240od_c_a ) )
             => ~ ( accp_P9111333681250203678_b_nat @ paths_169346709130657029_a_b_c @ ( produc2836127534114196229_b_nat @ X3 @ ( produc4539511198857703066_b_nat @ Xa @ ( produc6126712305228101945_b_nat @ Xb @ ( produc4973803328240822360_b_nat @ Xc @ zero_zero_nat ) ) ) ) ) ) )
         => ~ ! [K: nat] :
                ( ( Xd
                  = ( suc @ K ) )
               => ( ( Y3
                    = ( comple2269964827201748105od_c_a
                      @ ( image_8047758555021968235od_c_a
                        @ ^ [T3: produc2669932201312201782od_c_a] : ( paths_2590340026247609854_a_b_c @ ( append3884164603879732689od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ T3 @ nil_Pr1342775757158464060od_c_a ) ) @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ Xb @ Xc @ K )
                        @ ( transi7772877549217426849_a_b_c @ Xb @ Xc @ Xa ) ) ) )
                 => ~ ( accp_P9111333681250203678_b_nat @ paths_169346709130657029_a_b_c @ ( produc2836127534114196229_b_nat @ X3 @ ( produc4539511198857703066_b_nat @ Xa @ ( produc6126712305228101945_b_nat @ Xb @ ( produc4973803328240822360_b_nat @ Xc @ ( suc @ K ) ) ) ) ) ) ) ) ) ) ) ).

% paths_of_length'.pelims
thf(fact_1060_set__Cons__sing__Nil,axiom,
    ! [A3: set_Pr8357624050790843030od_c_a] :
      ( ( set_Co7906722538203907820od_c_a @ A3 @ ( insert4789241225314331020od_c_a @ nil_Pr1342775757158464060od_c_a @ bot_bo6236370880139903240od_c_a ) )
      = ( image_4166002503168551563od_c_a
        @ ^ [X4: produc2669932201312201782od_c_a] : ( cons_P8659537020926305772od_c_a @ X4 @ nil_Pr1342775757158464060od_c_a )
        @ A3 ) ) ).

% set_Cons_sing_Nil
thf(fact_1061_transitions__from_H_Oelims,axiom,
    ! [X3: produc2669932201312201782od_c_a > set_Product_prod_c_a,Xa: set_Pr272002107461315949od_c_a,Xb: a,Y3: set_Pr3884473935879652232od_c_a] :
      ( ( ( transi497389729532700819_c_a_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( comple7374806110032816117od_c_a
          @ ( image_5794031730513730368od_c_a
            @ ^ [X4: produc2133359284458116493od_c_a] :
                ( image_8309937651756752840od_c_a
                @ ( produc4169981485741864190od_c_a
                  @ ^ [Y4: c,Q7: a] : ( produc8441414538239260312od_c_a @ Xb @ ( produc1679040934828577519od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) )
                @ ( X3 @ ( produc5762801557034676134od_c_a @ Xb @ X4 ) ) )
            @ Xa ) ) ) ) ).

% transitions_from'.elims
thf(fact_1062_transitions__from_H_Oelims,axiom,
    ! [X3: produc5771957243474376000_c_a_a > set_Pr272002107461315949od_c_a,Xa: set_a,Xb: product_prod_c_a,Y3: set_Pr2640428967771533528od_c_a] :
      ( ( ( transi8535254487975430280_a_a_b @ X3 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( comple6130761141924697413od_c_a
          @ ( image_5994712598688360031od_c_a
            @ ^ [X4: a] :
                ( image_6325614729775467824od_c_a
                @ ( produc7024320895520758758od_c_a
                  @ ^ [Y4: b,Q7: product_prod_c_a] : ( produc7911353075612250984od_c_a @ Xb @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) )
                @ ( X3 @ ( produc7661602042108997680_c_a_a @ Xb @ X4 ) ) )
            @ Xa ) ) ) ) ).

% transitions_from'.elims
thf(fact_1063_transitions__from_H_Oelims,axiom,
    ! [X3: product_prod_a_b > set_Product_prod_c_a,Xa: set_b,Xb: a,Y3: set_Pr8357624050790843030od_c_a] :
      ( ( ( transi7772877549217426849_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( comple6722362287630405891od_c_a
          @ ( image_537995984993285980od_c_a
            @ ^ [X4: b] :
                ( image_3444923613702048726od_c_a
                @ ( produc3316798848510373132od_c_a
                  @ ^ [Y4: c,Q7: a] : ( produc5762801557034676134od_c_a @ Xb @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) )
                @ ( X3 @ ( product_Pair_a_b @ Xb @ X4 ) ) )
            @ Xa ) ) ) ) ).

% transitions_from'.elims
thf(fact_1064_transitions__from_H_Osimps,axiom,
    ( transi497389729532700819_c_a_c
    = ( ^ [HM2: produc2669932201312201782od_c_a > set_Product_prod_c_a,IM2: set_Pr272002107461315949od_c_a,Q6: a] :
          ( comple7374806110032816117od_c_a
          @ ( image_5794031730513730368od_c_a
            @ ^ [X4: produc2133359284458116493od_c_a] :
                ( image_8309937651756752840od_c_a
                @ ( produc4169981485741864190od_c_a
                  @ ^ [Y4: c,Q7: a] : ( produc8441414538239260312od_c_a @ Q6 @ ( produc1679040934828577519od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) )
                @ ( HM2 @ ( produc5762801557034676134od_c_a @ Q6 @ X4 ) ) )
            @ IM2 ) ) ) ) ).

% transitions_from'.simps
thf(fact_1065_transitions__from_H_Osimps,axiom,
    ( transi8535254487975430280_a_a_b
    = ( ^ [HM2: produc5771957243474376000_c_a_a > set_Pr272002107461315949od_c_a,IM2: set_a,Q6: product_prod_c_a] :
          ( comple6130761141924697413od_c_a
          @ ( image_5994712598688360031od_c_a
            @ ^ [X4: a] :
                ( image_6325614729775467824od_c_a
                @ ( produc7024320895520758758od_c_a
                  @ ^ [Y4: b,Q7: product_prod_c_a] : ( produc7911353075612250984od_c_a @ Q6 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) )
                @ ( HM2 @ ( produc7661602042108997680_c_a_a @ Q6 @ X4 ) ) )
            @ IM2 ) ) ) ) ).

% transitions_from'.simps
thf(fact_1066_transitions__from_H_Osimps,axiom,
    ( transi7772877549217426849_a_b_c
    = ( ^ [HM2: product_prod_a_b > set_Product_prod_c_a,IM2: set_b,Q6: a] :
          ( comple6722362287630405891od_c_a
          @ ( image_537995984993285980od_c_a
            @ ^ [X4: b] :
                ( image_3444923613702048726od_c_a
                @ ( produc3316798848510373132od_c_a
                  @ ^ [Y4: c,Q7: a] : ( produc5762801557034676134od_c_a @ Q6 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) )
                @ ( HM2 @ ( product_Pair_a_b @ Q6 @ X4 ) ) )
            @ IM2 ) ) ) ) ).

% transitions_from'.simps
thf(fact_1067_paths__up__to__length_H_Opelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xa: a,Xb: product_prod_a_b > set_Product_prod_c_a,Xc: set_b,Xd: nat,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( paths_5428161815106503551_a_b_c @ X3 @ Xa @ Xb @ Xc @ Xd )
        = Y3 )
     => ( ( accp_P9111333681250203678_b_nat @ paths_1230168996438096068_a_b_c @ ( produc2836127534114196229_b_nat @ X3 @ ( produc4539511198857703066_b_nat @ Xa @ ( produc6126712305228101945_b_nat @ Xb @ ( produc4973803328240822360_b_nat @ Xc @ Xd ) ) ) ) )
       => ( ( ( Xd = zero_zero_nat )
           => ( ( Y3
                = ( insert4789241225314331020od_c_a @ X3 @ bot_bo6236370880139903240od_c_a ) )
             => ~ ( accp_P9111333681250203678_b_nat @ paths_1230168996438096068_a_b_c @ ( produc2836127534114196229_b_nat @ X3 @ ( produc4539511198857703066_b_nat @ Xa @ ( produc6126712305228101945_b_nat @ Xb @ ( produc4973803328240822360_b_nat @ Xc @ zero_zero_nat ) ) ) ) ) ) )
         => ~ ! [K: nat] :
                ( ( Xd
                  = ( suc @ K ) )
               => ( ( Y3
                    = ( insert4789241225314331020od_c_a @ X3
                      @ ( comple2269964827201748105od_c_a
                        @ ( image_8047758555021968235od_c_a
                          @ ^ [T3: produc2669932201312201782od_c_a] : ( paths_5428161815106503551_a_b_c @ ( append3884164603879732689od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ T3 @ nil_Pr1342775757158464060od_c_a ) ) @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ Xb @ Xc @ K )
                          @ ( transi7772877549217426849_a_b_c @ Xb @ Xc @ Xa ) ) ) ) )
                 => ~ ( accp_P9111333681250203678_b_nat @ paths_1230168996438096068_a_b_c @ ( produc2836127534114196229_b_nat @ X3 @ ( produc4539511198857703066_b_nat @ Xa @ ( produc6126712305228101945_b_nat @ Xb @ ( produc4973803328240822360_b_nat @ Xc @ ( suc @ K ) ) ) ) ) ) ) ) ) ) ) ).

% paths_up_to_length'.pelims
thf(fact_1068_transitions__from_H_Opelims,axiom,
    ! [X3: produc2669932201312201782od_c_a > set_Product_prod_c_a,Xa: set_Pr272002107461315949od_c_a,Xb: a,Y3: set_Pr3884473935879652232od_c_a] :
      ( ( ( transi497389729532700819_c_a_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P6430246902678515312_c_a_a @ transi9121809870488969748_c_a_c @ ( produc1685102300155945451_c_a_a @ X3 @ ( produc6230263938701942376_c_a_a @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( comple7374806110032816117od_c_a
                @ ( image_5794031730513730368od_c_a
                  @ ^ [X4: produc2133359284458116493od_c_a] :
                      ( image_8309937651756752840od_c_a
                      @ ( produc4169981485741864190od_c_a
                        @ ^ [Y4: c,Q7: a] : ( produc8441414538239260312od_c_a @ Xb @ ( produc1679040934828577519od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) )
                      @ ( X3 @ ( produc5762801557034676134od_c_a @ Xb @ X4 ) ) )
                  @ Xa ) ) )
           => ~ ( accp_P6430246902678515312_c_a_a @ transi9121809870488969748_c_a_c @ ( produc1685102300155945451_c_a_a @ X3 @ ( produc6230263938701942376_c_a_a @ Xa @ Xb ) ) ) ) ) ) ).

% transitions_from'.pelims
thf(fact_1069_transitions__from_H_Opelims,axiom,
    ! [X3: produc5771957243474376000_c_a_a > set_Pr272002107461315949od_c_a,Xa: set_a,Xb: product_prod_c_a,Y3: set_Pr2640428967771533528od_c_a] :
      ( ( ( transi8535254487975430280_a_a_b @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P2765486758866033416od_c_a @ transi6137558656961800969_a_a_b @ ( produc2750636601947930499od_c_a @ X3 @ ( produc818365071461481310od_c_a @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( comple6130761141924697413od_c_a
                @ ( image_5994712598688360031od_c_a
                  @ ^ [X4: a] :
                      ( image_6325614729775467824od_c_a
                      @ ( produc7024320895520758758od_c_a
                        @ ^ [Y4: b,Q7: product_prod_c_a] : ( produc7911353075612250984od_c_a @ Xb @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) )
                      @ ( X3 @ ( produc7661602042108997680_c_a_a @ Xb @ X4 ) ) )
                  @ Xa ) ) )
           => ~ ( accp_P2765486758866033416od_c_a @ transi6137558656961800969_a_a_b @ ( produc2750636601947930499od_c_a @ X3 @ ( produc818365071461481310od_c_a @ Xa @ Xb ) ) ) ) ) ) ).

% transitions_from'.pelims
thf(fact_1070_transitions__from_H_Opelims,axiom,
    ! [X3: product_prod_a_b > set_Product_prod_c_a,Xa: set_b,Xb: a,Y3: set_Pr8357624050790843030od_c_a] :
      ( ( ( transi7772877549217426849_a_b_c @ X3 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P6074164402210365168et_b_a @ transi8192330815583697186_a_b_c @ ( produc3600278798717983083et_b_a @ X3 @ ( product_Pair_set_b_a @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( comple6722362287630405891od_c_a
                @ ( image_537995984993285980od_c_a
                  @ ^ [X4: b] :
                      ( image_3444923613702048726od_c_a
                      @ ( produc3316798848510373132od_c_a
                        @ ^ [Y4: c,Q7: a] : ( produc5762801557034676134od_c_a @ Xb @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) )
                      @ ( X3 @ ( product_Pair_a_b @ Xb @ X4 ) ) )
                  @ Xa ) ) )
           => ~ ( accp_P6074164402210365168et_b_a @ transi8192330815583697186_a_b_c @ ( produc3600278798717983083et_b_a @ X3 @ ( product_Pair_set_b_a @ Xa @ Xb ) ) ) ) ) ) ).

% transitions_from'.pelims
thf(fact_1071_acyclic__paths__up__to__length__code,axiom,
    ( acycli3978232057192500090_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a,K5: nat] :
          ( if_set5301400340663253090od_c_a @ ( member_a @ Q6 @ ( states_a_b_c @ M2 ) )
          @ ( acycli8838441729103940241_a_b_c @ nil_Pr1342775757158464060od_c_a @ Q6
            @ ^ [X4: a] :
                ( case_o5084979270738236875od_c_a @ bot_bo8402128546026624985od_c_a
                @ ^ [Xs3: set_Pr272002107461315949od_c_a] : Xs3
                @ ( set_as7394328699852255621od_c_a @ ( transitions_a_b_c @ M2 ) @ X4 ) )
            @ ( insert_a2 @ Q6 @ bot_bot_set_a )
            @ K5 )
          @ bot_bo6236370880139903240od_c_a ) ) ) ).

% acyclic_paths_up_to_length_code
thf(fact_1072_set__as__map__containment,axiom,
    ! [X3: a,Y3: produc2133359284458116493od_c_a,Zs: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) @ Zs )
     => ( member1160383752307813046od_c_a @ Y3
        @ ( case_o5084979270738236875od_c_a @ bot_bo8402128546026624985od_c_a
          @ ^ [Xs3: set_Pr272002107461315949od_c_a] : Xs3
          @ ( set_as7394328699852255621od_c_a @ Zs @ X3 ) ) ) ) ).

% set_as_map_containment
thf(fact_1073_set__as__map__containment,axiom,
    ! [X3: b,Y3: product_prod_c_a,Zs: set_Pr272002107461315949od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) @ Zs )
     => ( member5074992350434858958od_c_a @ Y3
        @ ( case_o3111718015729425867od_c_a @ bot_bo2086078286244720881od_c_a
          @ ^ [Xs3: set_Product_prod_c_a] : Xs3
          @ ( set_as4883611149566849244od_c_a @ Zs @ X3 ) ) ) ) ).

% set_as_map_containment
thf(fact_1074_set__as__map__containment,axiom,
    ! [X3: c,Y3: a,Zs: set_Product_prod_c_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) @ Zs )
     => ( member_a @ Y3
        @ ( case_o5082855197222185995_set_a @ bot_bot_set_a
          @ ^ [Xs3: set_a] : Xs3
          @ ( set_as_map_c_a @ Zs @ X3 ) ) ) ) ).

% set_as_map_containment
thf(fact_1075_set__as__map__elem,axiom,
    ! [Y3: produc2133359284458116493od_c_a,Xs: set_Pr8357624050790843030od_c_a,X3: a] :
      ( ( member1160383752307813046od_c_a @ Y3
        @ ( case_o5084979270738236875od_c_a @ bot_bo8402128546026624985od_c_a
          @ ^ [Xs3: set_Pr272002107461315949od_c_a] : Xs3
          @ ( set_as7394328699852255621od_c_a @ Xs @ X3 ) ) )
     => ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) @ Xs ) ) ).

% set_as_map_elem
thf(fact_1076_set__as__map__elem,axiom,
    ! [Y3: product_prod_c_a,Xs: set_Pr272002107461315949od_c_a,X3: b] :
      ( ( member5074992350434858958od_c_a @ Y3
        @ ( case_o3111718015729425867od_c_a @ bot_bo2086078286244720881od_c_a
          @ ^ [Xs3: set_Product_prod_c_a] : Xs3
          @ ( set_as4883611149566849244od_c_a @ Xs @ X3 ) ) )
     => ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) @ Xs ) ) ).

% set_as_map_elem
thf(fact_1077_set__as__map__elem,axiom,
    ! [Y3: a,Xs: set_Product_prod_c_a,X3: c] :
      ( ( member_a @ Y3
        @ ( case_o5082855197222185995_set_a @ bot_bot_set_a
          @ ^ [Xs3: set_a] : Xs3
          @ ( set_as_map_c_a @ Xs @ X3 ) ) )
     => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) @ Xs ) ) ).

% set_as_map_elem
thf(fact_1078_h__from,axiom,
    ( h_from_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q6: a] :
          ( case_o5084979270738236875od_c_a @ bot_bo8402128546026624985od_c_a
          @ ^ [Yqs: set_Pr272002107461315949od_c_a] : Yqs
          @ ( set_as7394328699852255621od_c_a @ ( transitions_a_b_c @ M2 ) @ Q6 ) ) ) ) ).

% h_from
thf(fact_1079_h__out__impl__helper,axiom,
    ! [A3: set_Pr8176813492957540505od_c_a] :
      ( ( produc2035080022814879075_set_a
        @ ^ [Q6: a,X4: produc2133359284458116493od_c_a] :
            ( collect_a
            @ ^ [Y4: a] :
              ? [Q7: produc2133359284458116493od_c_a] : ( member6342456451825786850od_c_a @ ( produc4548423997264226473od_c_a @ Q6 @ ( produc3761921832363547008od_c_a @ X4 @ ( produc5762801557034676134od_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: produc2669932201312201782od_c_a] :
            ( case_o5082855197222185995_set_a @ bot_bot_set_a
            @ ^ [Yqs: set_a] : Yqs
            @ ( set_as335611262650259134_c_a_a
              @ ( image_6345107340833294715_c_a_a
                @ ( produc8224644082662394289_c_a_a
                  @ ^ [Q6: a] :
                      ( produc6697903666491073690_c_a_a
                      @ ^ [X4: produc2133359284458116493od_c_a] :
                          ( produc2555251204939300852_c_a_a
                          @ ^ [Y4: a,Q7: produc2133359284458116493od_c_a] : ( produc174534808117382111_c_a_a @ ( produc5762801557034676134od_c_a @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1080_h__out__impl__helper,axiom,
    ! [A3: set_Pr1212000722004284992od_c_a] :
      ( ( produc711231852148568268_set_a
        @ ^ [Q6: b,X4: product_prod_c_a] :
            ( collect_a
            @ ^ [Y4: a] :
              ? [Q7: produc2133359284458116493od_c_a] : ( member5418238279316946185od_c_a @ ( produc8889421724123298768od_c_a @ Q6 @ ( produc7911353075612250984od_c_a @ X4 @ ( produc5762801557034676134od_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: produc2133359284458116493od_c_a] :
            ( case_o5082855197222185995_set_a @ bot_bot_set_a
            @ ^ [Yqs: set_a] : Yqs
            @ ( set_as4630677545522536487_c_a_a
              @ ( image_7500720858548113277_c_a_a
                @ ( produc2966705023942427827_c_a_a
                  @ ^ [Q6: b] :
                      ( produc2907555017552795419_c_a_a
                      @ ^ [X4: product_prod_c_a] :
                          ( produc3930913629802858589_c_a_a
                          @ ^ [Y4: a,Q7: produc2133359284458116493od_c_a] : ( produc2999150402704957000_c_a_a @ ( produc3440865914000881661od_c_a @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1081_h__out__impl__helper,axiom,
    ! [A3: set_Pr4854443895542973158od_c_a] :
      ( ( produc2545871480639997748_set_a
        @ ^ [Q6: c,X4: a] :
            ( collect_a
            @ ^ [Y4: a] :
              ? [Q7: produc2133359284458116493od_c_a] : ( member3093337089038594223od_c_a @ ( produc3346642955089884406od_c_a @ Q6 @ ( produc4410401784823481679od_c_a @ X4 @ ( produc5762801557034676134od_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: product_prod_c_a] :
            ( case_o5082855197222185995_set_a @ bot_bot_set_a
            @ ^ [Yqs: set_a] : Yqs
            @ ( set_as9104347277674965263_c_a_a
              @ ( image_8960197114356911167_c_a_a
                @ ( produc3769388632746313461_c_a_a
                  @ ^ [Q6: c] :
                      ( produc399306904482573532_c_a_a
                      @ ^ [X4: a] :
                          ( produc6810792263997984837_c_a_a
                          @ ^ [Y4: a,Q7: produc2133359284458116493od_c_a] : ( produc7661602042108997680_c_a_a @ ( product_Pair_c_a @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1082_h__out__impl__helper,axiom,
    ! [A3: set_Pr6352661749802654064od_c_a] :
      ( ( produc2035080027118107876_set_b
        @ ^ [Q6: a,X4: produc2133359284458116493od_c_a] :
            ( collect_b
            @ ^ [Y4: b] :
              ? [Q7: product_prod_c_a] : ( member4558601409277499449od_c_a @ ( produc8456968922050256512od_c_a @ Q6 @ ( produc9198192719383092439od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: produc2669932201312201782od_c_a] :
            ( case_o6570817238820213323_set_b @ bot_bot_set_b
            @ ^ [Yqs: set_b] : Yqs
            @ ( set_as335611262650259135_c_a_b
              @ ( image_1078689763015575397_c_a_b
                @ ( produc8373501312963702555_c_a_b
                  @ ^ [Q6: a] :
                      ( produc5792289196193782916_c_a_b
                      @ ^ [X4: produc2133359284458116493od_c_a] :
                          ( produc2024197684756935390_c_a_b
                          @ ^ [Y4: b,Q7: product_prod_c_a] : ( produc174534808117382112_c_a_b @ ( produc5762801557034676134od_c_a @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1083_h__out__impl__helper,axiom,
    ! [A3: set_Pr4299858919719910935od_c_a] :
      ( ( produc711231856451797069_set_b
        @ ^ [Q6: b,X4: product_prod_c_a] :
            ( collect_b
            @ ^ [Y4: b] :
              ? [Q7: product_prod_c_a] : ( member4343506560499364960od_c_a @ ( produc6426366379692732583od_c_a @ Q6 @ ( produc5244405752496070335od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: produc2133359284458116493od_c_a] :
            ( case_o6570817238820213323_set_b @ bot_bot_set_b
            @ ^ [Yqs: set_b] : Yqs
            @ ( set_as4630677545522536488_c_a_b
              @ ( image_7455466272679862119_c_a_b
                @ ( produc7701701250271345693_c_a_b
                  @ ^ [Q6: b] :
                      ( produc3540201283022206981_c_a_b
                      @ ^ [X4: product_prod_c_a] :
                          ( produc269508830457007175_c_a_b
                          @ ^ [Y4: b,Q7: product_prod_c_a] : ( produc2999150402704957001_c_a_b @ ( produc3440865914000881661od_c_a @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1084_h__out__impl__helper,axiom,
    ! [A3: set_Pr3884473935879652232od_c_a] :
      ( ( produc2035080031421336677_set_c
        @ ^ [Q6: a,X4: produc2133359284458116493od_c_a] :
            ( collect_c
            @ ^ [Y4: c] :
              ? [Q7: a] : ( member8587115632268026449od_c_a @ ( produc8441414538239260312od_c_a @ Q6 @ ( produc1679040934828577519od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: produc2669932201312201782od_c_a] :
            ( case_o8058779280418240651_set_c @ bot_bot_set_c
            @ ^ [Yqs: set_c] : Yqs
            @ ( set_as335611262650259136_c_a_c
              @ ( image_342861010296867278_c_a_c
                @ ( produc4454128318592592004_c_a_c
                  @ ^ [Q6: a] :
                      ( produc3750301517533083629_c_a_c
                      @ ^ [X4: produc2133359284458116493od_c_a] :
                          ( produc6995391536104085703_c_a_c
                          @ ^ [Y4: c,Q7: a] : ( produc174534808117382113_c_a_c @ ( produc5762801557034676134od_c_a @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1085_h__out__impl__helper,axiom,
    ! [A3: set_Pr5338671032280311599od_c_a] :
      ( ( produc711231860755025870_set_c
        @ ^ [Q6: b,X4: product_prod_c_a] :
            ( collect_c
            @ ^ [Y4: c] :
              ? [Q7: a] : ( member4823365305010683256od_c_a @ ( produc4639949295917647807od_c_a @ Q6 @ ( produc968124180156873943od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: produc2133359284458116493od_c_a] :
            ( case_o8058779280418240651_set_c @ bot_bot_set_c
            @ ^ [Yqs: set_c] : Yqs
            @ ( set_as4630677545522536489_c_a_c
              @ ( image_2763971867084589776_c_a_c
                @ ( produc7205819850556792454_c_a_c
                  @ ^ [Q6: b] :
                      ( produc6051657911608284014_c_a_c
                      @ ^ [X4: product_prod_c_a] :
                          ( produc2979329190755719728_c_a_c
                          @ ^ [Y4: c,Q7: a] : ( produc2999150402704957002_c_a_c @ ( produc3440865914000881661od_c_a @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1086_h__out__impl__helper,axiom,
    ! [A3: set_Pr6912582227606077397od_c_a] :
      ( ( produc2545871489246455350_set_c
        @ ^ [Q6: c,X4: a] :
            ( collect_c
            @ ^ [Y4: c] :
              ? [Q7: a] : ( member7958828110252711198od_c_a @ ( produc7522979079457647845od_c_a @ Q6 @ ( produc4080306214332697790od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: product_prod_c_a] :
            ( case_o8058779280418240651_set_c @ bot_bot_set_c
            @ ^ [Yqs: set_c] : Yqs
            @ ( set_as9104347277674965265_c_a_c
              @ ( image_6304947530448639122_c_a_c
                @ ( produc6958763622796094920_c_a_c
                  @ ^ [Q6: c] :
                      ( produc3975476122118186031_c_a_c
                      @ ^ [X4: a] :
                          ( produc7775365209255533336_c_a_c
                          @ ^ [Y4: c,Q7: a] : ( produc7661602042108997682_c_a_c @ ( product_Pair_c_a @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1087_h__out__impl__helper,axiom,
    ! [A3: set_Pr1963666890401038525od_c_a] :
      ( ( produc2545871484943226549_set_b
        @ ^ [Q6: c,X4: a] :
            ( collect_b
            @ ^ [Y4: b] :
              ? [Q7: product_prod_c_a] : ( member558622287379480070od_c_a @ ( produc8442212246646489037od_c_a @ Q6 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: product_prod_c_a] :
            ( case_o6570817238820213323_set_b @ bot_bot_set_b
            @ ^ [Yqs: set_b] : Yqs
            @ ( set_as9104347277674965264_c_a_b
              @ ( image_3230742632371525417_c_a_b
                @ ( produc3711305473767700831_c_a_b
                  @ ^ [Q6: c] :
                      ( produc6810792268301213638_c_a_b
                      @ ^ [X4: a] :
                          ( produc1270752430231315759_c_a_b
                          @ ^ [Y4: b,Q7: product_prod_c_a] : ( produc7661602042108997681_c_a_b @ ( product_Pair_c_a @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1088_h__out__impl__helper,axiom,
    ! [A3: set_Pr8357624050790843030od_c_a] :
      ( ( produc1338550831314148723_set_c
        @ ^ [Q6: a,X4: b] :
            ( collect_c
            @ ^ [Y4: c] :
              ? [Q7: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q6 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y4 @ Q7 ) ) ) @ A3 ) ) )
      = ( ^ [Qx: product_prod_a_b] :
            ( case_o8058779280418240651_set_c @ bot_bot_set_c
            @ ^ [Yqs: set_c] : Yqs
            @ ( set_as6499633083548493006_a_b_c
              @ ( image_1678017663536505166_a_b_c
                @ ( produc4206078078477970180_a_b_c
                  @ ^ [Q6: a] :
                      ( produc7889410277262848109_a_b_c
                      @ ^ [X4: b] :
                          ( produc5170651015129061077_a_b_c
                          @ ^ [Y4: c,Q7: a] : ( produc5056887847982525423_a_b_c @ ( product_Pair_a_b @ Q6 @ X4 ) @ Y4 ) ) ) )
                @ A3 )
              @ Qx ) ) ) ) ).

% h_out_impl_helper
thf(fact_1089_h__out__code,axiom,
    ( h_out_a_b_c
    = ( ^ [M2: fsm_a_b_c,Qx: product_prod_a_b] :
          ( case_o8058779280418240651_set_c @ bot_bot_set_c
          @ ^ [Yqs: set_c] : Yqs
          @ ( set_as6499633083548493006_a_b_c
            @ ( image_1678017663536505166_a_b_c
              @ ( produc4206078078477970180_a_b_c
                @ ^ [Q6: a] :
                    ( produc7889410277262848109_a_b_c
                    @ ^ [X4: b] :
                        ( produc5170651015129061077_a_b_c
                        @ ^ [Y4: c,Q7: a] : ( produc5056887847982525423_a_b_c @ ( product_Pair_a_b @ Q6 @ X4 ) @ Y4 ) ) ) )
              @ ( transitions_a_b_c @ M2 ) )
            @ Qx ) ) ) ) ).

% h_out_code
thf(fact_1090_set__remove1__eq,axiom,
    ! [Xs: list_a,X3: a] :
      ( ( distinct_a @ Xs )
     => ( ( set_a2 @ ( remove1_a @ X3 @ Xs ) )
        = ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).

% set_remove1_eq
thf(fact_1091_in__set__remove1,axiom,
    ! [A: a,B: a,Xs: list_a] :
      ( ( A != B )
     => ( ( member_a @ A @ ( set_a2 @ ( remove1_a @ B @ Xs ) ) )
        = ( member_a @ A @ ( set_a2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1092_in__set__remove1,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( A != B )
     => ( ( member5502040268208861279od_c_a @ A @ ( set_Pr5664641634098845387od_c_a @ ( remove2102263955212515282od_c_a @ B @ Xs ) ) )
        = ( member5502040268208861279od_c_a @ A @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1093_bot__empty__eq2,axiom,
    ( bot_bo2582192026207158283_c_a_o
    = ( ^ [X4: a,Y4: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y4 ) @ bot_bo267491471446728962od_c_a ) ) ) ).

% bot_empty_eq2
thf(fact_1094_bot__empty__eq2,axiom,
    ( bot_bo970769885656619636_c_a_o
    = ( ^ [X4: b,Y4: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y4 ) @ bot_bo8402128546026624985od_c_a ) ) ) ).

% bot_empty_eq2
thf(fact_1095_bot__empty__eq2,axiom,
    ( bot_bot_c_a_o
    = ( ^ [X4: c,Y4: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y4 ) @ bot_bo2086078286244720881od_c_a ) ) ) ).

% bot_empty_eq2
thf(fact_1096_remove1_Osimps_I1_J,axiom,
    ! [X3: produc2669932201312201782od_c_a] :
      ( ( remove2102263955212515282od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% remove1.simps(1)
thf(fact_1097_remove1__idem,axiom,
    ! [X3: a,Xs: list_a] :
      ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ( remove1_a @ X3 @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1098_remove1__idem,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( remove2102263955212515282od_c_a @ X3 @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1099_notin__set__remove1,axiom,
    ! [X3: a,Xs: list_a,Y3: a] :
      ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ~ ( member_a @ X3 @ ( set_a2 @ ( remove1_a @ Y3 @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1100_notin__set__remove1,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( remove2102263955212515282od_c_a @ Y3 @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1101_distinct__remove1,axiom,
    ! [Xs: list_a,X3: a] :
      ( ( distinct_a @ Xs )
     => ( distinct_a @ ( remove1_a @ X3 @ Xs ) ) ) ).

% distinct_remove1
thf(fact_1102_remove1_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Y3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( X3 = Y3 )
       => ( ( remove2102263955212515282od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ Y3 @ Xs ) )
          = Xs ) )
      & ( ( X3 != Y3 )
       => ( ( remove2102263955212515282od_c_a @ X3 @ ( cons_P8659537020926305772od_c_a @ Y3 @ Xs ) )
          = ( cons_P8659537020926305772od_c_a @ Y3 @ ( remove2102263955212515282od_c_a @ X3 @ Xs ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_1103_remove1__append,axiom,
    ! [X3: a,Xs: list_a,Ys: list_a] :
      ( ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ( ( remove1_a @ X3 @ ( append_a @ Xs @ Ys ) )
          = ( append_a @ ( remove1_a @ X3 @ Xs ) @ Ys ) ) )
      & ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ( ( remove1_a @ X3 @ ( append_a @ Xs @ Ys ) )
          = ( append_a @ Xs @ ( remove1_a @ X3 @ Ys ) ) ) ) ) ).

% remove1_append
thf(fact_1104_remove1__append,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
       => ( ( remove2102263955212515282od_c_a @ X3 @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
          = ( append3884164603879732689od_c_a @ ( remove2102263955212515282od_c_a @ X3 @ Xs ) @ Ys ) ) )
      & ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
       => ( ( remove2102263955212515282od_c_a @ X3 @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
          = ( append3884164603879732689od_c_a @ Xs @ ( remove2102263955212515282od_c_a @ X3 @ Ys ) ) ) ) ) ).

% remove1_append
thf(fact_1105_distinct__remove1__removeAll,axiom,
    ! [Xs: list_a,X3: a] :
      ( ( distinct_a @ Xs )
     => ( ( remove1_a @ X3 @ Xs )
        = ( removeAll_a @ X3 @ Xs ) ) ) ).

% distinct_remove1_removeAll
thf(fact_1106_remove1__split,axiom,
    ! [A: a,Xs: list_a,Ys: list_a] :
      ( ( member_a @ A @ ( set_a2 @ Xs ) )
     => ( ( ( remove1_a @ A @ Xs )
          = Ys )
        = ( ? [Ls: list_a,Rs: list_a] :
              ( ( Xs
                = ( append_a @ Ls @ ( cons_a @ A @ Rs ) ) )
              & ~ ( member_a @ A @ ( set_a2 @ Ls ) )
              & ( Ys
                = ( append_a @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1107_remove1__split,axiom,
    ! [A: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( ( remove2102263955212515282od_c_a @ A @ Xs )
          = Ys )
        = ( ? [Ls: list_P6327159017948738492od_c_a,Rs: list_P6327159017948738492od_c_a] :
              ( ( Xs
                = ( append3884164603879732689od_c_a @ Ls @ ( cons_P8659537020926305772od_c_a @ A @ Rs ) ) )
              & ~ ( member5502040268208861279od_c_a @ A @ ( set_Pr5664641634098845387od_c_a @ Ls ) )
              & ( Ys
                = ( append3884164603879732689od_c_a @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1108_Id__onI,axiom,
    ! [A: a,A3: set_a] :
      ( ( member_a @ A @ A3 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( id_on_a @ A3 ) ) ) ).

% Id_onI
thf(fact_1109_Id__onI,axiom,
    ! [A: produc2669932201312201782od_c_a,A3: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ A3 )
     => ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ A ) @ ( id_on_5114218928270390227od_c_a @ A3 ) ) ) ).

% Id_onI
thf(fact_1110_set__append,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
      = ( sup_su7950883165052597354od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs ) @ ( set_Pr5664641634098845387od_c_a @ Ys ) ) ) ).

% set_append
thf(fact_1111_Id__on__iff,axiom,
    ! [X3: a,Y3: a,A3: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( id_on_a @ A3 ) )
      = ( ( X3 = Y3 )
        & ( member_a @ X3 @ A3 ) ) ) ).

% Id_on_iff
thf(fact_1112_Id__on__iff,axiom,
    ! [X3: produc2669932201312201782od_c_a,Y3: produc2669932201312201782od_c_a,A3: set_Pr8357624050790843030od_c_a] :
      ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X3 @ Y3 ) @ ( id_on_5114218928270390227od_c_a @ A3 ) )
      = ( ( X3 = Y3 )
        & ( member5502040268208861279od_c_a @ X3 @ A3 ) ) ) ).

% Id_on_iff
thf(fact_1113_Id__on__eqI,axiom,
    ! [A: a,B: a,A3: set_a] :
      ( ( A = B )
     => ( ( member_a @ A @ A3 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( id_on_a @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_1114_Id__on__eqI,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a,A3: set_Pr8357624050790843030od_c_a] :
      ( ( A = B )
     => ( ( member5502040268208861279od_c_a @ A @ A3 )
       => ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ B ) @ ( id_on_5114218928270390227od_c_a @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_1115_Id__onE,axiom,
    ! [C: product_prod_a_a,A3: set_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( id_on_a @ A3 ) )
     => ~ ! [X: a] :
            ( ( member_a @ X @ A3 )
           => ( C
             != ( product_Pair_a_a @ X @ X ) ) ) ) ).

% Id_onE
thf(fact_1116_Id__onE,axiom,
    ! [C: produc1647090505208597031od_c_a,A3: set_Pr8357624050790843030od_c_a] :
      ( ( member3417889999246264912od_c_a @ C @ ( id_on_5114218928270390227od_c_a @ A3 ) )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ A3 )
           => ( C
             != ( produc4004402368211057175od_c_a @ X @ X ) ) ) ) ).

% Id_onE
thf(fact_1117_set__prefixes__append,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) )
      = ( sup_su500200128730103920od_c_a @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) )
        @ ( collec6273869032445462695od_c_a
          @ ^ [Uu3: list_P6327159017948738492od_c_a] :
            ? [Ys4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( append3884164603879732689od_c_a @ Xs @ Ys4 ) )
              & ( member7410604586820865893od_c_a @ Ys4 @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Ys ) ) ) ) ) ) ) ).

% set_prefixes_append
thf(fact_1118_set__suffixes__append,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( set_li8663184538779258833od_c_a @ ( suffix7794987338261656196od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) )
      = ( sup_su500200128730103920od_c_a @ ( set_li8663184538779258833od_c_a @ ( suffix7794987338261656196od_c_a @ Ys ) )
        @ ( collec6273869032445462695od_c_a
          @ ^ [Uu3: list_P6327159017948738492od_c_a] :
            ? [Xs4: list_P6327159017948738492od_c_a] :
              ( ( Uu3
                = ( append3884164603879732689od_c_a @ Xs4 @ Ys ) )
              & ( member7410604586820865893od_c_a @ Xs4 @ ( set_li8663184538779258833od_c_a @ ( suffix7794987338261656196od_c_a @ Xs ) ) ) ) ) ) ) ).

% set_suffixes_append
thf(fact_1119_fsm__impl__from__list_Oelims,axiom,
    ! [X3: a,Xa: list_P6327159017948738492od_c_a,Y3: fSM_fsm_impl_a_b_c] :
      ( ( ( fSM_fs5279388792078304906_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( ( ( Xa = nil_Pr1342775757158464060od_c_a )
         => ( Y3
           != ( fSM_fsm_FSMI_a_b_c @ X3 @ ( insert_a2 @ X3 @ bot_bot_set_a ) @ bot_bot_set_b @ bot_bot_set_c @ bot_bo267491471446728962od_c_a ) ) )
       => ~ ! [T2: produc2669932201312201782od_c_a,Ts2: list_P6327159017948738492od_c_a] :
              ( ( Xa
                = ( cons_P8659537020926305772od_c_a @ T2 @ Ts2 ) )
             => ( Y3
               != ( fSM_fsm_FSMI_a_b_c @ ( produc4225943634317630866od_c_a @ T2 )
                  @ ( sup_sup_set_a @ ( image_9171994905208062541_c_a_a @ produc4225943634317630866od_c_a @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T2 @ Ts2 ) ) )
                    @ ( image_9171994905208062541_c_a_a
                      @ ^ [A5: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A5 ) ) )
                      @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T2 @ Ts2 ) ) ) )
                  @ ( image_9171994905208062542_c_a_b
                    @ ^ [A5: produc2669932201312201782od_c_a] : ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ A5 ) )
                    @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T2 @ Ts2 ) ) )
                  @ ( image_9171994905208062543_c_a_c
                    @ ^ [A5: produc2669932201312201782od_c_a] : ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A5 ) ) )
                    @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T2 @ Ts2 ) ) )
                  @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T2 @ Ts2 ) ) ) ) ) ) ) ).

% fsm_impl_from_list.elims
thf(fact_1120_fsm__impl__from__list_Osimps_I1_J,axiom,
    ! [Q: a] :
      ( ( fSM_fs5279388792078304906_a_b_c @ Q @ nil_Pr1342775757158464060od_c_a )
      = ( fSM_fsm_FSMI_a_b_c @ Q @ ( insert_a2 @ Q @ bot_bot_set_a ) @ bot_bot_set_b @ bot_bot_set_c @ bot_bo267491471446728962od_c_a ) ) ).

% fsm_impl_from_list.simps(1)
thf(fact_1121_fsm__impl__from__list_Osimps_I2_J,axiom,
    ! [Q: a,T: produc2669932201312201782od_c_a,Ts: list_P6327159017948738492od_c_a] :
      ( ( fSM_fs5279388792078304906_a_b_c @ Q @ ( cons_P8659537020926305772od_c_a @ T @ Ts ) )
      = ( fSM_fsm_FSMI_a_b_c @ ( produc4225943634317630866od_c_a @ T )
        @ ( sup_sup_set_a @ ( image_9171994905208062541_c_a_a @ produc4225943634317630866od_c_a @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T @ Ts ) ) )
          @ ( image_9171994905208062541_c_a_a
            @ ^ [A5: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A5 ) ) )
            @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T @ Ts ) ) ) )
        @ ( image_9171994905208062542_c_a_b
          @ ^ [A5: produc2669932201312201782od_c_a] : ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ A5 ) )
          @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T @ Ts ) ) )
        @ ( image_9171994905208062543_c_a_c
          @ ^ [A5: produc2669932201312201782od_c_a] : ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A5 ) ) )
          @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T @ Ts ) ) )
        @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ T @ Ts ) ) ) ) ).

% fsm_impl_from_list.simps(2)
thf(fact_1122_fsm__impl__from__list_H_Osimps_I1_J,axiom,
    ! [Q: a] :
      ( ( fSM_fs5857938762048732577_a_b_c @ Q @ nil_Pr1342775757158464060od_c_a )
      = ( fSM_fsm_FSMI_a_b_c @ Q @ ( insert_a2 @ Q @ bot_bot_set_a ) @ bot_bot_set_b @ bot_bot_set_c @ bot_bo267491471446728962od_c_a ) ) ).

% fsm_impl_from_list'.simps(1)
thf(fact_1123_remdups__eq__nil__right__iff,axiom,
    ! [X3: list_P6327159017948738492od_c_a] :
      ( ( nil_Pr1342775757158464060od_c_a
        = ( remdup5562453435236534737od_c_a @ X3 ) )
      = ( X3 = nil_Pr1342775757158464060od_c_a ) ) ).

% remdups_eq_nil_right_iff
thf(fact_1124_remdups__eq__nil__iff,axiom,
    ! [X3: list_P6327159017948738492od_c_a] :
      ( ( ( remdup5562453435236534737od_c_a @ X3 )
        = nil_Pr1342775757158464060od_c_a )
      = ( X3 = nil_Pr1342775757158464060od_c_a ) ) ).

% remdups_eq_nil_iff
thf(fact_1125_remdups__id__iff__distinct,axiom,
    ! [Xs: list_a] :
      ( ( ( remdups_a @ Xs )
        = Xs )
      = ( distinct_a @ Xs ) ) ).

% remdups_id_iff_distinct
thf(fact_1126_distinct__remdups,axiom,
    ! [Xs: list_a] : ( distinct_a @ ( remdups_a @ Xs ) ) ).

% distinct_remdups
thf(fact_1127_remdups_Osimps_I2_J,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ( ( remdups_a @ ( cons_a @ X3 @ Xs ) )
          = ( remdups_a @ Xs ) ) )
      & ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ( ( remdups_a @ ( cons_a @ X3 @ Xs ) )
          = ( cons_a @ X3 @ ( remdups_a @ Xs ) ) ) ) ) ).

% remdups.simps(2)
thf(fact_1128_remdups_Osimps_I2_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
       => ( ( remdup5562453435236534737od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
          = ( remdup5562453435236534737od_c_a @ Xs ) ) )
      & ( ~ ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
       => ( ( remdup5562453435236534737od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
          = ( cons_P8659537020926305772od_c_a @ X3 @ ( remdup5562453435236534737od_c_a @ Xs ) ) ) ) ) ).

% remdups.simps(2)
thf(fact_1129_remdups__append2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( remdup5562453435236534737od_c_a @ ( append3884164603879732689od_c_a @ Xs @ ( remdup5562453435236534737od_c_a @ Ys ) ) )
      = ( remdup5562453435236534737od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) ) ).

% remdups_append2
thf(fact_1130_distinct__remdups__id,axiom,
    ! [Xs: list_a] :
      ( ( distinct_a @ Xs )
     => ( ( remdups_a @ Xs )
        = Xs ) ) ).

% distinct_remdups_id
thf(fact_1131_remdups_Osimps_I1_J,axiom,
    ( ( remdup5562453435236534737od_c_a @ nil_Pr1342775757158464060od_c_a )
    = nil_Pr1342775757158464060od_c_a ) ).

% remdups.simps(1)
thf(fact_1132_remove1__remdups,axiom,
    ! [Xs: list_a,X3: a] :
      ( ( distinct_a @ Xs )
     => ( ( remove1_a @ X3 @ ( remdups_a @ Xs ) )
        = ( remdups_a @ ( remove1_a @ X3 @ Xs ) ) ) ) ).

% remove1_remdups
thf(fact_1133_card__distinct,axiom,
    ! [Xs: list_a] :
      ( ( ( finite_card_a @ ( set_a2 @ Xs ) )
        = ( size_size_list_a @ Xs ) )
     => ( distinct_a @ Xs ) ) ).

% card_distinct
thf(fact_1134_distinct__card,axiom,
    ! [Xs: list_a] :
      ( ( distinct_a @ Xs )
     => ( ( finite_card_a @ ( set_a2 @ Xs ) )
        = ( size_size_list_a @ Xs ) ) ) ).

% distinct_card
thf(fact_1135_restr__simps_I1_J,axiom,
    ! [A3: set_a] :
      ( ( restri8266807399487178462od_c_a @ A3 @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% restr_simps(1)
thf(fact_1136_restr__empty_I2_J,axiom,
    ! [A3: set_a] :
      ( ( restri8266807399487178462od_c_a @ A3 @ nil_Pr1342775757158464060od_c_a )
      = nil_Pr1342775757158464060od_c_a ) ).

% restr_empty(2)
thf(fact_1137_restr__empty_I1_J,axiom,
    ! [Al: list_P6327159017948738492od_c_a] :
      ( ( restri8266807399487178462od_c_a @ bot_bot_set_a @ Al )
      = nil_Pr1342775757158464060od_c_a ) ).

% restr_empty(1)
thf(fact_1138_restr__simps_I2_J,axiom,
    ! [P5: produc2669932201312201782od_c_a,A3: set_a,Ps2: list_P6327159017948738492od_c_a] :
      ( ( ( member_a @ ( produc4225943634317630866od_c_a @ P5 ) @ A3 )
       => ( ( restri8266807399487178462od_c_a @ A3 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( cons_P8659537020926305772od_c_a @ P5 @ ( restri8266807399487178462od_c_a @ A3 @ Ps2 ) ) ) )
      & ( ~ ( member_a @ ( produc4225943634317630866od_c_a @ P5 ) @ A3 )
       => ( ( restri8266807399487178462od_c_a @ A3 @ ( cons_P8659537020926305772od_c_a @ P5 @ Ps2 ) )
          = ( restri8266807399487178462od_c_a @ A3 @ Ps2 ) ) ) ) ).

% restr_simps(2)
thf(fact_1139_FieldI1,axiom,
    ! [I: a,J2: a,R2: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J2 ) @ R2 )
     => ( member_a @ I @ ( field_a @ R2 ) ) ) ).

% FieldI1
thf(fact_1140_FieldI1,axiom,
    ! [I: produc2669932201312201782od_c_a,J2: produc2669932201312201782od_c_a,R2: set_Pr1454594181397332999od_c_a] :
      ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ I @ J2 ) @ R2 )
     => ( member5502040268208861279od_c_a @ I @ ( field_767294042308039811od_c_a @ R2 ) ) ) ).

% FieldI1
thf(fact_1141_FieldI2,axiom,
    ! [I: a,J2: a,R2: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J2 ) @ R2 )
     => ( member_a @ J2 @ ( field_a @ R2 ) ) ) ).

% FieldI2
thf(fact_1142_FieldI2,axiom,
    ! [I: produc2669932201312201782od_c_a,J2: produc2669932201312201782od_c_a,R2: set_Pr1454594181397332999od_c_a] :
      ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ I @ J2 ) @ R2 )
     => ( member5502040268208861279od_c_a @ J2 @ ( field_767294042308039811od_c_a @ R2 ) ) ) ).

% FieldI2
thf(fact_1143_shuffles_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Xa: list_P6327159017948738492od_c_a,Y3: set_li1159382662694783132od_c_a] :
      ( ( ( shuffl6718010903913732049od_c_a @ X3 @ Xa )
        = Y3 )
     => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
         => ( Y3
           != ( insert4789241225314331020od_c_a @ Xa @ bot_bo6236370880139903240od_c_a ) ) )
       => ( ( ( Xa = nil_Pr1342775757158464060od_c_a )
           => ( Y3
             != ( insert4789241225314331020od_c_a @ X3 @ bot_bo6236370880139903240od_c_a ) ) )
         => ~ ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
                ( ( X3
                  = ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
               => ! [Y: produc2669932201312201782od_c_a,Ys2: list_P6327159017948738492od_c_a] :
                    ( ( Xa
                      = ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) )
                   => ( Y3
                     != ( sup_su500200128730103920od_c_a @ ( image_1061481830026363525od_c_a @ ( cons_P8659537020926305772od_c_a @ X ) @ ( shuffl6718010903913732049od_c_a @ Xs2 @ ( cons_P8659537020926305772od_c_a @ Y @ Ys2 ) ) ) @ ( image_1061481830026363525od_c_a @ ( cons_P8659537020926305772od_c_a @ Y ) @ ( shuffl6718010903913732049od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) @ Ys2 ) ) ) ) ) ) ) ) ) ).

% shuffles.elims
thf(fact_1144_Nil__in__shuffles,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ nil_Pr1342775757158464060od_c_a @ ( shuffl6718010903913732049od_c_a @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr1342775757158464060od_c_a )
        & ( Ys = nil_Pr1342775757158464060od_c_a ) ) ) ).

% Nil_in_shuffles
thf(fact_1145_Nil__in__shufflesI,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( Xs = nil_Pr1342775757158464060od_c_a )
     => ( ( Ys = nil_Pr1342775757158464060od_c_a )
       => ( member7410604586820865893od_c_a @ nil_Pr1342775757158464060od_c_a @ ( shuffl6718010903913732049od_c_a @ Xs @ Ys ) ) ) ) ).

% Nil_in_shufflesI
thf(fact_1146_Cons__in__shuffles__rightI,axiom,
    ! [Zs: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Z: produc2669932201312201782od_c_a] :
      ( ( member7410604586820865893od_c_a @ Zs @ ( shuffl6718010903913732049od_c_a @ Xs @ Ys ) )
     => ( member7410604586820865893od_c_a @ ( cons_P8659537020926305772od_c_a @ Z @ Zs ) @ ( shuffl6718010903913732049od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ Z @ Ys ) ) ) ) ).

% Cons_in_shuffles_rightI
thf(fact_1147_Cons__in__shuffles__leftI,axiom,
    ! [Zs: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Z: produc2669932201312201782od_c_a] :
      ( ( member7410604586820865893od_c_a @ Zs @ ( shuffl6718010903913732049od_c_a @ Xs @ Ys ) )
     => ( member7410604586820865893od_c_a @ ( cons_P8659537020926305772od_c_a @ Z @ Zs ) @ ( shuffl6718010903913732049od_c_a @ ( cons_P8659537020926305772od_c_a @ Z @ Xs ) @ Ys ) ) ) ).

% Cons_in_shuffles_leftI
thf(fact_1148_shufflesE,axiom,
    ! [Zs: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ Zs @ ( shuffl6718010903913732049od_c_a @ Xs @ Ys ) )
     => ( ( ( Zs = Xs )
         => ( Ys != nil_Pr1342775757158464060od_c_a ) )
       => ( ( ( Zs = Ys )
           => ( Xs != nil_Pr1342775757158464060od_c_a ) )
         => ( ! [X: produc2669932201312201782od_c_a,Xs5: list_P6327159017948738492od_c_a] :
                ( ( Xs
                  = ( cons_P8659537020926305772od_c_a @ X @ Xs5 ) )
               => ! [Z3: produc2669932201312201782od_c_a,Zs4: list_P6327159017948738492od_c_a] :
                    ( ( Zs
                      = ( cons_P8659537020926305772od_c_a @ Z3 @ Zs4 ) )
                   => ( ( X = Z3 )
                     => ~ ( member7410604586820865893od_c_a @ Zs4 @ ( shuffl6718010903913732049od_c_a @ Xs5 @ Ys ) ) ) ) )
           => ~ ! [Y: produc2669932201312201782od_c_a,Ys7: list_P6327159017948738492od_c_a] :
                  ( ( Ys
                    = ( cons_P8659537020926305772od_c_a @ Y @ Ys7 ) )
                 => ! [Z3: produc2669932201312201782od_c_a,Zs4: list_P6327159017948738492od_c_a] :
                      ( ( Zs
                        = ( cons_P8659537020926305772od_c_a @ Z3 @ Zs4 ) )
                     => ( ( Y = Z3 )
                       => ~ ( member7410604586820865893od_c_a @ Zs4 @ ( shuffl6718010903913732049od_c_a @ Xs @ Ys7 ) ) ) ) ) ) ) ) ) ).

% shufflesE
thf(fact_1149_Cons__shuffles__subset1,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] : ( ord_le2998388488506175548od_c_a @ ( image_1061481830026363525od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 ) @ ( shuffl6718010903913732049od_c_a @ Xs @ Ys ) ) @ ( shuffl6718010903913732049od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ Ys ) ) ).

% Cons_shuffles_subset1
thf(fact_1150_Cons__shuffles__subset2,axiom,
    ! [Y3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] : ( ord_le2998388488506175548od_c_a @ ( image_1061481830026363525od_c_a @ ( cons_P8659537020926305772od_c_a @ Y3 ) @ ( shuffl6718010903913732049od_c_a @ Xs @ Ys ) ) @ ( shuffl6718010903913732049od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) ) ) ).

% Cons_shuffles_subset2
thf(fact_1151_shuffles_Osimps_I3_J,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( shuffl6718010903913732049od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) )
      = ( sup_su500200128730103920od_c_a @ ( image_1061481830026363525od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 ) @ ( shuffl6718010903913732049od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ Y3 @ Ys ) ) ) @ ( image_1061481830026363525od_c_a @ ( cons_P8659537020926305772od_c_a @ Y3 ) @ ( shuffl6718010903913732049od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) @ Ys ) ) ) ) ).

% shuffles.simps(3)
thf(fact_1152_shuffles_Osimps_I1_J,axiom,
    ! [Ys: list_P6327159017948738492od_c_a] :
      ( ( shuffl6718010903913732049od_c_a @ nil_Pr1342775757158464060od_c_a @ Ys )
      = ( insert4789241225314331020od_c_a @ Ys @ bot_bo6236370880139903240od_c_a ) ) ).

% shuffles.simps(1)
thf(fact_1153_shuffles_Osimps_I2_J,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( shuffl6718010903913732049od_c_a @ Xs @ nil_Pr1342775757158464060od_c_a )
      = ( insert4789241225314331020od_c_a @ Xs @ bot_bo6236370880139903240od_c_a ) ) ).

% shuffles.simps(2)
thf(fact_1154_distinct__disjoint__shuffles,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( distinct_a @ Xs )
     => ( ( distinct_a @ Ys )
       => ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
            = bot_bot_set_a )
         => ( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
           => ( distinct_a @ Zs ) ) ) ) ) ).

% distinct_disjoint_shuffles
thf(fact_1155_prefixes__butlast,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( set_li8663184538779258833od_c_a @ ( butlas5589356188836753934od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) ) )
      = ( collec6273869032445462695od_c_a
        @ ^ [Ys3: list_P6327159017948738492od_c_a] :
          ? [Zs3: list_P6327159017948738492od_c_a] :
            ( ( ( append3884164603879732689od_c_a @ Ys3 @ Zs3 )
              = Xs )
            & ( Zs3 != nil_Pr1342775757158464060od_c_a ) ) ) ) ).

% prefixes_butlast
thf(fact_1156_butlast__snoc,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
      ( ( butlas2723625970585074568od_c_a @ ( append3884164603879732689od_c_a @ Xs @ ( cons_P8659537020926305772od_c_a @ X3 @ nil_Pr1342775757158464060od_c_a ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_1157_merge__empty,axiom,
    ! [Ys: list_P6327159017948738492od_c_a] :
      ( ( map_of6394555147874259954od_c_a @ ( merge_433830206917135860od_c_a @ nil_Pr1342775757158464060od_c_a @ Ys ) )
      = ( map_of6394555147874259954od_c_a @ Ys ) ) ).

% merge_empty
thf(fact_1158_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
      ( ( ( Xs = nil_Pr1342775757158464060od_c_a )
       => ( ( butlas2723625970585074568od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
          = nil_Pr1342775757158464060od_c_a ) )
      & ( ( Xs != nil_Pr1342775757158464060od_c_a )
       => ( ( butlas2723625970585074568od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
          = ( cons_P8659537020926305772od_c_a @ X3 @ ( butlas2723625970585074568od_c_a @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1159_merge__append,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( map_of6394555147874259954od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
      = ( map_of6394555147874259954od_c_a @ ( merge_433830206917135860od_c_a @ Ys @ Xs ) ) ) ).

% merge_append
thf(fact_1160_distinct__butlast,axiom,
    ! [Xs: list_a] :
      ( ( distinct_a @ Xs )
     => ( distinct_a @ ( butlast_a @ Xs ) ) ) ).

% distinct_butlast
thf(fact_1161_in__set__butlastD,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
     => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_1162_in__set__butlastD,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( butlas2723625970585074568od_c_a @ Xs ) ) )
     => ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_1163_butlast_Osimps_I1_J,axiom,
    ( ( butlas2723625970585074568od_c_a @ nil_Pr1342775757158464060od_c_a )
    = nil_Pr1342775757158464060od_c_a ) ).

% butlast.simps(1)
thf(fact_1164_in__set__butlast__appendI,axiom,
    ! [X3: a,Xs: list_a,Ys: list_a] :
      ( ( ( member_a @ X3 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
        | ( member_a @ X3 @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
     => ( member_a @ X3 @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_1165_in__set__butlast__appendI,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( butlas2723625970585074568od_c_a @ Xs ) ) )
        | ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( butlas2723625970585074568od_c_a @ Ys ) ) ) )
     => ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ ( butlas2723625970585074568od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_1166_butlast__append,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( Ys = nil_Pr1342775757158464060od_c_a )
       => ( ( butlas2723625970585074568od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
          = ( butlas2723625970585074568od_c_a @ Xs ) ) )
      & ( ( Ys != nil_Pr1342775757158464060od_c_a )
       => ( ( butlas2723625970585074568od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
          = ( append3884164603879732689od_c_a @ Xs @ ( butlas2723625970585074568od_c_a @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_1167_prefix__pairs_Osimps_I2_J,axiom,
    ! [V2: produc2669932201312201782od_c_a,Va: list_P6327159017948738492od_c_a] :
      ( ( prefix1345720000202332220od_c_a @ ( cons_P8659537020926305772od_c_a @ V2 @ Va ) )
      = ( append5502610681069433026od_c_a @ ( prefix1345720000202332220od_c_a @ ( butlas2723625970585074568od_c_a @ ( cons_P8659537020926305772od_c_a @ V2 @ Va ) ) )
        @ ( map_li1057820921152479279od_c_a
          @ ^ [Ys3: list_P6327159017948738492od_c_a] : ( produc8447801329164731415od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ V2 @ Va ) )
          @ ( butlas5589356188836753934od_c_a @ ( prefix8878233912999784899od_c_a @ ( cons_P8659537020926305772od_c_a @ V2 @ Va ) ) ) ) ) ) ).

% prefix_pairs.simps(2)
thf(fact_1168_prefix__pairs_Oelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P6842272289507149357od_c_a] :
      ( ( ( prefix1345720000202332220od_c_a @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
         => ( Y3 != nil_Pr7268063790293894701od_c_a ) )
       => ~ ! [V: produc2669932201312201782od_c_a,Va2: list_P6327159017948738492od_c_a] :
              ( ( X3
                = ( cons_P8659537020926305772od_c_a @ V @ Va2 ) )
             => ( Y3
               != ( append5502610681069433026od_c_a @ ( prefix1345720000202332220od_c_a @ ( butlas2723625970585074568od_c_a @ ( cons_P8659537020926305772od_c_a @ V @ Va2 ) ) )
                  @ ( map_li1057820921152479279od_c_a
                    @ ^ [Ys3: list_P6327159017948738492od_c_a] : ( produc8447801329164731415od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ V @ Va2 ) )
                    @ ( butlas5589356188836753934od_c_a @ ( prefix8878233912999784899od_c_a @ ( cons_P8659537020926305772od_c_a @ V @ Va2 ) ) ) ) ) ) ) ) ) ).

% prefix_pairs.elims
thf(fact_1169_prefix__pairs_Opelims,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P6842272289507149357od_c_a] :
      ( ( ( prefix1345720000202332220od_c_a @ X3 )
        = Y3 )
     => ( ( accp_l2553908896995535941od_c_a @ prefix4448639506701771893od_c_a @ X3 )
       => ( ( ( X3 = nil_Pr1342775757158464060od_c_a )
           => ( ( Y3 = nil_Pr7268063790293894701od_c_a )
             => ~ ( accp_l2553908896995535941od_c_a @ prefix4448639506701771893od_c_a @ nil_Pr1342775757158464060od_c_a ) ) )
         => ~ ! [V: produc2669932201312201782od_c_a,Va2: list_P6327159017948738492od_c_a] :
                ( ( X3
                  = ( cons_P8659537020926305772od_c_a @ V @ Va2 ) )
               => ( ( Y3
                    = ( append5502610681069433026od_c_a @ ( prefix1345720000202332220od_c_a @ ( butlas2723625970585074568od_c_a @ ( cons_P8659537020926305772od_c_a @ V @ Va2 ) ) )
                      @ ( map_li1057820921152479279od_c_a
                        @ ^ [Ys3: list_P6327159017948738492od_c_a] : ( produc8447801329164731415od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ V @ Va2 ) )
                        @ ( butlas5589356188836753934od_c_a @ ( prefix8878233912999784899od_c_a @ ( cons_P8659537020926305772od_c_a @ V @ Va2 ) ) ) ) ) )
                 => ~ ( accp_l2553908896995535941od_c_a @ prefix4448639506701771893od_c_a @ ( cons_P8659537020926305772od_c_a @ V @ Va2 ) ) ) ) ) ) ) ).

% prefix_pairs.pelims
thf(fact_1170_updates__simps_I3_J,axiom,
    ! [K3: produc2669932201312201782od_c_a,Ks: list_P6327159017948738492od_c_a,V2: produc2669932201312201782od_c_a,Vs: list_P6327159017948738492od_c_a,Ps2: list_P4800044111251741997od_c_a] :
      ( ( update4093864162837753335od_c_a @ ( cons_P8659537020926305772od_c_a @ K3 @ Ks ) @ ( cons_P8659537020926305772od_c_a @ V2 @ Vs ) @ Ps2 )
      = ( update4093864162837753335od_c_a @ Ks @ Vs @ ( update9203396048450369148od_c_a @ K3 @ V2 @ Ps2 ) ) ) ).

% updates_simps(3)
thf(fact_1171_map__of__eq__Some__iff,axiom,
    ! [Xys2: list_P6327159017948738492od_c_a,X3: a,Y3: produc2133359284458116493od_c_a] :
      ( ( distinct_a @ ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Xys2 ) )
     => ( ( ( map_of6394555147874259954od_c_a @ Xys2 @ X3 )
          = ( some_P7527605245950437490od_c_a @ Y3 ) )
        = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) @ ( set_Pr5664641634098845387od_c_a @ Xys2 ) ) ) ) ).

% map_of_eq_Some_iff
thf(fact_1172_map__of__eq__Some__iff,axiom,
    ! [Xys2: list_P5767727266746070163od_c_a,X3: b,Y3: product_prod_c_a] :
      ( ( distinct_b @ ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Xys2 ) )
     => ( ( ( map_of4626007997345826889od_c_a @ Xys2 @ X3 )
          = ( some_P7240528168104800650od_c_a @ Y3 ) )
        = ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) @ ( set_Pr5098181003905080610od_c_a @ Xys2 ) ) ) ) ).

% map_of_eq_Some_iff
thf(fact_1173_map__of__eq__Some__iff,axiom,
    ! [Xys2: list_P125642481956313003od_c_a,X3: c,Y3: a] :
      ( ( distinct_c @ ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Xys2 ) )
     => ( ( ( map_of_c_a @ Xys2 @ X3 )
          = ( some_a @ Y3 ) )
        = ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) @ ( set_Product_prod_c_a2 @ Xys2 ) ) ) ) ).

% map_of_eq_Some_iff
thf(fact_1174_map__of__is__SomeI,axiom,
    ! [Xys2: list_P6327159017948738492od_c_a,X3: a,Y3: produc2133359284458116493od_c_a] :
      ( ( distinct_a @ ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Xys2 ) )
     => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) @ ( set_Pr5664641634098845387od_c_a @ Xys2 ) )
       => ( ( map_of6394555147874259954od_c_a @ Xys2 @ X3 )
          = ( some_P7527605245950437490od_c_a @ Y3 ) ) ) ) ).

% map_of_is_SomeI
thf(fact_1175_map__of__is__SomeI,axiom,
    ! [Xys2: list_P5767727266746070163od_c_a,X3: b,Y3: product_prod_c_a] :
      ( ( distinct_b @ ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Xys2 ) )
     => ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) @ ( set_Pr5098181003905080610od_c_a @ Xys2 ) )
       => ( ( map_of4626007997345826889od_c_a @ Xys2 @ X3 )
          = ( some_P7240528168104800650od_c_a @ Y3 ) ) ) ) ).

% map_of_is_SomeI
thf(fact_1176_map__of__is__SomeI,axiom,
    ! [Xys2: list_P125642481956313003od_c_a,X3: c,Y3: a] :
      ( ( distinct_c @ ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Xys2 ) )
     => ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) @ ( set_Product_prod_c_a2 @ Xys2 ) )
       => ( ( map_of_c_a @ Xys2 @ X3 )
          = ( some_a @ Y3 ) ) ) ) ).

% map_of_is_SomeI
thf(fact_1177_Some__eq__map__of__iff,axiom,
    ! [Xys2: list_P6327159017948738492od_c_a,Y3: produc2133359284458116493od_c_a,X3: a] :
      ( ( distinct_a @ ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Xys2 ) )
     => ( ( ( some_P7527605245950437490od_c_a @ Y3 )
          = ( map_of6394555147874259954od_c_a @ Xys2 @ X3 ) )
        = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) @ ( set_Pr5664641634098845387od_c_a @ Xys2 ) ) ) ) ).

% Some_eq_map_of_iff
thf(fact_1178_Some__eq__map__of__iff,axiom,
    ! [Xys2: list_P5767727266746070163od_c_a,Y3: product_prod_c_a,X3: b] :
      ( ( distinct_b @ ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Xys2 ) )
     => ( ( ( some_P7240528168104800650od_c_a @ Y3 )
          = ( map_of4626007997345826889od_c_a @ Xys2 @ X3 ) )
        = ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) @ ( set_Pr5098181003905080610od_c_a @ Xys2 ) ) ) ) ).

% Some_eq_map_of_iff
thf(fact_1179_Some__eq__map__of__iff,axiom,
    ! [Xys2: list_P125642481956313003od_c_a,Y3: a,X3: c] :
      ( ( distinct_c @ ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Xys2 ) )
     => ( ( ( some_a @ Y3 )
          = ( map_of_c_a @ Xys2 @ X3 ) )
        = ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) @ ( set_Product_prod_c_a2 @ Xys2 ) ) ) ) ).

% Some_eq_map_of_iff
thf(fact_1180_map__of__Cons__code_I2_J,axiom,
    ! [L: a,K3: a,V2: produc2133359284458116493od_c_a,Ps2: list_P6327159017948738492od_c_a] :
      ( ( ( L = K3 )
       => ( ( map_of6394555147874259954od_c_a @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ L @ V2 ) @ Ps2 ) @ K3 )
          = ( some_P7527605245950437490od_c_a @ V2 ) ) )
      & ( ( L != K3 )
       => ( ( map_of6394555147874259954od_c_a @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ L @ V2 ) @ Ps2 ) @ K3 )
          = ( map_of6394555147874259954od_c_a @ Ps2 @ K3 ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_1181_map__of__Cons__code_I2_J,axiom,
    ! [L: b,K3: b,V2: product_prod_c_a,Ps2: list_P5767727266746070163od_c_a] :
      ( ( ( L = K3 )
       => ( ( map_of4626007997345826889od_c_a @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ L @ V2 ) @ Ps2 ) @ K3 )
          = ( some_P7240528168104800650od_c_a @ V2 ) ) )
      & ( ( L != K3 )
       => ( ( map_of4626007997345826889od_c_a @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ L @ V2 ) @ Ps2 ) @ K3 )
          = ( map_of4626007997345826889od_c_a @ Ps2 @ K3 ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_1182_map__of__Cons__code_I2_J,axiom,
    ! [L: c,K3: c,V2: a,Ps2: list_P125642481956313003od_c_a] :
      ( ( ( L = K3 )
       => ( ( map_of_c_a @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ L @ V2 ) @ Ps2 ) @ K3 )
          = ( some_a @ V2 ) ) )
      & ( ( L != K3 )
       => ( ( map_of_c_a @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ L @ V2 ) @ Ps2 ) @ K3 )
          = ( map_of_c_a @ Ps2 @ K3 ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_1183_weak__map__of__SomeI,axiom,
    ! [K3: a,X3: produc2133359284458116493od_c_a,L: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ X3 ) @ ( set_Pr5664641634098845387od_c_a @ L ) )
     => ? [X: produc2133359284458116493od_c_a] :
          ( ( map_of6394555147874259954od_c_a @ L @ K3 )
          = ( some_P7527605245950437490od_c_a @ X ) ) ) ).

% weak_map_of_SomeI
thf(fact_1184_weak__map__of__SomeI,axiom,
    ! [K3: b,X3: product_prod_c_a,L: list_P5767727266746070163od_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ X3 ) @ ( set_Pr5098181003905080610od_c_a @ L ) )
     => ? [X: product_prod_c_a] :
          ( ( map_of4626007997345826889od_c_a @ L @ K3 )
          = ( some_P7240528168104800650od_c_a @ X ) ) ) ).

% weak_map_of_SomeI
thf(fact_1185_weak__map__of__SomeI,axiom,
    ! [K3: c,X3: a,L: list_P125642481956313003od_c_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ K3 @ X3 ) @ ( set_Product_prod_c_a2 @ L ) )
     => ? [X: a] :
          ( ( map_of_c_a @ L @ K3 )
          = ( some_a @ X ) ) ) ).

% weak_map_of_SomeI
thf(fact_1186_map__of__SomeD,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,K3: a,Y3: produc2133359284458116493od_c_a] :
      ( ( ( map_of6394555147874259954od_c_a @ Xs @ K3 )
        = ( some_P7527605245950437490od_c_a @ Y3 ) )
     => ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ K3 @ Y3 ) @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_1187_map__of__SomeD,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,K3: b,Y3: product_prod_c_a] :
      ( ( ( map_of4626007997345826889od_c_a @ Xs @ K3 )
        = ( some_P7240528168104800650od_c_a @ Y3 ) )
     => ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ K3 @ Y3 ) @ ( set_Pr5098181003905080610od_c_a @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_1188_map__of__SomeD,axiom,
    ! [Xs: list_P125642481956313003od_c_a,K3: c,Y3: a] :
      ( ( ( map_of_c_a @ Xs @ K3 )
        = ( some_a @ Y3 ) )
     => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ K3 @ Y3 ) @ ( set_Product_prod_c_a2 @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_1189_extract__Some__iff,axiom,
    ! [P7: produc2669932201312201782od_c_a > $o,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( extrac8890047888210018366od_c_a @ P7 @ Xs )
        = ( some_P5353416460050419325od_c_a @ ( produc3234295564627548168od_c_a @ Ys @ ( produc9095057293059448093od_c_a @ Y3 @ Zs ) ) ) )
      = ( ( Xs
          = ( append3884164603879732689od_c_a @ Ys @ ( cons_P8659537020926305772od_c_a @ Y3 @ Zs ) ) )
        & ( P7 @ Y3 )
        & ~ ? [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
              & ( P7 @ X4 ) ) ) ) ).

% extract_Some_iff
thf(fact_1190_extract__SomeE,axiom,
    ! [P7: produc2669932201312201782od_c_a > $o,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( extrac8890047888210018366od_c_a @ P7 @ Xs )
        = ( some_P5353416460050419325od_c_a @ ( produc3234295564627548168od_c_a @ Ys @ ( produc9095057293059448093od_c_a @ Y3 @ Zs ) ) ) )
     => ( ( Xs
          = ( append3884164603879732689od_c_a @ Ys @ ( cons_P8659537020926305772od_c_a @ Y3 @ Zs ) ) )
        & ( P7 @ Y3 )
        & ~ ? [X6: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X6 @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
              & ( P7 @ X6 ) ) ) ) ).

% extract_SomeE
thf(fact_1191_set__update__with__aux,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,V2: produc2133359284458116493od_c_a,K3: a,F2: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a] :
      ( ( distinct_a @ ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Xs ) )
     => ( ( set_Pr5664641634098845387od_c_a @ ( update3791493425278745523_c_a_a @ V2 @ K3 @ F2 @ Xs ) )
        = ( sup_su7950883165052597354od_c_a
          @ ( minus_2240829453383958269od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs )
            @ ( produc3776285048666693463od_c_a @ ( insert_a2 @ K3 @ bot_bot_set_a )
              @ ^ [Uu3: a] : top_to8238381149062888125od_c_a ) )
          @ ( insert8105206408023001606od_c_a
            @ ( produc5762801557034676134od_c_a @ K3
              @ ( F2
                @ ( case_o7016707287753590219od_c_a @ V2
                  @ ^ [V3: produc2133359284458116493od_c_a] : V3
                  @ ( map_of6394555147874259954od_c_a @ Xs @ K3 ) ) ) )
            @ bot_bo267491471446728962od_c_a ) ) ) ) ).

% set_update_with_aux
thf(fact_1192_set__update__with__aux,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,V2: product_prod_c_a,K3: b,F2: product_prod_c_a > product_prod_c_a] :
      ( ( distinct_b @ ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Xs ) )
     => ( ( set_Pr5098181003905080610od_c_a @ ( update338640773394215068_c_a_b @ V2 @ K3 @ F2 @ Xs ) )
        = ( sup_su5664318780038075201od_c_a
          @ ( minus_3945889481084070100od_c_a @ ( set_Pr5098181003905080610od_c_a @ Xs )
            @ ( produc127969556904116398od_c_a @ ( insert_b @ K3 @ bot_bot_set_b )
              @ ^ [Uu3: b] : top_to6792073431047673813od_c_a ) )
          @ ( insert4048788074769793885od_c_a
            @ ( produc3440865914000881661od_c_a @ K3
              @ ( F2
                @ ( case_o4475901460358254027od_c_a @ V2
                  @ ^ [V3: product_prod_c_a] : V3
                  @ ( map_of4626007997345826889od_c_a @ Xs @ K3 ) ) ) )
            @ bot_bo8402128546026624985od_c_a ) ) ) ) ).

% set_update_with_aux
thf(fact_1193_set__update__with__aux,axiom,
    ! [Xs: list_P125642481956313003od_c_a,V2: a,K3: c,F2: a > a] :
      ( ( distinct_c @ ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Xs ) )
     => ( ( set_Product_prod_c_a2 @ ( update_with_aux_a_c @ V2 @ K3 @ F2 @ Xs ) )
        = ( sup_su1776960780389684313od_c_a
          @ ( minus_5545738918597211372od_c_a @ ( set_Product_prod_c_a2 @ Xs )
            @ ( product_Sigma_c_a @ ( insert_c @ K3 @ bot_bot_set_c )
              @ ^ [Uu3: c] : top_top_set_a ) )
          @ ( insert8183397254950580085od_c_a
            @ ( product_Pair_c_a @ K3
              @ ( F2
                @ ( case_option_a_a @ V2
                  @ ^ [V3: a] : V3
                  @ ( map_of_c_a @ Xs @ K3 ) ) ) )
            @ bot_bo2086078286244720881od_c_a ) ) ) ) ).

% set_update_with_aux
thf(fact_1194_mem__Sigma__iff,axiom,
    ! [A: a,B: a,A3: set_a,B5: a > set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A3 @ B5 ) )
      = ( ( member_a @ A @ A3 )
        & ( member_a @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1195_mem__Sigma__iff,axiom,
    ! [A: a,B: produc2669932201312201782od_c_a,A3: set_a,B5: a > set_Pr8357624050790843030od_c_a] :
      ( ( member5750183862411248520od_c_a @ ( produc4410401784823481679od_c_a @ A @ B ) @ ( produc1654692551143595008od_c_a @ A3 @ B5 ) )
      = ( ( member_a @ A @ A3 )
        & ( member5502040268208861279od_c_a @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1196_mem__Sigma__iff,axiom,
    ! [A: produc2669932201312201782od_c_a,B: a,A3: set_Pr8357624050790843030od_c_a,B5: produc2669932201312201782od_c_a > set_a] :
      ( ( member3614124511450460696_c_a_a @ ( produc174534808117382111_c_a_a @ A @ B ) @ ( produc6642197611292271248_c_a_a @ A3 @ B5 ) )
      = ( ( member5502040268208861279od_c_a @ A @ A3 )
        & ( member_a @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1197_mem__Sigma__iff,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a,A3: set_Pr8357624050790843030od_c_a,B5: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a] :
      ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ B ) @ ( produc7033095811749742024od_c_a @ A3 @ B5 ) )
      = ( ( member5502040268208861279od_c_a @ A @ A3 )
        & ( member5502040268208861279od_c_a @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1198_mem__Sigma__iff,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A3: set_a,B5: a > set_Pr272002107461315949od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ ( produc3776285048666693463od_c_a @ A3 @ B5 ) )
      = ( ( member_a @ A @ A3 )
        & ( member1160383752307813046od_c_a @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1199_mem__Sigma__iff,axiom,
    ! [A: b,B: product_prod_c_a,A3: set_b,B5: b > set_Product_prod_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ ( produc127969556904116398od_c_a @ A3 @ B5 ) )
      = ( ( member_b @ A @ A3 )
        & ( member5074992350434858958od_c_a @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1200_mem__Sigma__iff,axiom,
    ! [A: c,B: a,A3: set_c,B5: c > set_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ ( product_Sigma_c_a @ A3 @ B5 ) )
      = ( ( member_c @ A @ A3 )
        & ( member_a @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1201_SigmaI,axiom,
    ! [A: a,A3: set_a,B: a,B5: a > set_a] :
      ( ( member_a @ A @ A3 )
     => ( ( member_a @ B @ ( B5 @ A ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1202_SigmaI,axiom,
    ! [A: a,A3: set_a,B: produc2669932201312201782od_c_a,B5: a > set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ A @ A3 )
     => ( ( member5502040268208861279od_c_a @ B @ ( B5 @ A ) )
       => ( member5750183862411248520od_c_a @ ( produc4410401784823481679od_c_a @ A @ B ) @ ( produc1654692551143595008od_c_a @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1203_SigmaI,axiom,
    ! [A: produc2669932201312201782od_c_a,A3: set_Pr8357624050790843030od_c_a,B: a,B5: produc2669932201312201782od_c_a > set_a] :
      ( ( member5502040268208861279od_c_a @ A @ A3 )
     => ( ( member_a @ B @ ( B5 @ A ) )
       => ( member3614124511450460696_c_a_a @ ( produc174534808117382111_c_a_a @ A @ B ) @ ( produc6642197611292271248_c_a_a @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1204_SigmaI,axiom,
    ! [A: produc2669932201312201782od_c_a,A3: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a,B5: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ A3 )
     => ( ( member5502040268208861279od_c_a @ B @ ( B5 @ A ) )
       => ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ B ) @ ( produc7033095811749742024od_c_a @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1205_SigmaI,axiom,
    ! [A: a,A3: set_a,B: produc2133359284458116493od_c_a,B5: a > set_Pr272002107461315949od_c_a] :
      ( ( member_a @ A @ A3 )
     => ( ( member1160383752307813046od_c_a @ B @ ( B5 @ A ) )
       => ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ ( produc3776285048666693463od_c_a @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1206_SigmaI,axiom,
    ! [A: b,A3: set_b,B: product_prod_c_a,B5: b > set_Product_prod_c_a] :
      ( ( member_b @ A @ A3 )
     => ( ( member5074992350434858958od_c_a @ B @ ( B5 @ A ) )
       => ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ ( produc127969556904116398od_c_a @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1207_SigmaI,axiom,
    ! [A: c,A3: set_c,B: a,B5: c > set_a] :
      ( ( member_c @ A @ A3 )
     => ( ( member_a @ B @ ( B5 @ A ) )
       => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ ( product_Sigma_c_a @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1208_insert__Times__insert,axiom,
    ! [A: a,A3: set_a,B: produc2133359284458116493od_c_a,B5: set_Pr272002107461315949od_c_a] :
      ( ( produc3776285048666693463od_c_a @ ( insert_a2 @ A @ A3 )
        @ ^ [Uu3: a] : ( insert4048788074769793885od_c_a @ B @ B5 ) )
      = ( insert8105206408023001606od_c_a @ ( produc5762801557034676134od_c_a @ A @ B )
        @ ( sup_su7950883165052597354od_c_a
          @ ( produc3776285048666693463od_c_a @ A3
            @ ^ [Uu3: a] : ( insert4048788074769793885od_c_a @ B @ B5 ) )
          @ ( produc3776285048666693463od_c_a @ ( insert_a2 @ A @ A3 )
            @ ^ [Uu3: a] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1209_insert__Times__insert,axiom,
    ! [A: b,A3: set_b,B: product_prod_c_a,B5: set_Product_prod_c_a] :
      ( ( produc127969556904116398od_c_a @ ( insert_b @ A @ A3 )
        @ ^ [Uu3: b] : ( insert8183397254950580085od_c_a @ B @ B5 ) )
      = ( insert4048788074769793885od_c_a @ ( produc3440865914000881661od_c_a @ A @ B )
        @ ( sup_su5664318780038075201od_c_a
          @ ( produc127969556904116398od_c_a @ A3
            @ ^ [Uu3: b] : ( insert8183397254950580085od_c_a @ B @ B5 ) )
          @ ( produc127969556904116398od_c_a @ ( insert_b @ A @ A3 )
            @ ^ [Uu3: b] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1210_insert__Times__insert,axiom,
    ! [A: c,A3: set_c,B: a,B5: set_a] :
      ( ( product_Sigma_c_a @ ( insert_c @ A @ A3 )
        @ ^ [Uu3: c] : ( insert_a2 @ B @ B5 ) )
      = ( insert8183397254950580085od_c_a @ ( product_Pair_c_a @ A @ B )
        @ ( sup_su1776960780389684313od_c_a
          @ ( product_Sigma_c_a @ A3
            @ ^ [Uu3: c] : ( insert_a2 @ B @ B5 ) )
          @ ( product_Sigma_c_a @ ( insert_c @ A @ A3 )
            @ ^ [Uu3: c] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1211_member__product,axiom,
    ! [X3: produc2669932201312201782od_c_a,A3: set_a,B5: set_Pr272002107461315949od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( produc4596061662451530337od_c_a @ A3 @ B5 ) )
      = ( member5502040268208861279od_c_a @ X3
        @ ( produc3776285048666693463od_c_a @ A3
          @ ^ [Uu3: a] : B5 ) ) ) ).

% member_product
thf(fact_1212_mem__Times__iff,axiom,
    ! [X3: product_prod_a_a,A3: set_a,B5: set_a] :
      ( ( member1426531477525435216od_a_a @ X3
        @ ( product_Sigma_a_a @ A3
          @ ^ [Uu3: a] : B5 ) )
      = ( ( member_a @ ( product_fst_a_a @ X3 ) @ A3 )
        & ( member_a @ ( product_snd_a_a @ X3 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_1213_mem__Times__iff,axiom,
    ! [X3: produc871828282075189343od_c_a,A3: set_a,B5: set_Pr8357624050790843030od_c_a] :
      ( ( member5750183862411248520od_c_a @ X3
        @ ( produc1654692551143595008od_c_a @ A3
          @ ^ [Uu3: a] : B5 ) )
      = ( ( member_a @ ( produc3141450672998377275od_c_a @ X3 ) @ A3 )
        & ( member5502040268208861279od_c_a @ ( produc367105772464697469od_c_a @ X3 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_1214_mem__Times__iff,axiom,
    ! [X3: produc7959140967969177327_c_a_a,A3: set_Pr8357624050790843030od_c_a,B5: set_a] :
      ( ( member3614124511450460696_c_a_a @ X3
        @ ( produc6642197611292271248_c_a_a @ A3
          @ ^ [Uu3: produc2669932201312201782od_c_a] : B5 ) )
      = ( ( member5502040268208861279od_c_a @ ( produc8128955733147053515_c_a_a @ X3 ) @ A3 )
        & ( member_a @ ( produc5354610832613373709_c_a_a @ X3 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_1215_mem__Times__iff,axiom,
    ! [X3: produc1647090505208597031od_c_a,A3: set_Pr8357624050790843030od_c_a,B5: set_Pr8357624050790843030od_c_a] :
      ( ( member3417889999246264912od_c_a @ X3
        @ ( produc7033095811749742024od_c_a @ A3
          @ ^ [Uu3: produc2669932201312201782od_c_a] : B5 ) )
      = ( ( member5502040268208861279od_c_a @ ( produc2189173959066220547od_c_a @ X3 ) @ A3 )
        & ( member5502040268208861279od_c_a @ ( produc3204662071173827397od_c_a @ X3 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_1216_mem__Times__iff,axiom,
    ! [X3: produc2669932201312201782od_c_a,A3: set_a,B5: set_Pr272002107461315949od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3
        @ ( produc3776285048666693463od_c_a @ A3
          @ ^ [Uu3: a] : B5 ) )
      = ( ( member_a @ ( produc4225943634317630866od_c_a @ X3 ) @ A3 )
        & ( member1160383752307813046od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_1217_clearjunk_Osimps_I1_J,axiom,
    ( ( clearj517495611650374705od_c_a @ nil_Pr1342775757158464060od_c_a )
    = nil_Pr1342775757158464060od_c_a ) ).

% clearjunk.simps(1)
thf(fact_1218_SigmaE2,axiom,
    ! [A: a,B: a,A3: set_a,B5: a > set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A3 @ B5 ) )
     => ~ ( ( member_a @ A @ A3 )
         => ~ ( member_a @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1219_SigmaE2,axiom,
    ! [A: a,B: produc2669932201312201782od_c_a,A3: set_a,B5: a > set_Pr8357624050790843030od_c_a] :
      ( ( member5750183862411248520od_c_a @ ( produc4410401784823481679od_c_a @ A @ B ) @ ( produc1654692551143595008od_c_a @ A3 @ B5 ) )
     => ~ ( ( member_a @ A @ A3 )
         => ~ ( member5502040268208861279od_c_a @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1220_SigmaE2,axiom,
    ! [A: produc2669932201312201782od_c_a,B: a,A3: set_Pr8357624050790843030od_c_a,B5: produc2669932201312201782od_c_a > set_a] :
      ( ( member3614124511450460696_c_a_a @ ( produc174534808117382111_c_a_a @ A @ B ) @ ( produc6642197611292271248_c_a_a @ A3 @ B5 ) )
     => ~ ( ( member5502040268208861279od_c_a @ A @ A3 )
         => ~ ( member_a @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1221_SigmaE2,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a,A3: set_Pr8357624050790843030od_c_a,B5: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a] :
      ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ B ) @ ( produc7033095811749742024od_c_a @ A3 @ B5 ) )
     => ~ ( ( member5502040268208861279od_c_a @ A @ A3 )
         => ~ ( member5502040268208861279od_c_a @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1222_SigmaE2,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A3: set_a,B5: a > set_Pr272002107461315949od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ ( produc3776285048666693463od_c_a @ A3 @ B5 ) )
     => ~ ( ( member_a @ A @ A3 )
         => ~ ( member1160383752307813046od_c_a @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1223_SigmaE2,axiom,
    ! [A: b,B: product_prod_c_a,A3: set_b,B5: b > set_Product_prod_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ ( produc127969556904116398od_c_a @ A3 @ B5 ) )
     => ~ ( ( member_b @ A @ A3 )
         => ~ ( member5074992350434858958od_c_a @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1224_SigmaE2,axiom,
    ! [A: c,B: a,A3: set_c,B5: c > set_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ ( product_Sigma_c_a @ A3 @ B5 ) )
     => ~ ( ( member_c @ A @ A3 )
         => ~ ( member_a @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1225_SigmaD2,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A3: set_a,B5: a > set_Pr272002107461315949od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ ( produc3776285048666693463od_c_a @ A3 @ B5 ) )
     => ( member1160383752307813046od_c_a @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_1226_SigmaD2,axiom,
    ! [A: b,B: product_prod_c_a,A3: set_b,B5: b > set_Product_prod_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ ( produc127969556904116398od_c_a @ A3 @ B5 ) )
     => ( member5074992350434858958od_c_a @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_1227_SigmaD2,axiom,
    ! [A: c,B: a,A3: set_c,B5: c > set_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ ( product_Sigma_c_a @ A3 @ B5 ) )
     => ( member_a @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_1228_SigmaD1,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A3: set_a,B5: a > set_Pr272002107461315949od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ ( produc3776285048666693463od_c_a @ A3 @ B5 ) )
     => ( member_a @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1229_SigmaD1,axiom,
    ! [A: b,B: product_prod_c_a,A3: set_b,B5: b > set_Product_prod_c_a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ ( produc127969556904116398od_c_a @ A3 @ B5 ) )
     => ( member_b @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1230_SigmaD1,axiom,
    ! [A: c,B: a,A3: set_c,B5: c > set_a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ ( product_Sigma_c_a @ A3 @ B5 ) )
     => ( member_c @ A @ A3 ) ) ).

% SigmaD1
thf(fact_1231_SigmaE,axiom,
    ! [C: product_prod_a_a,A3: set_a,B5: a > set_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( product_Sigma_a_a @ A3 @ B5 ) )
     => ~ ! [X: a] :
            ( ( member_a @ X @ A3 )
           => ! [Y: a] :
                ( ( member_a @ Y @ ( B5 @ X ) )
               => ( C
                 != ( product_Pair_a_a @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_1232_SigmaE,axiom,
    ! [C: produc871828282075189343od_c_a,A3: set_a,B5: a > set_Pr8357624050790843030od_c_a] :
      ( ( member5750183862411248520od_c_a @ C @ ( produc1654692551143595008od_c_a @ A3 @ B5 ) )
     => ~ ! [X: a] :
            ( ( member_a @ X @ A3 )
           => ! [Y: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y @ ( B5 @ X ) )
               => ( C
                 != ( produc4410401784823481679od_c_a @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_1233_SigmaE,axiom,
    ! [C: produc7959140967969177327_c_a_a,A3: set_Pr8357624050790843030od_c_a,B5: produc2669932201312201782od_c_a > set_a] :
      ( ( member3614124511450460696_c_a_a @ C @ ( produc6642197611292271248_c_a_a @ A3 @ B5 ) )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ A3 )
           => ! [Y: a] :
                ( ( member_a @ Y @ ( B5 @ X ) )
               => ( C
                 != ( produc174534808117382111_c_a_a @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_1234_SigmaE,axiom,
    ! [C: produc1647090505208597031od_c_a,A3: set_Pr8357624050790843030od_c_a,B5: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a] :
      ( ( member3417889999246264912od_c_a @ C @ ( produc7033095811749742024od_c_a @ A3 @ B5 ) )
     => ~ ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ A3 )
           => ! [Y: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y @ ( B5 @ X ) )
               => ( C
                 != ( produc4004402368211057175od_c_a @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_1235_SigmaE,axiom,
    ! [C: produc2669932201312201782od_c_a,A3: set_a,B5: a > set_Pr272002107461315949od_c_a] :
      ( ( member5502040268208861279od_c_a @ C @ ( produc3776285048666693463od_c_a @ A3 @ B5 ) )
     => ~ ! [X: a] :
            ( ( member_a @ X @ A3 )
           => ! [Y: produc2133359284458116493od_c_a] :
                ( ( member1160383752307813046od_c_a @ Y @ ( B5 @ X ) )
               => ( C
                 != ( produc5762801557034676134od_c_a @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_1236_SigmaE,axiom,
    ! [C: produc2133359284458116493od_c_a,A3: set_b,B5: b > set_Product_prod_c_a] :
      ( ( member1160383752307813046od_c_a @ C @ ( produc127969556904116398od_c_a @ A3 @ B5 ) )
     => ~ ! [X: b] :
            ( ( member_b @ X @ A3 )
           => ! [Y: product_prod_c_a] :
                ( ( member5074992350434858958od_c_a @ Y @ ( B5 @ X ) )
               => ( C
                 != ( produc3440865914000881661od_c_a @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_1237_SigmaE,axiom,
    ! [C: product_prod_c_a,A3: set_c,B5: c > set_a] :
      ( ( member5074992350434858958od_c_a @ C @ ( product_Sigma_c_a @ A3 @ B5 ) )
     => ~ ! [X: c] :
            ( ( member_c @ X @ A3 )
           => ! [Y: a] :
                ( ( member_a @ Y @ ( B5 @ X ) )
               => ( C
                 != ( product_Pair_c_a @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_1238_swap__product,axiom,
    ! [A3: set_Pr272002107461315949od_c_a,B5: set_a] :
      ( ( image_6327630916522945251od_c_a
        @ ( produc3213192282681773593od_c_a
          @ ^ [I3: produc2133359284458116493od_c_a,J: a] : ( produc5762801557034676134od_c_a @ J @ I3 ) )
        @ ( produc1012633894336974329_c_a_a @ A3
          @ ^ [Uu3: produc2133359284458116493od_c_a] : B5 ) )
      = ( produc3776285048666693463od_c_a @ B5
        @ ^ [Uu3: a] : A3 ) ) ).

% swap_product
thf(fact_1239_swap__product,axiom,
    ! [A3: set_Product_prod_c_a,B5: set_b] :
      ( ( image_3580325567873573649od_c_a
        @ ( produc2689806817602706119od_c_a
          @ ^ [I3: product_prod_c_a,J: b] : ( produc3440865914000881661od_c_a @ J @ I3 ) )
        @ ( produc4348705685012232418_c_a_b @ A3
          @ ^ [Uu3: product_prod_c_a] : B5 ) )
      = ( produc127969556904116398od_c_a @ B5
        @ ^ [Uu3: b] : A3 ) ) ).

% swap_product
thf(fact_1240_swap__product,axiom,
    ! [A3: set_a,B5: set_c] :
      ( ( image_5613013798208914369od_c_a
        @ ( produc2777847913367752695od_c_a
          @ ^ [I3: a,J: c] : ( product_Pair_c_a @ J @ I3 ) )
        @ ( product_Sigma_a_c @ A3
          @ ^ [Uu3: a] : B5 ) )
      = ( product_Sigma_c_a @ B5
        @ ^ [Uu3: c] : A3 ) ) ).

% swap_product
thf(fact_1241_lists__length__Suc__eq,axiom,
    ! [A3: set_Pr8357624050790843030od_c_a,N: nat] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [Xs3: list_P6327159017948738492od_c_a] :
            ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs3 ) @ A3 )
            & ( ( size_s3386368156187063848od_c_a @ Xs3 )
              = ( suc @ N ) ) ) )
      = ( image_5676263954566856032od_c_a
        @ ( produc7017227194855770646od_c_a
          @ ^ [Xs3: list_P6327159017948738492od_c_a,N3: produc2669932201312201782od_c_a] : ( cons_P8659537020926305772od_c_a @ N3 @ Xs3 ) )
        @ ( produc2650631260902794562od_c_a
          @ ( collec6273869032445462695od_c_a
            @ ^ [Xs3: list_P6327159017948738492od_c_a] :
                ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs3 ) @ A3 )
                & ( ( size_s3386368156187063848od_c_a @ Xs3 )
                  = N ) ) )
          @ ^ [Uu3: list_P6327159017948738492od_c_a] : A3 ) ) ) ).

% lists_length_Suc_eq
thf(fact_1242_Sigma__def,axiom,
    ( produc3776285048666693463od_c_a
    = ( ^ [A6: set_a,B7: a > set_Pr272002107461315949od_c_a] :
          ( comple6722362287630405891od_c_a
          @ ( image_2946797484331234845od_c_a
            @ ^ [X4: a] :
                ( comple6722362287630405891od_c_a
                @ ( image_3137457815373564494od_c_a
                  @ ^ [Y4: produc2133359284458116493od_c_a] : ( insert8105206408023001606od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y4 ) @ bot_bo267491471446728962od_c_a )
                  @ ( B7 @ X4 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1243_Sigma__def,axiom,
    ( produc127969556904116398od_c_a
    = ( ^ [A6: set_b,B7: b > set_Product_prod_c_a] :
          ( comple6943640584117399770od_c_a
          @ ( image_3253712296381159475od_c_a
            @ ^ [X4: b] :
                ( comple6943640584117399770od_c_a
                @ ( image_3866828679125363981od_c_a
                  @ ^ [Y4: product_prod_c_a] : ( insert4048788074769793885od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y4 ) @ bot_bo8402128546026624985od_c_a )
                  @ ( B7 @ X4 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1244_Sigma__def,axiom,
    ( product_Sigma_c_a
    = ( ^ [A6: set_c,B7: c > set_a] :
          ( comple7150381169481872114od_c_a
          @ ( image_5632462938777209482od_c_a
            @ ^ [X4: c] :
                ( comple7150381169481872114od_c_a
                @ ( image_3150212591781473292od_c_a
                  @ ^ [Y4: a] : ( insert8183397254950580085od_c_a @ ( product_Pair_c_a @ X4 @ Y4 ) @ bot_bo2086078286244720881od_c_a )
                  @ ( B7 @ X4 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1245_extract__Cons__code,axiom,
    ! [P7: produc2669932201312201782od_c_a > $o,X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( P7 @ X3 )
       => ( ( extrac8890047888210018366od_c_a @ P7 @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
          = ( some_P5353416460050419325od_c_a @ ( produc3234295564627548168od_c_a @ nil_Pr1342775757158464060od_c_a @ ( produc9095057293059448093od_c_a @ X3 @ Xs ) ) ) ) )
      & ( ~ ( P7 @ X3 )
       => ( ( extrac8890047888210018366od_c_a @ P7 @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
          = ( case_o8062426749775953093od_c_a @ none_P309522368803027705od_c_a
            @ ( produc4798796627778141057od_c_a
              @ ^ [Ys3: list_P6327159017948738492od_c_a] :
                  ( produc8182279139527428908od_c_a
                  @ ^ [Y4: produc2669932201312201782od_c_a,Zs3: list_P6327159017948738492od_c_a] : ( some_P5353416460050419325od_c_a @ ( produc3234295564627548168od_c_a @ ( cons_P8659537020926305772od_c_a @ X3 @ Ys3 ) @ ( produc9095057293059448093od_c_a @ Y4 @ Zs3 ) ) ) ) )
            @ ( extrac8890047888210018366od_c_a @ P7 @ Xs ) ) ) ) ) ).

% extract_Cons_code
thf(fact_1246_map__of__eq__empty__iff,axiom,
    ! [Xys2: list_P6327159017948738492od_c_a] :
      ( ( ( map_of6394555147874259954od_c_a @ Xys2 )
        = ( ^ [X4: a] : none_P6162255670594677998od_c_a ) )
      = ( Xys2 = nil_Pr1342775757158464060od_c_a ) ) ).

% map_of_eq_empty_iff
thf(fact_1247_empty__eq__map__of__iff,axiom,
    ! [Xys2: list_P6327159017948738492od_c_a] :
      ( ( ( ^ [X4: a] : none_P6162255670594677998od_c_a )
        = ( map_of6394555147874259954od_c_a @ Xys2 ) )
      = ( Xys2 = nil_Pr1342775757158464060od_c_a ) ) ).

% empty_eq_map_of_iff
thf(fact_1248_top__empty__eq2,axiom,
    ( top_to1994068701486593775_c_a_o
    = ( ^ [X4: a,Y4: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y4 ) @ top_to7710394033828511718od_c_a ) ) ) ).

% top_empty_eq2
thf(fact_1249_top__empty__eq2,axiom,
    ( top_to3665770462873310552_c_a_o
    = ( ^ [X4: b,Y4: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y4 ) @ top_to8238381149062888125od_c_a ) ) ) ).

% top_empty_eq2
thf(fact_1250_top__empty__eq2,axiom,
    ( top_top_c_a_o
    = ( ^ [X4: c,Y4: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y4 ) @ top_to6792073431047673813od_c_a ) ) ) ).

% top_empty_eq2
thf(fact_1251_find_Osimps_I2_J,axiom,
    ! [P7: produc2669932201312201782od_c_a > $o,X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( P7 @ X3 )
       => ( ( find_P7062026247731025490od_c_a @ P7 @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
          = ( some_P4170542693840572955od_c_a @ X3 ) ) )
      & ( ~ ( P7 @ X3 )
       => ( ( find_P7062026247731025490od_c_a @ P7 @ ( cons_P8659537020926305772od_c_a @ X3 @ Xs ) )
          = ( find_P7062026247731025490od_c_a @ P7 @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_1252_find__set,axiom,
    ! [P7: a > $o,Xs: list_a,X3: a] :
      ( ( ( find_a @ P7 @ Xs )
        = ( some_a @ X3 ) )
     => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).

% find_set
thf(fact_1253_find__set,axiom,
    ! [P7: produc2669932201312201782od_c_a > $o,Xs: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
      ( ( ( find_P7062026247731025490od_c_a @ P7 @ Xs )
        = ( some_P4170542693840572955od_c_a @ X3 ) )
     => ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ).

% find_set
thf(fact_1254_extract__Nil__code,axiom,
    ! [P7: produc2669932201312201782od_c_a > $o] :
      ( ( extrac8890047888210018366od_c_a @ P7 @ nil_Pr1342775757158464060od_c_a )
      = none_P309522368803027705od_c_a ) ).

% extract_Nil_code
thf(fact_1255_find__remove__2_H__set__rev,axiom,
    ! [X3: a,Xs: list_a,Y3: a,Ys: list_a,P7: a > a > $o,Prev2: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ( member_a @ Y3 @ ( set_a2 @ Ys ) )
       => ( ( P7 @ X3 @ Y3 )
         => ( ( find_remove_2_a_a @ P7 @ Xs @ Ys @ Prev2 )
           != none_P7310308508618416183list_a ) ) ) ) ).

% find_remove_2'_set_rev
thf(fact_1256_find__remove__2_H__set__rev,axiom,
    ! [X3: a,Xs: list_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a,P7: a > produc2669932201312201782od_c_a > $o,Prev2: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ( ( P7 @ X3 @ Y3 )
         => ( ( find_r934662375107687272od_c_a @ P7 @ Xs @ Ys @ Prev2 )
           != none_P8512630930997372159list_a ) ) ) ) ).

% find_remove_2'_set_rev
thf(fact_1257_find__remove__2_H__set__rev,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: a,Ys: list_a,P7: produc2669932201312201782od_c_a > a > $o,Prev2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( member_a @ Y3 @ ( set_a2 @ Ys ) )
       => ( ( P7 @ X3 @ Y3 )
         => ( ( find_r5922167435256363512_c_a_a @ P7 @ Xs @ Ys @ Prev2 )
           != none_P8903023469769883703od_c_a ) ) ) ) ).

% find_remove_2'_set_rev
thf(fact_1258_find__remove__2_H__set__rev,axiom,
    ! [X3: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Y3: produc2669932201312201782od_c_a,Ys: list_P6327159017948738492od_c_a,P7: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o,Prev2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ( member5502040268208861279od_c_a @ Y3 @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ( ( P7 @ X3 @ Y3 )
         => ( ( find_r6904947931933041968od_c_a @ P7 @ Xs @ Ys @ Prev2 )
           != none_P8214577678242665727od_c_a ) ) ) ) ).

% find_remove_2'_set_rev

% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X3: list_a,Y3: list_a] :
      ( ( if_list_a @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X3: list_a,Y3: list_a] :
      ( ( if_list_a @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_T,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( if_lis1795051829816687490od_c_a @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_T,axiom,
    ! [X3: list_P6327159017948738492od_c_a,Y3: list_P6327159017948738492od_c_a] :
      ( ( if_lis1795051829816687490od_c_a @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_3_1_If_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_T,axiom,
    ! [P7: $o] :
      ( ( P7 = $true )
      | ( P7 = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_T,axiom,
    ! [X3: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
      ( ( if_set5301400340663253090od_c_a @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_T,axiom,
    ! [X3: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
      ( ( if_set5301400340663253090od_c_a @ $true @ X3 @ Y3 )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    $false ).

%------------------------------------------------------------------------------