TPTP Problem File: SLH0285^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FSM_Tests/0054_Test_Suite_IO/prob_00179_007506__20400934_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2206 ( 560 unt; 929 typ;   0 def)
%            Number of atoms       : 3427 (1389 equ;   0 cnn)
%            Maximal formula atoms :   14 (   2 avg)
%            Number of connectives : 14764 ( 350   ~;  34   |; 409   &;12553   @)
%                                         (   0 <=>;1418  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   30 (   7 avg)
%            Number of types       :  245 ( 244 usr)
%            Number of type conns  : 2165 (2165   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  688 ( 685 usr;  26 con; 0-6 aty)
%            Number of variables   : 4620 ( 707   ^;3694   !; 219   ?;4620   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:29:20.364
%------------------------------------------------------------------------------
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thf(sy_c_Adaptive__Test__Case_Oatc__to__io__set_001tf__a_001tf__b_001tf__c_001tf__a,type,
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thf(sy_c_FSM_Ofrom__FSM_001tf__a_001tf__a_001t__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J,type,
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thf(sy_c_FSM_Oinitial_001t__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J_001tf__b_001tf__c,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    member1533839430763132368od_b_c: produc5878855887281848999od_b_c > set_Pr3842837961795859335od_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__a_J,type,
    member8373667307600438368_b_c_a: produc2267687419657828279_b_c_a > set_Pr3924463358016699415_b_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__FSM__Ofsm_Itf__d_Mtf__b_Mtf__c_J_Mt__Product____Type__Oprod_Itf__d_Mtf__d_J_J_Mt__Product____Type__Oprod_It__FSM__Ofsm_Itf__d_Mtf__b_Mtf__c_J_Mt__Product____Type__Oprod_Itf__d_Mtf__d_J_J_J,type,
    member6383636634632210000od_d_d: produc8430528367412472615od_d_d > set_Pr5335332815339889671od_d_d > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
    member1214319639984524048_set_a: produc913200677138634471_set_a > set_Pr3538216614922818759_set_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_Mtf__a_J,type,
    member8769086359091386215et_a_a: produc774156279037233726et_a_a > set_Pr1224833622948082078et_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J_J_J,type,
    member289795379306782288_a_b_c: produc4599717558597148967_a_b_c > set_Pr454692187318948871_a_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J_J_Mtf__a_J,type,
    member3104622976533456642_b_c_a: produc7890929710919849305_b_c_a > set_Pr7394172031124433337_b_c_a > $o ).

thf(sy_c_member_001t__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,type,
    member6508725278956418977od_c_a: produc3205353004677322104od_c_a > set_Pr2640428967771533528od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
    member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J_J,type,
    member997585926662434933_a_b_c: produc8838969699183119820_a_b_c > set_Pr2640454425625433516_a_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__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,type,
    member3484049051949443214od_c_a: produc3220416805844688357od_c_a > set_Pr5230154892316275013od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    member8874627675880996032od_b_c: produc2768647787938385943od_b_c > set_Pr6531061718924289143od_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
    member7961485381496024569_set_a: produc9189927338296647888_set_a > set_Pr5883750332947488560_set_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_Mtf__c_J,type,
    member7862447941013992593od_b_c: product_prod_b_c > set_Product_prod_b_c > $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__d_Mtf__d_J,type,
    member2287536781371869328od_d_d: product_prod_d_d > set_Product_prod_d_d > $o ).

thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    member6985331446368301687od_b_c: set_li6436108459499378894od_b_c > set_se3924713247505902254od_b_c > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $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_c_member_001tf__d,type,
    member_d: d > set_d > $o ).

thf(sy_v_A____,type,
    a2: fsm_d_b_c ).

thf(sy_v_M,type,
    m: fsm_a_b_c ).

thf(sy_v_P____,type,
    p: fsm_a_b_c ).

thf(sy_v_T,type,
    t: test_t899597172106206243_b_c_d ).

thf(sy_v_atcs____,type,
    atcs: product_prod_a_a > set_Pr9130538228769002824od_d_d ).

thf(sy_v_d____,type,
    d2: produc1703568184450464039_set_a ).

thf(sy_v_ioA____,type,
    ioA: list_P903359562653991662od_b_c ).

thf(sy_v_io____,type,
    io: list_P903359562653991662od_b_c ).

thf(sy_v_m,type,
    m2: nat ).

thf(sy_v_p2____,type,
    p2: list_P6327159017948738492od_c_a ).

thf(sy_v_p____,type,
    p3: list_P6327159017948738492od_c_a ).

thf(sy_v_prs____,type,
    prs: set_Pr2640454425625433516_a_b_c ).

thf(sy_v_pt____,type,
    pt: list_P6327159017948738492od_c_a ).

thf(sy_v_q_H____,type,
    q: a ).

thf(sy_v_q____,type,
    q2: a ).

thf(sy_v_rd__targets____,type,
    rd_targets: produc3220416805844688357od_c_a > set_a ).

thf(sy_v_repetition__sets____,type,
    repetition_sets: list_P3660316430366008877_set_a ).

thf(sy_v_t1____,type,
    t1: d ).

thf(sy_v_t2____,type,
    t2: d ).

thf(sy_v_tps____,type,
    tps: a > set_li1159382662694783132od_c_a ).

% Relevant facts (1267)
thf(fact_0__092_060open_062path_AM_A_IFSM_Oinitial_AM_J_A_Ip_A_064_Apt_J_092_060close_062,axiom,
    path_a_b_c @ m @ ( initial_a_b_c @ m ) @ ( append3884164603879732689od_c_a @ p3 @ pt ) ).

% \<open>path M (FSM.initial M) (p @ pt)\<close>
thf(fact_1__092_060open_062target_Aq_Apt_A_061_Atarget_A_IFSM_Oinitial_AM_J_A_Ip_A_064_Apt_J_092_060close_062,axiom,
    ( ( target_a_b_c @ q2 @ pt )
    = ( target_a_b_c @ ( initial_a_b_c @ m ) @ ( append3884164603879732689od_c_a @ p3 @ pt ) ) ) ).

% \<open>target q pt = target (FSM.initial M) (p @ pt)\<close>
thf(fact_2__092_060open_062pt_A_092_060in_062_Atps_Aq_092_060close_062,axiom,
    member7410604586820865893od_c_a @ pt @ ( tps @ q2 ) ).

% \<open>pt \<in> tps q\<close>
thf(fact_3__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 ) @ p3 )
    = q2 ) ).

% \<open>target (FSM.initial M) p = q\<close>
thf(fact_4__092_060open_062target_A_IFSM_Oinitial_AP_J_Ap_A_061_Aq_092_060close_062,axiom,
    ( ( target_a_b_c @ ( initial_a_b_c @ p ) @ p3 )
    = q2 ) ).

% \<open>target (FSM.initial P) p = q\<close>
thf(fact_5__092_060open_062path_AM_Aq_A_Ipt_A_064_Ap2_J_092_060close_062,axiom,
    path_a_b_c @ m @ q2 @ ( append3884164603879732689od_c_a @ pt @ p2 ) ).

% \<open>path M q (pt @ p2)\<close>
thf(fact_6__092_060open_062path_AM_A_IFSM_Oinitial_AM_J_Ap_092_060close_062,axiom,
    path_a_b_c @ m @ ( initial_a_b_c @ m ) @ p3 ).

% \<open>path M (FSM.initial M) p\<close>
thf(fact_7__092_060open_062FSM_Oinitial_AP_A_061_AFSM_Oinitial_AM_092_060close_062,axiom,
    ( ( initial_a_b_c @ p )
    = ( initial_a_b_c @ m ) ) ).

% \<open>FSM.initial P = FSM.initial M\<close>
thf(fact_8__092_060open_062ioA_A_092_060in_062_Aatc__to__io__set_A_Ifrom__FSM_AM_A_Itarget_Aq_Apt_J_J_AA_092_060close_062,axiom,
    member6330420149250801815od_b_c @ ioA @ ( adapti7464086540814349463_b_c_d @ ( from_FSM_a_b_c @ m @ ( target_a_b_c @ q2 @ pt ) ) @ a2 ) ).

% \<open>ioA \<in> atc_to_io_set (from_FSM M (target q pt)) A\<close>
thf(fact_9__092_060open_062q_H_A_092_060in_062_Ard__targets_A_Iq_M_Apt_J_092_060close_062,axiom,
    member_a @ q @ ( rd_targets @ ( produc3580641852002036821od_c_a @ q2 @ pt ) ) ).

% \<open>q' \<in> rd_targets (q, pt)\<close>
thf(fact_10__092_060open_062io_A_092_060in_062_Atest__suite__to__io_AM_AT_092_060close_062,axiom,
    member6330420149250801815od_b_c @ io @ ( test_S7399740988479946633_b_c_d @ m @ t ) ).

% \<open>io \<in> test_suite_to_io M T\<close>
thf(fact_11_path__target__is__state,axiom,
    ! [M: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P )
     => ( member_a @ ( target_a_b_c @ Q @ P ) @ ( states_a_b_c @ M ) ) ) ).

% path_target_is_state
thf(fact_12_assms,axiom,
    test_i4204698558891958671_b_c_d @ t @ m @ m2 ).

% assms
thf(fact_13__092_060open_062ioA_A_092_060in_062_AL_A_Ifrom__FSM_AM_A_Itarget_Aq_Apt_J_J_092_060close_062,axiom,
    member6330420149250801815od_b_c @ ioA @ ( lS_a_b_c @ ( from_FSM_a_b_c @ m @ ( target_a_b_c @ q2 @ pt ) ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ m @ ( target_a_b_c @ q2 @ pt ) ) ) ) ).

% \<open>ioA \<in> L (from_FSM M (target q pt))\<close>
thf(fact_14_fsm__initial,axiom,
    ! [M: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M ) @ ( states_a_b_c @ M ) ) ).

% fsm_initial
thf(fact_15__092_060open_062is__submachine_AP_AM_092_060close_062,axiom,
    is_submachine_a_b_c @ p @ m ).

% \<open>is_submachine P M\<close>
thf(fact_16_t5,axiom,
    ! [Q: a] :
      ( ( member_a @ Q @ ( states_a_b_c @ m ) )
     => ? [X: produc1703568184450464039_set_a] :
          ( ( member7983343339038529360_set_a @ X @ ( set_Pr7878197726666307004_set_a @ repetition_sets ) )
          & ( member_a @ Q @ ( produc9088895665703139587_set_a @ X ) ) ) ) ).

% t5
thf(fact_17__092_060open_062path_AP_A_IFSM_Oinitial_AP_J_Ap_092_060close_062,axiom,
    path_a_b_c @ p @ ( initial_a_b_c @ p ) @ p3 ).

% \<open>path P (FSM.initial P) p\<close>
thf(fact_18_from__from,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q12: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q12 @ ( states_a_b_c @ M ) )
       => ( ( from_FSM_a_b_c @ ( from_FSM_a_b_c @ M @ Q1 ) @ Q12 )
          = ( from_FSM_a_b_c @ M @ Q12 ) ) ) ) ).

% from_from
thf(fact_19_from__FSM__simps_I5_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( states_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = ( states_a_b_c @ M ) ) ) ).

% from_FSM_simps(5)
thf(fact_20_from__FSM__simps_I1_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = Q ) ) ).

% from_FSM_simps(1)
thf(fact_21_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_22_submachine__simps_I1_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( initial_a_b_c @ A )
        = ( initial_a_b_c @ B ) ) ) ).

% submachine_simps(1)
thf(fact_23_p__source_Ocases,axiom,
    ! [X2: produc3220416805844688357od_c_a] :
      ~ ! [Q2: a,P3: list_P6327159017948738492od_c_a] :
          ( X2
         != ( produc3580641852002036821od_c_a @ Q2 @ P3 ) ) ).

% p_source.cases
thf(fact_24_visited__states_Ocases,axiom,
    ! [X2: produc3220416805844688357od_c_a] :
      ~ ! [Q2: a,P3: list_P6327159017948738492od_c_a] :
          ( X2
         != ( produc3580641852002036821od_c_a @ Q2 @ P3 ) ) ).

% visited_states.cases
thf(fact_25_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_26_language__prefix,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M @ Q ) )
     => ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M @ Q ) ) ) ).

% language_prefix
thf(fact_27_submachine__from,axiom,
    ! [S: fsm_a_b_c,M: fsm_a_b_c,Q: a] :
      ( ( is_submachine_a_b_c @ S @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ S ) )
       => ( is_submachine_a_b_c @ ( from_FSM_a_b_c @ S @ Q ) @ ( from_FSM_a_b_c @ M @ Q ) ) ) ) ).

% submachine_from
thf(fact_28_submachine__path,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( path_a_b_c @ A @ Q @ P )
       => ( path_a_b_c @ B @ Q @ P ) ) ) ).

% submachine_path
thf(fact_29_from__FSM__language,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( lS_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) )
        = ( lS_a_b_c @ M @ Q ) ) ) ).

% from_FSM_language
thf(fact_30_from__FSM__path__initial,axiom,
    ! [Q: a,M: fsm_a_b_c,P: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( path_a_b_c @ M @ Q @ P )
        = ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) @ P ) ) ) ).

% from_FSM_path_initial
thf(fact_31_submachine__transitive,axiom,
    ! [S: fsm_a_b_c,M: fsm_a_b_c,S2: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ S @ M )
     => ( ( is_submachine_a_b_c @ S2 @ S )
       => ( is_submachine_a_b_c @ S2 @ M ) ) ) ).

% submachine_transitive
thf(fact_32_submachine__path__initial,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c,P: list_P6327159017948738492od_c_a] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( path_a_b_c @ A @ ( initial_a_b_c @ A ) @ P )
       => ( path_a_b_c @ B @ ( initial_a_b_c @ B ) @ P ) ) ) ).

% submachine_path_initial
thf(fact_33_from__FSM__path__rev__initial,axiom,
    ! [M: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P )
     => ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ Q @ P ) ) ).

% from_FSM_path_rev_initial
thf(fact_34_from__FSM__path,axiom,
    ! [Q: a,M: fsm_a_b_c,Q3: a,P: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ Q3 @ P )
       => ( path_a_b_c @ M @ Q3 @ P ) ) ) ).

% from_FSM_path
thf(fact_35_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_36_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_37_path__begin__state,axiom,
    ! [M: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P )
     => ( member_a @ Q @ ( states_a_b_c @ M ) ) ) ).

% path_begin_state
thf(fact_38_path__target__append,axiom,
    ! [Q1: a,P1: list_P6327159017948738492od_c_a,Q22: a,P2: list_P6327159017948738492od_c_a,Q32: a] :
      ( ( ( target_a_b_c @ Q1 @ P1 )
        = Q22 )
     => ( ( ( target_a_b_c @ Q22 @ P2 )
          = Q32 )
       => ( ( target_a_b_c @ Q1 @ ( append3884164603879732689od_c_a @ P1 @ P2 ) )
          = Q32 ) ) ) ).

% path_target_append
thf(fact_39_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_40_reachable__def,axiom,
    ( reachable_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q4: 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 )
            = Q4 ) ) ) ) ).

% reachable_def
thf(fact_41__092_060open_062_Ipt_A_064_Ap2_M_Ad_J_A_092_060in_062_Am__traversal__paths__with__witness_AM_Aq_Arepetition__sets_Am_092_060close_062,axiom,
    member6493638300815972283_set_a @ ( produc6508649944902238466_set_a @ ( append3884164603879732689od_c_a @ pt @ p2 ) @ d2 ) @ ( traver6421790936065540696_a_b_c @ m @ q2 @ repetition_sets @ m2 ) ).

% \<open>(pt @ p2, d) \<in> m_traversal_paths_with_witness M q repetition_sets m\<close>
thf(fact_42__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p2_Ad_O_A_Ipt_A_064_Ap2_M_Ad_J_A_092_060in_062_Am__traversal__paths__with__witness_AM_Aq_Arepetition__sets_Am_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [P22: list_P6327159017948738492od_c_a,D: produc1703568184450464039_set_a] :
        ~ ( member6493638300815972283_set_a @ ( produc6508649944902238466_set_a @ ( append3884164603879732689od_c_a @ pt @ P22 ) @ D ) @ ( traver6421790936065540696_a_b_c @ m @ q2 @ repetition_sets @ m2 ) ) ).

% \<open>\<And>thesis. (\<And>p2 d. (pt @ p2, d) \<in> m_traversal_paths_with_witness M q repetition_sets m \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_43_mem__Collect__eq,axiom,
    ! [A2: produc4859908366599477352od_d_d,P5: produc4859908366599477352od_d_d > $o] :
      ( ( member73601632213084689od_d_d @ A2 @ ( collec5362705567740981075od_d_d @ P5 ) )
      = ( P5 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
    ! [A2: produc8838969699183119820_a_b_c,P5: produc8838969699183119820_a_b_c > $o] :
      ( ( member997585926662434933_a_b_c @ A2 @ ( collec3867759823522760631_a_b_c @ P5 ) )
      = ( P5 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
    ! [A2: list_P903359562653991662od_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ( member6330420149250801815od_b_c @ A2 @ ( collec2280997390073109977od_b_c @ P5 ) )
      = ( P5 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
    ! [A2: a,P5: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P5 ) )
      = ( P5 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A2: list_P6327159017948738492od_c_a,P5: list_P6327159017948738492od_c_a > $o] :
      ( ( member7410604586820865893od_c_a @ A2 @ ( collec6273869032445462695od_c_a @ P5 ) )
      = ( P5 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A2: set_li6436108459499378894od_b_c,P5: set_li6436108459499378894od_b_c > $o] :
      ( ( member6985331446368301687od_b_c @ A2 @ ( collec1823247286932916153od_b_c @ P5 ) )
      = ( P5 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_49_Collect__mem__eq,axiom,
    ! [A: set_Pr9130538228769002824od_d_d] :
      ( ( collec5362705567740981075od_d_d
        @ ^ [X3: produc4859908366599477352od_d_d] : ( member73601632213084689od_d_d @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
    ! [A: set_Pr2640454425625433516_a_b_c] :
      ( ( collec3867759823522760631_a_b_c
        @ ^ [X3: produc8838969699183119820_a_b_c] : ( member997585926662434933_a_b_c @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X3: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X3: a] : ( member_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
    ! [A: set_li1159382662694783132od_c_a] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [X3: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
    ! [A: set_se3924713247505902254od_b_c] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [X3: set_li6436108459499378894od_b_c] : ( member6985331446368301687od_b_c @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_55_Collect__cong,axiom,
    ! [P5: list_P6327159017948738492od_c_a > $o,Q5: list_P6327159017948738492od_c_a > $o] :
      ( ! [X: list_P6327159017948738492od_c_a] :
          ( ( P5 @ X )
          = ( Q5 @ X ) )
     => ( ( collec6273869032445462695od_c_a @ P5 )
        = ( collec6273869032445462695od_c_a @ Q5 ) ) ) ).

% Collect_cong
thf(fact_56_Collect__cong,axiom,
    ! [P5: set_li6436108459499378894od_b_c > $o,Q5: set_li6436108459499378894od_b_c > $o] :
      ( ! [X: set_li6436108459499378894od_b_c] :
          ( ( P5 @ X )
          = ( Q5 @ X ) )
     => ( ( collec1823247286932916153od_b_c @ P5 )
        = ( collec1823247286932916153od_b_c @ Q5 ) ) ) ).

% Collect_cong
thf(fact_57_append_Oassoc,axiom,
    ! [A2: list_P6327159017948738492od_c_a,B2: list_P6327159017948738492od_c_a,C: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ ( append3884164603879732689od_c_a @ A2 @ B2 ) @ C )
      = ( append3884164603879732689od_c_a @ A2 @ ( append3884164603879732689od_c_a @ B2 @ C ) ) ) ).

% append.assoc
thf(fact_58_append_Oassoc,axiom,
    ! [A2: list_P903359562653991662od_b_c,B2: list_P903359562653991662od_b_c,C: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ ( append2547753245680614915od_b_c @ A2 @ B2 ) @ C )
      = ( append2547753245680614915od_b_c @ A2 @ ( append2547753245680614915od_b_c @ B2 @ C ) ) ) ).

% append.assoc
thf(fact_59_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_60_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_61_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_62_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_63_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_64_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_65_prod_Oinject,axiom,
    ! [X1: d,X22: d,Y1: d,Y2: d] :
      ( ( ( product_Pair_d_d @ X1 @ X22 )
        = ( product_Pair_d_d @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_66_prod_Oinject,axiom,
    ! [X1: b,X22: c,Y1: b,Y2: c] :
      ( ( ( product_Pair_b_c @ X1 @ X22 )
        = ( product_Pair_b_c @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_67_prod_Oinject,axiom,
    ! [X1: a,X22: list_P6327159017948738492od_c_a,Y1: a,Y2: list_P6327159017948738492od_c_a] :
      ( ( ( produc3580641852002036821od_c_a @ X1 @ X22 )
        = ( produc3580641852002036821od_c_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_68_prod_Oinject,axiom,
    ! [X1: a,X22: fsm_a_b_c,Y1: a,Y2: fsm_a_b_c] :
      ( ( ( produc2795441082327197628_a_b_c @ X1 @ X22 )
        = ( produc2795441082327197628_a_b_c @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_69_prod_Oinject,axiom,
    ! [X1: a,X22: a,Y1: a,Y2: a] :
      ( ( ( product_Pair_a_a @ X1 @ X22 )
        = ( product_Pair_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_70_old_Oprod_Oinject,axiom,
    ! [A2: d,B2: d,A3: d,B3: d] :
      ( ( ( product_Pair_d_d @ A2 @ B2 )
        = ( product_Pair_d_d @ A3 @ B3 ) )
      = ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_71_old_Oprod_Oinject,axiom,
    ! [A2: b,B2: c,A3: b,B3: c] :
      ( ( ( product_Pair_b_c @ A2 @ B2 )
        = ( product_Pair_b_c @ A3 @ B3 ) )
      = ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_72_old_Oprod_Oinject,axiom,
    ! [A2: a,B2: list_P6327159017948738492od_c_a,A3: a,B3: list_P6327159017948738492od_c_a] :
      ( ( ( produc3580641852002036821od_c_a @ A2 @ B2 )
        = ( produc3580641852002036821od_c_a @ A3 @ B3 ) )
      = ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_73_old_Oprod_Oinject,axiom,
    ! [A2: a,B2: fsm_a_b_c,A3: a,B3: fsm_a_b_c] :
      ( ( ( produc2795441082327197628_a_b_c @ A2 @ B2 )
        = ( produc2795441082327197628_a_b_c @ A3 @ B3 ) )
      = ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_74_old_Oprod_Oinject,axiom,
    ! [A2: a,B2: a,A3: a,B3: a] :
      ( ( ( product_Pair_a_a @ A2 @ B2 )
        = ( product_Pair_a_a @ A3 @ B3 ) )
      = ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_75_list__set__sym,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Y: list_P6327159017948738492od_c_a] :
      ( ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ X2 @ Y ) )
      = ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Y @ X2 ) ) ) ).

% list_set_sym
thf(fact_76_list__set__sym,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ X2 @ Y ) )
      = ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Y @ X2 ) ) ) ).

% list_set_sym
thf(fact_77_list__set__sym,axiom,
    ! [X2: list_P3660316430366008877_set_a,Y: list_P3660316430366008877_set_a] :
      ( ( set_Pr7878197726666307004_set_a @ ( append6456390900168921282_set_a @ X2 @ Y ) )
      = ( set_Pr7878197726666307004_set_a @ ( append6456390900168921282_set_a @ Y @ X2 ) ) ) ).

% list_set_sym
thf(fact_78_list__set__sym,axiom,
    ! [X2: list_l8907847357763382004od_b_c,Y: list_l8907847357763382004od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ X2 @ Y ) )
      = ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Y @ X2 ) ) ) ).

% list_set_sym
thf(fact_79_list__prefix__elem,axiom,
    ! [X2: produc4859908366599477352od_d_d,Xs: list_P4069430723938770798od_d_d,Ys: list_P4069430723938770798od_d_d] :
      ( ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ ( append6772029687554554499od_d_d @ Xs @ Ys ) ) )
     => ( ~ ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ Ys ) )
       => ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_80_list__prefix__elem,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Xs: list_P5219093185649440210_a_b_c,Ys: list_P5219093185649440210_a_b_c] :
      ( ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ ( append2545967182438564583_a_b_c @ Xs @ Ys ) ) )
     => ( ~ ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Ys ) )
       => ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_81_list__prefix__elem,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a] :
      ( ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ ( append2943488786620085207od_c_a @ Xs @ Ys ) ) )
     => ( ~ ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ Ys ) )
       => ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_82_list__prefix__elem,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a] :
      ( ( member_a @ X2 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) )
     => ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
       => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_83_list__prefix__elem,axiom,
    ! [X2: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X2 @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) )
     => ( ~ ( member5502040268208861279od_c_a @ X2 @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ( member5502040268208861279od_c_a @ X2 @ ( set_Pr5664641634098845387od_c_a @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_84_list__prefix__elem,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs @ Ys ) ) )
     => ( ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Ys ) )
       => ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_85_list__prefix__elem,axiom,
    ! [X2: produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a,Ys: list_P3660316430366008877_set_a] :
      ( ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ ( append6456390900168921282_set_a @ Xs @ Ys ) ) )
     => ( ~ ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Ys ) )
       => ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_86_list__prefix__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs @ Ys ) ) )
     => ( ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Ys ) )
       => ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_87_list__concat__non__elem,axiom,
    ! [X2: produc4859908366599477352od_d_d,Xs: list_P4069430723938770798od_d_d,Ys: list_P4069430723938770798od_d_d] :
      ( ~ ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
     => ( ~ ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ Ys ) )
       => ~ ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ ( append6772029687554554499od_d_d @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_88_list__concat__non__elem,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Xs: list_P5219093185649440210_a_b_c,Ys: list_P5219093185649440210_a_b_c] :
      ( ~ ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
     => ( ~ ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Ys ) )
       => ~ ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ ( append2545967182438564583_a_b_c @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_89_list__concat__non__elem,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a,Ys: list_l2826717958560394946od_c_a] :
      ( ~ ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ Xs ) )
     => ( ~ ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ Ys ) )
       => ~ ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ ( append2943488786620085207od_c_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_90_list__concat__non__elem,axiom,
    ! [X2: a,Xs: list_a,Ys: list_a] :
      ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
       => ~ ( member_a @ X2 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_91_list__concat__non__elem,axiom,
    ! [X2: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X2 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
     => ( ~ ( member5502040268208861279od_c_a @ X2 @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ~ ( member5502040268208861279od_c_a @ X2 @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_92_list__concat__non__elem,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ Ys ) )
       => ~ ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_93_list__concat__non__elem,axiom,
    ! [X2: produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a,Ys: list_P3660316430366008877_set_a] :
      ( ~ ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Xs ) )
     => ( ~ ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Ys ) )
       => ~ ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ ( append6456390900168921282_set_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_94_list__concat__non__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Ys ) )
       => ~ ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_95__092_060open_062_Iq_M_AP_J_A_092_060in_062_Aprs_092_060close_062,axiom,
    member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ q2 @ p ) @ prs ).

% \<open>(q, P) \<in> prs\<close>
thf(fact_96_does__distinguish_Ocases,axiom,
    ! [X2: produc7320101719827038340od_c_a] :
      ~ ! [M3: fsm_a_1173558889237794545od_c_a,Q13: a,Q23: a,Io: list_P6327159017948738492od_c_a] :
          ( X2
         != ( produc6043144016332479348od_c_a @ M3 @ ( produc2817528682913273214od_c_a @ Q13 @ ( produc3580641852002036821od_c_a @ Q23 @ Io ) ) ) ) ).

% does_distinguish.cases
thf(fact_97_completed__path_Ocases,axiom,
    ! [X2: produc2300901281425032105od_c_a] :
      ~ ! [M3: fsm_a_b_c,Q2: a,P3: list_P6327159017948738492od_c_a] :
          ( X2
         != ( produc5669890062478367513od_c_a @ M3 @ ( produc3580641852002036821od_c_a @ Q2 @ P3 ) ) ) ).

% completed_path.cases
thf(fact_98_Pair__inject,axiom,
    ! [A2: d,B2: d,A3: d,B3: d] :
      ( ( ( product_Pair_d_d @ A2 @ B2 )
        = ( product_Pair_d_d @ A3 @ B3 ) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_99_Pair__inject,axiom,
    ! [A2: b,B2: c,A3: b,B3: c] :
      ( ( ( product_Pair_b_c @ A2 @ B2 )
        = ( product_Pair_b_c @ A3 @ B3 ) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_100_Pair__inject,axiom,
    ! [A2: a,B2: list_P6327159017948738492od_c_a,A3: a,B3: list_P6327159017948738492od_c_a] :
      ( ( ( produc3580641852002036821od_c_a @ A2 @ B2 )
        = ( produc3580641852002036821od_c_a @ A3 @ B3 ) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_101_Pair__inject,axiom,
    ! [A2: a,B2: fsm_a_b_c,A3: a,B3: fsm_a_b_c] :
      ( ( ( produc2795441082327197628_a_b_c @ A2 @ B2 )
        = ( produc2795441082327197628_a_b_c @ A3 @ B3 ) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_102_Pair__inject,axiom,
    ! [A2: a,B2: a,A3: a,B3: a] :
      ( ( ( product_Pair_a_a @ A2 @ B2 )
        = ( product_Pair_a_a @ A3 @ B3 ) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_103_prod__cases,axiom,
    ! [P5: product_prod_d_d > $o,P: product_prod_d_d] :
      ( ! [A4: d,B4: d] : ( P5 @ ( product_Pair_d_d @ A4 @ B4 ) )
     => ( P5 @ P ) ) ).

% prod_cases
thf(fact_104_prod__cases,axiom,
    ! [P5: product_prod_b_c > $o,P: product_prod_b_c] :
      ( ! [A4: b,B4: c] : ( P5 @ ( product_Pair_b_c @ A4 @ B4 ) )
     => ( P5 @ P ) ) ).

% prod_cases
thf(fact_105_prod__cases,axiom,
    ! [P5: produc3220416805844688357od_c_a > $o,P: produc3220416805844688357od_c_a] :
      ( ! [A4: a,B4: list_P6327159017948738492od_c_a] : ( P5 @ ( produc3580641852002036821od_c_a @ A4 @ B4 ) )
     => ( P5 @ P ) ) ).

% prod_cases
thf(fact_106_prod__cases,axiom,
    ! [P5: produc8838969699183119820_a_b_c > $o,P: produc8838969699183119820_a_b_c] :
      ( ! [A4: a,B4: fsm_a_b_c] : ( P5 @ ( produc2795441082327197628_a_b_c @ A4 @ B4 ) )
     => ( P5 @ P ) ) ).

% prod_cases
thf(fact_107_prod__cases,axiom,
    ! [P5: product_prod_a_a > $o,P: product_prod_a_a] :
      ( ! [A4: a,B4: a] : ( P5 @ ( product_Pair_a_a @ A4 @ B4 ) )
     => ( P5 @ P ) ) ).

% prod_cases
thf(fact_108_surj__pair,axiom,
    ! [P: product_prod_d_d] :
    ? [X: d,Y3: d] :
      ( P
      = ( product_Pair_d_d @ X @ Y3 ) ) ).

% surj_pair
thf(fact_109_surj__pair,axiom,
    ! [P: product_prod_b_c] :
    ? [X: b,Y3: c] :
      ( P
      = ( product_Pair_b_c @ X @ Y3 ) ) ).

% surj_pair
thf(fact_110_surj__pair,axiom,
    ! [P: produc3220416805844688357od_c_a] :
    ? [X: a,Y3: list_P6327159017948738492od_c_a] :
      ( P
      = ( produc3580641852002036821od_c_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_111_surj__pair,axiom,
    ! [P: produc8838969699183119820_a_b_c] :
    ? [X: a,Y3: fsm_a_b_c] :
      ( P
      = ( produc2795441082327197628_a_b_c @ X @ Y3 ) ) ).

% surj_pair
thf(fact_112_surj__pair,axiom,
    ! [P: product_prod_a_a] :
    ? [X: a,Y3: a] :
      ( P
      = ( product_Pair_a_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_113_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_d_d] :
      ~ ! [A4: d,B4: d] :
          ( Y
         != ( product_Pair_d_d @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_114_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_b_c] :
      ~ ! [A4: b,B4: c] :
          ( Y
         != ( product_Pair_b_c @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_115_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3220416805844688357od_c_a] :
      ~ ! [A4: a,B4: list_P6327159017948738492od_c_a] :
          ( Y
         != ( produc3580641852002036821od_c_a @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_116_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8838969699183119820_a_b_c] :
      ~ ! [A4: a,B4: fsm_a_b_c] :
          ( Y
         != ( produc2795441082327197628_a_b_c @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_117_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_a] :
      ~ ! [A4: a,B4: a] :
          ( Y
         != ( product_Pair_a_a @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_118_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_119_append__eq__append__conv2,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,Ts: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Ys )
        = ( append2547753245680614915od_b_c @ Zs @ Ts ) )
      = ( ? [Us: list_P903359562653991662od_b_c] :
            ( ( ( Xs
                = ( append2547753245680614915od_b_c @ Zs @ Us ) )
              & ( ( append2547753245680614915od_b_c @ Us @ Ys )
                = Ts ) )
            | ( ( ( append2547753245680614915od_b_c @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append2547753245680614915od_b_c @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_120_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_121_append__eq__appendI,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Xs1: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Us2: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append2547753245680614915od_b_c @ Xs1 @ Us2 ) )
       => ( ( append2547753245680614915od_b_c @ Xs @ Ys )
          = ( append2547753245680614915od_b_c @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_122_fst__conv,axiom,
    ! [X1: d,X22: d] :
      ( ( product_fst_d_d @ ( product_Pair_d_d @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_123_fst__conv,axiom,
    ! [X1: b,X22: c] :
      ( ( product_fst_b_c @ ( product_Pair_b_c @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_124_fst__conv,axiom,
    ! [X1: a,X22: list_P6327159017948738492od_c_a] :
      ( ( produc4637894536740104257od_c_a @ ( produc3580641852002036821od_c_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_125_fst__conv,axiom,
    ! [X1: a,X22: a] :
      ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_126_fst__conv,axiom,
    ! [X1: set_a,X22: set_a] :
      ( ( produc9088895665703139587_set_a @ ( produc9088192753505129239_set_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_127_fst__conv,axiom,
    ! [X1: a,X22: fsm_a_b_c] :
      ( ( produc2620512763296580520_a_b_c @ ( produc2795441082327197628_a_b_c @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_128_fst__conv,axiom,
    ! [X1: b,X22: product_prod_c_a] :
      ( ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_129_fst__conv,axiom,
    ! [X1: c,X22: a] :
      ( ( product_fst_c_a @ ( product_Pair_c_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_130_fst__conv,axiom,
    ! [X1: list_P6327159017948738492od_c_a,X22: produc1703568184450464039_set_a] :
      ( ( produc3242630202250842350_set_a @ ( produc6508649944902238466_set_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_131_fst__eqD,axiom,
    ! [X2: d,Y: d,A2: d] :
      ( ( ( product_fst_d_d @ ( product_Pair_d_d @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_132_fst__eqD,axiom,
    ! [X2: b,Y: c,A2: b] :
      ( ( ( product_fst_b_c @ ( product_Pair_b_c @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_133_fst__eqD,axiom,
    ! [X2: a,Y: list_P6327159017948738492od_c_a,A2: a] :
      ( ( ( produc4637894536740104257od_c_a @ ( produc3580641852002036821od_c_a @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_134_fst__eqD,axiom,
    ! [X2: a,Y: a,A2: a] :
      ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_135_fst__eqD,axiom,
    ! [X2: set_a,Y: set_a,A2: set_a] :
      ( ( ( produc9088895665703139587_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_136_fst__eqD,axiom,
    ! [X2: a,Y: fsm_a_b_c,A2: a] :
      ( ( ( produc2620512763296580520_a_b_c @ ( produc2795441082327197628_a_b_c @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_137_fst__eqD,axiom,
    ! [X2: b,Y: product_prod_c_a,A2: b] :
      ( ( ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_138_fst__eqD,axiom,
    ! [X2: c,Y: a,A2: c] :
      ( ( ( product_fst_c_a @ ( product_Pair_c_a @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_139_fst__eqD,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Y: produc1703568184450464039_set_a,A2: list_P6327159017948738492od_c_a] :
      ( ( ( produc3242630202250842350_set_a @ ( produc6508649944902238466_set_a @ X2 @ Y ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_140__092_060open_062_IA_M_At1_M_At2_J_A_092_060in_062_Aatcs_A_Itarget_Aq_Apt_M_Aq_H_J_092_060close_062,axiom,
    member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ a2 @ ( product_Pair_d_d @ t1 @ t2 ) ) @ ( atcs @ ( product_Pair_a_a @ ( target_a_b_c @ q2 @ pt ) @ q ) ) ).

% \<open>(A, t1, t2) \<in> atcs (target q pt, q')\<close>
thf(fact_141_eq__fst__iff,axiom,
    ! [A2: d,P: product_prod_d_d] :
      ( ( A2
        = ( product_fst_d_d @ P ) )
      = ( ? [B5: d] :
            ( P
            = ( product_Pair_d_d @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_142_eq__fst__iff,axiom,
    ! [A2: b,P: product_prod_b_c] :
      ( ( A2
        = ( product_fst_b_c @ P ) )
      = ( ? [B5: c] :
            ( P
            = ( product_Pair_b_c @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_143_eq__fst__iff,axiom,
    ! [A2: a,P: produc3220416805844688357od_c_a] :
      ( ( A2
        = ( produc4637894536740104257od_c_a @ P ) )
      = ( ? [B5: list_P6327159017948738492od_c_a] :
            ( P
            = ( produc3580641852002036821od_c_a @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_144_eq__fst__iff,axiom,
    ! [A2: a,P: product_prod_a_a] :
      ( ( A2
        = ( product_fst_a_a @ P ) )
      = ( ? [B5: a] :
            ( P
            = ( product_Pair_a_a @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_145_eq__fst__iff,axiom,
    ! [A2: set_a,P: produc1703568184450464039_set_a] :
      ( ( A2
        = ( produc9088895665703139587_set_a @ P ) )
      = ( ? [B5: set_a] :
            ( P
            = ( produc9088192753505129239_set_a @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_146_eq__fst__iff,axiom,
    ! [A2: a,P: produc8838969699183119820_a_b_c] :
      ( ( A2
        = ( produc2620512763296580520_a_b_c @ P ) )
      = ( ? [B5: fsm_a_b_c] :
            ( P
            = ( produc2795441082327197628_a_b_c @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_147_eq__fst__iff,axiom,
    ! [A2: b,P: produc2133359284458116493od_c_a] :
      ( ( A2
        = ( produc301949074746585577od_c_a @ P ) )
      = ( ? [B5: product_prod_c_a] :
            ( P
            = ( produc3440865914000881661od_c_a @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_148_eq__fst__iff,axiom,
    ! [A2: c,P: product_prod_c_a] :
      ( ( A2
        = ( product_fst_c_a @ P ) )
      = ( ? [B5: a] :
            ( P
            = ( product_Pair_c_a @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_149_eq__fst__iff,axiom,
    ! [A2: list_P6327159017948738492od_c_a,P: produc7049025859877086738_set_a] :
      ( ( A2
        = ( produc3242630202250842350_set_a @ P ) )
      = ( ? [B5: produc1703568184450464039_set_a] :
            ( P
            = ( produc6508649944902238466_set_a @ A2 @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_150_fstI,axiom,
    ! [X2: product_prod_d_d,Y: d,Z: d] :
      ( ( X2
        = ( product_Pair_d_d @ Y @ Z ) )
     => ( ( product_fst_d_d @ X2 )
        = Y ) ) ).

% fstI
thf(fact_151_fstI,axiom,
    ! [X2: product_prod_b_c,Y: b,Z: c] :
      ( ( X2
        = ( product_Pair_b_c @ Y @ Z ) )
     => ( ( product_fst_b_c @ X2 )
        = Y ) ) ).

% fstI
thf(fact_152_fstI,axiom,
    ! [X2: produc3220416805844688357od_c_a,Y: a,Z: list_P6327159017948738492od_c_a] :
      ( ( X2
        = ( produc3580641852002036821od_c_a @ Y @ Z ) )
     => ( ( produc4637894536740104257od_c_a @ X2 )
        = Y ) ) ).

% fstI
thf(fact_153_fstI,axiom,
    ! [X2: product_prod_a_a,Y: a,Z: a] :
      ( ( X2
        = ( product_Pair_a_a @ Y @ Z ) )
     => ( ( product_fst_a_a @ X2 )
        = Y ) ) ).

% fstI
thf(fact_154_fstI,axiom,
    ! [X2: produc1703568184450464039_set_a,Y: set_a,Z: set_a] :
      ( ( X2
        = ( produc9088192753505129239_set_a @ Y @ Z ) )
     => ( ( produc9088895665703139587_set_a @ X2 )
        = Y ) ) ).

% fstI
thf(fact_155_fstI,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Y: a,Z: fsm_a_b_c] :
      ( ( X2
        = ( produc2795441082327197628_a_b_c @ Y @ Z ) )
     => ( ( produc2620512763296580520_a_b_c @ X2 )
        = Y ) ) ).

% fstI
thf(fact_156_fstI,axiom,
    ! [X2: produc2133359284458116493od_c_a,Y: b,Z: product_prod_c_a] :
      ( ( X2
        = ( produc3440865914000881661od_c_a @ Y @ Z ) )
     => ( ( produc301949074746585577od_c_a @ X2 )
        = Y ) ) ).

% fstI
thf(fact_157_fstI,axiom,
    ! [X2: product_prod_c_a,Y: c,Z: a] :
      ( ( X2
        = ( product_Pair_c_a @ Y @ Z ) )
     => ( ( product_fst_c_a @ X2 )
        = Y ) ) ).

% fstI
thf(fact_158_fstI,axiom,
    ! [X2: produc7049025859877086738_set_a,Y: list_P6327159017948738492od_c_a,Z: produc1703568184450464039_set_a] :
      ( ( X2
        = ( produc6508649944902238466_set_a @ Y @ Z ) )
     => ( ( produc3242630202250842350_set_a @ X2 )
        = Y ) ) ).

% fstI
thf(fact_159__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062prs_Atps_Ard__targets_Aatcs_O_AT_A_061_ATest__Suite_Aprs_Atps_Ard__targets_Aatcs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Prs: set_Pr2640454425625433516_a_b_c,Tps: a > set_li1159382662694783132od_c_a,Rd_targets: produc3220416805844688357od_c_a > set_a,Atcs: product_prod_a_a > set_Pr9130538228769002824od_d_d] :
        ( t
       != ( test_t5871526445387177708_b_c_d @ Prs @ Tps @ Rd_targets @ Atcs ) ) ).

% \<open>\<And>thesis. (\<And>prs tps rd_targets atcs. T = Test_Suite prs tps rd_targets atcs \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_160_list__ordered__pairs__set__containment,axiom,
    ! [X2: produc4859908366599477352od_d_d,Xs: list_P4069430723938770798od_d_d,Y: produc4859908366599477352od_d_d] :
      ( ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
     => ( ( member73601632213084689od_d_d @ Y @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member6383636634632210000od_d_d @ ( produc1795809570569509655od_d_d @ X2 @ Y ) @ ( set_Pr5589546020064100540od_d_d @ ( list_o6941861607552427311od_d_d @ Xs ) ) )
            | ( member6383636634632210000od_d_d @ ( produc1795809570569509655od_d_d @ Y @ X2 ) @ ( set_Pr5589546020064100540od_d_d @ ( list_o6941861607552427311od_d_d @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_161_list__ordered__pairs__set__containment,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Xs: list_P5219093185649440210_a_b_c,Y: produc8838969699183119820_a_b_c] :
      ( ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
     => ( ( member997585926662434933_a_b_c @ Y @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member289795379306782288_a_b_c @ ( produc7246914118821965079_a_b_c @ X2 @ Y ) @ ( set_Pr4521536175723917500_a_b_c @ ( list_o2413796117712206227_a_b_c @ Xs ) ) )
            | ( member289795379306782288_a_b_c @ ( produc7246914118821965079_a_b_c @ Y @ X2 ) @ ( set_Pr4521536175723917500_a_b_c @ ( list_o2413796117712206227_a_b_c @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_162_list__ordered__pairs__set__containment,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a,Y: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ Xs ) )
     => ( ( member7410604586820865893od_c_a @ Y @ ( set_li8663184538779258833od_c_a @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ X2 @ Y ) @ ( set_Pr1402144133550743484od_c_a @ ( list_o7432265490351511171od_c_a @ Xs ) ) )
            | ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Y @ X2 ) @ ( set_Pr1402144133550743484od_c_a @ ( list_o7432265490351511171od_c_a @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_163_list__ordered__pairs__set__containment,axiom,
    ! [X2: set_a,Xs: list_set_a,Y: set_a] :
      ( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
     => ( ( member_set_a @ Y @ ( set_set_a2 @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( list_o7130288553133997989_set_a @ Xs ) ) )
            | ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y @ X2 ) @ ( set_Pr7878197726666307004_set_a @ ( list_o7130288553133997989_set_a @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_164_list__ordered__pairs__set__containment,axiom,
    ! [X2: produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a,Y: produc1703568184450464039_set_a] :
      ( ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Xs ) )
     => ( ( member7983343339038529360_set_a @ Y @ ( set_Pr7878197726666307004_set_a @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member1214319639984524048_set_a @ ( produc4984519302771176279_set_a @ X2 @ Y ) @ ( set_Pr6500651403171443068_set_a @ ( list_o549372535332925806_set_a @ Xs ) ) )
            | ( member1214319639984524048_set_a @ ( produc4984519302771176279_set_a @ Y @ X2 ) @ ( set_Pr6500651403171443068_set_a @ ( list_o549372535332925806_set_a @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_165_list__ordered__pairs__set__containment,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ X2 @ Y ) @ ( set_Pr292865031892111420od_b_c @ ( list_o8363625500407931829od_b_c @ Xs ) ) )
            | ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ Y @ X2 ) @ ( set_Pr292865031892111420od_b_c @ ( list_o8363625500407931829od_b_c @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_166_list__ordered__pairs__set__containment,axiom,
    ! [X2: d,Xs: list_d,Y: d] :
      ( ( member_d @ X2 @ ( set_d2 @ Xs ) )
     => ( ( member_d @ Y @ ( set_d2 @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( list_ordered_pairs_d @ Xs ) ) )
            | ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ Y @ X2 ) @ ( set_Product_prod_d_d2 @ ( list_ordered_pairs_d @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_167_list__ordered__pairs__set__containment,axiom,
    ! [X2: a,Xs: list_a,Y: a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( member_a @ Y @ ( set_a2 @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( list_ordered_pairs_a @ Xs ) ) )
            | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ ( set_Product_prod_a_a2 @ ( list_ordered_pairs_a @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_168_t8,axiom,
    ! [D2: produc1703568184450464039_set_a] :
      ( ( member7983343339038529360_set_a @ D2 @ ( set_Pr7878197726666307004_set_a @ repetition_sets ) )
     => ( ord_less_eq_set_a @ ( produc1983107199584856133_set_a @ D2 ) @ ( produc9088895665703139587_set_a @ D2 ) ) ) ).

% t8
thf(fact_169_non__sym__dist__pairs__set__iff,axiom,
    ! [X2: produc4859908366599477352od_d_d,Y: produc4859908366599477352od_d_d,Xs: list_P4069430723938770798od_d_d] :
      ( ( member6383636634632210000od_d_d @ ( produc1795809570569509655od_d_d @ X2 @ Y ) @ ( set_Pr5589546020064100540od_d_d @ ( non_sy4332508533479320540od_d_d @ Xs ) ) )
      = ( ( X2 != Y )
        & ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
        & ( member73601632213084689od_d_d @ Y @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
        & ~ ( member6383636634632210000od_d_d @ ( produc1795809570569509655od_d_d @ Y @ X2 ) @ ( set_Pr5589546020064100540od_d_d @ ( non_sy4332508533479320540od_d_d @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs_set_iff
thf(fact_170_non__sym__dist__pairs__set__iff,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Y: produc8838969699183119820_a_b_c,Xs: list_P5219093185649440210_a_b_c] :
      ( ( member289795379306782288_a_b_c @ ( produc7246914118821965079_a_b_c @ X2 @ Y ) @ ( set_Pr4521536175723917500_a_b_c @ ( non_sy4469082471694185792_a_b_c @ Xs ) ) )
      = ( ( X2 != Y )
        & ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
        & ( member997585926662434933_a_b_c @ Y @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
        & ~ ( member289795379306782288_a_b_c @ ( produc7246914118821965079_a_b_c @ Y @ X2 ) @ ( set_Pr4521536175723917500_a_b_c @ ( non_sy4469082471694185792_a_b_c @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs_set_iff
thf(fact_171_non__sym__dist__pairs__set__iff,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Y: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ X2 @ Y ) @ ( set_Pr1402144133550743484od_c_a @ ( non_sy8103353363212570416od_c_a @ Xs ) ) )
      = ( ( X2 != Y )
        & ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ Xs ) )
        & ( member7410604586820865893od_c_a @ Y @ ( set_li8663184538779258833od_c_a @ Xs ) )
        & ~ ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Y @ X2 ) @ ( set_Pr1402144133550743484od_c_a @ ( non_sy8103353363212570416od_c_a @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs_set_iff
thf(fact_172_non__sym__dist__pairs__set__iff,axiom,
    ! [X2: produc1703568184450464039_set_a,Y: produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a] :
      ( ( member1214319639984524048_set_a @ ( produc4984519302771176279_set_a @ X2 @ Y ) @ ( set_Pr6500651403171443068_set_a @ ( non_sy5028023902500071067_set_a @ Xs ) ) )
      = ( ( X2 != Y )
        & ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Xs ) )
        & ( member7983343339038529360_set_a @ Y @ ( set_Pr7878197726666307004_set_a @ Xs ) )
        & ~ ( member1214319639984524048_set_a @ ( produc4984519302771176279_set_a @ Y @ X2 ) @ ( set_Pr6500651403171443068_set_a @ ( non_sy5028023902500071067_set_a @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs_set_iff
thf(fact_173_non__sym__dist__pairs__set__iff,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ X2 @ Y ) @ ( set_Pr292865031892111420od_b_c @ ( non_sy2897533375561516642od_b_c @ Xs ) ) )
      = ( ( X2 != Y )
        & ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
        & ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ Xs ) )
        & ~ ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ Y @ X2 ) @ ( set_Pr292865031892111420od_b_c @ ( non_sy2897533375561516642od_b_c @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs_set_iff
thf(fact_174_non__sym__dist__pairs__set__iff,axiom,
    ! [X2: set_a,Y: set_a,Xs: list_set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( non_sy7997271732749704658_set_a @ Xs ) ) )
      = ( ( X2 != Y )
        & ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
        & ( member_set_a @ Y @ ( set_set_a2 @ Xs ) )
        & ~ ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y @ X2 ) @ ( set_Pr7878197726666307004_set_a @ ( non_sy7997271732749704658_set_a @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs_set_iff
thf(fact_175_non__sym__dist__pairs__set__iff,axiom,
    ! [X2: d,Y: d,Xs: list_d] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( non_sym_dist_pairs_d @ Xs ) ) )
      = ( ( X2 != Y )
        & ( member_d @ X2 @ ( set_d2 @ Xs ) )
        & ( member_d @ Y @ ( set_d2 @ Xs ) )
        & ~ ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ Y @ X2 ) @ ( set_Product_prod_d_d2 @ ( non_sym_dist_pairs_d @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs_set_iff
thf(fact_176_non__sym__dist__pairs__set__iff,axiom,
    ! [X2: a,Y: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
      = ( ( X2 != Y )
        & ( member_a @ X2 @ ( set_a2 @ Xs ) )
        & ( member_a @ Y @ ( set_a2 @ Xs ) )
        & ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs_set_iff
thf(fact_177__092_060open_062T_A_061_ATest__Suite_Aprs_Atps_Ard__targets_Aatcs_092_060close_062,axiom,
    ( t
    = ( test_t5871526445387177708_b_c_d @ prs @ tps @ rd_targets @ atcs ) ) ).

% \<open>T = Test_Suite prs tps rd_targets atcs\<close>
thf(fact_178_prod_Ocollapse,axiom,
    ! [Prod: product_prod_d_d] :
      ( ( product_Pair_d_d @ ( product_fst_d_d @ Prod ) @ ( product_snd_d_d @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_179_prod_Ocollapse,axiom,
    ! [Prod: product_prod_b_c] :
      ( ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_180_prod_Ocollapse,axiom,
    ! [Prod: produc3220416805844688357od_c_a] :
      ( ( produc3580641852002036821od_c_a @ ( produc4637894536740104257od_c_a @ Prod ) @ ( produc5527568727691618435od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_181_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_a] :
      ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_182_prod_Ocollapse,axiom,
    ! [Prod: produc8838969699183119820_a_b_c] :
      ( ( produc2795441082327197628_a_b_c @ ( produc2620512763296580520_a_b_c @ Prod ) @ ( produc4728557313310798826_a_b_c @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_183_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_184_prod_Ocollapse,axiom,
    ! [Prod: produc7049025859877086738_set_a] :
      ( ( produc6508649944902238466_set_a @ ( produc3242630202250842350_set_a @ Prod ) @ ( produc2451121746714373936_set_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_185_prod_Ocollapse,axiom,
    ! [Prod: produc1703568184450464039_set_a] :
      ( ( produc9088192753505129239_set_a @ ( produc9088895665703139587_set_a @ Prod ) @ ( produc1983107199584856133_set_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_186_prod_Ocollapse,axiom,
    ! [Prod: produc2669932201312201782od_c_a] :
      ( ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ Prod ) @ ( produc8088130389382527188od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_187_prod_Ocollapse,axiom,
    ! [Prod: produc2133359284458116493od_c_a] :
      ( ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ Prod ) @ ( produc5221446324160585515od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_188__092_060open_062implies__completeness_A_ITest__Suite_Aprs_Atps_Ard__targets_Aatcs_J_AM_Am_092_060close_062,axiom,
    test_i4204698558891958671_b_c_d @ ( test_t5871526445387177708_b_c_d @ prs @ tps @ rd_targets @ atcs ) @ m @ m2 ).

% \<open>implies_completeness (Test_Suite prs tps rd_targets atcs) M m\<close>
thf(fact_189_repetition__sets__def,axiom,
    test_i4756896328399799264_b_c_d @ ( test_t5871526445387177708_b_c_d @ prs @ tps @ rd_targets @ atcs ) @ m @ m2 @ repetition_sets ).

% repetition_sets_def
thf(fact_190_cyclic__path__transition__states__property_I2_J,axiom,
    ! [P: list_P6327159017948738492od_c_a,P5: a > $o,M: fsm_a_b_c,Q: a] :
      ( ? [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ P ) )
          & ( P5 @ ( produc4225943634317630866od_c_a @ X4 ) ) )
     => ( ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ P ) )
           => ( ( P5 @ ( produc4225943634317630866od_c_a @ X ) )
             => ( P5 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) ) ) ) )
       => ( ( path_a_b_c @ M @ Q @ P )
         => ( ( ( target_a_b_c @ Q @ P )
              = Q )
           => ! [X4: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ P ) )
               => ( P5 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(2)
thf(fact_191_cyclic__path__transition__states__property_I2_J,axiom,
    ! [P: list_P1562850763667305982od_c_a,P5: product_prod_c_a > $o,M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a] :
      ( ? [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( set_Pr6987804667698513421od_c_a @ P ) )
          & ( P5 @ ( produc8171714036265584980od_c_a @ X4 ) ) )
     => ( ! [X: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X @ ( set_Pr6987804667698513421od_c_a @ P ) )
           => ( ( P5 @ ( produc8171714036265584980od_c_a @ X ) )
             => ( P5 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) ) ) ) )
       => ( ( path_P1107610407223809260_a_a_b @ M @ Q @ P )
         => ( ( ( target6303326113586172192_a_a_b @ Q @ P )
              = Q )
           => ! [X4: produc3205353004677322104od_c_a] :
                ( ( member6508725278956418977od_c_a @ X4 @ ( set_Pr6987804667698513421od_c_a @ P ) )
               => ( P5 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(2)
thf(fact_192_cyclic__path__transition__states__property_I1_J,axiom,
    ! [P: list_P6327159017948738492od_c_a,P5: a > $o,M: fsm_a_b_c,Q: a] :
      ( ? [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ P ) )
          & ( P5 @ ( produc4225943634317630866od_c_a @ X4 ) ) )
     => ( ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ P ) )
           => ( ( P5 @ ( produc4225943634317630866od_c_a @ X ) )
             => ( P5 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) ) ) ) )
       => ( ( path_a_b_c @ M @ Q @ P )
         => ( ( ( target_a_b_c @ Q @ P )
              = Q )
           => ! [X4: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ P ) )
               => ( P5 @ ( produc4225943634317630866od_c_a @ X4 ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(1)
thf(fact_193_cyclic__path__transition__states__property_I1_J,axiom,
    ! [P: list_P1562850763667305982od_c_a,P5: product_prod_c_a > $o,M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a] :
      ( ? [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( set_Pr6987804667698513421od_c_a @ P ) )
          & ( P5 @ ( produc8171714036265584980od_c_a @ X4 ) ) )
     => ( ! [X: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X @ ( set_Pr6987804667698513421od_c_a @ P ) )
           => ( ( P5 @ ( produc8171714036265584980od_c_a @ X ) )
             => ( P5 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) ) ) ) )
       => ( ( path_P1107610407223809260_a_a_b @ M @ Q @ P )
         => ( ( ( target6303326113586172192_a_a_b @ Q @ P )
              = Q )
           => ! [X4: produc3205353004677322104od_c_a] :
                ( ( member6508725278956418977od_c_a @ X4 @ ( set_Pr6987804667698513421od_c_a @ P ) )
               => ( P5 @ ( produc8171714036265584980od_c_a @ X4 ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(1)
thf(fact_194_test__suite__to__io_Ocases,axiom,
    ! [X2: produc1521100015819645301_b_c_d] :
      ~ ! [M3: fsm_a_b_c,Prs: set_Pr2640454425625433516_a_b_c,Tps: a > set_li1159382662694783132od_c_a,Rd_targets: produc3220416805844688357od_c_a > set_a,Atcs: product_prod_a_a > set_Pr9130538228769002824od_d_d] :
          ( X2
         != ( produc6507248844927732207_b_c_d @ M3 @ ( test_t5871526445387177708_b_c_d @ Prs @ Tps @ Rd_targets @ Atcs ) ) ) ).

% test_suite_to_io.cases
thf(fact_195_sndI,axiom,
    ! [X2: product_prod_d_d,Y: d,Z: d] :
      ( ( X2
        = ( product_Pair_d_d @ Y @ Z ) )
     => ( ( product_snd_d_d @ X2 )
        = Z ) ) ).

% sndI
thf(fact_196_sndI,axiom,
    ! [X2: product_prod_b_c,Y: b,Z: c] :
      ( ( X2
        = ( product_Pair_b_c @ Y @ Z ) )
     => ( ( product_snd_b_c @ X2 )
        = Z ) ) ).

% sndI
thf(fact_197_sndI,axiom,
    ! [X2: produc3220416805844688357od_c_a,Y: a,Z: list_P6327159017948738492od_c_a] :
      ( ( X2
        = ( produc3580641852002036821od_c_a @ Y @ Z ) )
     => ( ( produc5527568727691618435od_c_a @ X2 )
        = Z ) ) ).

% sndI
thf(fact_198_sndI,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Y: a,Z: fsm_a_b_c] :
      ( ( X2
        = ( produc2795441082327197628_a_b_c @ Y @ Z ) )
     => ( ( produc4728557313310798826_a_b_c @ X2 )
        = Z ) ) ).

% sndI
thf(fact_199_sndI,axiom,
    ! [X2: product_prod_a_a,Y: a,Z: a] :
      ( ( X2
        = ( product_Pair_a_a @ Y @ Z ) )
     => ( ( product_snd_a_a @ X2 )
        = Z ) ) ).

% sndI
thf(fact_200_sndI,axiom,
    ! [X2: produc1703568184450464039_set_a,Y: set_a,Z: set_a] :
      ( ( X2
        = ( produc9088192753505129239_set_a @ Y @ Z ) )
     => ( ( produc1983107199584856133_set_a @ X2 )
        = Z ) ) ).

% sndI
thf(fact_201_sndI,axiom,
    ! [X2: produc2669932201312201782od_c_a,Y: a,Z: produc2133359284458116493od_c_a] :
      ( ( X2
        = ( produc5762801557034676134od_c_a @ Y @ Z ) )
     => ( ( produc8088130389382527188od_c_a @ X2 )
        = Z ) ) ).

% sndI
thf(fact_202_sndI,axiom,
    ! [X2: produc2133359284458116493od_c_a,Y: b,Z: product_prod_c_a] :
      ( ( X2
        = ( produc3440865914000881661od_c_a @ Y @ Z ) )
     => ( ( produc5221446324160585515od_c_a @ X2 )
        = Z ) ) ).

% sndI
thf(fact_203_eq__snd__iff,axiom,
    ! [B2: d,P: product_prod_d_d] :
      ( ( B2
        = ( product_snd_d_d @ P ) )
      = ( ? [A5: d] :
            ( P
            = ( product_Pair_d_d @ A5 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_204_eq__snd__iff,axiom,
    ! [B2: c,P: product_prod_b_c] :
      ( ( B2
        = ( product_snd_b_c @ P ) )
      = ( ? [A5: b] :
            ( P
            = ( product_Pair_b_c @ A5 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_205_eq__snd__iff,axiom,
    ! [B2: list_P6327159017948738492od_c_a,P: produc3220416805844688357od_c_a] :
      ( ( B2
        = ( produc5527568727691618435od_c_a @ P ) )
      = ( ? [A5: a] :
            ( P
            = ( produc3580641852002036821od_c_a @ A5 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_206_eq__snd__iff,axiom,
    ! [B2: fsm_a_b_c,P: produc8838969699183119820_a_b_c] :
      ( ( B2
        = ( produc4728557313310798826_a_b_c @ P ) )
      = ( ? [A5: a] :
            ( P
            = ( produc2795441082327197628_a_b_c @ A5 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_207_eq__snd__iff,axiom,
    ! [B2: a,P: product_prod_a_a] :
      ( ( B2
        = ( product_snd_a_a @ P ) )
      = ( ? [A5: a] :
            ( P
            = ( product_Pair_a_a @ A5 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_208_eq__snd__iff,axiom,
    ! [B2: set_a,P: produc1703568184450464039_set_a] :
      ( ( B2
        = ( produc1983107199584856133_set_a @ P ) )
      = ( ? [A5: set_a] :
            ( P
            = ( produc9088192753505129239_set_a @ A5 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_209_eq__snd__iff,axiom,
    ! [B2: produc2133359284458116493od_c_a,P: produc2669932201312201782od_c_a] :
      ( ( B2
        = ( produc8088130389382527188od_c_a @ P ) )
      = ( ? [A5: a] :
            ( P
            = ( produc5762801557034676134od_c_a @ A5 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_210_eq__snd__iff,axiom,
    ! [B2: product_prod_c_a,P: produc2133359284458116493od_c_a] :
      ( ( B2
        = ( produc5221446324160585515od_c_a @ P ) )
      = ( ? [A5: b] :
            ( P
            = ( produc3440865914000881661od_c_a @ A5 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_211_snd__eqD,axiom,
    ! [X2: d,Y: d,A2: d] :
      ( ( ( product_snd_d_d @ ( product_Pair_d_d @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_212_snd__eqD,axiom,
    ! [X2: b,Y: c,A2: c] :
      ( ( ( product_snd_b_c @ ( product_Pair_b_c @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_213_snd__eqD,axiom,
    ! [X2: a,Y: list_P6327159017948738492od_c_a,A2: list_P6327159017948738492od_c_a] :
      ( ( ( produc5527568727691618435od_c_a @ ( produc3580641852002036821od_c_a @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_214_snd__eqD,axiom,
    ! [X2: a,Y: fsm_a_b_c,A2: fsm_a_b_c] :
      ( ( ( produc4728557313310798826_a_b_c @ ( produc2795441082327197628_a_b_c @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_215_snd__eqD,axiom,
    ! [X2: a,Y: a,A2: a] :
      ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_216_snd__eqD,axiom,
    ! [X2: set_a,Y: set_a,A2: set_a] :
      ( ( ( produc1983107199584856133_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_217_snd__eqD,axiom,
    ! [X2: a,Y: produc2133359284458116493od_c_a,A2: produc2133359284458116493od_c_a] :
      ( ( ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_218_snd__eqD,axiom,
    ! [X2: b,Y: product_prod_c_a,A2: product_prod_c_a] :
      ( ( ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X2 @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_219_snd__conv,axiom,
    ! [X1: d,X22: d] :
      ( ( product_snd_d_d @ ( product_Pair_d_d @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_220_snd__conv,axiom,
    ! [X1: b,X22: c] :
      ( ( product_snd_b_c @ ( product_Pair_b_c @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_221_snd__conv,axiom,
    ! [X1: a,X22: list_P6327159017948738492od_c_a] :
      ( ( produc5527568727691618435od_c_a @ ( produc3580641852002036821od_c_a @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_222_snd__conv,axiom,
    ! [X1: a,X22: fsm_a_b_c] :
      ( ( produc4728557313310798826_a_b_c @ ( produc2795441082327197628_a_b_c @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_223_snd__conv,axiom,
    ! [X1: a,X22: a] :
      ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_224_snd__conv,axiom,
    ! [X1: set_a,X22: set_a] :
      ( ( produc1983107199584856133_set_a @ ( produc9088192753505129239_set_a @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_225_snd__conv,axiom,
    ! [X1: a,X22: produc2133359284458116493od_c_a] :
      ( ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_226_snd__conv,axiom,
    ! [X1: b,X22: product_prod_c_a] :
      ( ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_227_prod_Oexpand,axiom,
    ! [Prod: produc8838969699183119820_a_b_c,Prod2: produc8838969699183119820_a_b_c] :
      ( ( ( ( produc2620512763296580520_a_b_c @ Prod )
          = ( produc2620512763296580520_a_b_c @ Prod2 ) )
        & ( ( produc4728557313310798826_a_b_c @ Prod )
          = ( produc4728557313310798826_a_b_c @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_228_prod_Oexpand,axiom,
    ! [Prod: product_prod_c_a,Prod2: product_prod_c_a] :
      ( ( ( ( product_fst_c_a @ Prod )
          = ( product_fst_c_a @ Prod2 ) )
        & ( ( product_snd_c_a @ Prod )
          = ( product_snd_c_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_229_prod_Oexpand,axiom,
    ! [Prod: produc7049025859877086738_set_a,Prod2: produc7049025859877086738_set_a] :
      ( ( ( ( produc3242630202250842350_set_a @ Prod )
          = ( produc3242630202250842350_set_a @ Prod2 ) )
        & ( ( produc2451121746714373936_set_a @ Prod )
          = ( produc2451121746714373936_set_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_230_prod_Oexpand,axiom,
    ! [Prod: produc1703568184450464039_set_a,Prod2: produc1703568184450464039_set_a] :
      ( ( ( ( produc9088895665703139587_set_a @ Prod )
          = ( produc9088895665703139587_set_a @ Prod2 ) )
        & ( ( produc1983107199584856133_set_a @ Prod )
          = ( produc1983107199584856133_set_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_231_prod_Oexpand,axiom,
    ! [Prod: produc2669932201312201782od_c_a,Prod2: produc2669932201312201782od_c_a] :
      ( ( ( ( produc4225943634317630866od_c_a @ Prod )
          = ( produc4225943634317630866od_c_a @ Prod2 ) )
        & ( ( produc8088130389382527188od_c_a @ Prod )
          = ( produc8088130389382527188od_c_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_232_prod_Oexpand,axiom,
    ! [Prod: produc2133359284458116493od_c_a,Prod2: produc2133359284458116493od_c_a] :
      ( ( ( ( produc301949074746585577od_c_a @ Prod )
          = ( produc301949074746585577od_c_a @ Prod2 ) )
        & ( ( produc5221446324160585515od_c_a @ Prod )
          = ( produc5221446324160585515od_c_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_233_prod__eqI,axiom,
    ! [P: produc8838969699183119820_a_b_c,Q: produc8838969699183119820_a_b_c] :
      ( ( ( produc2620512763296580520_a_b_c @ P )
        = ( produc2620512763296580520_a_b_c @ Q ) )
     => ( ( ( produc4728557313310798826_a_b_c @ P )
          = ( produc4728557313310798826_a_b_c @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_234_prod__eqI,axiom,
    ! [P: product_prod_c_a,Q: product_prod_c_a] :
      ( ( ( product_fst_c_a @ P )
        = ( product_fst_c_a @ Q ) )
     => ( ( ( product_snd_c_a @ P )
          = ( product_snd_c_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_235_prod__eqI,axiom,
    ! [P: produc7049025859877086738_set_a,Q: produc7049025859877086738_set_a] :
      ( ( ( produc3242630202250842350_set_a @ P )
        = ( produc3242630202250842350_set_a @ Q ) )
     => ( ( ( produc2451121746714373936_set_a @ P )
          = ( produc2451121746714373936_set_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_236_prod__eqI,axiom,
    ! [P: produc1703568184450464039_set_a,Q: produc1703568184450464039_set_a] :
      ( ( ( produc9088895665703139587_set_a @ P )
        = ( produc9088895665703139587_set_a @ Q ) )
     => ( ( ( produc1983107199584856133_set_a @ P )
          = ( produc1983107199584856133_set_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_237_prod__eqI,axiom,
    ! [P: produc2669932201312201782od_c_a,Q: produc2669932201312201782od_c_a] :
      ( ( ( produc4225943634317630866od_c_a @ P )
        = ( produc4225943634317630866od_c_a @ Q ) )
     => ( ( ( produc8088130389382527188od_c_a @ P )
          = ( produc8088130389382527188od_c_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_238_prod__eqI,axiom,
    ! [P: produc2133359284458116493od_c_a,Q: produc2133359284458116493od_c_a] :
      ( ( ( produc301949074746585577od_c_a @ P )
        = ( produc301949074746585577od_c_a @ Q ) )
     => ( ( ( produc5221446324160585515od_c_a @ P )
          = ( produc5221446324160585515od_c_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_239_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc8838969699183119820_a_b_c,Z2: produc8838969699183119820_a_b_c] : ( Y4 = Z2 ) )
    = ( ^ [S3: produc8838969699183119820_a_b_c,T: produc8838969699183119820_a_b_c] :
          ( ( ( produc2620512763296580520_a_b_c @ S3 )
            = ( produc2620512763296580520_a_b_c @ T ) )
          & ( ( produc4728557313310798826_a_b_c @ S3 )
            = ( produc4728557313310798826_a_b_c @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_240_prod__eq__iff,axiom,
    ( ( ^ [Y4: product_prod_c_a,Z2: product_prod_c_a] : ( Y4 = Z2 ) )
    = ( ^ [S3: product_prod_c_a,T: product_prod_c_a] :
          ( ( ( product_fst_c_a @ S3 )
            = ( product_fst_c_a @ T ) )
          & ( ( product_snd_c_a @ S3 )
            = ( product_snd_c_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_241_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc7049025859877086738_set_a,Z2: produc7049025859877086738_set_a] : ( Y4 = Z2 ) )
    = ( ^ [S3: produc7049025859877086738_set_a,T: produc7049025859877086738_set_a] :
          ( ( ( produc3242630202250842350_set_a @ S3 )
            = ( produc3242630202250842350_set_a @ T ) )
          & ( ( produc2451121746714373936_set_a @ S3 )
            = ( produc2451121746714373936_set_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_242_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc1703568184450464039_set_a,Z2: produc1703568184450464039_set_a] : ( Y4 = Z2 ) )
    = ( ^ [S3: produc1703568184450464039_set_a,T: produc1703568184450464039_set_a] :
          ( ( ( produc9088895665703139587_set_a @ S3 )
            = ( produc9088895665703139587_set_a @ T ) )
          & ( ( produc1983107199584856133_set_a @ S3 )
            = ( produc1983107199584856133_set_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_243_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc2669932201312201782od_c_a,Z2: produc2669932201312201782od_c_a] : ( Y4 = Z2 ) )
    = ( ^ [S3: produc2669932201312201782od_c_a,T: produc2669932201312201782od_c_a] :
          ( ( ( produc4225943634317630866od_c_a @ S3 )
            = ( produc4225943634317630866od_c_a @ T ) )
          & ( ( produc8088130389382527188od_c_a @ S3 )
            = ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_244_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc2133359284458116493od_c_a,Z2: produc2133359284458116493od_c_a] : ( Y4 = Z2 ) )
    = ( ^ [S3: produc2133359284458116493od_c_a,T: produc2133359284458116493od_c_a] :
          ( ( ( produc301949074746585577od_c_a @ S3 )
            = ( produc301949074746585577od_c_a @ T ) )
          & ( ( produc5221446324160585515od_c_a @ S3 )
            = ( produc5221446324160585515od_c_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_245_subset__code_I1_J,axiom,
    ! [Xs: list_P4069430723938770798od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ( ord_le7464911323401878760od_d_d @ ( set_Pr5977834532684968573od_d_d @ Xs ) @ B )
      = ( ! [X3: produc4859908366599477352od_d_d] :
            ( ( member73601632213084689od_d_d @ X3 @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
           => ( member73601632213084689od_d_d @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_246_subset__code_I1_J,axiom,
    ! [Xs: list_P5219093185649440210_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( ord_le6086132928344890700_a_b_c @ ( set_Pr3403252632855053537_a_b_c @ Xs ) @ B )
      = ( ! [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
           => ( member997585926662434933_a_b_c @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_247_subset__code_I1_J,axiom,
    ! [Xs: list_P3660316430366008877_set_a,B: set_Pr5845495582615845127_set_a] :
      ( ( ord_le8376522849517564071_set_a @ ( set_Pr7878197726666307004_set_a @ Xs ) @ B )
      = ( ! [X3: produc1703568184450464039_set_a] :
            ( ( member7983343339038529360_set_a @ X3 @ ( set_Pr7878197726666307004_set_a @ Xs ) )
           => ( member7983343339038529360_set_a @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_248_subset__code_I1_J,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ B )
      = ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( set_li4480668622519654659od_b_c @ Xs ) )
           => ( member6330420149250801815od_b_c @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_249_subset__code_I1_J,axiom,
    ! [Xs: list_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
      = ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ( member_a @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_250_subset__code_I1_J,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( set_li8663184538779258833od_c_a @ Xs ) @ B )
      = ( ! [X3: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X3 @ ( set_li8663184538779258833od_c_a @ Xs ) )
           => ( member7410604586820865893od_c_a @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_251_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_d_d] :
      ( Prod
      = ( product_Pair_d_d @ ( product_fst_d_d @ Prod ) @ ( product_snd_d_d @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_252_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_b_c] :
      ( Prod
      = ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_253_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3220416805844688357od_c_a] :
      ( Prod
      = ( produc3580641852002036821od_c_a @ ( produc4637894536740104257od_c_a @ Prod ) @ ( produc5527568727691618435od_c_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_254_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_a] :
      ( Prod
      = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_255_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8838969699183119820_a_b_c] :
      ( Prod
      = ( produc2795441082327197628_a_b_c @ ( produc2620512763296580520_a_b_c @ Prod ) @ ( produc4728557313310798826_a_b_c @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_256_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_257_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7049025859877086738_set_a] :
      ( Prod
      = ( produc6508649944902238466_set_a @ ( produc3242630202250842350_set_a @ Prod ) @ ( produc2451121746714373936_set_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_258_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1703568184450464039_set_a] :
      ( Prod
      = ( produc9088192753505129239_set_a @ ( produc9088895665703139587_set_a @ Prod ) @ ( produc1983107199584856133_set_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_259_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_260_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_261_surjective__pairing,axiom,
    ! [T2: product_prod_d_d] :
      ( T2
      = ( product_Pair_d_d @ ( product_fst_d_d @ T2 ) @ ( product_snd_d_d @ T2 ) ) ) ).

% surjective_pairing
thf(fact_262_surjective__pairing,axiom,
    ! [T2: product_prod_b_c] :
      ( T2
      = ( product_Pair_b_c @ ( product_fst_b_c @ T2 ) @ ( product_snd_b_c @ T2 ) ) ) ).

% surjective_pairing
thf(fact_263_surjective__pairing,axiom,
    ! [T2: produc3220416805844688357od_c_a] :
      ( T2
      = ( produc3580641852002036821od_c_a @ ( produc4637894536740104257od_c_a @ T2 ) @ ( produc5527568727691618435od_c_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_264_surjective__pairing,axiom,
    ! [T2: product_prod_a_a] :
      ( T2
      = ( product_Pair_a_a @ ( product_fst_a_a @ T2 ) @ ( product_snd_a_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_265_surjective__pairing,axiom,
    ! [T2: produc8838969699183119820_a_b_c] :
      ( T2
      = ( produc2795441082327197628_a_b_c @ ( produc2620512763296580520_a_b_c @ T2 ) @ ( produc4728557313310798826_a_b_c @ T2 ) ) ) ).

% surjective_pairing
thf(fact_266_surjective__pairing,axiom,
    ! [T2: product_prod_c_a] :
      ( T2
      = ( product_Pair_c_a @ ( product_fst_c_a @ T2 ) @ ( product_snd_c_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_267_surjective__pairing,axiom,
    ! [T2: produc7049025859877086738_set_a] :
      ( T2
      = ( produc6508649944902238466_set_a @ ( produc3242630202250842350_set_a @ T2 ) @ ( produc2451121746714373936_set_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_268_surjective__pairing,axiom,
    ! [T2: produc1703568184450464039_set_a] :
      ( T2
      = ( produc9088192753505129239_set_a @ ( produc9088895665703139587_set_a @ T2 ) @ ( produc1983107199584856133_set_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_269_surjective__pairing,axiom,
    ! [T2: produc2669932201312201782od_c_a] :
      ( T2
      = ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ T2 ) @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_270_surjective__pairing,axiom,
    ! [T2: produc2133359284458116493od_c_a] :
      ( T2
      = ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ T2 ) @ ( produc5221446324160585515od_c_a @ T2 ) ) ) ).

% surjective_pairing
thf(fact_271_list__prefix__subset,axiom,
    ! [Ts: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ? [Ys2: list_P6327159017948738492od_c_a] :
          ( Ts
          = ( append3884164603879732689od_c_a @ Xs @ Ys2 ) )
     => ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs ) @ ( set_Pr5664641634098845387od_c_a @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_272_list__prefix__subset,axiom,
    ! [Ts: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ? [Ys2: list_P903359562653991662od_b_c] :
          ( Ts
          = ( append2547753245680614915od_b_c @ Xs @ Ys2 ) )
     => ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_273_list__prefix__subset,axiom,
    ! [Ts: list_P3660316430366008877_set_a,Xs: list_P3660316430366008877_set_a] :
      ( ? [Ys2: list_P3660316430366008877_set_a] :
          ( Ts
          = ( append6456390900168921282_set_a @ Xs @ Ys2 ) )
     => ( ord_le8376522849517564071_set_a @ ( set_Pr7878197726666307004_set_a @ Xs ) @ ( set_Pr7878197726666307004_set_a @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_274_list__prefix__subset,axiom,
    ! [Ts: list_l8907847357763382004od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ? [Ys2: list_l8907847357763382004od_b_c] :
          ( Ts
          = ( append6992831555715836169od_b_c @ Xs @ Ys2 ) )
     => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_275_list__prefix__subset,axiom,
    ! [Ts: list_a,Xs: list_a] :
      ( ? [Ys2: list_a] :
          ( Ts
          = ( append_a @ Xs @ Ys2 ) )
     => ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_276_list__prefix__subset,axiom,
    ! [Ts: list_l2826717958560394946od_c_a,Xs: list_l2826717958560394946od_c_a] :
      ( ? [Ys2: list_l2826717958560394946od_c_a] :
          ( Ts
          = ( append2943488786620085207od_c_a @ Xs @ Ys2 ) )
     => ( ord_le2998388488506175548od_c_a @ ( set_li8663184538779258833od_c_a @ Xs ) @ ( set_li8663184538779258833od_c_a @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_277_list__append__subset3,axiom,
    ! [Xs1: list_P6327159017948738492od_c_a,Ys1: list_P6327159017948738492od_c_a,Xs2: list_P6327159017948738492od_c_a,Ys22: list_P6327159017948738492od_c_a,Xs3: list_P6327159017948738492od_c_a,Ys3: 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 @ Xs2 ) @ ( set_Pr5664641634098845387od_c_a @ Ys22 ) )
       => ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs3 ) @ ( set_Pr5664641634098845387od_c_a @ Ys3 ) )
         => ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs1 @ ( append3884164603879732689od_c_a @ Xs2 @ Xs3 ) ) ) @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Ys1 @ ( append3884164603879732689od_c_a @ Ys22 @ Ys3 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_278_list__append__subset3,axiom,
    ! [Xs1: list_P903359562653991662od_b_c,Ys1: list_P903359562653991662od_b_c,Xs2: list_P903359562653991662od_b_c,Ys22: list_P903359562653991662od_b_c,Xs3: list_P903359562653991662od_b_c,Ys3: list_P903359562653991662od_b_c] :
      ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs1 ) @ ( set_Product_prod_b_c2 @ Ys1 ) )
     => ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs2 ) @ ( set_Product_prod_b_c2 @ Ys22 ) )
       => ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs3 ) @ ( set_Product_prod_b_c2 @ Ys3 ) )
         => ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs1 @ ( append2547753245680614915od_b_c @ Xs2 @ Xs3 ) ) ) @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Ys1 @ ( append2547753245680614915od_b_c @ Ys22 @ Ys3 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_279_list__append__subset3,axiom,
    ! [Xs1: list_P3660316430366008877_set_a,Ys1: list_P3660316430366008877_set_a,Xs2: list_P3660316430366008877_set_a,Ys22: list_P3660316430366008877_set_a,Xs3: list_P3660316430366008877_set_a,Ys3: list_P3660316430366008877_set_a] :
      ( ( ord_le8376522849517564071_set_a @ ( set_Pr7878197726666307004_set_a @ Xs1 ) @ ( set_Pr7878197726666307004_set_a @ Ys1 ) )
     => ( ( ord_le8376522849517564071_set_a @ ( set_Pr7878197726666307004_set_a @ Xs2 ) @ ( set_Pr7878197726666307004_set_a @ Ys22 ) )
       => ( ( ord_le8376522849517564071_set_a @ ( set_Pr7878197726666307004_set_a @ Xs3 ) @ ( set_Pr7878197726666307004_set_a @ Ys3 ) )
         => ( ord_le8376522849517564071_set_a @ ( set_Pr7878197726666307004_set_a @ ( append6456390900168921282_set_a @ Xs1 @ ( append6456390900168921282_set_a @ Xs2 @ Xs3 ) ) ) @ ( set_Pr7878197726666307004_set_a @ ( append6456390900168921282_set_a @ Ys1 @ ( append6456390900168921282_set_a @ Ys22 @ Ys3 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_280_list__append__subset3,axiom,
    ! [Xs1: list_l8907847357763382004od_b_c,Ys1: list_l8907847357763382004od_b_c,Xs2: list_l8907847357763382004od_b_c,Ys22: list_l8907847357763382004od_b_c,Xs3: list_l8907847357763382004od_b_c,Ys3: list_l8907847357763382004od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs1 ) @ ( set_li4480668622519654659od_b_c @ Ys1 ) )
     => ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs2 ) @ ( set_li4480668622519654659od_b_c @ Ys22 ) )
       => ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs3 ) @ ( set_li4480668622519654659od_b_c @ Ys3 ) )
         => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs1 @ ( append6992831555715836169od_b_c @ Xs2 @ Xs3 ) ) ) @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Ys1 @ ( append6992831555715836169od_b_c @ Ys22 @ Ys3 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_281_list__append__subset3,axiom,
    ! [Xs1: list_a,Ys1: list_a,Xs2: list_a,Ys22: list_a,Xs3: list_a,Ys3: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs1 ) @ ( set_a2 @ Ys1 ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys22 ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ ( set_a2 @ Ys3 ) )
         => ( ord_less_eq_set_a @ ( set_a2 @ ( append_a @ Xs1 @ ( append_a @ Xs2 @ Xs3 ) ) ) @ ( set_a2 @ ( append_a @ Ys1 @ ( append_a @ Ys22 @ Ys3 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_282_list__append__subset3,axiom,
    ! [Xs1: list_l2826717958560394946od_c_a,Ys1: list_l2826717958560394946od_c_a,Xs2: list_l2826717958560394946od_c_a,Ys22: list_l2826717958560394946od_c_a,Xs3: list_l2826717958560394946od_c_a,Ys3: list_l2826717958560394946od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( set_li8663184538779258833od_c_a @ Xs1 ) @ ( set_li8663184538779258833od_c_a @ Ys1 ) )
     => ( ( ord_le2998388488506175548od_c_a @ ( set_li8663184538779258833od_c_a @ Xs2 ) @ ( set_li8663184538779258833od_c_a @ Ys22 ) )
       => ( ( ord_le2998388488506175548od_c_a @ ( set_li8663184538779258833od_c_a @ Xs3 ) @ ( set_li8663184538779258833od_c_a @ Ys3 ) )
         => ( ord_le2998388488506175548od_c_a @ ( set_li8663184538779258833od_c_a @ ( append2943488786620085207od_c_a @ Xs1 @ ( append2943488786620085207od_c_a @ Xs2 @ Xs3 ) ) ) @ ( set_li8663184538779258833od_c_a @ ( append2943488786620085207od_c_a @ Ys1 @ ( append2943488786620085207od_c_a @ Ys22 @ Ys3 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_283_submachine__simps_I2_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ord_less_eq_set_a @ ( states_a_b_c @ A ) @ ( states_a_b_c @ B ) ) ) ).

% submachine_simps(2)
thf(fact_284_non__sym__dist__pairs__elems__distinct_I3_J,axiom,
    ! [X2: set_a,Y: set_a,Xs: list_set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( non_sy7997271732749704658_set_a @ Xs ) ) )
     => ( X2 != Y ) ) ).

% non_sym_dist_pairs_elems_distinct(3)
thf(fact_285_non__sym__dist__pairs__elems__distinct_I3_J,axiom,
    ! [X2: d,Y: d,Xs: list_d] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( non_sym_dist_pairs_d @ Xs ) ) )
     => ( X2 != Y ) ) ).

% non_sym_dist_pairs_elems_distinct(3)
thf(fact_286_non__sym__dist__pairs__elems__distinct_I3_J,axiom,
    ! [X2: a,Y: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
     => ( X2 != Y ) ) ).

% non_sym_dist_pairs_elems_distinct(3)
thf(fact_287_non__sym__dist__pairs__elems__non__refl,axiom,
    ! [X2: set_a,Y: set_a,Xs: list_set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( non_sy7997271732749704658_set_a @ Xs ) ) )
     => ~ ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y @ X2 ) @ ( set_Pr7878197726666307004_set_a @ ( non_sy7997271732749704658_set_a @ Xs ) ) ) ) ).

% non_sym_dist_pairs_elems_non_refl
thf(fact_288_non__sym__dist__pairs__elems__non__refl,axiom,
    ! [X2: d,Y: d,Xs: list_d] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( non_sym_dist_pairs_d @ Xs ) ) )
     => ~ ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ Y @ X2 ) @ ( set_Product_prod_d_d2 @ ( non_sym_dist_pairs_d @ Xs ) ) ) ) ).

% non_sym_dist_pairs_elems_non_refl
thf(fact_289_non__sym__dist__pairs__elems__non__refl,axiom,
    ! [X2: a,Y: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
     => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) ) ) ).

% non_sym_dist_pairs_elems_non_refl
thf(fact_290_non__sym__dist__pairs__elems__distinct_I2_J,axiom,
    ! [X2: produc4859908366599477352od_d_d,Y: produc4859908366599477352od_d_d,Xs: list_P4069430723938770798od_d_d] :
      ( ( member6383636634632210000od_d_d @ ( produc1795809570569509655od_d_d @ X2 @ Y ) @ ( set_Pr5589546020064100540od_d_d @ ( non_sy4332508533479320540od_d_d @ Xs ) ) )
     => ( member73601632213084689od_d_d @ Y @ ( set_Pr5977834532684968573od_d_d @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(2)
thf(fact_291_non__sym__dist__pairs__elems__distinct_I2_J,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Y: produc8838969699183119820_a_b_c,Xs: list_P5219093185649440210_a_b_c] :
      ( ( member289795379306782288_a_b_c @ ( produc7246914118821965079_a_b_c @ X2 @ Y ) @ ( set_Pr4521536175723917500_a_b_c @ ( non_sy4469082471694185792_a_b_c @ Xs ) ) )
     => ( member997585926662434933_a_b_c @ Y @ ( set_Pr3403252632855053537_a_b_c @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(2)
thf(fact_292_non__sym__dist__pairs__elems__distinct_I2_J,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Y: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ X2 @ Y ) @ ( set_Pr1402144133550743484od_c_a @ ( non_sy8103353363212570416od_c_a @ Xs ) ) )
     => ( member7410604586820865893od_c_a @ Y @ ( set_li8663184538779258833od_c_a @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(2)
thf(fact_293_non__sym__dist__pairs__elems__distinct_I2_J,axiom,
    ! [X2: produc1703568184450464039_set_a,Y: produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a] :
      ( ( member1214319639984524048_set_a @ ( produc4984519302771176279_set_a @ X2 @ Y ) @ ( set_Pr6500651403171443068_set_a @ ( non_sy5028023902500071067_set_a @ Xs ) ) )
     => ( member7983343339038529360_set_a @ Y @ ( set_Pr7878197726666307004_set_a @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(2)
thf(fact_294_non__sym__dist__pairs__elems__distinct_I2_J,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ X2 @ Y ) @ ( set_Pr292865031892111420od_b_c @ ( non_sy2897533375561516642od_b_c @ Xs ) ) )
     => ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(2)
thf(fact_295_non__sym__dist__pairs__elems__distinct_I2_J,axiom,
    ! [X2: set_a,Y: set_a,Xs: list_set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( non_sy7997271732749704658_set_a @ Xs ) ) )
     => ( member_set_a @ Y @ ( set_set_a2 @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(2)
thf(fact_296_non__sym__dist__pairs__elems__distinct_I2_J,axiom,
    ! [X2: d,Y: d,Xs: list_d] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( non_sym_dist_pairs_d @ Xs ) ) )
     => ( member_d @ Y @ ( set_d2 @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(2)
thf(fact_297_non__sym__dist__pairs__elems__distinct_I2_J,axiom,
    ! [X2: a,Y: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
     => ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(2)
thf(fact_298_non__sym__dist__pairs__elems__distinct_I1_J,axiom,
    ! [X2: produc4859908366599477352od_d_d,Y: produc4859908366599477352od_d_d,Xs: list_P4069430723938770798od_d_d] :
      ( ( member6383636634632210000od_d_d @ ( produc1795809570569509655od_d_d @ X2 @ Y ) @ ( set_Pr5589546020064100540od_d_d @ ( non_sy4332508533479320540od_d_d @ Xs ) ) )
     => ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(1)
thf(fact_299_non__sym__dist__pairs__elems__distinct_I1_J,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Y: produc8838969699183119820_a_b_c,Xs: list_P5219093185649440210_a_b_c] :
      ( ( member289795379306782288_a_b_c @ ( produc7246914118821965079_a_b_c @ X2 @ Y ) @ ( set_Pr4521536175723917500_a_b_c @ ( non_sy4469082471694185792_a_b_c @ Xs ) ) )
     => ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(1)
thf(fact_300_non__sym__dist__pairs__elems__distinct_I1_J,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Y: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ X2 @ Y ) @ ( set_Pr1402144133550743484od_c_a @ ( non_sy8103353363212570416od_c_a @ Xs ) ) )
     => ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(1)
thf(fact_301_non__sym__dist__pairs__elems__distinct_I1_J,axiom,
    ! [X2: produc1703568184450464039_set_a,Y: produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a] :
      ( ( member1214319639984524048_set_a @ ( produc4984519302771176279_set_a @ X2 @ Y ) @ ( set_Pr6500651403171443068_set_a @ ( non_sy5028023902500071067_set_a @ Xs ) ) )
     => ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(1)
thf(fact_302_non__sym__dist__pairs__elems__distinct_I1_J,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ X2 @ Y ) @ ( set_Pr292865031892111420od_b_c @ ( non_sy2897533375561516642od_b_c @ Xs ) ) )
     => ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(1)
thf(fact_303_non__sym__dist__pairs__elems__distinct_I1_J,axiom,
    ! [X2: set_a,Y: set_a,Xs: list_set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( non_sy7997271732749704658_set_a @ Xs ) ) )
     => ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(1)
thf(fact_304_non__sym__dist__pairs__elems__distinct_I1_J,axiom,
    ! [X2: d,Y: d,Xs: list_d] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( non_sym_dist_pairs_d @ Xs ) ) )
     => ( member_d @ X2 @ ( set_d2 @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(1)
thf(fact_305_non__sym__dist__pairs__elems__distinct_I1_J,axiom,
    ! [X2: a,Y: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% non_sym_dist_pairs_elems_distinct(1)
thf(fact_306_non__sym__dist__pairs__elems,axiom,
    ! [X2: produc4859908366599477352od_d_d,Xs: list_P4069430723938770798od_d_d,Y: produc4859908366599477352od_d_d] :
      ( ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
     => ( ( member73601632213084689od_d_d @ Y @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member6383636634632210000od_d_d @ ( produc1795809570569509655od_d_d @ X2 @ Y ) @ ( set_Pr5589546020064100540od_d_d @ ( non_sy4332508533479320540od_d_d @ Xs ) ) )
            | ( member6383636634632210000od_d_d @ ( produc1795809570569509655od_d_d @ Y @ X2 ) @ ( set_Pr5589546020064100540od_d_d @ ( non_sy4332508533479320540od_d_d @ Xs ) ) ) ) ) ) ) ).

% non_sym_dist_pairs_elems
thf(fact_307_non__sym__dist__pairs__elems,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Xs: list_P5219093185649440210_a_b_c,Y: produc8838969699183119820_a_b_c] :
      ( ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
     => ( ( member997585926662434933_a_b_c @ Y @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member289795379306782288_a_b_c @ ( produc7246914118821965079_a_b_c @ X2 @ Y ) @ ( set_Pr4521536175723917500_a_b_c @ ( non_sy4469082471694185792_a_b_c @ Xs ) ) )
            | ( member289795379306782288_a_b_c @ ( produc7246914118821965079_a_b_c @ Y @ X2 ) @ ( set_Pr4521536175723917500_a_b_c @ ( non_sy4469082471694185792_a_b_c @ Xs ) ) ) ) ) ) ) ).

% non_sym_dist_pairs_elems
thf(fact_308_non__sym__dist__pairs__elems,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a,Y: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ Xs ) )
     => ( ( member7410604586820865893od_c_a @ Y @ ( set_li8663184538779258833od_c_a @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ X2 @ Y ) @ ( set_Pr1402144133550743484od_c_a @ ( non_sy8103353363212570416od_c_a @ Xs ) ) )
            | ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ Y @ X2 ) @ ( set_Pr1402144133550743484od_c_a @ ( non_sy8103353363212570416od_c_a @ Xs ) ) ) ) ) ) ) ).

% non_sym_dist_pairs_elems
thf(fact_309_non__sym__dist__pairs__elems,axiom,
    ! [X2: set_a,Xs: list_set_a,Y: set_a] :
      ( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
     => ( ( member_set_a @ Y @ ( set_set_a2 @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( non_sy7997271732749704658_set_a @ Xs ) ) )
            | ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y @ X2 ) @ ( set_Pr7878197726666307004_set_a @ ( non_sy7997271732749704658_set_a @ Xs ) ) ) ) ) ) ) ).

% non_sym_dist_pairs_elems
thf(fact_310_non__sym__dist__pairs__elems,axiom,
    ! [X2: produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a,Y: produc1703568184450464039_set_a] :
      ( ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Xs ) )
     => ( ( member7983343339038529360_set_a @ Y @ ( set_Pr7878197726666307004_set_a @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member1214319639984524048_set_a @ ( produc4984519302771176279_set_a @ X2 @ Y ) @ ( set_Pr6500651403171443068_set_a @ ( non_sy5028023902500071067_set_a @ Xs ) ) )
            | ( member1214319639984524048_set_a @ ( produc4984519302771176279_set_a @ Y @ X2 ) @ ( set_Pr6500651403171443068_set_a @ ( non_sy5028023902500071067_set_a @ Xs ) ) ) ) ) ) ) ).

% non_sym_dist_pairs_elems
thf(fact_311_non__sym__dist__pairs__elems,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ X2 @ Y ) @ ( set_Pr292865031892111420od_b_c @ ( non_sy2897533375561516642od_b_c @ Xs ) ) )
            | ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ Y @ X2 ) @ ( set_Pr292865031892111420od_b_c @ ( non_sy2897533375561516642od_b_c @ Xs ) ) ) ) ) ) ) ).

% non_sym_dist_pairs_elems
thf(fact_312_non__sym__dist__pairs__elems,axiom,
    ! [X2: d,Xs: list_d,Y: d] :
      ( ( member_d @ X2 @ ( set_d2 @ Xs ) )
     => ( ( member_d @ Y @ ( set_d2 @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( non_sym_dist_pairs_d @ Xs ) ) )
            | ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ Y @ X2 ) @ ( set_Product_prod_d_d2 @ ( non_sym_dist_pairs_d @ Xs ) ) ) ) ) ) ) ).

% non_sym_dist_pairs_elems
thf(fact_313_non__sym__dist__pairs__elems,axiom,
    ! [X2: a,Xs: list_a,Y: a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( member_a @ Y @ ( set_a2 @ Xs ) )
       => ( ( X2 != Y )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
            | ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X2 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) ) ) ) ) ) ).

% non_sym_dist_pairs_elems
thf(fact_314__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062repetition__sets_O_Aimplies__completeness__for__repetition__sets_A_ITest__Suite_Aprs_Atps_Ard__targets_Aatcs_J_AM_Am_Arepetition__sets_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Repetition_sets: list_P3660316430366008877_set_a] :
        ~ ( test_i4756896328399799264_b_c_d @ ( test_t5871526445387177708_b_c_d @ prs @ tps @ rd_targets @ atcs ) @ m @ m2 @ Repetition_sets ) ).

% \<open>\<And>thesis. (\<And>repetition_sets. implies_completeness_for_repetition_sets (Test_Suite prs tps rd_targets atcs) M m repetition_sets \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_315_t2,axiom,
    ! [Q: a,P5: fsm_a_b_c] :
      ( ( member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ Q @ P5 ) @ prs )
     => ( state_3011419026027373943_a_b_c @ P5 @ m @ Q ) ) ).

% t2
thf(fact_316_exI__realizer,axiom,
    ! [P5: d > d > $o,Y: d,X2: d] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( product_snd_d_d @ ( product_Pair_d_d @ X2 @ Y ) ) @ ( product_fst_d_d @ ( product_Pair_d_d @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_317_exI__realizer,axiom,
    ! [P5: c > b > $o,Y: c,X2: b] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( product_snd_b_c @ ( product_Pair_b_c @ X2 @ Y ) ) @ ( product_fst_b_c @ ( product_Pair_b_c @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_318_exI__realizer,axiom,
    ! [P5: list_P6327159017948738492od_c_a > a > $o,Y: list_P6327159017948738492od_c_a,X2: a] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( produc5527568727691618435od_c_a @ ( produc3580641852002036821od_c_a @ X2 @ Y ) ) @ ( produc4637894536740104257od_c_a @ ( produc3580641852002036821od_c_a @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_319_exI__realizer,axiom,
    ! [P5: a > a > $o,Y: a,X2: a] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( product_snd_a_a @ ( product_Pair_a_a @ X2 @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_320_exI__realizer,axiom,
    ! [P5: fsm_a_b_c > a > $o,Y: fsm_a_b_c,X2: a] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( produc4728557313310798826_a_b_c @ ( produc2795441082327197628_a_b_c @ X2 @ Y ) ) @ ( produc2620512763296580520_a_b_c @ ( produc2795441082327197628_a_b_c @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_321_exI__realizer,axiom,
    ! [P5: a > c > $o,Y: a,X2: c] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( product_snd_c_a @ ( product_Pair_c_a @ X2 @ Y ) ) @ ( product_fst_c_a @ ( product_Pair_c_a @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_322_exI__realizer,axiom,
    ! [P5: produc1703568184450464039_set_a > list_P6327159017948738492od_c_a > $o,Y: produc1703568184450464039_set_a,X2: list_P6327159017948738492od_c_a] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( produc2451121746714373936_set_a @ ( produc6508649944902238466_set_a @ X2 @ Y ) ) @ ( produc3242630202250842350_set_a @ ( produc6508649944902238466_set_a @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_323_exI__realizer,axiom,
    ! [P5: set_a > set_a > $o,Y: set_a,X2: set_a] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( produc1983107199584856133_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) ) @ ( produc9088895665703139587_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_324_exI__realizer,axiom,
    ! [P5: produc2133359284458116493od_c_a > a > $o,Y: produc2133359284458116493od_c_a,X2: a] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X2 @ Y ) ) @ ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_325_exI__realizer,axiom,
    ! [P5: product_prod_c_a > b > $o,Y: product_prod_c_a,X2: b] :
      ( ( P5 @ Y @ X2 )
     => ( P5 @ ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X2 @ Y ) ) @ ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_326_conjI__realizer,axiom,
    ! [P5: d > $o,P: d,Q5: d > $o,Q: d] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( product_fst_d_d @ ( product_Pair_d_d @ P @ Q ) ) )
          & ( Q5 @ ( product_snd_d_d @ ( product_Pair_d_d @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_327_conjI__realizer,axiom,
    ! [P5: b > $o,P: b,Q5: c > $o,Q: c] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( product_fst_b_c @ ( product_Pair_b_c @ P @ Q ) ) )
          & ( Q5 @ ( product_snd_b_c @ ( product_Pair_b_c @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_328_conjI__realizer,axiom,
    ! [P5: a > $o,P: a,Q5: list_P6327159017948738492od_c_a > $o,Q: list_P6327159017948738492od_c_a] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( produc4637894536740104257od_c_a @ ( produc3580641852002036821od_c_a @ P @ Q ) ) )
          & ( Q5 @ ( produc5527568727691618435od_c_a @ ( produc3580641852002036821od_c_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_329_conjI__realizer,axiom,
    ! [P5: a > $o,P: a,Q5: a > $o,Q: a] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q ) ) )
          & ( Q5 @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_330_conjI__realizer,axiom,
    ! [P5: a > $o,P: a,Q5: fsm_a_b_c > $o,Q: fsm_a_b_c] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( produc2620512763296580520_a_b_c @ ( produc2795441082327197628_a_b_c @ P @ Q ) ) )
          & ( Q5 @ ( produc4728557313310798826_a_b_c @ ( produc2795441082327197628_a_b_c @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_331_conjI__realizer,axiom,
    ! [P5: c > $o,P: c,Q5: a > $o,Q: a] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( product_fst_c_a @ ( product_Pair_c_a @ P @ Q ) ) )
          & ( Q5 @ ( product_snd_c_a @ ( product_Pair_c_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_332_conjI__realizer,axiom,
    ! [P5: list_P6327159017948738492od_c_a > $o,P: list_P6327159017948738492od_c_a,Q5: produc1703568184450464039_set_a > $o,Q: produc1703568184450464039_set_a] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( produc3242630202250842350_set_a @ ( produc6508649944902238466_set_a @ P @ Q ) ) )
          & ( Q5 @ ( produc2451121746714373936_set_a @ ( produc6508649944902238466_set_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_333_conjI__realizer,axiom,
    ! [P5: set_a > $o,P: set_a,Q5: set_a > $o,Q: set_a] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( produc9088895665703139587_set_a @ ( produc9088192753505129239_set_a @ P @ Q ) ) )
          & ( Q5 @ ( produc1983107199584856133_set_a @ ( produc9088192753505129239_set_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_334_conjI__realizer,axiom,
    ! [P5: a > $o,P: a,Q5: produc2133359284458116493od_c_a > $o,Q: produc2133359284458116493od_c_a] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ P @ Q ) ) )
          & ( Q5 @ ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_335_conjI__realizer,axiom,
    ! [P5: b > $o,P: b,Q5: product_prod_c_a > $o,Q: product_prod_c_a] :
      ( ( P5 @ P )
     => ( ( Q5 @ Q )
       => ( ( P5 @ ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ P @ Q ) ) )
          & ( Q5 @ ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_336_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: d > d > $o,X2: d,Y: d,A2: product_prod_d_d] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( product_Pair_d_d @ X2 @ Y ) )
       => ( P5 @ ( product_fst_d_d @ A2 ) @ ( product_snd_d_d @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_337_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: b > c > $o,X2: b,Y: c,A2: product_prod_b_c] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( product_Pair_b_c @ X2 @ Y ) )
       => ( P5 @ ( product_fst_b_c @ A2 ) @ ( product_snd_b_c @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_338_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: a > list_P6327159017948738492od_c_a > $o,X2: a,Y: list_P6327159017948738492od_c_a,A2: produc3220416805844688357od_c_a] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( produc3580641852002036821od_c_a @ X2 @ Y ) )
       => ( P5 @ ( produc4637894536740104257od_c_a @ A2 ) @ ( produc5527568727691618435od_c_a @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_339_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: a > a > $o,X2: a,Y: a,A2: product_prod_a_a] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( product_Pair_a_a @ X2 @ Y ) )
       => ( P5 @ ( product_fst_a_a @ A2 ) @ ( product_snd_a_a @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_340_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: a > fsm_a_b_c > $o,X2: a,Y: fsm_a_b_c,A2: produc8838969699183119820_a_b_c] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( produc2795441082327197628_a_b_c @ X2 @ Y ) )
       => ( P5 @ ( produc2620512763296580520_a_b_c @ A2 ) @ ( produc4728557313310798826_a_b_c @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_341_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: c > a > $o,X2: c,Y: a,A2: product_prod_c_a] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( product_Pair_c_a @ X2 @ Y ) )
       => ( P5 @ ( product_fst_c_a @ A2 ) @ ( product_snd_c_a @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_342_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: list_P6327159017948738492od_c_a > produc1703568184450464039_set_a > $o,X2: list_P6327159017948738492od_c_a,Y: produc1703568184450464039_set_a,A2: produc7049025859877086738_set_a] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( produc6508649944902238466_set_a @ X2 @ Y ) )
       => ( P5 @ ( produc3242630202250842350_set_a @ A2 ) @ ( produc2451121746714373936_set_a @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_343_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: set_a > set_a > $o,X2: set_a,Y: set_a,A2: produc1703568184450464039_set_a] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( produc9088192753505129239_set_a @ X2 @ Y ) )
       => ( P5 @ ( produc9088895665703139587_set_a @ A2 ) @ ( produc1983107199584856133_set_a @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_344_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: a > produc2133359284458116493od_c_a > $o,X2: a,Y: produc2133359284458116493od_c_a,A2: produc2669932201312201782od_c_a] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( produc5762801557034676134od_c_a @ X2 @ Y ) )
       => ( P5 @ ( produc4225943634317630866od_c_a @ A2 ) @ ( produc8088130389382527188od_c_a @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_345_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P5: b > product_prod_c_a > $o,X2: b,Y: product_prod_c_a,A2: produc2133359284458116493od_c_a] :
      ( ( P5 @ X2 @ Y )
     => ( ( A2
          = ( produc3440865914000881661od_c_a @ X2 @ Y ) )
       => ( P5 @ ( produc301949074746585577od_c_a @ A2 ) @ ( produc5221446324160585515od_c_a @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_346_test__suite_Oinject,axiom,
    ! [X1: set_Pr2640454425625433516_a_b_c,X22: a > set_li1159382662694783132od_c_a,X32: produc3220416805844688357od_c_a > set_a,X42: product_prod_a_a > set_Pr9130538228769002824od_d_d,Y1: set_Pr2640454425625433516_a_b_c,Y2: a > set_li1159382662694783132od_c_a,Y32: produc3220416805844688357od_c_a > set_a,Y42: product_prod_a_a > set_Pr9130538228769002824od_d_d] :
      ( ( ( test_t5871526445387177708_b_c_d @ X1 @ X22 @ X32 @ X42 )
        = ( test_t5871526445387177708_b_c_d @ Y1 @ Y2 @ Y32 @ Y42 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 )
        & ( X32 = Y32 )
        & ( X42 = Y42 ) ) ) ).

% test_suite.inject
thf(fact_347__092_060open_062q_A_092_060in_062_Afst_A_096_Aprs_092_060close_062,axiom,
    member_a @ q2 @ ( image_3273489647386116407_b_c_a @ produc2620512763296580520_a_b_c @ prs ) ).

% \<open>q \<in> fst ` prs\<close>
thf(fact_348_subsetI,axiom,
    ! [A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ! [X: produc4859908366599477352od_d_d] :
          ( ( member73601632213084689od_d_d @ X @ A )
         => ( member73601632213084689od_d_d @ X @ B ) )
     => ( ord_le7464911323401878760od_d_d @ A @ B ) ) ).

% subsetI
thf(fact_349_subsetI,axiom,
    ! [A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ! [X: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ X @ A )
         => ( member997585926662434933_a_b_c @ X @ B ) )
     => ( ord_le6086132928344890700_a_b_c @ A @ B ) ) ).

% subsetI
thf(fact_350_subsetI,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ A )
         => ( member6330420149250801815od_b_c @ X @ B ) )
     => ( ord_le282488521294790766od_b_c @ A @ B ) ) ).

% subsetI
thf(fact_351_subsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_a @ X @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% subsetI
thf(fact_352_subsetI,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ! [X: list_P6327159017948738492od_c_a] :
          ( ( member7410604586820865893od_c_a @ X @ A )
         => ( member7410604586820865893od_c_a @ X @ B ) )
     => ( ord_le2998388488506175548od_c_a @ A @ B ) ) ).

% subsetI
thf(fact_353_subset__antisym,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_354_subset__antisym,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A @ B )
     => ( ( ord_le2998388488506175548od_c_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_355_order__refl,axiom,
    ! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).

% order_refl
thf(fact_356_order__refl,axiom,
    ! [X2: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ X2 @ X2 ) ).

% order_refl
thf(fact_357_dual__order_Orefl,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_358_dual__order_Orefl,axiom,
    ! [A2: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_359_image__eqI,axiom,
    ! [B2: a,F: a > a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_360_image__eqI,axiom,
    ! [B2: a,F: list_P903359562653991662od_b_c > a,X2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member6330420149250801815od_b_c @ X2 @ A )
       => ( member_a @ B2 @ ( image_3173854338463926933_b_c_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_361_image__eqI,axiom,
    ! [B2: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member6330420149250801815od_b_c @ B2 @ ( image_8656872703560917237od_b_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_362_image__eqI,axiom,
    ! [B2: a,F: produc8838969699183119820_a_b_c > a,X2: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member997585926662434933_a_b_c @ X2 @ A )
       => ( member_a @ B2 @ ( image_3273489647386116407_b_c_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_363_image__eqI,axiom,
    ! [B2: produc8838969699183119820_a_b_c,F: a > produc8838969699183119820_a_b_c,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member997585926662434933_a_b_c @ B2 @ ( image_6580063982545254099_a_b_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_364_image__eqI,axiom,
    ! [B2: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,X2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member6330420149250801815od_b_c @ X2 @ A )
       => ( member6330420149250801815od_b_c @ B2 @ ( image_2850395784482391045od_b_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_365_image__eqI,axiom,
    ! [B2: a,F: produc4859908366599477352od_d_d > a,X2: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member73601632213084689od_d_d @ X2 @ A )
       => ( member_a @ B2 @ ( image_3229965221763006363_d_d_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_366_image__eqI,axiom,
    ! [B2: a,F: list_P6327159017948738492od_c_a > a,X2: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member7410604586820865893od_c_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_8824809819145125191_c_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_367_image__eqI,axiom,
    ! [B2: produc4859908366599477352od_d_d,F: a > produc4859908366599477352od_d_d,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member73601632213084689od_d_d @ B2 @ ( image_2487513010242200175od_d_d @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_368_image__eqI,axiom,
    ! [B2: list_P6327159017948738492od_c_a,F: a > list_P6327159017948738492od_c_a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member7410604586820865893od_c_a @ B2 @ ( image_3129299617955238851od_c_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_369_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: a,F: a > a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_370_rev__image__eqI,axiom,
    ! [X2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B2: a,F: list_P903359562653991662od_b_c > a] :
      ( ( member6330420149250801815od_b_c @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_3173854338463926933_b_c_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_371_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member6330420149250801815od_b_c @ B2 @ ( image_8656872703560917237od_b_c @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_372_rev__image__eqI,axiom,
    ! [X2: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B2: a,F: produc8838969699183119820_a_b_c > a] :
      ( ( member997585926662434933_a_b_c @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_3273489647386116407_b_c_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_373_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: produc8838969699183119820_a_b_c,F: a > produc8838969699183119820_a_b_c] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member997585926662434933_a_b_c @ B2 @ ( image_6580063982545254099_a_b_c @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_374_rev__image__eqI,axiom,
    ! [X2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B2: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member6330420149250801815od_b_c @ B2 @ ( image_2850395784482391045od_b_c @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_375_rev__image__eqI,axiom,
    ! [X2: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B2: a,F: produc4859908366599477352od_d_d > a] :
      ( ( member73601632213084689od_d_d @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_3229965221763006363_d_d_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_376_rev__image__eqI,axiom,
    ! [X2: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B2: a,F: list_P6327159017948738492od_c_a > a] :
      ( ( member7410604586820865893od_c_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_8824809819145125191_c_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_377_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: produc4859908366599477352od_d_d,F: a > produc4859908366599477352od_d_d] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member73601632213084689od_d_d @ B2 @ ( image_2487513010242200175od_d_d @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_378_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: list_P6327159017948738492od_c_a,F: a > list_P6327159017948738492od_c_a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member7410604586820865893od_c_a @ B2 @ ( image_3129299617955238851od_c_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_379_ball__imageD,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c,P5: a > $o] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( image_3273489647386116407_b_c_a @ F @ A ) )
         => ( P5 @ X ) )
     => ! [X4: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ X4 @ A )
         => ( P5 @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_380_ball__imageD,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ! [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ ( image_2850395784482391045od_b_c @ F @ A ) )
         => ( P5 @ X ) )
     => ! [X4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X4 @ A )
         => ( P5 @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_381_ball__imageD,axiom,
    ! [F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,A: set_Pr3779000724142394866_set_a,P5: list_P6327159017948738492od_c_a > $o] :
      ( ! [X: list_P6327159017948738492od_c_a] :
          ( ( member7410604586820865893od_c_a @ X @ ( image_865596521259543215od_c_a @ F @ A ) )
         => ( P5 @ X ) )
     => ! [X4: produc7049025859877086738_set_a] :
          ( ( member6493638300815972283_set_a @ X4 @ A )
         => ( P5 @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_382_ball__imageD,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,P5: set_li6436108459499378894od_b_c > $o] :
      ( ! [X: set_li6436108459499378894od_b_c] :
          ( ( member6985331446368301687od_b_c @ X @ ( image_2344789136157180295od_b_c @ F @ A ) )
         => ( P5 @ X ) )
     => ! [X4: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ X4 @ A )
         => ( P5 @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_383_image__cong,axiom,
    ! [M: set_Pr3779000724142394866_set_a,N: set_Pr3779000724142394866_set_a,F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,G: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a] :
      ( ( M = N )
     => ( ! [X: produc7049025859877086738_set_a] :
            ( ( member6493638300815972283_set_a @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_865596521259543215od_c_a @ F @ M )
          = ( image_865596521259543215od_c_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_384_image__cong,axiom,
    ! [M: set_Pr2640454425625433516_a_b_c,N: set_Pr2640454425625433516_a_b_c,F: produc8838969699183119820_a_b_c > a,G: produc8838969699183119820_a_b_c > a] :
      ( ( M = N )
     => ( ! [X: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_3273489647386116407_b_c_a @ F @ M )
          = ( image_3273489647386116407_b_c_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_385_image__cong,axiom,
    ! [M: set_Pr2640454425625433516_a_b_c,N: set_Pr2640454425625433516_a_b_c,F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,G: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c] :
      ( ( M = N )
     => ( ! [X: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_2344789136157180295od_b_c @ F @ M )
          = ( image_2344789136157180295od_b_c @ G @ N ) ) ) ) ).

% image_cong
thf(fact_386_image__cong,axiom,
    ! [M: set_li6436108459499378894od_b_c,N: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,G: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( M = N )
     => ( ! [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ N )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( image_2850395784482391045od_b_c @ F @ M )
          = ( image_2850395784482391045od_b_c @ G @ N ) ) ) ) ).

% image_cong
thf(fact_387_bex__imageD,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c,P5: a > $o] :
      ( ? [X4: a] :
          ( ( member_a @ X4 @ ( image_3273489647386116407_b_c_a @ F @ A ) )
          & ( P5 @ X4 ) )
     => ? [X: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ X @ A )
          & ( P5 @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_388_bex__imageD,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ? [X4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X4 @ ( image_2850395784482391045od_b_c @ F @ A ) )
          & ( P5 @ X4 ) )
     => ? [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ A )
          & ( P5 @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_389_bex__imageD,axiom,
    ! [F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,A: set_Pr3779000724142394866_set_a,P5: list_P6327159017948738492od_c_a > $o] :
      ( ? [X4: list_P6327159017948738492od_c_a] :
          ( ( member7410604586820865893od_c_a @ X4 @ ( image_865596521259543215od_c_a @ F @ A ) )
          & ( P5 @ X4 ) )
     => ? [X: produc7049025859877086738_set_a] :
          ( ( member6493638300815972283_set_a @ X @ A )
          & ( P5 @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_390_bex__imageD,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,P5: set_li6436108459499378894od_b_c > $o] :
      ( ? [X4: set_li6436108459499378894od_b_c] :
          ( ( member6985331446368301687od_b_c @ X4 @ ( image_2344789136157180295od_b_c @ F @ A ) )
          & ( P5 @ X4 ) )
     => ? [X: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ X @ A )
          & ( P5 @ ( F @ X ) ) ) ) ).

% bex_imageD
thf(fact_391_image__iff,axiom,
    ! [Z: set_li6436108459499378894od_b_c,F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( member6985331446368301687od_b_c @ Z @ ( image_2344789136157180295od_b_c @ F @ A ) )
      = ( ? [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_392_image__iff,axiom,
    ! [Z: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ Z @ ( image_2850395784482391045od_b_c @ F @ A ) )
      = ( ? [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_393_image__iff,axiom,
    ! [Z: list_P6327159017948738492od_c_a,F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,A: set_Pr3779000724142394866_set_a] :
      ( ( member7410604586820865893od_c_a @ Z @ ( image_865596521259543215od_c_a @ F @ A ) )
      = ( ? [X3: produc7049025859877086738_set_a] :
            ( ( member6493638300815972283_set_a @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_394_image__iff,axiom,
    ! [Z: a,F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c] :
      ( ( member_a @ Z @ ( image_3273489647386116407_b_c_a @ F @ A ) )
      = ( ? [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_395_imageI,axiom,
    ! [X2: a,A: set_a,F: a > a] :
      ( ( member_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_396_imageI,axiom,
    ! [X2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > a] :
      ( ( member6330420149250801815od_b_c @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_3173854338463926933_b_c_a @ F @ A ) ) ) ).

% imageI
thf(fact_397_imageI,axiom,
    ! [X2: a,A: set_a,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ A )
     => ( member6330420149250801815od_b_c @ ( F @ X2 ) @ ( image_8656872703560917237od_b_c @ F @ A ) ) ) ).

% imageI
thf(fact_398_imageI,axiom,
    ! [X2: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,F: produc8838969699183119820_a_b_c > a] :
      ( ( member997585926662434933_a_b_c @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_3273489647386116407_b_c_a @ F @ A ) ) ) ).

% imageI
thf(fact_399_imageI,axiom,
    ! [X2: a,A: set_a,F: a > produc8838969699183119820_a_b_c] :
      ( ( member_a @ X2 @ A )
     => ( member997585926662434933_a_b_c @ ( F @ X2 ) @ ( image_6580063982545254099_a_b_c @ F @ A ) ) ) ).

% imageI
thf(fact_400_imageI,axiom,
    ! [X2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ A )
     => ( member6330420149250801815od_b_c @ ( F @ X2 ) @ ( image_2850395784482391045od_b_c @ F @ A ) ) ) ).

% imageI
thf(fact_401_imageI,axiom,
    ! [X2: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,F: produc4859908366599477352od_d_d > a] :
      ( ( member73601632213084689od_d_d @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_3229965221763006363_d_d_a @ F @ A ) ) ) ).

% imageI
thf(fact_402_imageI,axiom,
    ! [X2: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,F: list_P6327159017948738492od_c_a > a] :
      ( ( member7410604586820865893od_c_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_8824809819145125191_c_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_403_imageI,axiom,
    ! [X2: a,A: set_a,F: a > produc4859908366599477352od_d_d] :
      ( ( member_a @ X2 @ A )
     => ( member73601632213084689od_d_d @ ( F @ X2 ) @ ( image_2487513010242200175od_d_d @ F @ A ) ) ) ).

% imageI
thf(fact_404_imageI,axiom,
    ! [X2: a,A: set_a,F: a > list_P6327159017948738492od_c_a] :
      ( ( member_a @ X2 @ A )
     => ( member7410604586820865893od_c_a @ ( F @ X2 ) @ ( image_3129299617955238851od_c_a @ F @ A ) ) ) ).

% imageI
thf(fact_405_subset__image__iff,axiom,
    ! [B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ ( image_2850395784482391045od_b_c @ F @ A ) )
      = ( ? [AA: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ AA @ A )
            & ( B
              = ( image_2850395784482391045od_b_c @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_406_subset__image__iff,axiom,
    ! [B: set_se3924713247505902254od_b_c,F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( ord_le6656836712342966862od_b_c @ B @ ( image_2344789136157180295od_b_c @ F @ A ) )
      = ( ? [AA: set_Pr2640454425625433516_a_b_c] :
            ( ( ord_le6086132928344890700_a_b_c @ AA @ A )
            & ( B
              = ( image_2344789136157180295od_b_c @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_407_subset__image__iff,axiom,
    ! [B: set_a,F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c] :
      ( ( ord_less_eq_set_a @ B @ ( image_3273489647386116407_b_c_a @ F @ A ) )
      = ( ? [AA: set_Pr2640454425625433516_a_b_c] :
            ( ( ord_le6086132928344890700_a_b_c @ AA @ A )
            & ( B
              = ( image_3273489647386116407_b_c_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_408_subset__image__iff,axiom,
    ! [B: set_a,F: a > a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_409_subset__image__iff,axiom,
    ! [B: set_a,F: list_P6327159017948738492od_c_a > a,A: set_li1159382662694783132od_c_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_8824809819145125191_c_a_a @ F @ A ) )
      = ( ? [AA: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ AA @ A )
            & ( B
              = ( image_8824809819145125191_c_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_410_subset__image__iff,axiom,
    ! [B: set_li1159382662694783132od_c_a,F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,A: set_Pr3779000724142394866_set_a] :
      ( ( ord_le2998388488506175548od_c_a @ B @ ( image_865596521259543215od_c_a @ F @ A ) )
      = ( ? [AA: set_Pr3779000724142394866_set_a] :
            ( ( ord_le1057066820810438034_set_a @ AA @ A )
            & ( B
              = ( image_865596521259543215od_c_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_411_subset__image__iff,axiom,
    ! [B: set_li1159382662694783132od_c_a,F: a > list_P6327159017948738492od_c_a,A: set_a] :
      ( ( ord_le2998388488506175548od_c_a @ B @ ( image_3129299617955238851od_c_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_3129299617955238851od_c_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_412_subset__image__iff,axiom,
    ! [B: set_li1159382662694783132od_c_a,F: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B @ ( image_1061481830026363525od_c_a @ F @ A ) )
      = ( ? [AA: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ AA @ A )
            & ( B
              = ( image_1061481830026363525od_c_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_413_image__subset__iff,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_se3924713247505902254od_b_c] :
      ( ( ord_le6656836712342966862od_b_c @ ( image_2344789136157180295od_b_c @ F @ A ) @ B )
      = ( ! [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
           => ( member6985331446368301687od_b_c @ ( F @ X3 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_414_image__subset__iff,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( image_2850395784482391045od_b_c @ F @ A ) @ B )
      = ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A )
           => ( member6330420149250801815od_b_c @ ( F @ X3 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_415_image__subset__iff,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c,B: set_a] :
      ( ( ord_less_eq_set_a @ ( image_3273489647386116407_b_c_a @ F @ A ) @ B )
      = ( ! [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
           => ( member_a @ ( F @ X3 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_416_image__subset__iff,axiom,
    ! [F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,A: set_Pr3779000724142394866_set_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( image_865596521259543215od_c_a @ F @ A ) @ B )
      = ( ! [X3: produc7049025859877086738_set_a] :
            ( ( member6493638300815972283_set_a @ X3 @ A )
           => ( member7410604586820865893od_c_a @ ( F @ X3 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_417_subset__imageE,axiom,
    ! [B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ ( image_2850395784482391045od_b_c @ F @ A ) )
     => ~ ! [C2: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ C2 @ A )
           => ( B
             != ( image_2850395784482391045od_b_c @ F @ C2 ) ) ) ) ).

% subset_imageE
thf(fact_418_subset__imageE,axiom,
    ! [B: set_se3924713247505902254od_b_c,F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( ord_le6656836712342966862od_b_c @ B @ ( image_2344789136157180295od_b_c @ F @ A ) )
     => ~ ! [C2: set_Pr2640454425625433516_a_b_c] :
            ( ( ord_le6086132928344890700_a_b_c @ C2 @ A )
           => ( B
             != ( image_2344789136157180295od_b_c @ F @ C2 ) ) ) ) ).

% subset_imageE
thf(fact_419_subset__imageE,axiom,
    ! [B: set_a,F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c] :
      ( ( ord_less_eq_set_a @ B @ ( image_3273489647386116407_b_c_a @ F @ A ) )
     => ~ ! [C2: set_Pr2640454425625433516_a_b_c] :
            ( ( ord_le6086132928344890700_a_b_c @ C2 @ A )
           => ( B
             != ( image_3273489647386116407_b_c_a @ F @ C2 ) ) ) ) ).

% subset_imageE
thf(fact_420_subset__imageE,axiom,
    ! [B: set_a,F: a > a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
     => ~ ! [C2: set_a] :
            ( ( ord_less_eq_set_a @ C2 @ A )
           => ( B
             != ( image_a_a @ F @ C2 ) ) ) ) ).

% subset_imageE
thf(fact_421_subset__imageE,axiom,
    ! [B: set_a,F: list_P6327159017948738492od_c_a > a,A: set_li1159382662694783132od_c_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_8824809819145125191_c_a_a @ F @ A ) )
     => ~ ! [C2: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ C2 @ A )
           => ( B
             != ( image_8824809819145125191_c_a_a @ F @ C2 ) ) ) ) ).

% subset_imageE
thf(fact_422_subset__imageE,axiom,
    ! [B: set_li1159382662694783132od_c_a,F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,A: set_Pr3779000724142394866_set_a] :
      ( ( ord_le2998388488506175548od_c_a @ B @ ( image_865596521259543215od_c_a @ F @ A ) )
     => ~ ! [C2: set_Pr3779000724142394866_set_a] :
            ( ( ord_le1057066820810438034_set_a @ C2 @ A )
           => ( B
             != ( image_865596521259543215od_c_a @ F @ C2 ) ) ) ) ).

% subset_imageE
thf(fact_423_subset__imageE,axiom,
    ! [B: set_li1159382662694783132od_c_a,F: a > list_P6327159017948738492od_c_a,A: set_a] :
      ( ( ord_le2998388488506175548od_c_a @ B @ ( image_3129299617955238851od_c_a @ F @ A ) )
     => ~ ! [C2: set_a] :
            ( ( ord_less_eq_set_a @ C2 @ A )
           => ( B
             != ( image_3129299617955238851od_c_a @ F @ C2 ) ) ) ) ).

% subset_imageE
thf(fact_424_subset__imageE,axiom,
    ! [B: set_li1159382662694783132od_c_a,F: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B @ ( image_1061481830026363525od_c_a @ F @ A ) )
     => ~ ! [C2: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ C2 @ A )
           => ( B
             != ( image_1061481830026363525od_c_a @ F @ C2 ) ) ) ) ).

% subset_imageE
thf(fact_425_image__subsetI,axiom,
    ! [A: set_a,F: a > a,B: set_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_426_image__subsetI,axiom,
    ! [A: set_a,F: a > list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member6330420149250801815od_b_c @ ( F @ X ) @ B ) )
     => ( ord_le282488521294790766od_b_c @ ( image_8656872703560917237od_b_c @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_427_image__subsetI,axiom,
    ! [A: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > a,B: set_a] :
      ( ! [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_3173854338463926933_b_c_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_428_image__subsetI,axiom,
    ! [A: set_a,F: a > produc8838969699183119820_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member997585926662434933_a_b_c @ ( F @ X ) @ B ) )
     => ( ord_le6086132928344890700_a_b_c @ ( image_6580063982545254099_a_b_c @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_429_image__subsetI,axiom,
    ! [A: set_Pr2640454425625433516_a_b_c,F: produc8838969699183119820_a_b_c > a,B: set_a] :
      ( ! [X: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_3273489647386116407_b_c_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_430_image__subsetI,axiom,
    ! [A: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ A )
         => ( member6330420149250801815od_b_c @ ( F @ X ) @ B ) )
     => ( ord_le282488521294790766od_b_c @ ( image_2850395784482391045od_b_c @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_431_image__subsetI,axiom,
    ! [A: set_a,F: a > produc4859908366599477352od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member73601632213084689od_d_d @ ( F @ X ) @ B ) )
     => ( ord_le7464911323401878760od_d_d @ ( image_2487513010242200175od_d_d @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_432_image__subsetI,axiom,
    ! [A: set_Pr9130538228769002824od_d_d,F: produc4859908366599477352od_d_d > a,B: set_a] :
      ( ! [X: produc4859908366599477352od_d_d] :
          ( ( member73601632213084689od_d_d @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_3229965221763006363_d_d_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_433_image__subsetI,axiom,
    ! [A: set_li1159382662694783132od_c_a,F: list_P6327159017948738492od_c_a > a,B: set_a] :
      ( ! [X: list_P6327159017948738492od_c_a] :
          ( ( member7410604586820865893od_c_a @ X @ A )
         => ( member_a @ ( F @ X ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_8824809819145125191_c_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_434_image__subsetI,axiom,
    ! [A: set_a,F: a > list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ! [X: a] :
          ( ( member_a @ X @ A )
         => ( member7410604586820865893od_c_a @ ( F @ X ) @ B ) )
     => ( ord_le2998388488506175548od_c_a @ ( image_3129299617955238851od_c_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_435_image__mono,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ord_le282488521294790766od_b_c @ ( image_2850395784482391045od_b_c @ F @ A ) @ ( image_2850395784482391045od_b_c @ F @ B ) ) ) ).

% image_mono
thf(fact_436_image__mono,axiom,
    ! [A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c,F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c] :
      ( ( ord_le6086132928344890700_a_b_c @ A @ B )
     => ( ord_le6656836712342966862od_b_c @ ( image_2344789136157180295od_b_c @ F @ A ) @ ( image_2344789136157180295od_b_c @ F @ B ) ) ) ).

% image_mono
thf(fact_437_image__mono,axiom,
    ! [A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c,F: produc8838969699183119820_a_b_c > a] :
      ( ( ord_le6086132928344890700_a_b_c @ A @ B )
     => ( ord_less_eq_set_a @ ( image_3273489647386116407_b_c_a @ F @ A ) @ ( image_3273489647386116407_b_c_a @ F @ B ) ) ) ).

% image_mono
thf(fact_438_image__mono,axiom,
    ! [A: set_Pr3779000724142394866_set_a,B: set_Pr3779000724142394866_set_a,F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a] :
      ( ( ord_le1057066820810438034_set_a @ A @ B )
     => ( ord_le2998388488506175548od_c_a @ ( image_865596521259543215od_c_a @ F @ A ) @ ( image_865596521259543215od_c_a @ F @ B ) ) ) ).

% image_mono
thf(fact_439_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_440_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > list_P6327159017948738492od_c_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le2998388488506175548od_c_a @ ( image_3129299617955238851od_c_a @ F @ A ) @ ( image_3129299617955238851od_c_a @ F @ B ) ) ) ).

% image_mono
thf(fact_441_image__mono,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,F: list_P6327159017948738492od_c_a > a] :
      ( ( ord_le2998388488506175548od_c_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_8824809819145125191_c_a_a @ F @ A ) @ ( image_8824809819145125191_c_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_442_image__mono,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,F: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A @ B )
     => ( ord_le2998388488506175548od_c_a @ ( image_1061481830026363525od_c_a @ F @ A ) @ ( image_1061481830026363525od_c_a @ F @ B ) ) ) ).

% image_mono
thf(fact_443_implies__completeness__def,axiom,
    ( test_i4204698558891958671_b_c_d
    = ( ^ [T3: test_t899597172106206243_b_c_d,M2: fsm_a_b_c,M4: nat] :
        ? [X5: list_P3660316430366008877_set_a] : ( test_i4756896328399799264_b_c_d @ T3 @ M2 @ M4 @ X5 ) ) ) ).

% implies_completeness_def
thf(fact_444_order__antisym__conv,axiom,
    ! [Y: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X2 )
     => ( ( ord_less_eq_set_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_445_order__antisym__conv,axiom,
    ! [Y: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ Y @ X2 )
     => ( ( ord_le2998388488506175548od_c_a @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_446_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_447_ord__le__eq__subst,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le2998388488506175548od_c_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2998388488506175548od_c_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_448_ord__le__eq__subst,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,F: set_li1159382662694783132od_c_a > set_a,C: set_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_449_ord__le__eq__subst,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,F: set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ X @ Y3 )
             => ( ord_le2998388488506175548od_c_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2998388488506175548od_c_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_450_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_451_ord__eq__le__subst,axiom,
    ! [A2: set_li1159382662694783132od_c_a,F: set_a > set_li1159382662694783132od_c_a,B2: set_a,C: set_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le2998388488506175548od_c_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2998388488506175548od_c_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_452_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: set_li1159382662694783132od_c_a > set_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ C )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_453_ord__eq__le__subst,axiom,
    ! [A2: set_li1159382662694783132od_c_a,F: set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ C )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ X @ Y3 )
             => ( ord_le2998388488506175548od_c_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2998388488506175548od_c_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_454_order__eq__refl,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( X2 = Y )
     => ( ord_less_eq_set_a @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_455_order__eq__refl,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] :
      ( ( X2 = Y )
     => ( ord_le2998388488506175548od_c_a @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_456_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_457_order__subst2,axiom,
    ! [A2: set_a,B2: set_a,F: set_a > set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_le2998388488506175548od_c_a @ ( F @ B2 ) @ C )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le2998388488506175548od_c_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2998388488506175548od_c_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_458_order__subst2,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,F: set_li1159382662694783132od_c_a > set_a,C: set_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_459_order__subst2,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,F: set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( ord_le2998388488506175548od_c_a @ ( F @ B2 ) @ C )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ X @ Y3 )
             => ( ord_le2998388488506175548od_c_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2998388488506175548od_c_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_460_order__subst1,axiom,
    ! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_461_order__subst1,axiom,
    ! [A2: set_a,F: set_li1159382662694783132od_c_a > set_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ C )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ X @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_462_order__subst1,axiom,
    ! [A2: set_li1159382662694783132od_c_a,F: set_a > set_li1159382662694783132od_c_a,B2: set_a,C: set_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ord_le2998388488506175548od_c_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2998388488506175548od_c_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_463_order__subst1,axiom,
    ! [A2: set_li1159382662694783132od_c_a,F: set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ ( F @ B2 ) )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ C )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ X @ Y3 )
             => ( ord_le2998388488506175548od_c_a @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_le2998388488506175548od_c_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_464_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
    = ( ^ [A5: set_a,B5: set_a] :
          ( ( ord_less_eq_set_a @ A5 @ B5 )
          & ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_465_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_li1159382662694783132od_c_a,Z2: set_li1159382662694783132od_c_a] : ( Y4 = Z2 ) )
    = ( ^ [A5: set_li1159382662694783132od_c_a,B5: set_li1159382662694783132od_c_a] :
          ( ( ord_le2998388488506175548od_c_a @ A5 @ B5 )
          & ( ord_le2998388488506175548od_c_a @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_466_antisym,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_467_antisym,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_468_dual__order_Otrans,axiom,
    ! [B2: set_a,A2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ C @ B2 )
       => ( ord_less_eq_set_a @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_469_dual__order_Otrans,axiom,
    ! [B2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B2 @ A2 )
     => ( ( ord_le2998388488506175548od_c_a @ C @ B2 )
       => ( ord_le2998388488506175548od_c_a @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_470_dual__order_Oantisym,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_471_dual__order_Oantisym,axiom,
    ! [B2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B2 @ A2 )
     => ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_472_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
    = ( ^ [A5: set_a,B5: set_a] :
          ( ( ord_less_eq_set_a @ B5 @ A5 )
          & ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_473_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_li1159382662694783132od_c_a,Z2: set_li1159382662694783132od_c_a] : ( Y4 = Z2 ) )
    = ( ^ [A5: set_li1159382662694783132od_c_a,B5: set_li1159382662694783132od_c_a] :
          ( ( ord_le2998388488506175548od_c_a @ B5 @ A5 )
          & ( ord_le2998388488506175548od_c_a @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_474_order__trans,axiom,
    ! [X2: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( ord_less_eq_set_a @ Y @ Z )
       => ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_475_order__trans,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a,Z: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ X2 @ Y )
     => ( ( ord_le2998388488506175548od_c_a @ Y @ Z )
       => ( ord_le2998388488506175548od_c_a @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_476_order_Otrans,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% order.trans
thf(fact_477_order_Otrans,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ C )
       => ( ord_le2998388488506175548od_c_a @ A2 @ C ) ) ) ).

% order.trans
thf(fact_478_order__antisym,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( ord_less_eq_set_a @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_479_order__antisym,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ X2 @ Y )
     => ( ( ord_le2998388488506175548od_c_a @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_480_ord__le__eq__trans,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_481_ord__le__eq__trans,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_le2998388488506175548od_c_a @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_482_ord__eq__le__trans,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_483_ord__eq__le__trans,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( A2 = B2 )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ C )
       => ( ord_le2998388488506175548od_c_a @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_484_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
    = ( ^ [X3: set_a,Y5: set_a] :
          ( ( ord_less_eq_set_a @ X3 @ Y5 )
          & ( ord_less_eq_set_a @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_485_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_li1159382662694783132od_c_a,Z2: set_li1159382662694783132od_c_a] : ( Y4 = Z2 ) )
    = ( ^ [X3: set_li1159382662694783132od_c_a,Y5: set_li1159382662694783132od_c_a] :
          ( ( ord_le2998388488506175548od_c_a @ X3 @ Y5 )
          & ( ord_le2998388488506175548od_c_a @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_486_Collect__mono__iff,axiom,
    ! [P5: set_li6436108459499378894od_b_c > $o,Q5: set_li6436108459499378894od_b_c > $o] :
      ( ( ord_le6656836712342966862od_b_c @ ( collec1823247286932916153od_b_c @ P5 ) @ ( collec1823247286932916153od_b_c @ Q5 ) )
      = ( ! [X3: set_li6436108459499378894od_b_c] :
            ( ( P5 @ X3 )
           => ( Q5 @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_487_Collect__mono__iff,axiom,
    ! [P5: a > $o,Q5: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P5 ) @ ( collect_a @ Q5 ) )
      = ( ! [X3: a] :
            ( ( P5 @ X3 )
           => ( Q5 @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_488_Collect__mono__iff,axiom,
    ! [P5: list_P6327159017948738492od_c_a > $o,Q5: list_P6327159017948738492od_c_a > $o] :
      ( ( ord_le2998388488506175548od_c_a @ ( collec6273869032445462695od_c_a @ P5 ) @ ( collec6273869032445462695od_c_a @ Q5 ) )
      = ( ! [X3: list_P6327159017948738492od_c_a] :
            ( ( P5 @ X3 )
           => ( Q5 @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_489_set__eq__subset,axiom,
    ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
    = ( ^ [A6: set_a,B6: set_a] :
          ( ( ord_less_eq_set_a @ A6 @ B6 )
          & ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_490_set__eq__subset,axiom,
    ( ( ^ [Y4: set_li1159382662694783132od_c_a,Z2: set_li1159382662694783132od_c_a] : ( Y4 = Z2 ) )
    = ( ^ [A6: set_li1159382662694783132od_c_a,B6: set_li1159382662694783132od_c_a] :
          ( ( ord_le2998388488506175548od_c_a @ A6 @ B6 )
          & ( ord_le2998388488506175548od_c_a @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_491_subset__trans,axiom,
    ! [A: set_a,B: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ C3 )
       => ( ord_less_eq_set_a @ A @ C3 ) ) ) ).

% subset_trans
thf(fact_492_subset__trans,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,C3: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A @ B )
     => ( ( ord_le2998388488506175548od_c_a @ B @ C3 )
       => ( ord_le2998388488506175548od_c_a @ A @ C3 ) ) ) ).

% subset_trans
thf(fact_493_Collect__mono,axiom,
    ! [P5: set_li6436108459499378894od_b_c > $o,Q5: set_li6436108459499378894od_b_c > $o] :
      ( ! [X: set_li6436108459499378894od_b_c] :
          ( ( P5 @ X )
         => ( Q5 @ X ) )
     => ( ord_le6656836712342966862od_b_c @ ( collec1823247286932916153od_b_c @ P5 ) @ ( collec1823247286932916153od_b_c @ Q5 ) ) ) ).

% Collect_mono
thf(fact_494_Collect__mono,axiom,
    ! [P5: a > $o,Q5: a > $o] :
      ( ! [X: a] :
          ( ( P5 @ X )
         => ( Q5 @ X ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P5 ) @ ( collect_a @ Q5 ) ) ) ).

% Collect_mono
thf(fact_495_Collect__mono,axiom,
    ! [P5: list_P6327159017948738492od_c_a > $o,Q5: list_P6327159017948738492od_c_a > $o] :
      ( ! [X: list_P6327159017948738492od_c_a] :
          ( ( P5 @ X )
         => ( Q5 @ X ) )
     => ( ord_le2998388488506175548od_c_a @ ( collec6273869032445462695od_c_a @ P5 ) @ ( collec6273869032445462695od_c_a @ Q5 ) ) ) ).

% Collect_mono
thf(fact_496_subset__refl,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% subset_refl
thf(fact_497_subset__refl,axiom,
    ! [A: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ A @ A ) ).

% subset_refl
thf(fact_498_subset__iff,axiom,
    ( ord_le7464911323401878760od_d_d
    = ( ^ [A6: set_Pr9130538228769002824od_d_d,B6: set_Pr9130538228769002824od_d_d] :
        ! [T: produc4859908366599477352od_d_d] :
          ( ( member73601632213084689od_d_d @ T @ A6 )
         => ( member73601632213084689od_d_d @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_499_subset__iff,axiom,
    ( ord_le6086132928344890700_a_b_c
    = ( ^ [A6: set_Pr2640454425625433516_a_b_c,B6: set_Pr2640454425625433516_a_b_c] :
        ! [T: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ T @ A6 )
         => ( member997585926662434933_a_b_c @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_500_subset__iff,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A6: set_li6436108459499378894od_b_c,B6: set_li6436108459499378894od_b_c] :
        ! [T: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ T @ A6 )
         => ( member6330420149250801815od_b_c @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_501_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B6: set_a] :
        ! [T: a] :
          ( ( member_a @ T @ A6 )
         => ( member_a @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_502_subset__iff,axiom,
    ( ord_le2998388488506175548od_c_a
    = ( ^ [A6: set_li1159382662694783132od_c_a,B6: set_li1159382662694783132od_c_a] :
        ! [T: list_P6327159017948738492od_c_a] :
          ( ( member7410604586820865893od_c_a @ T @ A6 )
         => ( member7410604586820865893od_c_a @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_503_equalityD2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ B @ A ) ) ).

% equalityD2
thf(fact_504_equalityD2,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( A = B )
     => ( ord_le2998388488506175548od_c_a @ B @ A ) ) ).

% equalityD2
thf(fact_505_equalityD1,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% equalityD1
thf(fact_506_equalityD1,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( A = B )
     => ( ord_le2998388488506175548od_c_a @ A @ B ) ) ).

% equalityD1
thf(fact_507_subset__eq,axiom,
    ( ord_le7464911323401878760od_d_d
    = ( ^ [A6: set_Pr9130538228769002824od_d_d,B6: set_Pr9130538228769002824od_d_d] :
        ! [X3: produc4859908366599477352od_d_d] :
          ( ( member73601632213084689od_d_d @ X3 @ A6 )
         => ( member73601632213084689od_d_d @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_508_subset__eq,axiom,
    ( ord_le6086132928344890700_a_b_c
    = ( ^ [A6: set_Pr2640454425625433516_a_b_c,B6: set_Pr2640454425625433516_a_b_c] :
        ! [X3: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ X3 @ A6 )
         => ( member997585926662434933_a_b_c @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_509_subset__eq,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A6: set_li6436108459499378894od_b_c,B6: set_li6436108459499378894od_b_c] :
        ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A6 )
         => ( member6330420149250801815od_b_c @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_510_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B6: set_a] :
        ! [X3: a] :
          ( ( member_a @ X3 @ A6 )
         => ( member_a @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_511_subset__eq,axiom,
    ( ord_le2998388488506175548od_c_a
    = ( ^ [A6: set_li1159382662694783132od_c_a,B6: set_li1159382662694783132od_c_a] :
        ! [X3: list_P6327159017948738492od_c_a] :
          ( ( member7410604586820865893od_c_a @ X3 @ A6 )
         => ( member7410604586820865893od_c_a @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_512_equalityE,axiom,
    ! [A: set_a,B: set_a] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_a @ A @ B )
         => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).

% equalityE
thf(fact_513_equalityE,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( A = B )
     => ~ ( ( ord_le2998388488506175548od_c_a @ A @ B )
         => ~ ( ord_le2998388488506175548od_c_a @ B @ A ) ) ) ).

% equalityE
thf(fact_514_subsetD,axiom,
    ! [A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d,C: produc4859908366599477352od_d_d] :
      ( ( ord_le7464911323401878760od_d_d @ A @ B )
     => ( ( member73601632213084689od_d_d @ C @ A )
       => ( member73601632213084689od_d_d @ C @ B ) ) ) ).

% subsetD
thf(fact_515_subsetD,axiom,
    ! [A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c,C: produc8838969699183119820_a_b_c] :
      ( ( ord_le6086132928344890700_a_b_c @ A @ B )
     => ( ( member997585926662434933_a_b_c @ C @ A )
       => ( member997585926662434933_a_b_c @ C @ B ) ) ) ).

% subsetD
thf(fact_516_subsetD,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( member6330420149250801815od_b_c @ C @ A )
       => ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% subsetD
thf(fact_517_subsetD,axiom,
    ! [A: set_a,B: set_a,C: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ C @ A )
       => ( member_a @ C @ B ) ) ) ).

% subsetD
thf(fact_518_subsetD,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,C: list_P6327159017948738492od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A @ B )
     => ( ( member7410604586820865893od_c_a @ C @ A )
       => ( member7410604586820865893od_c_a @ C @ B ) ) ) ).

% subsetD
thf(fact_519_in__mono,axiom,
    ! [A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d,X2: produc4859908366599477352od_d_d] :
      ( ( ord_le7464911323401878760od_d_d @ A @ B )
     => ( ( member73601632213084689od_d_d @ X2 @ A )
       => ( member73601632213084689od_d_d @ X2 @ B ) ) ) ).

% in_mono
thf(fact_520_in__mono,axiom,
    ! [A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c,X2: produc8838969699183119820_a_b_c] :
      ( ( ord_le6086132928344890700_a_b_c @ A @ B )
     => ( ( member997585926662434933_a_b_c @ X2 @ A )
       => ( member997585926662434933_a_b_c @ X2 @ B ) ) ) ).

% in_mono
thf(fact_521_in__mono,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,X2: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( member6330420149250801815od_b_c @ X2 @ A )
       => ( member6330420149250801815od_b_c @ X2 @ B ) ) ) ).

% in_mono
thf(fact_522_in__mono,axiom,
    ! [A: set_a,B: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( member_a @ X2 @ A )
       => ( member_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_523_in__mono,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,X2: list_P6327159017948738492od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A @ B )
     => ( ( member7410604586820865893od_c_a @ X2 @ A )
       => ( member7410604586820865893od_c_a @ X2 @ B ) ) ) ).

% in_mono
thf(fact_524_submachine__language,axiom,
    ! [S: fsm_a_b_c,M: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ S @ M )
     => ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ S @ ( initial_a_b_c @ S ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ) ).

% submachine_language
thf(fact_525_is__finite__test__suite_Ocases,axiom,
    ! [X2: test_t899597172106206243_b_c_d] :
      ~ ! [Prs: set_Pr2640454425625433516_a_b_c,Tps: a > set_li1159382662694783132od_c_a,Rd_targets: produc3220416805844688357od_c_a > set_a,Separators: product_prod_a_a > set_Pr9130538228769002824od_d_d] :
          ( X2
         != ( test_t5871526445387177708_b_c_d @ Prs @ Tps @ Rd_targets @ Separators ) ) ).

% is_finite_test_suite.cases
thf(fact_526_test__suite_Oexhaust,axiom,
    ! [Y: test_t899597172106206243_b_c_d] :
      ~ ! [X12: set_Pr2640454425625433516_a_b_c,X23: a > set_li1159382662694783132od_c_a,X33: produc3220416805844688357od_c_a > set_a,X43: product_prod_a_a > set_Pr9130538228769002824od_d_d] :
          ( Y
         != ( test_t5871526445387177708_b_c_d @ X12 @ X23 @ X33 @ X43 ) ) ).

% test_suite.exhaust
thf(fact_527_implies__completeness__for__repetition__sets_Ocases,axiom,
    ! [X2: produc1983128955183120424_set_a] :
      ~ ! [Prs: set_Pr2640454425625433516_a_b_c,Tps: a > set_li1159382662694783132od_c_a,Rd_targets: produc3220416805844688357od_c_a > set_a,Separators: product_prod_a_a > set_Pr9130538228769002824od_d_d,M3: fsm_a_b_c,M5: nat,Repetition_sets: list_P3660316430366008877_set_a] :
          ( X2
         != ( produc2074771797241578208_set_a @ ( test_t5871526445387177708_b_c_d @ Prs @ Tps @ Rd_targets @ Separators ) @ ( produc7030729324366460446_set_a @ M3 @ ( produc2982395248235039108_set_a @ M5 @ Repetition_sets ) ) ) ) ).

% implies_completeness_for_repetition_sets.cases
thf(fact_528_implies__completeness__for__repetition__sets__simps_I4_J,axiom,
    ! [Prs2: set_Pr2640454425625433516_a_b_c,Tps2: a > set_li1159382662694783132od_c_a,Rd_targets2: produc3220416805844688357od_c_a > set_a,Separators2: product_prod_a_a > set_Pr9130538228769002824od_d_d,M: fsm_a_b_c,M6: nat,Repetition_sets2: list_P3660316430366008877_set_a,Q: a] :
      ( ( test_i4756896328399799264_b_c_d @ ( test_t5871526445387177708_b_c_d @ Prs2 @ Tps2 @ Rd_targets2 @ Separators2 ) @ M @ M6 @ Repetition_sets2 )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ? [X: produc1703568184450464039_set_a] :
            ( ( member7983343339038529360_set_a @ X @ ( set_Pr7878197726666307004_set_a @ Repetition_sets2 ) )
            & ( member_a @ Q @ ( produc9088895665703139587_set_a @ X ) ) ) ) ) ).

% implies_completeness_for_repetition_sets_simps(4)
thf(fact_529_exE__realizer_H,axiom,
    ! [P5: fsm_a_b_c > a > $o,P: produc8838969699183119820_a_b_c] :
      ( ( P5 @ ( produc4728557313310798826_a_b_c @ P ) @ ( produc2620512763296580520_a_b_c @ P ) )
     => ~ ! [X: a,Y3: fsm_a_b_c] :
            ~ ( P5 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_530_exE__realizer_H,axiom,
    ! [P5: a > c > $o,P: product_prod_c_a] :
      ( ( P5 @ ( product_snd_c_a @ P ) @ ( product_fst_c_a @ P ) )
     => ~ ! [X: c,Y3: a] :
            ~ ( P5 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_531_exE__realizer_H,axiom,
    ! [P5: produc1703568184450464039_set_a > list_P6327159017948738492od_c_a > $o,P: produc7049025859877086738_set_a] :
      ( ( P5 @ ( produc2451121746714373936_set_a @ P ) @ ( produc3242630202250842350_set_a @ P ) )
     => ~ ! [X: list_P6327159017948738492od_c_a,Y3: produc1703568184450464039_set_a] :
            ~ ( P5 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_532_exE__realizer_H,axiom,
    ! [P5: set_a > set_a > $o,P: produc1703568184450464039_set_a] :
      ( ( P5 @ ( produc1983107199584856133_set_a @ P ) @ ( produc9088895665703139587_set_a @ P ) )
     => ~ ! [X: set_a,Y3: set_a] :
            ~ ( P5 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_533_exE__realizer_H,axiom,
    ! [P5: produc2133359284458116493od_c_a > a > $o,P: produc2669932201312201782od_c_a] :
      ( ( P5 @ ( produc8088130389382527188od_c_a @ P ) @ ( produc4225943634317630866od_c_a @ P ) )
     => ~ ! [X: a,Y3: produc2133359284458116493od_c_a] :
            ~ ( P5 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_534_exE__realizer_H,axiom,
    ! [P5: product_prod_c_a > b > $o,P: produc2133359284458116493od_c_a] :
      ( ( P5 @ ( produc5221446324160585515od_c_a @ P ) @ ( produc301949074746585577od_c_a @ P ) )
     => ~ ! [X: b,Y3: product_prod_c_a] :
            ~ ( P5 @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_535_implies__completeness__for__repetition__sets__simps_I8_J,axiom,
    ! [Prs2: set_Pr2640454425625433516_a_b_c,Tps2: a > set_li1159382662694783132od_c_a,Rd_targets2: produc3220416805844688357od_c_a > set_a,Separators2: product_prod_a_a > set_Pr9130538228769002824od_d_d,M: fsm_a_b_c,M6: nat,Repetition_sets2: list_P3660316430366008877_set_a,Q: a,P: list_P6327159017948738492od_c_a,D2: produc1703568184450464039_set_a,P1: list_P6327159017948738492od_c_a,P2: list_P6327159017948738492od_c_a,P32: list_P6327159017948738492od_c_a] :
      ( ( test_i4756896328399799264_b_c_d @ ( test_t5871526445387177708_b_c_d @ Prs2 @ Tps2 @ Rd_targets2 @ Separators2 ) @ M @ M6 @ Repetition_sets2 )
     => ( ( member_a @ Q @ ( image_3273489647386116407_b_c_a @ produc2620512763296580520_a_b_c @ Prs2 ) )
       => ( ( member6493638300815972283_set_a @ ( produc6508649944902238466_set_a @ P @ D2 ) @ ( traver6421790936065540696_a_b_c @ M @ Q @ Repetition_sets2 @ M6 ) )
         => ( ( P
              = ( append3884164603879732689od_c_a @ P1 @ ( append3884164603879732689od_c_a @ P2 @ P32 ) ) )
           => ( ( P2 != nil_Pr1342775757158464060od_c_a )
             => ( ( member_a @ ( target_a_b_c @ Q @ P1 ) @ ( produc9088895665703139587_set_a @ D2 ) )
               => ( ( member_a @ ( target_a_b_c @ Q @ ( append3884164603879732689od_c_a @ P1 @ P2 ) ) @ ( produc9088895665703139587_set_a @ D2 ) )
                 => ( ( ( target_a_b_c @ Q @ P1 )
                     != ( target_a_b_c @ Q @ ( append3884164603879732689od_c_a @ P1 @ P2 ) ) )
                   => ( ( member7410604586820865893od_c_a @ P1 @ ( Tps2 @ Q ) )
                      & ( member7410604586820865893od_c_a @ ( append3884164603879732689od_c_a @ P1 @ P2 ) @ ( Tps2 @ Q ) )
                      & ( member_a @ ( target_a_b_c @ Q @ P1 ) @ ( Rd_targets2 @ ( produc3580641852002036821od_c_a @ Q @ ( append3884164603879732689od_c_a @ P1 @ P2 ) ) ) )
                      & ( member_a @ ( target_a_b_c @ Q @ ( append3884164603879732689od_c_a @ P1 @ P2 ) ) @ ( Rd_targets2 @ ( produc3580641852002036821od_c_a @ Q @ P1 ) ) ) ) ) ) ) ) ) ) ) ) ).

% implies_completeness_for_repetition_sets_simps(8)
thf(fact_536_implies__completeness__for__repetition__sets__simps_I9_J,axiom,
    ! [Prs2: set_Pr2640454425625433516_a_b_c,Tps2: a > set_li1159382662694783132od_c_a,Rd_targets2: produc3220416805844688357od_c_a > set_a,Separators2: product_prod_a_a > set_Pr9130538228769002824od_d_d,M: fsm_a_b_c,M6: nat,Repetition_sets2: list_P3660316430366008877_set_a,Q: a,P: list_P6327159017948738492od_c_a,D2: produc1703568184450464039_set_a,P1: list_P6327159017948738492od_c_a,P2: list_P6327159017948738492od_c_a,Q3: a] :
      ( ( test_i4756896328399799264_b_c_d @ ( test_t5871526445387177708_b_c_d @ Prs2 @ Tps2 @ Rd_targets2 @ Separators2 ) @ M @ M6 @ Repetition_sets2 )
     => ( ( member_a @ Q @ ( image_3273489647386116407_b_c_a @ produc2620512763296580520_a_b_c @ Prs2 ) )
       => ( ( member6493638300815972283_set_a @ ( produc6508649944902238466_set_a @ P @ D2 ) @ ( traver6421790936065540696_a_b_c @ M @ Q @ Repetition_sets2 @ M6 ) )
         => ( ( P
              = ( append3884164603879732689od_c_a @ P1 @ P2 ) )
           => ( ( member_a @ Q3 @ ( image_3273489647386116407_b_c_a @ produc2620512763296580520_a_b_c @ Prs2 ) )
             => ( ( member_a @ ( target_a_b_c @ Q @ P1 ) @ ( produc9088895665703139587_set_a @ D2 ) )
               => ( ( member_a @ Q3 @ ( produc9088895665703139587_set_a @ D2 ) )
                 => ( ( ( target_a_b_c @ Q @ P1 )
                     != Q3 )
                   => ( ( member7410604586820865893od_c_a @ P1 @ ( Tps2 @ Q ) )
                      & ( member7410604586820865893od_c_a @ nil_Pr1342775757158464060od_c_a @ ( Tps2 @ Q3 ) )
                      & ( member_a @ ( target_a_b_c @ Q @ P1 ) @ ( Rd_targets2 @ ( produc3580641852002036821od_c_a @ Q3 @ nil_Pr1342775757158464060od_c_a ) ) )
                      & ( member_a @ Q3 @ ( Rd_targets2 @ ( produc3580641852002036821od_c_a @ Q @ P1 ) ) ) ) ) ) ) ) ) ) ) ) ).

% implies_completeness_for_repetition_sets_simps(9)
thf(fact_537_implies__completeness__for__repetition__sets__simps_I5_J,axiom,
    ! [Prs2: set_Pr2640454425625433516_a_b_c,Tps2: a > set_li1159382662694783132od_c_a,Rd_targets2: produc3220416805844688357od_c_a > set_a,Separators2: product_prod_a_a > set_Pr9130538228769002824od_d_d,M: fsm_a_b_c,M6: nat,Repetition_sets2: list_P3660316430366008877_set_a,D2: produc1703568184450464039_set_a] :
      ( ( test_i4756896328399799264_b_c_d @ ( test_t5871526445387177708_b_c_d @ Prs2 @ Tps2 @ Rd_targets2 @ Separators2 ) @ M @ M6 @ Repetition_sets2 )
     => ( ( member7983343339038529360_set_a @ D2 @ ( set_Pr7878197726666307004_set_a @ Repetition_sets2 ) )
       => ( ( ord_less_eq_set_a @ ( produc9088895665703139587_set_a @ D2 ) @ ( states_a_b_c @ M ) )
          & ( ( produc1983107199584856133_set_a @ D2 )
            = ( inf_inf_set_a @ ( produc9088895665703139587_set_a @ D2 ) @ ( image_3273489647386116407_b_c_a @ produc2620512763296580520_a_b_c @ Prs2 ) ) ) ) ) ) ).

% implies_completeness_for_repetition_sets_simps(5)
thf(fact_538_implies__completeness__for__repetition__sets__simps_I10_J,axiom,
    ! [Prs2: set_Pr2640454425625433516_a_b_c,Tps2: a > set_li1159382662694783132od_c_a,Rd_targets2: produc3220416805844688357od_c_a > set_a,Separators2: product_prod_a_a > set_Pr9130538228769002824od_d_d,M: fsm_a_b_c,M6: nat,Repetition_sets2: list_P3660316430366008877_set_a,Q: a,P: list_P6327159017948738492od_c_a,D2: produc1703568184450464039_set_a,Q1: a,Q22: a] :
      ( ( test_i4756896328399799264_b_c_d @ ( test_t5871526445387177708_b_c_d @ Prs2 @ Tps2 @ Rd_targets2 @ Separators2 ) @ M @ M6 @ Repetition_sets2 )
     => ( ( member_a @ Q @ ( image_3273489647386116407_b_c_a @ produc2620512763296580520_a_b_c @ Prs2 ) )
       => ( ( member6493638300815972283_set_a @ ( produc6508649944902238466_set_a @ P @ D2 ) @ ( traver6421790936065540696_a_b_c @ M @ Q @ Repetition_sets2 @ M6 ) )
         => ( ( Q1 != Q22 )
           => ( ( member_a @ Q1 @ ( produc1983107199584856133_set_a @ D2 ) )
             => ( ( member_a @ Q22 @ ( produc1983107199584856133_set_a @ D2 ) )
               => ( ( member7410604586820865893od_c_a @ nil_Pr1342775757158464060od_c_a @ ( Tps2 @ Q1 ) )
                  & ( member7410604586820865893od_c_a @ nil_Pr1342775757158464060od_c_a @ ( Tps2 @ Q22 ) )
                  & ( member_a @ Q1 @ ( Rd_targets2 @ ( produc3580641852002036821od_c_a @ Q22 @ nil_Pr1342775757158464060od_c_a ) ) )
                  & ( member_a @ Q22 @ ( Rd_targets2 @ ( produc3580641852002036821od_c_a @ Q1 @ nil_Pr1342775757158464060od_c_a ) ) ) ) ) ) ) ) ) ) ).

% implies_completeness_for_repetition_sets_simps(10)
thf(fact_539_cycle__incoming__transition__ex,axiom,
    ! [M: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a,T2: produc2669932201312201782od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P )
     => ( ( P != nil_Pr1342775757158464060od_c_a )
       => ( ( ( target_a_b_c @ Q @ P )
            = Q )
         => ( ( member5502040268208861279od_c_a @ T2 @ ( set_Pr5664641634098845387od_c_a @ P ) )
           => ? [X: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ P ) )
                & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                  = ( produc4225943634317630866od_c_a @ T2 ) ) ) ) ) ) ) ).

% cycle_incoming_transition_ex
thf(fact_540_cycle__incoming__transition__ex,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P: list_P1562850763667305982od_c_a,T2: produc3205353004677322104od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ M @ Q @ P )
     => ( ( P != nil_Pr7913020553331505022od_c_a )
       => ( ( ( target6303326113586172192_a_a_b @ Q @ P )
            = Q )
         => ( ( member6508725278956418977od_c_a @ T2 @ ( set_Pr6987804667698513421od_c_a @ P ) )
           => ? [X: produc3205353004677322104od_c_a] :
                ( ( member6508725278956418977od_c_a @ X @ ( set_Pr6987804667698513421od_c_a @ P ) )
                & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
                  = ( produc8171714036265584980od_c_a @ T2 ) ) ) ) ) ) ) ).

% cycle_incoming_transition_ex
thf(fact_541_is__preamble__is__state,axiom,
    ! [P5: fsm_a_b_c,M: fsm_a_b_c,Q: a] :
      ( ( state_3011419026027373943_a_b_c @ P5 @ M @ Q )
     => ( member_a @ Q @ ( states_a_b_c @ M ) ) ) ).

% is_preamble_is_state
thf(fact_542__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_Apt_Aq_AA_AP_Aq_H_At1_At2_O_A_092_060lbrakk_062io_A_092_060in_062_A_I_092_060lambda_062io__atc_O_Ap__io_Ap_A_064_Ap__io_Apt_A_064_Aio__atc_J_A_096_Aatc__to__io__set_A_Ifrom__FSM_AM_A_Itarget_Aq_Apt_J_J_AA_059_A_Iq_M_AP_J_A_092_060in_062_Aprs_059_Apath_AP_A_IFSM_Oinitial_AP_J_Ap_059_Atarget_A_IFSM_Oinitial_AP_J_Ap_A_061_Aq_059_Apt_A_092_060in_062_Atps_Aq_059_Aq_H_A_092_060in_062_Ard__targets_A_Iq_M_Apt_J_059_A_IA_M_At1_M_At2_J_A_092_060in_062_Aatcs_A_Itarget_Aq_Apt_M_Aq_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [P3: list_P6327159017948738492od_c_a,Pt: list_P6327159017948738492od_c_a,Q2: a,A7: fsm_d_b_c] :
        ( ( member6330420149250801815od_b_c @ io
          @ ( image_2850395784482391045od_b_c
            @ ^ [Io_atc: list_P903359562653991662od_b_c] :
                ( append2547753245680614915od_b_c
                @ ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P3 )
                @ ( append2547753245680614915od_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                    @ Pt )
                  @ Io_atc ) )
            @ ( adapti7464086540814349463_b_c_d @ ( from_FSM_a_b_c @ m @ ( target_a_b_c @ Q2 @ Pt ) ) @ A7 ) ) )
       => ! [P6: fsm_a_b_c] :
            ( ( member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ Q2 @ P6 ) @ prs )
           => ( ( path_a_b_c @ P6 @ ( initial_a_b_c @ P6 ) @ P3 )
             => ( ( ( target_a_b_c @ ( initial_a_b_c @ P6 ) @ P3 )
                  = Q2 )
               => ( ( member7410604586820865893od_c_a @ Pt @ ( tps @ Q2 ) )
                 => ! [Q6: a] :
                      ( ( member_a @ Q6 @ ( rd_targets @ ( produc3580641852002036821od_c_a @ Q2 @ Pt ) ) )
                     => ! [T1: d,T22: d] :
                          ~ ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ A7 @ ( product_Pair_d_d @ T1 @ T22 ) ) @ ( atcs @ ( product_Pair_a_a @ ( target_a_b_c @ Q2 @ Pt ) @ Q6 ) ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>p pt q A P q' t1 t2. \<lbrakk>io \<in> (\<lambda>io_atc. p_io p @ p_io pt @ io_atc) ` atc_to_io_set (from_FSM M (target q pt)) A; (q, P) \<in> prs; path P (FSM.initial P) p; target (FSM.initial P) p = q; pt \<in> tps q; q' \<in> rd_targets (q, pt); (A, t1, t2) \<in> atcs (target q pt, q')\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_543_all__subset__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,P5: set_li6436108459499378894od_b_c > $o] :
      ( ( ! [B6: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ B6 @ ( image_2850395784482391045od_b_c @ F @ A ) )
           => ( P5 @ B6 ) ) )
      = ( ! [B6: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ B6 @ A )
           => ( P5 @ ( image_2850395784482391045od_b_c @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_544_all__subset__image,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,P5: set_se3924713247505902254od_b_c > $o] :
      ( ( ! [B6: set_se3924713247505902254od_b_c] :
            ( ( ord_le6656836712342966862od_b_c @ B6 @ ( image_2344789136157180295od_b_c @ F @ A ) )
           => ( P5 @ B6 ) ) )
      = ( ! [B6: set_Pr2640454425625433516_a_b_c] :
            ( ( ord_le6086132928344890700_a_b_c @ B6 @ A )
           => ( P5 @ ( image_2344789136157180295od_b_c @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_545_all__subset__image,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c,P5: set_a > $o] :
      ( ( ! [B6: set_a] :
            ( ( ord_less_eq_set_a @ B6 @ ( image_3273489647386116407_b_c_a @ F @ A ) )
           => ( P5 @ B6 ) ) )
      = ( ! [B6: set_Pr2640454425625433516_a_b_c] :
            ( ( ord_le6086132928344890700_a_b_c @ B6 @ A )
           => ( P5 @ ( image_3273489647386116407_b_c_a @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_546_all__subset__image,axiom,
    ! [F: a > a,A: set_a,P5: set_a > $o] :
      ( ( ! [B6: set_a] :
            ( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A ) )
           => ( P5 @ B6 ) ) )
      = ( ! [B6: set_a] :
            ( ( ord_less_eq_set_a @ B6 @ A )
           => ( P5 @ ( image_a_a @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_547_all__subset__image,axiom,
    ! [F: list_P6327159017948738492od_c_a > a,A: set_li1159382662694783132od_c_a,P5: set_a > $o] :
      ( ( ! [B6: set_a] :
            ( ( ord_less_eq_set_a @ B6 @ ( image_8824809819145125191_c_a_a @ F @ A ) )
           => ( P5 @ B6 ) ) )
      = ( ! [B6: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ B6 @ A )
           => ( P5 @ ( image_8824809819145125191_c_a_a @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_548_all__subset__image,axiom,
    ! [F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,A: set_Pr3779000724142394866_set_a,P5: set_li1159382662694783132od_c_a > $o] :
      ( ( ! [B6: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ B6 @ ( image_865596521259543215od_c_a @ F @ A ) )
           => ( P5 @ B6 ) ) )
      = ( ! [B6: set_Pr3779000724142394866_set_a] :
            ( ( ord_le1057066820810438034_set_a @ B6 @ A )
           => ( P5 @ ( image_865596521259543215od_c_a @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_549_all__subset__image,axiom,
    ! [F: a > list_P6327159017948738492od_c_a,A: set_a,P5: set_li1159382662694783132od_c_a > $o] :
      ( ( ! [B6: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ B6 @ ( image_3129299617955238851od_c_a @ F @ A ) )
           => ( P5 @ B6 ) ) )
      = ( ! [B6: set_a] :
            ( ( ord_less_eq_set_a @ B6 @ A )
           => ( P5 @ ( image_3129299617955238851od_c_a @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_550_all__subset__image,axiom,
    ! [F: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,P5: set_li1159382662694783132od_c_a > $o] :
      ( ( ! [B6: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ B6 @ ( image_1061481830026363525od_c_a @ F @ A ) )
           => ( P5 @ B6 ) ) )
      = ( ! [B6: set_li1159382662694783132od_c_a] :
            ( ( ord_le2998388488506175548od_c_a @ B6 @ A )
           => ( P5 @ ( image_1061481830026363525od_c_a @ F @ B6 ) ) ) ) ) ).

% all_subset_image
thf(fact_551__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ioA_O_A_092_060lbrakk_062io_A_061_Ap__io_Ap_A_064_Ap__io_Apt_A_064_AioA_059_AioA_A_092_060in_062_Aatc__to__io__set_A_Ifrom__FSM_AM_A_Itarget_Aq_Apt_J_J_AA_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [IoA: list_P903359562653991662od_b_c] :
        ( ( io
          = ( append2547753245680614915od_b_c
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
              @ p3 )
            @ ( append2547753245680614915od_b_c
              @ ( map_Pr776894537227705078od_b_c
                @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                @ pt )
              @ IoA ) ) )
       => ~ ( member6330420149250801815od_b_c @ IoA @ ( adapti7464086540814349463_b_c_d @ ( from_FSM_a_b_c @ m @ ( target_a_b_c @ q2 @ pt ) ) @ a2 ) ) ) ).

% \<open>\<And>thesis. (\<And>ioA. \<lbrakk>io = p_io p @ p_io pt @ ioA; ioA \<in> atc_to_io_set (from_FSM M (target q pt)) A\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_552_Int__iff,axiom,
    ! [C: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ( member73601632213084689od_d_d @ C @ ( inf_in7322238654750965302od_d_d @ A @ B ) )
      = ( ( member73601632213084689od_d_d @ C @ A )
        & ( member73601632213084689od_d_d @ C @ B ) ) ) ).

% Int_iff
thf(fact_553_Int__iff,axiom,
    ! [C: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( member997585926662434933_a_b_c @ C @ ( inf_in4776242878804941978_a_b_c @ A @ B ) )
      = ( ( member997585926662434933_a_b_c @ C @ A )
        & ( member997585926662434933_a_b_c @ C @ B ) ) ) ).

% Int_iff
thf(fact_554_Int__iff,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
      = ( ( member6330420149250801815od_b_c @ C @ A )
        & ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% Int_iff
thf(fact_555_Int__iff,axiom,
    ! [C: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ ( inf_in980870363377917322od_c_a @ A @ B ) )
      = ( ( member7410604586820865893od_c_a @ C @ A )
        & ( member7410604586820865893od_c_a @ C @ B ) ) ) ).

% Int_iff
thf(fact_556_Int__iff,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
      = ( ( member_a @ C @ A )
        & ( member_a @ C @ B ) ) ) ).

% Int_iff
thf(fact_557_IntI,axiom,
    ! [C: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ( member73601632213084689od_d_d @ C @ A )
     => ( ( member73601632213084689od_d_d @ C @ B )
       => ( member73601632213084689od_d_d @ C @ ( inf_in7322238654750965302od_d_d @ A @ B ) ) ) ) ).

% IntI
thf(fact_558_IntI,axiom,
    ! [C: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( member997585926662434933_a_b_c @ C @ A )
     => ( ( member997585926662434933_a_b_c @ C @ B )
       => ( member997585926662434933_a_b_c @ C @ ( inf_in4776242878804941978_a_b_c @ A @ B ) ) ) ) ).

% IntI
thf(fact_559_IntI,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ A )
     => ( ( member6330420149250801815od_b_c @ C @ B )
       => ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% IntI
thf(fact_560_IntI,axiom,
    ! [C: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ A )
     => ( ( member7410604586820865893od_c_a @ C @ B )
       => ( member7410604586820865893od_c_a @ C @ ( inf_in980870363377917322od_c_a @ A @ B ) ) ) ) ).

% IntI
thf(fact_561_IntI,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ A )
     => ( ( member_a @ C @ B )
       => ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% IntI
thf(fact_562_image__ident,axiom,
    ! [Y6: set_li6436108459499378894od_b_c] :
      ( ( image_2850395784482391045od_b_c
        @ ^ [X3: list_P903359562653991662od_b_c] : X3
        @ Y6 )
      = Y6 ) ).

% image_ident
thf(fact_563_append_Oright__neutral,axiom,
    ! [A2: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ A2 @ nil_Pr1342775757158464060od_c_a )
      = A2 ) ).

% append.right_neutral
thf(fact_564_append_Oright__neutral,axiom,
    ! [A2: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ A2 @ nil_Product_prod_b_c )
      = A2 ) ).

% append.right_neutral
thf(fact_565_append__Nil2,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ Xs @ nil_Pr1342775757158464060od_c_a )
      = Xs ) ).

% append_Nil2
thf(fact_566_append__Nil2,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ Xs @ nil_Product_prod_b_c )
      = Xs ) ).

% append_Nil2
thf(fact_567_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_568_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_569_self__append__conv,axiom,
    ! [Y: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( Y
        = ( append3884164603879732689od_c_a @ Y @ Ys ) )
      = ( Ys = nil_Pr1342775757158464060od_c_a ) ) ).

% self_append_conv
thf(fact_570_self__append__conv,axiom,
    ! [Y: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( Y
        = ( append2547753245680614915od_b_c @ Y @ Ys ) )
      = ( Ys = nil_Product_prod_b_c ) ) ).

% self_append_conv
thf(fact_571_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_572_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_573_self__append__conv2,axiom,
    ! [Y: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( Y
        = ( append3884164603879732689od_c_a @ Xs @ Y ) )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% self_append_conv2
thf(fact_574_self__append__conv2,axiom,
    ! [Y: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( Y
        = ( append2547753245680614915od_b_c @ Xs @ Y ) )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% self_append_conv2
thf(fact_575_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_576_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_577_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_578_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_579_map__is__Nil__conv,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr776894537227705078od_b_c @ F @ Xs )
        = nil_Product_prod_b_c )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% map_is_Nil_conv
thf(fact_580_Nil__is__map__conv,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a] :
      ( ( nil_Product_prod_b_c
        = ( map_Pr776894537227705078od_b_c @ F @ Xs ) )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% Nil_is_map_conv
thf(fact_581_list_Omap__disc__iff,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,A2: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr776894537227705078od_b_c @ F @ A2 )
        = nil_Product_prod_b_c )
      = ( A2 = nil_Pr1342775757158464060od_c_a ) ) ).

% list.map_disc_iff
thf(fact_582_map__eq__conv,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a,G: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( ( map_Pr776894537227705078od_b_c @ F @ Xs )
        = ( map_Pr776894537227705078od_b_c @ G @ Xs ) )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) ) ) ) ).

% map_eq_conv
thf(fact_583_map__append,axiom,
    ! [F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( map_Pr6705208461955284164od_c_a @ F @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
      = ( append3884164603879732689od_c_a @ ( map_Pr6705208461955284164od_c_a @ F @ Xs ) @ ( map_Pr6705208461955284164od_c_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_584_map__append,axiom,
    ! [F: product_prod_b_c > produc2669932201312201782od_c_a,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( map_Pr7314640756821961490od_c_a @ F @ ( append2547753245680614915od_b_c @ Xs @ Ys ) )
      = ( append3884164603879732689od_c_a @ ( map_Pr7314640756821961490od_c_a @ F @ Xs ) @ ( map_Pr7314640756821961490od_c_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_585_map__append,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( map_Pr6384634858812015940od_b_c @ F @ ( append2547753245680614915od_b_c @ Xs @ Ys ) )
      = ( append2547753245680614915od_b_c @ ( map_Pr6384634858812015940od_b_c @ F @ Xs ) @ ( map_Pr6384634858812015940od_b_c @ F @ Ys ) ) ) ).

% map_append
thf(fact_586_map__append,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( map_Pr776894537227705078od_b_c @ F @ ( append3884164603879732689od_c_a @ Xs @ Ys ) )
      = ( append2547753245680614915od_b_c @ ( map_Pr776894537227705078od_b_c @ F @ Xs ) @ ( map_Pr776894537227705078od_b_c @ F @ Ys ) ) ) ).

% map_append
thf(fact_587_Int__subset__iff,axiom,
    ! [C3: set_a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A @ B ) )
      = ( ( ord_less_eq_set_a @ C3 @ A )
        & ( ord_less_eq_set_a @ C3 @ B ) ) ) ).

% Int_subset_iff
thf(fact_588_Int__subset__iff,axiom,
    ! [C3: set_li1159382662694783132od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ C3 @ ( inf_in980870363377917322od_c_a @ A @ B ) )
      = ( ( ord_le2998388488506175548od_c_a @ C3 @ A )
        & ( ord_le2998388488506175548od_c_a @ C3 @ B ) ) ) ).

% Int_subset_iff
thf(fact_589_target__nil,axiom,
    ! [Q: a] :
      ( ( target_a_b_c @ Q @ nil_Pr1342775757158464060od_c_a )
      = Q ) ).

% target_nil
thf(fact_590__092_060open_062io_A_061_Ap__io_Ap_A_064_Ap__io_Apt_A_064_AioA_092_060close_062,axiom,
    ( io
    = ( append2547753245680614915od_b_c
      @ ( map_Pr776894537227705078od_b_c
        @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
        @ p3 )
      @ ( append2547753245680614915od_b_c
        @ ( map_Pr776894537227705078od_b_c
          @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
          @ pt )
        @ ioA ) ) ) ).

% \<open>io = p_io p @ p_io pt @ ioA\<close>
thf(fact_591_list_Oset__map,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,V: list_P5219093185649440210_a_b_c] :
      ( ( set_a2 @ ( map_Pr811584189524267510_b_c_a @ F @ V ) )
      = ( image_3273489647386116407_b_c_a @ F @ ( set_Pr3403252632855053537_a_b_c @ V ) ) ) ).

% list.set_map
thf(fact_592_list_Oset__map,axiom,
    ! [F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,V: list_P4908235088432371736_set_a] :
      ( ( set_li8663184538779258833od_c_a @ ( map_Pr6055023778478330862od_c_a @ F @ V ) )
      = ( image_865596521259543215od_c_a @ F @ ( set_Pr1469968110526226471_set_a @ V ) ) ) ).

% list.set_map
thf(fact_593_list_Oset__map,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,V: list_P5219093185649440210_a_b_c] :
      ( ( set_se1205977669550889699od_b_c @ ( map_Pr4360419603926651846od_b_c @ F @ V ) )
      = ( image_2344789136157180295od_b_c @ F @ ( set_Pr3403252632855053537_a_b_c @ V ) ) ) ).

% list.set_map
thf(fact_594_list_Oset__map,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,V: list_P6327159017948738492od_c_a] :
      ( ( set_Product_prod_b_c2 @ ( map_Pr776894537227705078od_b_c @ F @ V ) )
      = ( image_4981958329637652151od_b_c @ F @ ( set_Pr5664641634098845387od_c_a @ V ) ) ) ).

% list.set_map
thf(fact_595_list_Oset__map,axiom,
    ! [F: produc1703568184450464039_set_a > produc1703568184450464039_set_a,V: list_P3660316430366008877_set_a] :
      ( ( set_Pr7878197726666307004_set_a @ ( map_Pr5757421898035411460_set_a @ F @ V ) )
      = ( image_3741791430265939909_set_a @ F @ ( set_Pr7878197726666307004_set_a @ V ) ) ) ).

% list.set_map
thf(fact_596_list_Oset__map,axiom,
    ! [F: list_P903359562653991662od_b_c > produc1703568184450464039_set_a,V: list_l8907847357763382004od_b_c] :
      ( ( set_Pr7878197726666307004_set_a @ ( map_li4890537941830539389_set_a @ F @ V ) )
      = ( image_244643190434904510_set_a @ F @ ( set_li4480668622519654659od_b_c @ V ) ) ) ).

% list.set_map
thf(fact_597_list_Oset__map,axiom,
    ! [F: produc1703568184450464039_set_a > list_P903359562653991662od_b_c,V: list_P3660316430366008877_set_a] :
      ( ( set_li4480668622519654659od_b_c @ ( map_Pr191379420489308491od_b_c @ F @ V ) )
      = ( image_4768856705948449420od_b_c @ F @ ( set_Pr7878197726666307004_set_a @ V ) ) ) ).

% list.set_map
thf(fact_598_list_Oset__map,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,V: list_l8907847357763382004od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ V ) )
      = ( image_2850395784482391045od_b_c @ F @ ( set_li4480668622519654659od_b_c @ V ) ) ) ).

% list.set_map
thf(fact_599__092_060open_062io_A_092_060in_062_A_I_092_060lambda_062io__atc_O_Ap__io_Ap_A_064_Ap__io_Apt_A_064_Aio__atc_J_A_096_Aatc__to__io__set_A_Ifrom__FSM_AM_A_Itarget_Aq_Apt_J_J_AA_092_060close_062,axiom,
    ( member6330420149250801815od_b_c @ io
    @ ( image_2850395784482391045od_b_c
      @ ^ [Io_atc: list_P903359562653991662od_b_c] :
          ( append2547753245680614915od_b_c
          @ ( map_Pr776894537227705078od_b_c
            @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
            @ p3 )
          @ ( append2547753245680614915od_b_c
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
              @ pt )
            @ Io_atc ) )
      @ ( adapti7464086540814349463_b_c_d @ ( from_FSM_a_b_c @ m @ ( target_a_b_c @ q2 @ pt ) ) @ a2 ) ) ) ).

% \<open>io \<in> (\<lambda>io_atc. p_io p @ p_io pt @ io_atc) ` atc_to_io_set (from_FSM M (target q pt)) A\<close>
thf(fact_600_map__pair__fst,axiom,
    ! [F: d > d,Xs: list_d] :
      ( ( map_Pr332336403249921950_d_d_d @ product_fst_d_d
        @ ( map_d_6802736565751565034od_d_d
          @ ^ [X3: d] : ( product_Pair_d_d @ X3 @ ( F @ X3 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_601_map__pair__fst,axiom,
    ! [F: b > c,Xs: list_b] :
      ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
        @ ( map_b_4433156289202544749od_b_c
          @ ^ [X3: b] : ( product_Pair_b_c @ X3 @ ( F @ X3 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_602_map__pair__fst,axiom,
    ! [F: a > list_P6327159017948738492od_c_a,Xs: list_a] :
      ( ( map_Pr7928508354609250845_c_a_a @ produc4637894536740104257od_c_a
        @ ( map_a_4329988354072741803od_c_a
          @ ^ [X3: a] : ( produc3580641852002036821od_c_a @ X3 @ ( F @ X3 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_603_map__pair__fst,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a
        @ ( map_a_7860052162900579309od_a_a
          @ ^ [X3: a] : ( product_Pair_a_a @ X3 @ ( F @ X3 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_604_map__pair__fst,axiom,
    ! [F: set_a > set_a,Xs: list_set_a] :
      ( ( map_Pr8866779738092938811_set_a @ produc9088895665703139587_set_a
        @ ( map_se9006238007647464077_set_a
          @ ^ [X3: set_a] : ( produc9088192753505129239_set_a @ X3 @ ( F @ X3 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_605_map__pair__fst,axiom,
    ! [F: a > fsm_a_b_c,Xs: list_a] :
      ( ( map_Pr811584189524267510_b_c_a @ produc2620512763296580520_a_b_c
        @ ( map_a_4118158524683405202_a_b_c
          @ ^ [X3: a] : ( produc2795441082327197628_a_b_c @ X3 @ ( F @ X3 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_606_map__pair__fst,axiom,
    ! [F: b > product_prod_c_a,Xs: list_b] :
      ( ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a
        @ ( map_b_1729217027404323474od_c_a
          @ ^ [X3: b] : ( produc3440865914000881661od_c_a @ X3 @ ( F @ X3 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_607_map__pair__fst,axiom,
    ! [F: c > a,Xs: list_c] :
      ( ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a
        @ ( map_c_1006260398291594985od_c_a
          @ ^ [X3: c] : ( product_Pair_c_a @ X3 @ ( F @ X3 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_608_map__pair__fst,axiom,
    ! [F: list_P6327159017948738492od_c_a > produc1703568184450464039_set_a,Xs: list_l2826717958560394946od_c_a] :
      ( ( map_Pr6055023778478330862od_c_a @ produc3242630202250842350_set_a
        @ ( map_li5011125751384388634_set_a
          @ ^ [X3: list_P6327159017948738492od_c_a] : ( produc6508649944902238466_set_a @ X3 @ ( F @ X3 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_609_Collect__subset,axiom,
    ! [A: set_Pr9130538228769002824od_d_d,P5: produc4859908366599477352od_d_d > $o] :
      ( ord_le7464911323401878760od_d_d
      @ ( collec5362705567740981075od_d_d
        @ ^ [X3: produc4859908366599477352od_d_d] :
            ( ( member73601632213084689od_d_d @ X3 @ A )
            & ( P5 @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_610_Collect__subset,axiom,
    ! [A: set_Pr2640454425625433516_a_b_c,P5: produc8838969699183119820_a_b_c > $o] :
      ( ord_le6086132928344890700_a_b_c
      @ ( collec3867759823522760631_a_b_c
        @ ^ [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
            & ( P5 @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_611_Collect__subset,axiom,
    ! [A: set_li6436108459499378894od_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ord_le282488521294790766od_b_c
      @ ( collec2280997390073109977od_b_c
        @ ^ [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A )
            & ( P5 @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_612_Collect__subset,axiom,
    ! [A: set_se3924713247505902254od_b_c,P5: set_li6436108459499378894od_b_c > $o] :
      ( ord_le6656836712342966862od_b_c
      @ ( collec1823247286932916153od_b_c
        @ ^ [X3: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X3 @ A )
            & ( P5 @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_613_Collect__subset,axiom,
    ! [A: set_a,P5: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X3: a] :
            ( ( member_a @ X3 @ A )
            & ( P5 @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_614_Collect__subset,axiom,
    ! [A: set_li1159382662694783132od_c_a,P5: list_P6327159017948738492od_c_a > $o] :
      ( ord_le2998388488506175548od_c_a
      @ ( collec6273869032445462695od_c_a
        @ ^ [X3: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X3 @ A )
            & ( P5 @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_615_less__eq__set__def,axiom,
    ( ord_le7464911323401878760od_d_d
    = ( ^ [A6: set_Pr9130538228769002824od_d_d,B6: set_Pr9130538228769002824od_d_d] :
          ( ord_le7046260657170184693_d_d_o
          @ ^ [X3: produc4859908366599477352od_d_d] : ( member73601632213084689od_d_d @ X3 @ A6 )
          @ ^ [X3: produc4859908366599477352od_d_d] : ( member73601632213084689od_d_d @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_616_less__eq__set__def,axiom,
    ( ord_le6086132928344890700_a_b_c
    = ( ^ [A6: set_Pr2640454425625433516_a_b_c,B6: set_Pr2640454425625433516_a_b_c] :
          ( ord_le792551613302170257_b_c_o
          @ ^ [X3: produc8838969699183119820_a_b_c] : ( member997585926662434933_a_b_c @ X3 @ A6 )
          @ ^ [X3: produc8838969699183119820_a_b_c] : ( member997585926662434933_a_b_c @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_617_less__eq__set__def,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A6: set_li6436108459499378894od_b_c,B6: set_li6436108459499378894od_b_c] :
          ( ord_le1261880702084804079_b_c_o
          @ ^ [X3: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X3 @ A6 )
          @ ^ [X3: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_618_less__eq__set__def,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B6: set_a] :
          ( ord_less_eq_a_o
          @ ^ [X3: a] : ( member_a @ X3 @ A6 )
          @ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_619_less__eq__set__def,axiom,
    ( ord_le2998388488506175548od_c_a
    = ( ^ [A6: set_li1159382662694783132od_c_a,B6: set_li1159382662694783132od_c_a] :
          ( ord_le2170622466739808161_c_a_o
          @ ^ [X3: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ X3 @ A6 )
          @ ^ [X3: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_620_subset__CollectI,axiom,
    ! [B: set_Pr9130538228769002824od_d_d,A: set_Pr9130538228769002824od_d_d,Q5: produc4859908366599477352od_d_d > $o,P5: produc4859908366599477352od_d_d > $o] :
      ( ( ord_le7464911323401878760od_d_d @ B @ A )
     => ( ! [X: produc4859908366599477352od_d_d] :
            ( ( member73601632213084689od_d_d @ X @ B )
           => ( ( Q5 @ X )
             => ( P5 @ X ) ) )
       => ( ord_le7464911323401878760od_d_d
          @ ( collec5362705567740981075od_d_d
            @ ^ [X3: produc4859908366599477352od_d_d] :
                ( ( member73601632213084689od_d_d @ X3 @ B )
                & ( Q5 @ X3 ) ) )
          @ ( collec5362705567740981075od_d_d
            @ ^ [X3: produc4859908366599477352od_d_d] :
                ( ( member73601632213084689od_d_d @ X3 @ A )
                & ( P5 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_621_subset__CollectI,axiom,
    ! [B: set_Pr2640454425625433516_a_b_c,A: set_Pr2640454425625433516_a_b_c,Q5: produc8838969699183119820_a_b_c > $o,P5: produc8838969699183119820_a_b_c > $o] :
      ( ( ord_le6086132928344890700_a_b_c @ B @ A )
     => ( ! [X: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X @ B )
           => ( ( Q5 @ X )
             => ( P5 @ X ) ) )
       => ( ord_le6086132928344890700_a_b_c
          @ ( collec3867759823522760631_a_b_c
            @ ^ [X3: produc8838969699183119820_a_b_c] :
                ( ( member997585926662434933_a_b_c @ X3 @ B )
                & ( Q5 @ X3 ) ) )
          @ ( collec3867759823522760631_a_b_c
            @ ^ [X3: produc8838969699183119820_a_b_c] :
                ( ( member997585926662434933_a_b_c @ X3 @ A )
                & ( P5 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_622_subset__CollectI,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,Q5: list_P903359562653991662od_b_c > $o,P5: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ B @ A )
     => ( ! [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ B )
           => ( ( Q5 @ X )
             => ( P5 @ X ) ) )
       => ( ord_le282488521294790766od_b_c
          @ ( collec2280997390073109977od_b_c
            @ ^ [X3: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X3 @ B )
                & ( Q5 @ X3 ) ) )
          @ ( collec2280997390073109977od_b_c
            @ ^ [X3: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X3 @ A )
                & ( P5 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_623_subset__CollectI,axiom,
    ! [B: set_se3924713247505902254od_b_c,A: set_se3924713247505902254od_b_c,Q5: set_li6436108459499378894od_b_c > $o,P5: set_li6436108459499378894od_b_c > $o] :
      ( ( ord_le6656836712342966862od_b_c @ B @ A )
     => ( ! [X: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X @ B )
           => ( ( Q5 @ X )
             => ( P5 @ X ) ) )
       => ( ord_le6656836712342966862od_b_c
          @ ( collec1823247286932916153od_b_c
            @ ^ [X3: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ X3 @ B )
                & ( Q5 @ X3 ) ) )
          @ ( collec1823247286932916153od_b_c
            @ ^ [X3: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ X3 @ A )
                & ( P5 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_624_subset__CollectI,axiom,
    ! [B: set_a,A: set_a,Q5: a > $o,P5: a > $o] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ! [X: a] :
            ( ( member_a @ X @ B )
           => ( ( Q5 @ X )
             => ( P5 @ X ) ) )
       => ( ord_less_eq_set_a
          @ ( collect_a
            @ ^ [X3: a] :
                ( ( member_a @ X3 @ B )
                & ( Q5 @ X3 ) ) )
          @ ( collect_a
            @ ^ [X3: a] :
                ( ( member_a @ X3 @ A )
                & ( P5 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_625_subset__CollectI,axiom,
    ! [B: set_li1159382662694783132od_c_a,A: set_li1159382662694783132od_c_a,Q5: list_P6327159017948738492od_c_a > $o,P5: list_P6327159017948738492od_c_a > $o] :
      ( ( ord_le2998388488506175548od_c_a @ B @ A )
     => ( ! [X: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X @ B )
           => ( ( Q5 @ X )
             => ( P5 @ X ) ) )
       => ( ord_le2998388488506175548od_c_a
          @ ( collec6273869032445462695od_c_a
            @ ^ [X3: list_P6327159017948738492od_c_a] :
                ( ( member7410604586820865893od_c_a @ X3 @ B )
                & ( Q5 @ X3 ) ) )
          @ ( collec6273869032445462695od_c_a
            @ ^ [X3: list_P6327159017948738492od_c_a] :
                ( ( member7410604586820865893od_c_a @ X3 @ A )
                & ( P5 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_626_subset__Collect__iff,axiom,
    ! [B: set_Pr9130538228769002824od_d_d,A: set_Pr9130538228769002824od_d_d,P5: produc4859908366599477352od_d_d > $o] :
      ( ( ord_le7464911323401878760od_d_d @ B @ A )
     => ( ( ord_le7464911323401878760od_d_d @ B
          @ ( collec5362705567740981075od_d_d
            @ ^ [X3: produc4859908366599477352od_d_d] :
                ( ( member73601632213084689od_d_d @ X3 @ A )
                & ( P5 @ X3 ) ) ) )
        = ( ! [X3: produc4859908366599477352od_d_d] :
              ( ( member73601632213084689od_d_d @ X3 @ B )
             => ( P5 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_627_subset__Collect__iff,axiom,
    ! [B: set_Pr2640454425625433516_a_b_c,A: set_Pr2640454425625433516_a_b_c,P5: produc8838969699183119820_a_b_c > $o] :
      ( ( ord_le6086132928344890700_a_b_c @ B @ A )
     => ( ( ord_le6086132928344890700_a_b_c @ B
          @ ( collec3867759823522760631_a_b_c
            @ ^ [X3: produc8838969699183119820_a_b_c] :
                ( ( member997585926662434933_a_b_c @ X3 @ A )
                & ( P5 @ X3 ) ) ) )
        = ( ! [X3: produc8838969699183119820_a_b_c] :
              ( ( member997585926662434933_a_b_c @ X3 @ B )
             => ( P5 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_628_subset__Collect__iff,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ B @ A )
     => ( ( ord_le282488521294790766od_b_c @ B
          @ ( collec2280997390073109977od_b_c
            @ ^ [X3: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X3 @ A )
                & ( P5 @ X3 ) ) ) )
        = ( ! [X3: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X3 @ B )
             => ( P5 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_629_subset__Collect__iff,axiom,
    ! [B: set_se3924713247505902254od_b_c,A: set_se3924713247505902254od_b_c,P5: set_li6436108459499378894od_b_c > $o] :
      ( ( ord_le6656836712342966862od_b_c @ B @ A )
     => ( ( ord_le6656836712342966862od_b_c @ B
          @ ( collec1823247286932916153od_b_c
            @ ^ [X3: set_li6436108459499378894od_b_c] :
                ( ( member6985331446368301687od_b_c @ X3 @ A )
                & ( P5 @ X3 ) ) ) )
        = ( ! [X3: set_li6436108459499378894od_b_c] :
              ( ( member6985331446368301687od_b_c @ X3 @ B )
             => ( P5 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_630_subset__Collect__iff,axiom,
    ! [B: set_a,A: set_a,P5: a > $o] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ( ord_less_eq_set_a @ B
          @ ( collect_a
            @ ^ [X3: a] :
                ( ( member_a @ X3 @ A )
                & ( P5 @ X3 ) ) ) )
        = ( ! [X3: a] :
              ( ( member_a @ X3 @ B )
             => ( P5 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_631_subset__Collect__iff,axiom,
    ! [B: set_li1159382662694783132od_c_a,A: set_li1159382662694783132od_c_a,P5: list_P6327159017948738492od_c_a > $o] :
      ( ( ord_le2998388488506175548od_c_a @ B @ A )
     => ( ( ord_le2998388488506175548od_c_a @ B
          @ ( collec6273869032445462695od_c_a
            @ ^ [X3: list_P6327159017948738492od_c_a] :
                ( ( member7410604586820865893od_c_a @ X3 @ A )
                & ( P5 @ X3 ) ) ) )
        = ( ! [X3: list_P6327159017948738492od_c_a] :
              ( ( member7410604586820865893od_c_a @ X3 @ B )
             => ( P5 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_632_Collect__conj__eq,axiom,
    ! [P5: list_P6327159017948738492od_c_a > $o,Q5: list_P6327159017948738492od_c_a > $o] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [X3: list_P6327159017948738492od_c_a] :
            ( ( P5 @ X3 )
            & ( Q5 @ X3 ) ) )
      = ( inf_in980870363377917322od_c_a @ ( collec6273869032445462695od_c_a @ P5 ) @ ( collec6273869032445462695od_c_a @ Q5 ) ) ) ).

% Collect_conj_eq
thf(fact_633_Collect__conj__eq,axiom,
    ! [P5: set_li6436108459499378894od_b_c > $o,Q5: set_li6436108459499378894od_b_c > $o] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [X3: set_li6436108459499378894od_b_c] :
            ( ( P5 @ X3 )
            & ( Q5 @ X3 ) ) )
      = ( inf_in5524221907227607964od_b_c @ ( collec1823247286932916153od_b_c @ P5 ) @ ( collec1823247286932916153od_b_c @ Q5 ) ) ) ).

% Collect_conj_eq
thf(fact_634_Compr__image__eq,axiom,
    ! [F: a > a,A: set_a,P5: a > $o] :
      ( ( collect_a
        @ ^ [X3: a] :
            ( ( member_a @ X3 @ ( image_a_a @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_a_a @ F
        @ ( collect_a
          @ ^ [X3: a] :
              ( ( member_a @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_635_Compr__image__eq,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A: set_a,P5: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( image_8656872703560917237od_b_c @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_8656872703560917237od_b_c @ F
        @ ( collect_a
          @ ^ [X3: a] :
              ( ( member_a @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_636_Compr__image__eq,axiom,
    ! [F: list_P903359562653991662od_b_c > a,A: set_li6436108459499378894od_b_c,P5: a > $o] :
      ( ( collect_a
        @ ^ [X3: a] :
            ( ( member_a @ X3 @ ( image_3173854338463926933_b_c_a @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_3173854338463926933_b_c_a @ F
        @ ( collec2280997390073109977od_b_c
          @ ^ [X3: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_637_Compr__image__eq,axiom,
    ! [F: set_li6436108459499378894od_b_c > a,A: set_se3924713247505902254od_b_c,P5: a > $o] :
      ( ( collect_a
        @ ^ [X3: a] :
            ( ( member_a @ X3 @ ( image_3292806214809998517_b_c_a @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_3292806214809998517_b_c_a @ F
        @ ( collec1823247286932916153od_b_c
          @ ^ [X3: set_li6436108459499378894od_b_c] :
              ( ( member6985331446368301687od_b_c @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_638_Compr__image__eq,axiom,
    ! [F: a > set_li6436108459499378894od_b_c,A: set_a,P5: set_li6436108459499378894od_b_c > $o] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [X3: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X3 @ ( image_6261743117854694101od_b_c @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_6261743117854694101od_b_c @ F
        @ ( collect_a
          @ ^ [X3: a] :
              ( ( member_a @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_639_Compr__image__eq,axiom,
    ! [F: a > produc8838969699183119820_a_b_c,A: set_a,P5: produc8838969699183119820_a_b_c > $o] :
      ( ( collec3867759823522760631_a_b_c
        @ ^ [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ ( image_6580063982545254099_a_b_c @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_6580063982545254099_a_b_c @ F
        @ ( collect_a
          @ ^ [X3: a] :
              ( ( member_a @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_640_Compr__image__eq,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c,P5: a > $o] :
      ( ( collect_a
        @ ^ [X3: a] :
            ( ( member_a @ X3 @ ( image_3273489647386116407_b_c_a @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_3273489647386116407_b_c_a @ F
        @ ( collec3867759823522760631_a_b_c
          @ ^ [X3: produc8838969699183119820_a_b_c] :
              ( ( member997585926662434933_a_b_c @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_641_Compr__image__eq,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( image_2850395784482391045od_b_c @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_2850395784482391045od_b_c @ F
        @ ( collec2280997390073109977od_b_c
          @ ^ [X3: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_642_Compr__image__eq,axiom,
    ! [F: a > produc4859908366599477352od_d_d,A: set_a,P5: produc4859908366599477352od_d_d > $o] :
      ( ( collec5362705567740981075od_d_d
        @ ^ [X3: produc4859908366599477352od_d_d] :
            ( ( member73601632213084689od_d_d @ X3 @ ( image_2487513010242200175od_d_d @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_2487513010242200175od_d_d @ F
        @ ( collect_a
          @ ^ [X3: a] :
              ( ( member_a @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_643_Compr__image__eq,axiom,
    ! [F: produc4859908366599477352od_d_d > a,A: set_Pr9130538228769002824od_d_d,P5: a > $o] :
      ( ( collect_a
        @ ^ [X3: a] :
            ( ( member_a @ X3 @ ( image_3229965221763006363_d_d_a @ F @ A ) )
            & ( P5 @ X3 ) ) )
      = ( image_3229965221763006363_d_d_a @ F
        @ ( collec5362705567740981075od_d_d
          @ ^ [X3: produc4859908366599477352od_d_d] :
              ( ( member73601632213084689od_d_d @ X3 @ A )
              & ( P5 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_644_image__image,axiom,
    ! [F: a > a,G: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c] :
      ( ( image_a_a @ F @ ( image_3273489647386116407_b_c_a @ G @ A ) )
      = ( image_3273489647386116407_b_c_a
        @ ^ [X3: produc8838969699183119820_a_b_c] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_645_image__image,axiom,
    ! [F: a > set_li6436108459499378894od_b_c,G: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c] :
      ( ( image_6261743117854694101od_b_c @ F @ ( image_3273489647386116407_b_c_a @ G @ A ) )
      = ( image_2344789136157180295od_b_c
        @ ^ [X3: produc8838969699183119820_a_b_c] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_646_image__image,axiom,
    ! [F: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a,G: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,A: set_Pr3779000724142394866_set_a] :
      ( ( image_1061481830026363525od_c_a @ F @ ( image_865596521259543215od_c_a @ G @ A ) )
      = ( image_865596521259543215od_c_a
        @ ^ [X3: produc7049025859877086738_set_a] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_647_image__image,axiom,
    ! [F: set_li6436108459499378894od_b_c > a,G: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( image_3292806214809998517_b_c_a @ F @ ( image_2344789136157180295od_b_c @ G @ A ) )
      = ( image_3273489647386116407_b_c_a
        @ ^ [X3: produc8838969699183119820_a_b_c] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_648_image__image,axiom,
    ! [F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,G: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( image_8181729759120431109od_b_c @ F @ ( image_2344789136157180295od_b_c @ G @ A ) )
      = ( image_2344789136157180295od_b_c
        @ ^ [X3: produc8838969699183119820_a_b_c] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_649_image__image,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,G: produc8838969699183119820_a_b_c > produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( image_3273489647386116407_b_c_a @ F @ ( image_8009415464282214277_a_b_c @ G @ A ) )
      = ( image_3273489647386116407_b_c_a
        @ ^ [X3: produc8838969699183119820_a_b_c] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_650_image__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,G: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( image_2850395784482391045od_b_c @ F @ ( image_2850395784482391045od_b_c @ G @ A ) )
      = ( image_2850395784482391045od_b_c
        @ ^ [X3: list_P903359562653991662od_b_c] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_651_image__image,axiom,
    ! [F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,G: produc7049025859877086738_set_a > produc7049025859877086738_set_a,A: set_Pr3779000724142394866_set_a] :
      ( ( image_865596521259543215od_c_a @ F @ ( image_5185499578167095045_set_a @ G @ A ) )
      = ( image_865596521259543215od_c_a
        @ ^ [X3: produc7049025859877086738_set_a] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_652_image__image,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,G: produc8838969699183119820_a_b_c > produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( image_2344789136157180295od_b_c @ F @ ( image_8009415464282214277_a_b_c @ G @ A ) )
      = ( image_2344789136157180295od_b_c
        @ ^ [X3: produc8838969699183119820_a_b_c] : ( F @ ( G @ X3 ) )
        @ A ) ) ).

% image_image
thf(fact_653_Int__Collect,axiom,
    ! [X2: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,P5: produc4859908366599477352od_d_d > $o] :
      ( ( member73601632213084689od_d_d @ X2 @ ( inf_in7322238654750965302od_d_d @ A @ ( collec5362705567740981075od_d_d @ P5 ) ) )
      = ( ( member73601632213084689od_d_d @ X2 @ A )
        & ( P5 @ X2 ) ) ) ).

% Int_Collect
thf(fact_654_Int__Collect,axiom,
    ! [X2: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,P5: produc8838969699183119820_a_b_c > $o] :
      ( ( member997585926662434933_a_b_c @ X2 @ ( inf_in4776242878804941978_a_b_c @ A @ ( collec3867759823522760631_a_b_c @ P5 ) ) )
      = ( ( member997585926662434933_a_b_c @ X2 @ A )
        & ( P5 @ X2 ) ) ) ).

% Int_Collect
thf(fact_655_Int__Collect,axiom,
    ! [X2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( inf_in4978071631833541052od_b_c @ A @ ( collec2280997390073109977od_b_c @ P5 ) ) )
      = ( ( member6330420149250801815od_b_c @ X2 @ A )
        & ( P5 @ X2 ) ) ) ).

% Int_Collect
thf(fact_656_Int__Collect,axiom,
    ! [X2: a,A: set_a,P5: a > $o] :
      ( ( member_a @ X2 @ ( inf_inf_set_a @ A @ ( collect_a @ P5 ) ) )
      = ( ( member_a @ X2 @ A )
        & ( P5 @ X2 ) ) ) ).

% Int_Collect
thf(fact_657_Int__Collect,axiom,
    ! [X2: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,P5: list_P6327159017948738492od_c_a > $o] :
      ( ( member7410604586820865893od_c_a @ X2 @ ( inf_in980870363377917322od_c_a @ A @ ( collec6273869032445462695od_c_a @ P5 ) ) )
      = ( ( member7410604586820865893od_c_a @ X2 @ A )
        & ( P5 @ X2 ) ) ) ).

% Int_Collect
thf(fact_658_Int__Collect,axiom,
    ! [X2: set_li6436108459499378894od_b_c,A: set_se3924713247505902254od_b_c,P5: set_li6436108459499378894od_b_c > $o] :
      ( ( member6985331446368301687od_b_c @ X2 @ ( inf_in5524221907227607964od_b_c @ A @ ( collec1823247286932916153od_b_c @ P5 ) ) )
      = ( ( member6985331446368301687od_b_c @ X2 @ A )
        & ( P5 @ X2 ) ) ) ).

% Int_Collect
thf(fact_659_Int__def,axiom,
    ( inf_in7322238654750965302od_d_d
    = ( ^ [A6: set_Pr9130538228769002824od_d_d,B6: set_Pr9130538228769002824od_d_d] :
          ( collec5362705567740981075od_d_d
          @ ^ [X3: produc4859908366599477352od_d_d] :
              ( ( member73601632213084689od_d_d @ X3 @ A6 )
              & ( member73601632213084689od_d_d @ X3 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_660_Int__def,axiom,
    ( inf_in4776242878804941978_a_b_c
    = ( ^ [A6: set_Pr2640454425625433516_a_b_c,B6: set_Pr2640454425625433516_a_b_c] :
          ( collec3867759823522760631_a_b_c
          @ ^ [X3: produc8838969699183119820_a_b_c] :
              ( ( member997585926662434933_a_b_c @ X3 @ A6 )
              & ( member997585926662434933_a_b_c @ X3 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_661_Int__def,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [A6: set_li6436108459499378894od_b_c,B6: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X3: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X3 @ A6 )
              & ( member6330420149250801815od_b_c @ X3 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_662_Int__def,axiom,
    ( inf_inf_set_a
    = ( ^ [A6: set_a,B6: set_a] :
          ( collect_a
          @ ^ [X3: a] :
              ( ( member_a @ X3 @ A6 )
              & ( member_a @ X3 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_663_Int__def,axiom,
    ( inf_in980870363377917322od_c_a
    = ( ^ [A6: set_li1159382662694783132od_c_a,B6: set_li1159382662694783132od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [X3: list_P6327159017948738492od_c_a] :
              ( ( member7410604586820865893od_c_a @ X3 @ A6 )
              & ( member7410604586820865893od_c_a @ X3 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_664_Int__def,axiom,
    ( inf_in5524221907227607964od_b_c
    = ( ^ [A6: set_se3924713247505902254od_b_c,B6: set_se3924713247505902254od_b_c] :
          ( collec1823247286932916153od_b_c
          @ ^ [X3: set_li6436108459499378894od_b_c] :
              ( ( member6985331446368301687od_b_c @ X3 @ A6 )
              & ( member6985331446368301687od_b_c @ X3 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_665_imageE,axiom,
    ! [B2: a,F: a > a,A: set_a] :
      ( ( member_a @ B2 @ ( image_a_a @ F @ A ) )
     => ~ ! [X: a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_a @ X @ A ) ) ) ).

% imageE
thf(fact_666_imageE,axiom,
    ! [B2: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c,A: set_a] :
      ( ( member6330420149250801815od_b_c @ B2 @ ( image_8656872703560917237od_b_c @ F @ A ) )
     => ~ ! [X: a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_a @ X @ A ) ) ) ).

% imageE
thf(fact_667_imageE,axiom,
    ! [B2: a,F: list_P903359562653991662od_b_c > a,A: set_li6436108459499378894od_b_c] :
      ( ( member_a @ B2 @ ( image_3173854338463926933_b_c_a @ F @ A ) )
     => ~ ! [X: list_P903359562653991662od_b_c] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member6330420149250801815od_b_c @ X @ A ) ) ) ).

% imageE
thf(fact_668_imageE,axiom,
    ! [B2: produc8838969699183119820_a_b_c,F: a > produc8838969699183119820_a_b_c,A: set_a] :
      ( ( member997585926662434933_a_b_c @ B2 @ ( image_6580063982545254099_a_b_c @ F @ A ) )
     => ~ ! [X: a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_a @ X @ A ) ) ) ).

% imageE
thf(fact_669_imageE,axiom,
    ! [B2: a,F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c] :
      ( ( member_a @ B2 @ ( image_3273489647386116407_b_c_a @ F @ A ) )
     => ~ ! [X: produc8838969699183119820_a_b_c] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member997585926662434933_a_b_c @ X @ A ) ) ) ).

% imageE
thf(fact_670_imageE,axiom,
    ! [B2: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ B2 @ ( image_2850395784482391045od_b_c @ F @ A ) )
     => ~ ! [X: list_P903359562653991662od_b_c] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member6330420149250801815od_b_c @ X @ A ) ) ) ).

% imageE
thf(fact_671_imageE,axiom,
    ! [B2: produc4859908366599477352od_d_d,F: a > produc4859908366599477352od_d_d,A: set_a] :
      ( ( member73601632213084689od_d_d @ B2 @ ( image_2487513010242200175od_d_d @ F @ A ) )
     => ~ ! [X: a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_a @ X @ A ) ) ) ).

% imageE
thf(fact_672_imageE,axiom,
    ! [B2: list_P6327159017948738492od_c_a,F: a > list_P6327159017948738492od_c_a,A: set_a] :
      ( ( member7410604586820865893od_c_a @ B2 @ ( image_3129299617955238851od_c_a @ F @ A ) )
     => ~ ! [X: a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member_a @ X @ A ) ) ) ).

% imageE
thf(fact_673_imageE,axiom,
    ! [B2: a,F: produc4859908366599477352od_d_d > a,A: set_Pr9130538228769002824od_d_d] :
      ( ( member_a @ B2 @ ( image_3229965221763006363_d_d_a @ F @ A ) )
     => ~ ! [X: produc4859908366599477352od_d_d] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member73601632213084689od_d_d @ X @ A ) ) ) ).

% imageE
thf(fact_674_imageE,axiom,
    ! [B2: a,F: list_P6327159017948738492od_c_a > a,A: set_li1159382662694783132od_c_a] :
      ( ( member_a @ B2 @ ( image_8824809819145125191_c_a_a @ F @ A ) )
     => ~ ! [X: list_P6327159017948738492od_c_a] :
            ( ( B2
              = ( F @ X ) )
           => ~ ( member7410604586820865893od_c_a @ X @ A ) ) ) ).

% imageE
thf(fact_675_IntD2,axiom,
    ! [C: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ( member73601632213084689od_d_d @ C @ ( inf_in7322238654750965302od_d_d @ A @ B ) )
     => ( member73601632213084689od_d_d @ C @ B ) ) ).

% IntD2
thf(fact_676_IntD2,axiom,
    ! [C: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( member997585926662434933_a_b_c @ C @ ( inf_in4776242878804941978_a_b_c @ A @ B ) )
     => ( member997585926662434933_a_b_c @ C @ B ) ) ).

% IntD2
thf(fact_677_IntD2,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
     => ( member6330420149250801815od_b_c @ C @ B ) ) ).

% IntD2
thf(fact_678_IntD2,axiom,
    ! [C: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ ( inf_in980870363377917322od_c_a @ A @ B ) )
     => ( member7410604586820865893od_c_a @ C @ B ) ) ).

% IntD2
thf(fact_679_IntD2,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
     => ( member_a @ C @ B ) ) ).

% IntD2
thf(fact_680_IntD1,axiom,
    ! [C: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ( member73601632213084689od_d_d @ C @ ( inf_in7322238654750965302od_d_d @ A @ B ) )
     => ( member73601632213084689od_d_d @ C @ A ) ) ).

% IntD1
thf(fact_681_IntD1,axiom,
    ! [C: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( member997585926662434933_a_b_c @ C @ ( inf_in4776242878804941978_a_b_c @ A @ B ) )
     => ( member997585926662434933_a_b_c @ C @ A ) ) ).

% IntD1
thf(fact_682_IntD1,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
     => ( member6330420149250801815od_b_c @ C @ A ) ) ).

% IntD1
thf(fact_683_IntD1,axiom,
    ! [C: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ ( inf_in980870363377917322od_c_a @ A @ B ) )
     => ( member7410604586820865893od_c_a @ C @ A ) ) ).

% IntD1
thf(fact_684_IntD1,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
     => ( member_a @ C @ A ) ) ).

% IntD1
thf(fact_685_IntE,axiom,
    ! [C: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ( member73601632213084689od_d_d @ C @ ( inf_in7322238654750965302od_d_d @ A @ B ) )
     => ~ ( ( member73601632213084689od_d_d @ C @ A )
         => ~ ( member73601632213084689od_d_d @ C @ B ) ) ) ).

% IntE
thf(fact_686_IntE,axiom,
    ! [C: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( member997585926662434933_a_b_c @ C @ ( inf_in4776242878804941978_a_b_c @ A @ B ) )
     => ~ ( ( member997585926662434933_a_b_c @ C @ A )
         => ~ ( member997585926662434933_a_b_c @ C @ B ) ) ) ).

% IntE
thf(fact_687_IntE,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
     => ~ ( ( member6330420149250801815od_b_c @ C @ A )
         => ~ ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% IntE
thf(fact_688_IntE,axiom,
    ! [C: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ ( inf_in980870363377917322od_c_a @ A @ B ) )
     => ~ ( ( member7410604586820865893od_c_a @ C @ A )
         => ~ ( member7410604586820865893od_c_a @ C @ B ) ) ) ).

% IntE
thf(fact_689_IntE,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
     => ~ ( ( member_a @ C @ A )
         => ~ ( member_a @ C @ B ) ) ) ).

% IntE
thf(fact_690_list_Osimps_I8_J,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( map_Pr776894537227705078od_b_c @ F @ nil_Pr1342775757158464060od_c_a )
      = nil_Product_prod_b_c ) ).

% list.simps(8)
thf(fact_691_list_Omap__cong,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Ya: list_P6327159017948738492od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c,G: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( X2 = Ya )
     => ( ! [Z3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ Z3 @ ( set_Pr5664641634098845387od_c_a @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_Pr776894537227705078od_b_c @ F @ X2 )
          = ( map_Pr776894537227705078od_b_c @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_692_list_Omap__cong0,axiom,
    ! [X2: list_P6327159017948738492od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c,G: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ! [Z3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ Z3 @ ( set_Pr5664641634098845387od_c_a @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_Pr776894537227705078od_b_c @ F @ X2 )
        = ( map_Pr776894537227705078od_b_c @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_693_list_Oinj__map__strong,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Xa: list_P6327159017948738492od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c,Fa: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ! [Z3: produc2669932201312201782od_c_a,Za: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ Z3 @ ( set_Pr5664641634098845387od_c_a @ X2 ) )
         => ( ( member5502040268208861279od_c_a @ Za @ ( set_Pr5664641634098845387od_c_a @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_Pr776894537227705078od_b_c @ F @ X2 )
          = ( map_Pr776894537227705078od_b_c @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_694_list_Omap__ident__strong,axiom,
    ! [T2: list_P4069430723938770798od_d_d,F: produc4859908366599477352od_d_d > produc4859908366599477352od_d_d] :
      ( ! [Z3: produc4859908366599477352od_d_d] :
          ( ( member73601632213084689od_d_d @ Z3 @ ( set_Pr5977834532684968573od_d_d @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr8512485071813346500od_d_d @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_695_list_Omap__ident__strong,axiom,
    ! [T2: list_P5219093185649440210_a_b_c,F: produc8838969699183119820_a_b_c > produc8838969699183119820_a_b_c] :
      ( ! [Z3: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ Z3 @ ( set_Pr3403252632855053537_a_b_c @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr9043562808214552772_a_b_c @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_696_list_Omap__ident__strong,axiom,
    ! [T2: list_l2826717958560394946od_c_a,F: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a] :
      ( ! [Z3: list_P6327159017948738492od_c_a] :
          ( ( member7410604586820865893od_c_a @ Z3 @ ( set_li8663184538779258833od_c_a @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_li335866320336015812od_c_a @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_697_list_Omap__ident__strong,axiom,
    ! [T2: list_a,F: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_a_a @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_698_list_Omap__ident__strong,axiom,
    ! [T2: list_P3660316430366008877_set_a,F: produc1703568184450464039_set_a > produc1703568184450464039_set_a] :
      ( ! [Z3: produc1703568184450464039_set_a] :
          ( ( member7983343339038529360_set_a @ Z3 @ ( set_Pr7878197726666307004_set_a @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr5757421898035411460_set_a @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_699_list_Omap__ident__strong,axiom,
    ! [T2: list_l8907847357763382004od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ! [Z3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ Z3 @ ( set_li4480668622519654659od_b_c @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_li8081432502061035332od_b_c @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_700_map__ext,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c,G: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ! [X: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( map_Pr776894537227705078od_b_c @ F @ Xs )
        = ( map_Pr776894537227705078od_b_c @ G @ Xs ) ) ) ).

% map_ext
thf(fact_701_map__idI,axiom,
    ! [Xs: list_P4069430723938770798od_d_d,F: produc4859908366599477352od_d_d > produc4859908366599477352od_d_d] :
      ( ! [X: produc4859908366599477352od_d_d] :
          ( ( member73601632213084689od_d_d @ X @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_Pr8512485071813346500od_d_d @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_702_map__idI,axiom,
    ! [Xs: list_P5219093185649440210_a_b_c,F: produc8838969699183119820_a_b_c > produc8838969699183119820_a_b_c] :
      ( ! [X: produc8838969699183119820_a_b_c] :
          ( ( member997585926662434933_a_b_c @ X @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_Pr9043562808214552772_a_b_c @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_703_map__idI,axiom,
    ! [Xs: list_l2826717958560394946od_c_a,F: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a] :
      ( ! [X: list_P6327159017948738492od_c_a] :
          ( ( member7410604586820865893od_c_a @ X @ ( set_li8663184538779258833od_c_a @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_li335866320336015812od_c_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_704_map__idI,axiom,
    ! [Xs: list_a,F: a > a] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_705_map__idI,axiom,
    ! [Xs: list_P3660316430366008877_set_a,F: produc1703568184450464039_set_a > produc1703568184450464039_set_a] :
      ( ! [X: produc1703568184450464039_set_a] :
          ( ( member7983343339038529360_set_a @ X @ ( set_Pr7878197726666307004_set_a @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_Pr5757421898035411460_set_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_706_map__idI,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ! [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_li8081432502061035332od_b_c @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_707_map__set,axiom,
    ! [X2: a,Xs: list_a,F: a > a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( member_a @ ( F @ X2 ) @ ( set_a2 @ ( map_a_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_708_map__set,axiom,
    ! [X2: a,Xs: list_a,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( member6330420149250801815od_b_c @ ( F @ X2 ) @ ( set_li4480668622519654659od_b_c @ ( map_a_6125361980051820212od_b_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_709_map__set,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,F: list_P903359562653991662od_b_c > a] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( member_a @ ( F @ X2 ) @ ( set_a2 @ ( map_li642343614954829908_b_c_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_710_map__set,axiom,
    ! [X2: a,Xs: list_a,F: a > produc1703568184450464039_set_a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( member7983343339038529360_set_a @ ( F @ X2 ) @ ( set_Pr7878197726666307004_set_a @ ( map_a_757648918575525869_set_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_711_map__set,axiom,
    ! [X2: produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a,F: produc1703568184450464039_set_a > a] :
      ( ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Xs ) )
     => ( member_a @ ( F @ X2 ) @ ( set_a2 @ ( map_Pr4616894539174211803et_a_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_712_map__set,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Xs: list_P5219093185649440210_a_b_c,F: produc8838969699183119820_a_b_c > a] :
      ( ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
     => ( member_a @ ( F @ X2 ) @ ( set_a2 @ ( map_Pr811584189524267510_b_c_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_713_map__set,axiom,
    ! [X2: a,Xs: list_a,F: a > produc8838969699183119820_a_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( member997585926662434933_a_b_c @ ( F @ X2 ) @ ( set_Pr3403252632855053537_a_b_c @ ( map_a_4118158524683405202_a_b_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_714_map__set,axiom,
    ! [X2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
     => ( member6330420149250801815od_b_c @ ( F @ X2 ) @ ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_715_map__set,axiom,
    ! [X2: produc4859908366599477352od_d_d,Xs: list_P4069430723938770798od_d_d,F: produc4859908366599477352od_d_d > a] :
      ( ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
     => ( member_a @ ( F @ X2 ) @ ( set_a2 @ ( map_Pr5972083880475703642_d_d_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_716_map__set,axiom,
    ! [X2: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a,F: list_P6327159017948738492od_c_a > a] :
      ( ( member7410604586820865893od_c_a @ X2 @ ( set_li8663184538779258833od_c_a @ Xs ) )
     => ( member_a @ ( F @ X2 ) @ ( set_a2 @ ( map_li6584926188472171782_c_a_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_717_map__cong,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c,G: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( Xs = Ys )
     => ( ! [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( map_Pr776894537227705078od_b_c @ F @ Xs )
          = ( map_Pr776894537227705078od_b_c @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_718_ex__map__conv,axiom,
    ! [Ys: list_P903359562653991662od_b_c,F: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( ? [Xs4: list_P6327159017948738492od_c_a] :
            ( Ys
            = ( map_Pr776894537227705078od_b_c @ F @ Xs4 ) ) )
      = ( ! [X3: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Ys ) )
           => ? [Y5: produc2669932201312201782od_c_a] :
                ( X3
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_719_set__map__elem,axiom,
    ! [Y: a,F: a > a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( map_a_a @ F @ Xs ) ) )
     => ~ ! [X: a] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_720_set__map__elem,axiom,
    ! [Y: a,F: list_P903359562653991662od_b_c > a,Xs: list_l8907847357763382004od_b_c] :
      ( ( member_a @ Y @ ( set_a2 @ ( map_li642343614954829908_b_c_a @ F @ Xs ) ) )
     => ~ ! [X: list_P903359562653991662od_b_c] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_721_set__map__elem,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c,Xs: list_a] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( map_a_6125361980051820212od_b_c @ F @ Xs ) ) )
     => ~ ! [X: a] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_722_set__map__elem,axiom,
    ! [Y: a,F: produc1703568184450464039_set_a > a,Xs: list_P3660316430366008877_set_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( map_Pr4616894539174211803et_a_a @ F @ Xs ) ) )
     => ~ ! [X: produc1703568184450464039_set_a] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member7983343339038529360_set_a @ X @ ( set_Pr7878197726666307004_set_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_723_set__map__elem,axiom,
    ! [Y: produc1703568184450464039_set_a,F: a > produc1703568184450464039_set_a,Xs: list_a] :
      ( ( member7983343339038529360_set_a @ Y @ ( set_Pr7878197726666307004_set_a @ ( map_a_757648918575525869_set_a @ F @ Xs ) ) )
     => ~ ! [X: a] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_724_set__map__elem,axiom,
    ! [Y: produc8838969699183119820_a_b_c,F: a > produc8838969699183119820_a_b_c,Xs: list_a] :
      ( ( member997585926662434933_a_b_c @ Y @ ( set_Pr3403252632855053537_a_b_c @ ( map_a_4118158524683405202_a_b_c @ F @ Xs ) ) )
     => ~ ! [X: a] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_725_set__map__elem,axiom,
    ! [Y: a,F: produc8838969699183119820_a_b_c > a,Xs: list_P5219093185649440210_a_b_c] :
      ( ( member_a @ Y @ ( set_a2 @ ( map_Pr811584189524267510_b_c_a @ F @ Xs ) ) )
     => ~ ! [X: produc8838969699183119820_a_b_c] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member997585926662434933_a_b_c @ X @ ( set_Pr3403252632855053537_a_b_c @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_726_set__map__elem,axiom,
    ! [Y: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) )
     => ~ ! [X: list_P903359562653991662od_b_c] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_727_set__map__elem,axiom,
    ! [Y: produc4859908366599477352od_d_d,F: a > produc4859908366599477352od_d_d,Xs: list_a] :
      ( ( member73601632213084689od_d_d @ Y @ ( set_Pr5977834532684968573od_d_d @ ( map_a_5229631668954897454od_d_d @ F @ Xs ) ) )
     => ~ ! [X: a] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_728_set__map__elem,axiom,
    ! [Y: list_P6327159017948738492od_c_a,F: a > list_P6327159017948738492od_c_a,Xs: list_a] :
      ( ( member7410604586820865893od_c_a @ Y @ ( set_li8663184538779258833od_c_a @ ( map_a_889415987282285442od_c_a @ F @ Xs ) ) )
     => ~ ! [X: a] :
            ( ( Y
              = ( F @ X ) )
           => ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_729_list__map__set__prop,axiom,
    ! [X2: product_prod_b_c,F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a,P5: product_prod_b_c > $o] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( map_Pr776894537227705078od_b_c @ F @ Xs ) ) )
     => ( ! [Y3: produc2669932201312201782od_c_a] : ( P5 @ ( F @ Y3 ) )
       => ( P5 @ X2 ) ) ) ).

% list_map_set_prop
thf(fact_730_list__map__source__elem,axiom,
    ! [X2: a,F: list_P903359562653991662od_b_c > a,Xs: list_l8907847357763382004od_b_c] :
      ( ( member_a @ X2 @ ( set_a2 @ ( map_li642343614954829908_b_c_a @ F @ Xs ) ) )
     => ? [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_731_list__map__source__elem,axiom,
    ! [X2: a,F: produc1703568184450464039_set_a > a,Xs: list_P3660316430366008877_set_a] :
      ( ( member_a @ X2 @ ( set_a2 @ ( map_Pr4616894539174211803et_a_a @ F @ Xs ) ) )
     => ? [X: produc1703568184450464039_set_a] :
          ( ( member7983343339038529360_set_a @ X @ ( set_Pr7878197726666307004_set_a @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_732_list__map__source__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) )
     => ? [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_733_list__map__source__elem,axiom,
    ! [X2: produc1703568184450464039_set_a,F: list_P903359562653991662od_b_c > produc1703568184450464039_set_a,Xs: list_l8907847357763382004od_b_c] :
      ( ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ ( map_li4890537941830539389_set_a @ F @ Xs ) ) )
     => ? [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_734_list__map__source__elem,axiom,
    ! [X2: list_P903359562653991662od_b_c,F: produc1703568184450464039_set_a > list_P903359562653991662od_b_c,Xs: list_P3660316430366008877_set_a] :
      ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ ( map_Pr191379420489308491od_b_c @ F @ Xs ) ) )
     => ? [X: produc1703568184450464039_set_a] :
          ( ( member7983343339038529360_set_a @ X @ ( set_Pr7878197726666307004_set_a @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_735_list__map__source__elem,axiom,
    ! [X2: product_prod_b_c,F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a] :
      ( ( member7862447941013992593od_b_c @ X2 @ ( set_Product_prod_b_c2 @ ( map_Pr776894537227705078od_b_c @ F @ Xs ) ) )
     => ? [X: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_736_list__map__source__elem,axiom,
    ! [X2: produc8838969699183119820_a_b_c,F: list_P903359562653991662od_b_c > produc8838969699183119820_a_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ ( map_li6967035672056657954_a_b_c @ F @ Xs ) ) )
     => ? [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_737_list__map__source__elem,axiom,
    ! [X2: produc1703568184450464039_set_a,F: produc1703568184450464039_set_a > produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a] :
      ( ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ ( map_Pr5757421898035411460_set_a @ F @ Xs ) ) )
     => ? [X: produc1703568184450464039_set_a] :
          ( ( member7983343339038529360_set_a @ X @ ( set_Pr7878197726666307004_set_a @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_738_list__map__source__elem,axiom,
    ! [X2: produc8838969699183119820_a_b_c,F: produc1703568184450464039_set_a > produc8838969699183119820_a_b_c,Xs: list_P3660316430366008877_set_a] :
      ( ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ ( map_Pr6057462688338226473_a_b_c @ F @ Xs ) ) )
     => ? [X: produc1703568184450464039_set_a] :
          ( ( member7983343339038529360_set_a @ X @ ( set_Pr7878197726666307004_set_a @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_739_list__map__source__elem,axiom,
    ! [X2: produc4859908366599477352od_d_d,F: list_P903359562653991662od_b_c > produc4859908366599477352od_d_d,Xs: list_l8907847357763382004od_b_c] :
      ( ( member73601632213084689od_d_d @ X2 @ ( set_Pr5977834532684968573od_d_d @ ( map_li605568857130319038od_d_d @ F @ Xs ) ) )
     => ? [X: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs ) )
          & ( X2
            = ( F @ X ) ) ) ) ).

% list_map_source_elem
thf(fact_740_append__eq__map__conv,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( append3884164603879732689od_c_a @ Ys @ Zs )
        = ( map_Pr6705208461955284164od_c_a @ F @ Xs ) )
      = ( ? [Us: list_P6327159017948738492od_c_a,Vs: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr6705208461955284164od_c_a @ F @ Us ) )
            & ( Zs
              = ( map_Pr6705208461955284164od_c_a @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_741_append__eq__map__conv,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a,F: product_prod_b_c > produc2669932201312201782od_c_a,Xs: list_P903359562653991662od_b_c] :
      ( ( ( append3884164603879732689od_c_a @ Ys @ Zs )
        = ( map_Pr7314640756821961490od_c_a @ F @ Xs ) )
      = ( ? [Us: list_P903359562653991662od_b_c,Vs: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr7314640756821961490od_c_a @ F @ Us ) )
            & ( Zs
              = ( map_Pr7314640756821961490od_c_a @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_742_append__eq__map__conv,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Ys @ Zs )
        = ( map_Pr6384634858812015940od_b_c @ F @ Xs ) )
      = ( ? [Us: list_P903359562653991662od_b_c,Vs: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr6384634858812015940od_b_c @ F @ Us ) )
            & ( Zs
              = ( map_Pr6384634858812015940od_b_c @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_743_append__eq__map__conv,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( append2547753245680614915od_b_c @ Ys @ Zs )
        = ( map_Pr776894537227705078od_b_c @ F @ Xs ) )
      = ( ? [Us: list_P6327159017948738492od_c_a,Vs: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr776894537227705078od_b_c @ F @ Us ) )
            & ( Zs
              = ( map_Pr776894537227705078od_b_c @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_744_map__eq__append__conv,axiom,
    ! [F: 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 @ F @ Xs )
        = ( append3884164603879732689od_c_a @ Ys @ Zs ) )
      = ( ? [Us: list_P6327159017948738492od_c_a,Vs: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr6705208461955284164od_c_a @ F @ Us ) )
            & ( Zs
              = ( map_Pr6705208461955284164od_c_a @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_745_map__eq__append__conv,axiom,
    ! [F: product_prod_b_c > produc2669932201312201782od_c_a,Xs: list_P903359562653991662od_b_c,Ys: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr7314640756821961490od_c_a @ F @ Xs )
        = ( append3884164603879732689od_c_a @ Ys @ Zs ) )
      = ( ? [Us: list_P903359562653991662od_b_c,Vs: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr7314640756821961490od_c_a @ F @ Us ) )
            & ( Zs
              = ( map_Pr7314640756821961490od_c_a @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_746_map__eq__append__conv,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr6384634858812015940od_b_c @ F @ Xs )
        = ( append2547753245680614915od_b_c @ Ys @ Zs ) )
      = ( ? [Us: list_P903359562653991662od_b_c,Vs: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr6384634858812015940od_b_c @ F @ Us ) )
            & ( Zs
              = ( map_Pr6384634858812015940od_b_c @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_747_map__eq__append__conv,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr776894537227705078od_b_c @ F @ Xs )
        = ( append2547753245680614915od_b_c @ Ys @ Zs ) )
      = ( ? [Us: list_P6327159017948738492od_c_a,Vs: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( append3884164603879732689od_c_a @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr776894537227705078od_b_c @ F @ Us ) )
            & ( Zs
              = ( map_Pr776894537227705078od_b_c @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_748_Int__mono,axiom,
    ! [A: set_a,C3: set_a,B: set_a,D3: set_a] :
      ( ( ord_less_eq_set_a @ A @ C3 )
     => ( ( ord_less_eq_set_a @ B @ D3 )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C3 @ D3 ) ) ) ) ).

% Int_mono
thf(fact_749_Int__mono,axiom,
    ! [A: set_li1159382662694783132od_c_a,C3: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,D3: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A @ C3 )
     => ( ( ord_le2998388488506175548od_c_a @ B @ D3 )
       => ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A @ B ) @ ( inf_in980870363377917322od_c_a @ C3 @ D3 ) ) ) ) ).

% Int_mono
thf(fact_750_Int__lower1,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_751_Int__lower1,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_752_Int__lower2,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_753_Int__lower2,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_754_Int__absorb1,axiom,
    ! [B: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ A )
     => ( ( inf_inf_set_a @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_755_Int__absorb1,axiom,
    ! [B: set_li1159382662694783132od_c_a,A: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B @ A )
     => ( ( inf_in980870363377917322od_c_a @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_756_Int__absorb2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( inf_inf_set_a @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_757_Int__absorb2,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A @ B )
     => ( ( inf_in980870363377917322od_c_a @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_758_Int__greatest,axiom,
    ! [C3: set_a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C3 @ A )
     => ( ( ord_less_eq_set_a @ C3 @ B )
       => ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_759_Int__greatest,axiom,
    ! [C3: set_li1159382662694783132od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ C3 @ A )
     => ( ( ord_le2998388488506175548od_c_a @ C3 @ B )
       => ( ord_le2998388488506175548od_c_a @ C3 @ ( inf_in980870363377917322od_c_a @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_760_Int__Collect__mono,axiom,
    ! [A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d,P5: produc4859908366599477352od_d_d > $o,Q5: produc4859908366599477352od_d_d > $o] :
      ( ( ord_le7464911323401878760od_d_d @ A @ B )
     => ( ! [X: produc4859908366599477352od_d_d] :
            ( ( member73601632213084689od_d_d @ X @ A )
           => ( ( P5 @ X )
             => ( Q5 @ X ) ) )
       => ( ord_le7464911323401878760od_d_d @ ( inf_in7322238654750965302od_d_d @ A @ ( collec5362705567740981075od_d_d @ P5 ) ) @ ( inf_in7322238654750965302od_d_d @ B @ ( collec5362705567740981075od_d_d @ Q5 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_761_Int__Collect__mono,axiom,
    ! [A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c,P5: produc8838969699183119820_a_b_c > $o,Q5: produc8838969699183119820_a_b_c > $o] :
      ( ( ord_le6086132928344890700_a_b_c @ A @ B )
     => ( ! [X: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X @ A )
           => ( ( P5 @ X )
             => ( Q5 @ X ) ) )
       => ( ord_le6086132928344890700_a_b_c @ ( inf_in4776242878804941978_a_b_c @ A @ ( collec3867759823522760631_a_b_c @ P5 ) ) @ ( inf_in4776242878804941978_a_b_c @ B @ ( collec3867759823522760631_a_b_c @ Q5 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_762_Int__Collect__mono,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,P5: list_P903359562653991662od_b_c > $o,Q5: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ! [X: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X @ A )
           => ( ( P5 @ X )
             => ( Q5 @ X ) ) )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ ( collec2280997390073109977od_b_c @ P5 ) ) @ ( inf_in4978071631833541052od_b_c @ B @ ( collec2280997390073109977od_b_c @ Q5 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_763_Int__Collect__mono,axiom,
    ! [A: set_se3924713247505902254od_b_c,B: set_se3924713247505902254od_b_c,P5: set_li6436108459499378894od_b_c > $o,Q5: set_li6436108459499378894od_b_c > $o] :
      ( ( ord_le6656836712342966862od_b_c @ A @ B )
     => ( ! [X: set_li6436108459499378894od_b_c] :
            ( ( member6985331446368301687od_b_c @ X @ A )
           => ( ( P5 @ X )
             => ( Q5 @ X ) ) )
       => ( ord_le6656836712342966862od_b_c @ ( inf_in5524221907227607964od_b_c @ A @ ( collec1823247286932916153od_b_c @ P5 ) ) @ ( inf_in5524221907227607964od_b_c @ B @ ( collec1823247286932916153od_b_c @ Q5 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_764_Int__Collect__mono,axiom,
    ! [A: set_a,B: set_a,P5: a > $o,Q5: a > $o] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ! [X: a] :
            ( ( member_a @ X @ A )
           => ( ( P5 @ X )
             => ( Q5 @ X ) ) )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P5 ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q5 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_765_Int__Collect__mono,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,P5: list_P6327159017948738492od_c_a > $o,Q5: list_P6327159017948738492od_c_a > $o] :
      ( ( ord_le2998388488506175548od_c_a @ A @ B )
     => ( ! [X: list_P6327159017948738492od_c_a] :
            ( ( member7410604586820865893od_c_a @ X @ A )
           => ( ( P5 @ X )
             => ( Q5 @ X ) ) )
       => ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A @ ( collec6273869032445462695od_c_a @ P5 ) ) @ ( inf_in980870363377917322od_c_a @ B @ ( collec6273869032445462695od_c_a @ Q5 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_766_append__Nil,axiom,
    ! [Ys: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ nil_Pr1342775757158464060od_c_a @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_767_append__Nil,axiom,
    ! [Ys: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_768_append_Oleft__neutral,axiom,
    ! [A2: list_P6327159017948738492od_c_a] :
      ( ( append3884164603879732689od_c_a @ nil_Pr1342775757158464060od_c_a @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_769_append_Oleft__neutral,axiom,
    ! [A2: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_770_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_771_eq__Nil__appendI,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_772_image__set,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,Xs: list_P5219093185649440210_a_b_c] :
      ( ( image_3273489647386116407_b_c_a @ F @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
      = ( set_a2 @ ( map_Pr811584189524267510_b_c_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_773_image__set,axiom,
    ! [F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,Xs: list_P4908235088432371736_set_a] :
      ( ( image_865596521259543215od_c_a @ F @ ( set_Pr1469968110526226471_set_a @ Xs ) )
      = ( set_li8663184538779258833od_c_a @ ( map_Pr6055023778478330862od_c_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_774_image__set,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,Xs: list_P5219093185649440210_a_b_c] :
      ( ( image_2344789136157180295od_b_c @ F @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
      = ( set_se1205977669550889699od_b_c @ ( map_Pr4360419603926651846od_b_c @ F @ Xs ) ) ) ).

% image_set
thf(fact_775_image__set,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a] :
      ( ( image_4981958329637652151od_b_c @ F @ ( set_Pr5664641634098845387od_c_a @ Xs ) )
      = ( set_Product_prod_b_c2 @ ( map_Pr776894537227705078od_b_c @ F @ Xs ) ) ) ).

% image_set
thf(fact_776_image__set,axiom,
    ! [F: produc1703568184450464039_set_a > produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a] :
      ( ( image_3741791430265939909_set_a @ F @ ( set_Pr7878197726666307004_set_a @ Xs ) )
      = ( set_Pr7878197726666307004_set_a @ ( map_Pr5757421898035411460_set_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_777_image__set,axiom,
    ! [F: produc1703568184450464039_set_a > list_P903359562653991662od_b_c,Xs: list_P3660316430366008877_set_a] :
      ( ( image_4768856705948449420od_b_c @ F @ ( set_Pr7878197726666307004_set_a @ Xs ) )
      = ( set_li4480668622519654659od_b_c @ ( map_Pr191379420489308491od_b_c @ F @ Xs ) ) ) ).

% image_set
thf(fact_778_image__set,axiom,
    ! [F: list_P903359562653991662od_b_c > produc1703568184450464039_set_a,Xs: list_l8907847357763382004od_b_c] :
      ( ( image_244643190434904510_set_a @ F @ ( set_li4480668622519654659od_b_c @ Xs ) )
      = ( set_Pr7878197726666307004_set_a @ ( map_li4890537941830539389_set_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_779_image__set,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( image_2850395784482391045od_b_c @ F @ ( set_li4480668622519654659od_b_c @ Xs ) )
      = ( set_li4480668622519654659od_b_c @ ( map_li8081432502061035332od_b_c @ F @ Xs ) ) ) ).

% image_set
thf(fact_780_pair__list__eqI,axiom,
    ! [Xs: list_P5219093185649440210_a_b_c,Ys: list_P5219093185649440210_a_b_c] :
      ( ( ( map_Pr811584189524267510_b_c_a @ produc2620512763296580520_a_b_c @ Xs )
        = ( map_Pr811584189524267510_b_c_a @ produc2620512763296580520_a_b_c @ Ys ) )
     => ( ( ( map_Pr6813647754772709915_a_b_c @ produc4728557313310798826_a_b_c @ Xs )
          = ( map_Pr6813647754772709915_a_b_c @ produc4728557313310798826_a_b_c @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_781_pair__list__eqI,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys: list_P125642481956313003od_c_a] :
      ( ( ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Xs )
        = ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Ys ) )
     => ( ( ( map_Pr5866436826731527133_c_a_a @ product_snd_c_a @ Xs )
          = ( map_Pr5866436826731527133_c_a_a @ product_snd_c_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_782_pair__list__eqI,axiom,
    ! [Xs: list_P4908235088432371736_set_a,Ys: list_P4908235088432371736_set_a] :
      ( ( ( map_Pr6055023778478330862od_c_a @ produc3242630202250842350_set_a @ Xs )
        = ( map_Pr6055023778478330862od_c_a @ produc3242630202250842350_set_a @ Ys ) )
     => ( ( ( map_Pr2257100352023006809_set_a @ produc2451121746714373936_set_a @ Xs )
          = ( map_Pr2257100352023006809_set_a @ produc2451121746714373936_set_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_783_pair__list__eqI,axiom,
    ! [Xs: list_P3660316430366008877_set_a,Ys: list_P3660316430366008877_set_a] :
      ( ( ( map_Pr8866779738092938811_set_a @ produc9088895665703139587_set_a @ Xs )
        = ( map_Pr8866779738092938811_set_a @ produc9088895665703139587_set_a @ Ys ) )
     => ( ( ( map_Pr8866779738092938811_set_a @ produc1983107199584856133_set_a @ Xs )
          = ( map_Pr8866779738092938811_set_a @ produc1983107199584856133_set_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_784_pair__list__eqI,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Xs )
        = ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Ys ) )
     => ( ( ( map_Pr3684547939035685915od_c_a @ produc8088130389382527188od_c_a @ Xs )
          = ( map_Pr3684547939035685915od_c_a @ produc8088130389382527188od_c_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_785_pair__list__eqI,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Xs )
        = ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Ys ) )
     => ( ( ( map_Pr6684165712643126428od_c_a @ produc5221446324160585515od_c_a @ Xs )
          = ( map_Pr6684165712643126428od_c_a @ produc5221446324160585515od_c_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_786_language__intro,axiom,
    ! [M: fsm_fsm_a_b_c_a_a,Q: fsm_a_b_c,P: list_P4545384517222041535_a_b_c] :
      ( ( path_fsm_a_b_c_a_a @ M @ Q @ P )
     => ( member6824001069763096534od_a_a
        @ ( map_Pr7384117626843903474od_a_a
          @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
          @ P )
        @ ( lS_fsm_a_b_c_a_a @ M @ Q ) ) ) ).

% language_intro
thf(fact_787_language__intro,axiom,
    ! [M: fsm_Pr306566304496394697od_c_a,Q: produc1703568184450464039_set_a,P: list_P5750286787369789953_set_a] :
      ( ( path_P8254327248518380459od_c_a @ M @ Q @ P )
     => ( member4479000366354228116od_c_a
        @ ( map_Pr2884798845580615278od_c_a
          @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
          @ P )
        @ ( lS_Pro8045283330902898057od_c_a @ M @ Q ) ) ) ).

% language_intro
thf(fact_788_language__intro,axiom,
    ! [M: fsm_se8204406047609318756_set_a,Q: set_a,P: list_P3716011504951620193_set_a] :
      ( ( path_s1167107641669997510_set_a @ M @ Q @ P )
     => ( member5048304341121814904_set_a
        @ ( map_Pr3507802118303906290_set_a
          @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
          @ P )
        @ ( lS_set5116714952275434404_set_a @ M @ Q ) ) ) ).

% language_intro
thf(fact_789_language__intro,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ M @ Q @ P )
     => ( member6895035109809442519od_a_b
        @ ( map_Pr3411285261028437108od_a_b
          @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
          @ P )
        @ ( lS_Pro5114676609533226954_a_a_b @ M @ Q ) ) ) ).

% language_intro
thf(fact_790_language__intro,axiom,
    ! [M: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P )
     => ( member6330420149250801815od_b_c
        @ ( map_Pr776894537227705078od_b_c
          @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
          @ P )
        @ ( lS_a_b_c @ M @ Q ) ) ) ).

% language_intro
thf(fact_791_language__state__containment,axiom,
    ! [M: fsm_fsm_a_b_c_a_a,Q: fsm_a_b_c,P: list_P4545384517222041535_a_b_c,Io3: list_P1396940483166286381od_a_a] :
      ( ( path_fsm_a_b_c_a_a @ M @ Q @ P )
     => ( ( ( map_Pr7384117626843903474od_a_a
            @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
            @ P )
          = Io3 )
       => ( member6824001069763096534od_a_a @ Io3 @ ( lS_fsm_a_b_c_a_a @ M @ Q ) ) ) ) ).

% language_state_containment
thf(fact_792_language__state__containment,axiom,
    ! [M: fsm_Pr306566304496394697od_c_a,Q: produc1703568184450464039_set_a,P: list_P5750286787369789953_set_a,Io3: list_P1175628092075131243od_c_a] :
      ( ( path_P8254327248518380459od_c_a @ M @ Q @ P )
     => ( ( ( map_Pr2884798845580615278od_c_a
            @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
            @ P )
          = Io3 )
       => ( member4479000366354228116od_c_a @ Io3 @ ( lS_Pro8045283330902898057od_c_a @ M @ Q ) ) ) ) ).

% language_state_containment
thf(fact_793_language__state__containment,axiom,
    ! [M: fsm_se8204406047609318756_set_a,Q: set_a,P: list_P3716011504951620193_set_a,Io3: list_P5262903567315784527_set_a] :
      ( ( path_s1167107641669997510_set_a @ M @ Q @ P )
     => ( ( ( map_Pr3507802118303906290_set_a
            @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
            @ P )
          = Io3 )
       => ( member5048304341121814904_set_a @ Io3 @ ( lS_set5116714952275434404_set_a @ M @ Q ) ) ) ) ).

% language_state_containment
thf(fact_794_language__state__containment,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P: list_P1562850763667305982od_c_a,Io3: list_P1467974523212632366od_a_b] :
      ( ( path_P1107610407223809260_a_a_b @ M @ Q @ P )
     => ( ( ( map_Pr3411285261028437108od_a_b
            @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
            @ P )
          = Io3 )
       => ( member6895035109809442519od_a_b @ Io3 @ ( lS_Pro5114676609533226954_a_a_b @ M @ Q ) ) ) ) ).

% language_state_containment
thf(fact_795_language__state__containment,axiom,
    ! [M: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a,Io3: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M @ Q @ P )
     => ( ( ( map_Pr776894537227705078od_b_c
            @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
            @ P )
          = Io3 )
       => ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) ) ) ) ).

% language_state_containment
thf(fact_796_image__Int__subset,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( image_2850395784482391045od_b_c @ F @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) @ ( inf_in4978071631833541052od_b_c @ ( image_2850395784482391045od_b_c @ F @ A ) @ ( image_2850395784482391045od_b_c @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_797_image__Int__subset,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] : ( ord_le6656836712342966862od_b_c @ ( image_2344789136157180295od_b_c @ F @ ( inf_in4776242878804941978_a_b_c @ A @ B ) ) @ ( inf_in5524221907227607964od_b_c @ ( image_2344789136157180295od_b_c @ F @ A ) @ ( image_2344789136157180295od_b_c @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_798_image__Int__subset,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] : ( ord_less_eq_set_a @ ( image_3273489647386116407_b_c_a @ F @ ( inf_in4776242878804941978_a_b_c @ A @ B ) ) @ ( inf_inf_set_a @ ( image_3273489647386116407_b_c_a @ F @ A ) @ ( image_3273489647386116407_b_c_a @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_799_image__Int__subset,axiom,
    ! [F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,A: set_Pr3779000724142394866_set_a,B: set_Pr3779000724142394866_set_a] : ( ord_le2998388488506175548od_c_a @ ( image_865596521259543215od_c_a @ F @ ( inf_in1791426075705210592_set_a @ A @ B ) ) @ ( inf_in980870363377917322od_c_a @ ( image_865596521259543215od_c_a @ F @ A ) @ ( image_865596521259543215od_c_a @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_800_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_801_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_802_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_803_language__path__append,axiom,
    ! [M1: fsm_fsm_a_b_c_a_a,Q1: fsm_a_b_c,P1: list_P4545384517222041535_a_b_c,Io3: list_P1396940483166286381od_a_a] :
      ( ( path_fsm_a_b_c_a_a @ M1 @ Q1 @ P1 )
     => ( ( member6824001069763096534od_a_a @ Io3 @ ( lS_fsm_a_b_c_a_a @ M1 @ ( target_fsm_a_b_c_a_a @ Q1 @ P1 ) ) )
       => ( member6824001069763096534od_a_a
          @ ( append5335208819046833346od_a_a
            @ ( map_Pr7384117626843903474od_a_a
              @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
              @ P1 )
            @ Io3 )
          @ ( lS_fsm_a_b_c_a_a @ M1 @ Q1 ) ) ) ) ).

% language_path_append
thf(fact_804_language__path__append,axiom,
    ! [M1: fsm_Pr306566304496394697od_c_a,Q1: produc1703568184450464039_set_a,P1: list_P5750286787369789953_set_a,Io3: list_P1175628092075131243od_c_a] :
      ( ( path_P8254327248518380459od_c_a @ M1 @ Q1 @ P1 )
     => ( ( member4479000366354228116od_c_a @ Io3 @ ( lS_Pro8045283330902898057od_c_a @ M1 @ ( target3445005149252956639od_c_a @ Q1 @ P1 ) ) )
       => ( member4479000366354228116od_c_a
          @ ( append2230762392551410432od_c_a
            @ ( map_Pr2884798845580615278od_c_a
              @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
              @ P1 )
            @ Io3 )
          @ ( lS_Pro8045283330902898057od_c_a @ M1 @ Q1 ) ) ) ) ).

% language_path_append
thf(fact_805_language__path__append,axiom,
    ! [M1: fsm_se8204406047609318756_set_a,Q1: set_a,P1: list_P3716011504951620193_set_a,Io3: list_P5262903567315784527_set_a] :
      ( ( path_s1167107641669997510_set_a @ M1 @ Q1 @ P1 )
     => ( ( member5048304341121814904_set_a @ Io3 @ ( lS_set5116714952275434404_set_a @ M1 @ ( target6955975396184022522_set_a @ Q1 @ P1 ) ) )
       => ( member5048304341121814904_set_a
          @ ( append1006059461745118948_set_a
            @ ( map_Pr3507802118303906290_set_a
              @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
              @ P1 )
            @ Io3 )
          @ ( lS_set5116714952275434404_set_a @ M1 @ Q1 ) ) ) ) ).

% language_path_append
thf(fact_806_language__path__append,axiom,
    ! [M1: fsm_Pr1886043760914421514_a_a_b,Q1: product_prod_c_a,P1: list_P1562850763667305982od_c_a,Io3: list_P1467974523212632366od_a_b] :
      ( ( path_P1107610407223809260_a_a_b @ M1 @ Q1 @ P1 )
     => ( ( member6895035109809442519od_a_b @ Io3 @ ( lS_Pro5114676609533226954_a_a_b @ M1 @ ( target6303326113586172192_a_a_b @ Q1 @ P1 ) ) )
       => ( member6895035109809442519od_a_b
          @ ( append5335208823350062147od_a_b
            @ ( map_Pr3411285261028437108od_a_b
              @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
              @ P1 )
            @ Io3 )
          @ ( lS_Pro5114676609533226954_a_a_b @ M1 @ Q1 ) ) ) ) ).

% language_path_append
thf(fact_807_language__path__append,axiom,
    ! [M1: fsm_a_b_c,Q1: a,P1: list_P6327159017948738492od_c_a,Io3: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M1 @ Q1 @ P1 )
     => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M1 @ ( target_a_b_c @ Q1 @ P1 ) ) )
       => ( member6330420149250801815od_b_c
          @ ( append2547753245680614915od_b_c
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
              @ P1 )
            @ Io3 )
          @ ( lS_a_b_c @ M1 @ Q1 ) ) ) ) ).

% language_path_append
thf(fact_808_language__state__split,axiom,
    ! [Io1: list_P1396940483166286381od_a_a,Io2: list_P1396940483166286381od_a_a,M: fsm_fsm_a_b_c_a_a,Q1: fsm_a_b_c] :
      ( ( member6824001069763096534od_a_a @ ( append5335208819046833346od_a_a @ Io1 @ Io2 ) @ ( lS_fsm_a_b_c_a_a @ M @ Q1 ) )
     => ~ ! [P12: list_P4545384517222041535_a_b_c] :
            ( ( path_fsm_a_b_c_a_a @ M @ Q1 @ P12 )
           => ! [P22: list_P4545384517222041535_a_b_c] :
                ( ( path_fsm_a_b_c_a_a @ M @ ( target_fsm_a_b_c_a_a @ Q1 @ P12 ) @ P22 )
               => ( ( ( map_Pr7384117626843903474od_a_a
                      @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                      @ P12 )
                    = Io1 )
                 => ( ( map_Pr7384117626843903474od_a_a
                      @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                      @ P22 )
                   != Io2 ) ) ) ) ) ).

% language_state_split
thf(fact_809_language__state__split,axiom,
    ! [Io1: list_P1175628092075131243od_c_a,Io2: list_P1175628092075131243od_c_a,M: fsm_Pr306566304496394697od_c_a,Q1: produc1703568184450464039_set_a] :
      ( ( member4479000366354228116od_c_a @ ( append2230762392551410432od_c_a @ Io1 @ Io2 ) @ ( lS_Pro8045283330902898057od_c_a @ M @ Q1 ) )
     => ~ ! [P12: list_P5750286787369789953_set_a] :
            ( ( path_P8254327248518380459od_c_a @ M @ Q1 @ P12 )
           => ! [P22: list_P5750286787369789953_set_a] :
                ( ( path_P8254327248518380459od_c_a @ M @ ( target3445005149252956639od_c_a @ Q1 @ P12 ) @ P22 )
               => ( ( ( map_Pr2884798845580615278od_c_a
                      @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                      @ P12 )
                    = Io1 )
                 => ( ( map_Pr2884798845580615278od_c_a
                      @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                      @ P22 )
                   != Io2 ) ) ) ) ) ).

% language_state_split
thf(fact_810_language__state__split,axiom,
    ! [Io1: list_P5262903567315784527_set_a,Io2: list_P5262903567315784527_set_a,M: fsm_se8204406047609318756_set_a,Q1: set_a] :
      ( ( member5048304341121814904_set_a @ ( append1006059461745118948_set_a @ Io1 @ Io2 ) @ ( lS_set5116714952275434404_set_a @ M @ Q1 ) )
     => ~ ! [P12: list_P3716011504951620193_set_a] :
            ( ( path_s1167107641669997510_set_a @ M @ Q1 @ P12 )
           => ! [P22: list_P3716011504951620193_set_a] :
                ( ( path_s1167107641669997510_set_a @ M @ ( target6955975396184022522_set_a @ Q1 @ P12 ) @ P22 )
               => ( ( ( map_Pr3507802118303906290_set_a
                      @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                      @ P12 )
                    = Io1 )
                 => ( ( map_Pr3507802118303906290_set_a
                      @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                      @ P22 )
                   != Io2 ) ) ) ) ) ).

% language_state_split
thf(fact_811_language__state__split,axiom,
    ! [Io1: list_P1467974523212632366od_a_b,Io2: list_P1467974523212632366od_a_b,M: fsm_Pr1886043760914421514_a_a_b,Q1: product_prod_c_a] :
      ( ( member6895035109809442519od_a_b @ ( append5335208823350062147od_a_b @ Io1 @ Io2 ) @ ( lS_Pro5114676609533226954_a_a_b @ M @ Q1 ) )
     => ~ ! [P12: list_P1562850763667305982od_c_a] :
            ( ( path_P1107610407223809260_a_a_b @ M @ Q1 @ P12 )
           => ! [P22: list_P1562850763667305982od_c_a] :
                ( ( path_P1107610407223809260_a_a_b @ M @ ( target6303326113586172192_a_a_b @ Q1 @ P12 ) @ P22 )
               => ( ( ( map_Pr3411285261028437108od_a_b
                      @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                      @ P12 )
                    = Io1 )
                 => ( ( map_Pr3411285261028437108od_a_b
                      @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                      @ P22 )
                   != Io2 ) ) ) ) ) ).

% language_state_split
thf(fact_812_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
                      @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                      @ P12 )
                    = Io1 )
                 => ( ( map_Pr776894537227705078od_b_c
                      @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                      @ P22 )
                   != Io2 ) ) ) ) ) ).

% language_state_split
thf(fact_813_language__split,axiom,
    ! [Io1: list_P1396940483166286381od_a_a,Io2: list_P1396940483166286381od_a_a,M: fsm_fsm_a_b_c_a_a] :
      ( ( member6824001069763096534od_a_a @ ( append5335208819046833346od_a_a @ Io1 @ Io2 ) @ ( lS_fsm_a_b_c_a_a @ M @ ( initia7499468085720305490_c_a_a @ M ) ) )
     => ~ ! [P12: list_P4545384517222041535_a_b_c,P22: list_P4545384517222041535_a_b_c] :
            ( ( path_fsm_a_b_c_a_a @ M @ ( initia7499468085720305490_c_a_a @ M ) @ ( append1656274458552408916_a_b_c @ P12 @ P22 ) )
           => ( ( ( map_Pr7384117626843903474od_a_a
                  @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                  @ P12 )
                = Io1 )
             => ( ( map_Pr7384117626843903474od_a_a
                  @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                  @ P22 )
               != Io2 ) ) ) ) ).

% language_split
thf(fact_814_language__split,axiom,
    ! [Io1: list_P1175628092075131243od_c_a,Io2: list_P1175628092075131243od_c_a,M: fsm_Pr306566304496394697od_c_a] :
      ( ( member4479000366354228116od_c_a @ ( append2230762392551410432od_c_a @ Io1 @ Io2 ) @ ( lS_Pro8045283330902898057od_c_a @ M @ ( initia7978570708967869844od_c_a @ M ) ) )
     => ~ ! [P12: list_P5750286787369789953_set_a,P22: list_P5750286787369789953_set_a] :
            ( ( path_P8254327248518380459od_c_a @ M @ ( initia7978570708967869844od_c_a @ M ) @ ( append881000372101963414_set_a @ P12 @ P22 ) )
           => ( ( ( map_Pr2884798845580615278od_c_a
                  @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                  @ P12 )
                = Io1 )
             => ( ( map_Pr2884798845580615278od_c_a
                  @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                  @ P22 )
               != Io2 ) ) ) ) ).

% language_split
thf(fact_815_language__split,axiom,
    ! [Io1: list_P5262903567315784527_set_a,Io2: list_P5262903567315784527_set_a,M: fsm_se8204406047609318756_set_a] :
      ( ( member5048304341121814904_set_a @ ( append1006059461745118948_set_a @ Io1 @ Io2 ) @ ( lS_set5116714952275434404_set_a @ M @ ( initia4238633132216284847_set_a @ M ) ) )
     => ~ ! [P12: list_P3716011504951620193_set_a,P22: list_P3716011504951620193_set_a] :
            ( ( path_s1167107641669997510_set_a @ M @ ( initia4238633132216284847_set_a @ M ) @ ( append6159960292548482806_set_a @ P12 @ P22 ) )
           => ( ( ( map_Pr3507802118303906290_set_a
                  @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                  @ P12 )
                = Io1 )
             => ( ( map_Pr3507802118303906290_set_a
                  @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                  @ P22 )
               != Io2 ) ) ) ) ).

% language_split
thf(fact_816_language__split,axiom,
    ! [Io1: list_P1467974523212632366od_a_b,Io2: list_P1467974523212632366od_a_b,M: fsm_Pr1886043760914421514_a_a_b] :
      ( ( member6895035109809442519od_a_b @ ( append5335208823350062147od_a_b @ Io1 @ Io2 ) @ ( lS_Pro5114676609533226954_a_a_b @ M @ ( initia681974817753967189_a_a_b @ M ) ) )
     => ~ ! [P12: list_P1562850763667305982od_c_a,P22: list_P1562850763667305982od_c_a] :
            ( ( path_P1107610407223809260_a_a_b @ M @ ( initia681974817753967189_a_a_b @ M ) @ ( append311360319448987923od_c_a @ P12 @ P22 ) )
           => ( ( ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P12 )
                = Io1 )
             => ( ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P22 )
               != Io2 ) ) ) ) ).

% language_split
thf(fact_817_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
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P12 )
                = Io1 )
             => ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P22 )
               != Io2 ) ) ) ) ).

% language_split
thf(fact_818_atc__to__io__set_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: fsm_a_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ( adapti7464086540814349460_b_c_a @ X2 @ Xa )
        = Y )
     => ( Y
        = ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X2 @ ( initial_a_b_c @ X2 ) ) @ ( lS_a_b_c @ Xa @ ( initial_a_b_c @ Xa ) ) ) ) ) ).

% atc_to_io_set.elims
thf(fact_819_atc__to__io__set_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: fsm_d_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ( adapti7464086540814349463_b_c_d @ X2 @ Xa )
        = Y )
     => ( Y
        = ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X2 @ ( initial_a_b_c @ X2 ) ) @ ( lS_d_b_c @ Xa @ ( initial_d_b_c @ Xa ) ) ) ) ) ).

% atc_to_io_set.elims
thf(fact_820_atc__to__io__set_Osimps,axiom,
    ( adapti7464086540814349460_b_c_a
    = ( ^ [M2: fsm_a_b_c,A6: fsm_a_b_c] : ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) @ ( lS_a_b_c @ A6 @ ( initial_a_b_c @ A6 ) ) ) ) ) ).

% atc_to_io_set.simps
thf(fact_821_atc__to__io__set_Osimps,axiom,
    ( adapti7464086540814349463_b_c_d
    = ( ^ [M2: fsm_a_b_c,A6: fsm_d_b_c] : ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) @ ( lS_d_b_c @ A6 @ ( initial_d_b_c @ A6 ) ) ) ) ) ).

% atc_to_io_set.simps
thf(fact_822_from__product__initial__paths__ex,axiom,
    ! [Q1: a,M: fsm_a_d_d,Q22: a] :
      ( ( member_a @ Q1 @ ( states_a_d_d @ M ) )
     => ( ( member_a @ Q22 @ ( states_a_d_d @ M ) )
       => ? [P12: list_P4703773164859162617od_d_a,P22: list_P4703773164859162617od_d_a] :
            ( ( path_a_d_d @ ( from_FSM_a_d_d @ M @ Q1 ) @ ( initial_a_d_d @ ( from_FSM_a_d_d @ M @ Q1 ) ) @ P12 )
            & ( path_a_d_d @ ( from_FSM_a_d_d @ M @ Q22 ) @ ( initial_a_d_d @ ( from_FSM_a_d_d @ M @ Q22 ) ) @ P22 )
            & ( ( target_a_d_d @ ( initial_a_d_d @ ( from_FSM_a_d_d @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target_a_d_d @ ( initial_a_d_d @ ( from_FSM_a_d_d @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr6689519438302151288od_d_d
                @ ^ [T: produc2648501391905408755od_d_a] : ( product_Pair_d_d @ ( produc5458984928965053094od_d_a @ ( produc6173600788113908241od_d_a @ T ) ) @ ( product_fst_d_a @ ( produc1155110141524277224od_d_a @ ( produc6173600788113908241od_d_a @ T ) ) ) )
                @ P12 )
              = ( map_Pr6689519438302151288od_d_d
                @ ^ [T: produc2648501391905408755od_d_a] : ( product_Pair_d_d @ ( produc5458984928965053094od_d_a @ ( produc6173600788113908241od_d_a @ T ) ) @ ( product_fst_d_a @ ( produc1155110141524277224od_d_a @ ( produc6173600788113908241od_d_a @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_823_from__product__initial__paths__ex,axiom,
    ! [Q1: a,M: fsm_a_a_a,Q22: a] :
      ( ( member_a @ Q1 @ ( states_a_a_a @ M ) )
     => ( ( member_a @ Q22 @ ( states_a_a_a @ M ) )
       => ? [P12: list_P7817932933098766335od_a_a,P22: list_P7817932933098766335od_a_a] :
            ( ( path_a_a_a @ ( from_FSM_a_a_a @ M @ Q1 ) @ ( initial_a_a_a @ ( from_FSM_a_a_a @ M @ Q1 ) ) @ P12 )
            & ( path_a_a_a @ ( from_FSM_a_a_a @ M @ Q22 ) @ ( initial_a_a_a @ ( from_FSM_a_a_a @ M @ Q22 ) ) @ P22 )
            & ( ( target_a_a_a @ ( initial_a_a_a @ ( from_FSM_a_a_a @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target_a_a_a @ ( initial_a_a_a @ ( from_FSM_a_a_a @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr6384380680725278130od_a_a
                @ ^ [T: produc7928013997122480633od_a_a] : ( product_Pair_a_a @ ( produc6516300539023753772od_a_a @ ( produc775496654069540887od_a_a @ T ) ) @ ( product_fst_a_a @ ( produc2212425751582977902od_a_a @ ( produc775496654069540887od_a_a @ T ) ) ) )
                @ P12 )
              = ( map_Pr6384380680725278130od_a_a
                @ ^ [T: produc7928013997122480633od_a_a] : ( product_Pair_a_a @ ( produc6516300539023753772od_a_a @ ( produc775496654069540887od_a_a @ T ) ) @ ( product_fst_a_a @ ( produc2212425751582977902od_a_a @ ( produc775496654069540887od_a_a @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_824_from__product__initial__paths__ex,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q22: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
       => ? [P12: list_P6327159017948738492od_c_a,P22: list_P6327159017948738492od_c_a] :
            ( ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q1 ) ) @ P12 )
            & ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q22 ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q22 ) ) @ P22 )
            & ( ( target_a_b_c @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target_a_b_c @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr776894537227705078od_b_c
                @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                @ P12 )
              = ( map_Pr776894537227705078od_b_c
                @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_825_from__product__initial__paths__ex,axiom,
    ! [Q1: product_prod_c_a,M: fsm_Pr1886043760914421514_a_a_b,Q22: product_prod_c_a] :
      ( ( member5074992350434858958od_c_a @ Q1 @ ( states367398909958982703_a_a_b @ M ) )
     => ( ( member5074992350434858958od_c_a @ Q22 @ ( states367398909958982703_a_a_b @ M ) )
       => ? [P12: list_P1562850763667305982od_c_a,P22: list_P1562850763667305982od_c_a] :
            ( ( path_P1107610407223809260_a_a_b @ ( from_F6727869081933152899_a_a_b @ M @ Q1 ) @ ( initia681974817753967189_a_a_b @ ( from_F6727869081933152899_a_a_b @ M @ Q1 ) ) @ P12 )
            & ( path_P1107610407223809260_a_a_b @ ( from_F6727869081933152899_a_a_b @ M @ Q22 ) @ ( initia681974817753967189_a_a_b @ ( from_F6727869081933152899_a_a_b @ M @ Q22 ) ) @ P22 )
            & ( ( target6303326113586172192_a_a_b @ ( initia681974817753967189_a_a_b @ ( from_F6727869081933152899_a_a_b @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target6303326113586172192_a_a_b @ ( initia681974817753967189_a_a_b @ ( from_F6727869081933152899_a_a_b @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr3411285261028437108od_a_b
                @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                @ P12 )
              = ( map_Pr3411285261028437108od_a_b
                @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_826_from__product__initial__paths__ex,axiom,
    ! [Q1: list_P903359562653991662od_b_c,M: fsm_li1226222605880383890_c_d_d,Q22: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Q1 @ ( states2106791756085916471_c_d_d @ M ) )
     => ( ( member6330420149250801815od_b_c @ Q22 @ ( states2106791756085916471_c_d_d @ M ) )
       => ? [P12: list_P4918516663698092921od_b_c,P22: list_P4918516663698092921od_b_c] :
            ( ( path_l1403052360814049652_c_d_d @ ( from_F8620888367019372427_c_d_d @ M @ Q1 ) @ ( initia7216034209257014877_c_d_d @ ( from_F8620888367019372427_c_d_d @ M @ Q1 ) ) @ P12 )
            & ( path_l1403052360814049652_c_d_d @ ( from_F8620888367019372427_c_d_d @ M @ Q22 ) @ ( initia7216034209257014877_c_d_d @ ( from_F8620888367019372427_c_d_d @ M @ Q22 ) ) @ P22 )
            & ( ( target9028246747236273064_c_d_d @ ( initia7216034209257014877_c_d_d @ ( from_F8620888367019372427_c_d_d @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target9028246747236273064_c_d_d @ ( initia7216034209257014877_c_d_d @ ( from_F8620888367019372427_c_d_d @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr7351454074379792120od_d_d
                @ ^ [T: produc7248869284116280179od_b_c] : ( product_Pair_d_d @ ( produc2098964072593918742od_b_c @ ( produc4952405269795728785od_b_c @ T ) ) @ ( produc9094947606930591920od_b_c @ ( produc792910155945188696od_b_c @ ( produc4952405269795728785od_b_c @ T ) ) ) )
                @ P12 )
              = ( map_Pr7351454074379792120od_d_d
                @ ^ [T: produc7248869284116280179od_b_c] : ( product_Pair_d_d @ ( produc2098964072593918742od_b_c @ ( produc4952405269795728785od_b_c @ T ) ) @ ( produc9094947606930591920od_b_c @ ( produc792910155945188696od_b_c @ ( produc4952405269795728785od_b_c @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_827_from__product__initial__paths__ex,axiom,
    ! [Q1: list_P903359562653991662od_b_c,M: fsm_li6801133765522507155_c_b_c,Q22: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Q1 @ ( states7681702920031268536_c_b_c @ M ) )
     => ( ( member6330420149250801815od_b_c @ Q22 @ ( states7681702920031268536_c_b_c @ M ) )
       => ? [P12: list_P8364599791545511612od_b_c,P22: list_P8364599791545511612od_b_c] :
            ( ( path_l6977963524759401717_c_b_c @ ( from_F4972427494109948684_c_b_c @ M @ Q1 ) @ ( initia3567573336347591134_c_b_c @ ( from_F4972427494109948684_c_b_c @ M @ Q1 ) ) @ P12 )
            & ( path_l6977963524759401717_c_b_c @ ( from_F4972427494109948684_c_b_c @ M @ Q22 ) @ ( initia3567573336347591134_c_b_c @ ( from_F4972427494109948684_c_b_c @ M @ Q22 ) ) @ P22 )
            & ( ( target5379785874326849321_c_b_c @ ( initia3567573336347591134_c_b_c @ ( from_F4972427494109948684_c_b_c @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target5379785874326849321_c_b_c @ ( initia3567573336347591134_c_b_c @ ( from_F4972427494109948684_c_b_c @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr4814182253694984182od_b_c
                @ ^ [T: produc596031391177498678od_b_c] : ( product_Pair_b_c @ ( produc4488652850390612441od_b_c @ ( produc829356566976367572od_b_c @ T ) ) @ ( produc5388473464130769137od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T ) ) ) )
                @ P12 )
              = ( map_Pr4814182253694984182od_b_c
                @ ^ [T: produc596031391177498678od_b_c] : ( product_Pair_b_c @ ( produc4488652850390612441od_b_c @ ( produc829356566976367572od_b_c @ T ) ) @ ( produc5388473464130769137od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_828_from__product__initial__paths__ex,axiom,
    ! [Q1: a,M: fsm_a_a_fsm_a_b_c,Q22: a] :
      ( ( member_a @ Q1 @ ( states_a_a_fsm_a_b_c @ M ) )
     => ( ( member_a @ Q22 @ ( states_a_a_fsm_a_b_c @ M ) )
       => ? [P12: list_P6783732442441166490_b_c_a,P22: list_P6783732442441166490_b_c_a] :
            ( ( path_a_a_fsm_a_b_c @ ( from_F740856572626617088_a_b_c @ M @ Q1 ) @ ( initia4382001536991938002_a_b_c @ ( from_F740856572626617088_a_b_c @ M @ Q1 ) ) @ P12 )
            & ( path_a_a_fsm_a_b_c @ ( from_F740856572626617088_a_b_c @ M @ Q22 ) @ ( initia4382001536991938002_a_b_c @ ( from_F740856572626617088_a_b_c @ M @ Q22 ) ) @ P22 )
            & ( ( target_a_a_fsm_a_b_c @ ( initia4382001536991938002_a_b_c @ ( from_F740856572626617088_a_b_c @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target_a_a_fsm_a_b_c @ ( initia4382001536991938002_a_b_c @ ( from_F740856572626617088_a_b_c @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr7502839659173251580_a_b_c
                @ ^ [T: produc2364950978236125332_b_c_a] : ( produc2795441082327197628_a_b_c @ ( produc5208970367483154503_b_c_a @ ( produc2604435702061889714_b_c_a @ T ) ) @ ( produc1155351766825858590_b_c_a @ ( produc75634468776750729_b_c_a @ ( produc2604435702061889714_b_c_a @ T ) ) ) )
                @ P12 )
              = ( map_Pr7502839659173251580_a_b_c
                @ ^ [T: produc2364950978236125332_b_c_a] : ( produc2795441082327197628_a_b_c @ ( produc5208970367483154503_b_c_a @ ( produc2604435702061889714_b_c_a @ T ) ) @ ( produc1155351766825858590_b_c_a @ ( produc75634468776750729_b_c_a @ ( produc2604435702061889714_b_c_a @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_829_from__product__initial__paths__ex,axiom,
    ! [Q1: list_P903359562653991662od_b_c,M: fsm_li365217302033949778_c_a_a,Q22: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Q1 @ ( states1245786465149168759_c_a_a @ M ) )
     => ( ( member6330420149250801815od_b_c @ Q22 @ ( states1245786465149168759_c_a_a @ M ) )
       => ? [P12: list_P8948690831120843391od_b_c,P22: list_P8948690831120843391od_b_c] :
            ( ( path_l542047069877301940_c_a_a @ ( from_F7759883076082624715_c_a_a @ M @ Q1 ) @ ( initia6355028918320267165_c_a_a @ ( from_F7759883076082624715_c_a_a @ M @ Q1 ) ) @ P12 )
            & ( path_l542047069877301940_c_a_a @ ( from_F7759883076082624715_c_a_a @ M @ Q22 ) @ ( initia6355028918320267165_c_a_a @ ( from_F7759883076082624715_c_a_a @ M @ Q22 ) ) @ P22 )
            & ( ( target8167241456299525352_c_a_a @ ( initia6355028918320267165_c_a_a @ ( from_F7759883076082624715_c_a_a @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target8167241456299525352_c_a_a @ ( initia6355028918320267165_c_a_a @ ( from_F7759883076082624715_c_a_a @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr3749974615233628978od_a_a
                @ ^ [T: produc188457162657408377od_b_c] : ( product_Pair_a_a @ ( produc4735472043516613020od_b_c @ ( produc6497444872579809943od_b_c @ T ) ) @ ( produc7198897215385899379od_b_c @ ( produc3429418126867882974od_b_c @ ( produc6497444872579809943od_b_c @ T ) ) ) )
                @ P12 )
              = ( map_Pr3749974615233628978od_a_a
                @ ^ [T: produc188457162657408377od_b_c] : ( product_Pair_a_a @ ( produc4735472043516613020od_b_c @ ( produc6497444872579809943od_b_c @ T ) ) @ ( produc7198897215385899379od_b_c @ ( produc3429418126867882974od_b_c @ ( produc6497444872579809943od_b_c @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_830_from__product__initial__paths__ex,axiom,
    ! [Q1: fsm_a_b_c,M: fsm_fsm_a_b_c_a_a,Q22: fsm_a_b_c] :
      ( ( member_fsm_a_b_c @ Q1 @ ( states_fsm_a_b_c_a_a @ M ) )
     => ( ( member_fsm_a_b_c @ Q22 @ ( states_fsm_a_b_c_a_a @ M ) )
       => ? [P12: list_P4545384517222041535_a_b_c,P22: list_P4545384517222041535_a_b_c] :
            ( ( path_fsm_a_b_c_a_a @ ( from_F3858323121354984576_c_a_a @ M @ Q1 ) @ ( initia7499468085720305490_c_a_a @ ( from_F3858323121354984576_c_a_a @ M @ Q1 ) ) @ P12 )
            & ( path_fsm_a_b_c_a_a @ ( from_F3858323121354984576_c_a_a @ M @ Q22 ) @ ( initia7499468085720305490_c_a_a @ ( from_F3858323121354984576_c_a_a @ M @ Q22 ) ) @ P22 )
            & ( ( target_fsm_a_b_c_a_a @ ( initia7499468085720305490_c_a_a @ ( from_F3858323121354984576_c_a_a @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target_fsm_a_b_c_a_a @ ( initia7499468085720305490_c_a_a @ ( from_F3858323121354984576_c_a_a @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr7384117626843903474od_a_a
                @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                @ P12 )
              = ( map_Pr7384117626843903474od_a_a
                @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_831_from__product__initial__paths__ex,axiom,
    ! [Q1: produc8838969699183119820_a_b_c,M: fsm_Pr833494485807227248_c_d_d,Q22: produc8838969699183119820_a_b_c] :
      ( ( member997585926662434933_a_b_c @ Q1 @ ( states3936357408656391189_c_d_d @ M ) )
     => ( ( member997585926662434933_a_b_c @ Q22 @ ( states3936357408656391189_c_d_d @ M ) )
       => ? [P12: list_P8957986552689298425_a_b_c,P22: list_P8957986552689298425_a_b_c] :
            ( ( path_P784432154605857618_c_d_d @ ( from_F5520467122992069737_c_d_d @ M @ Q1 ) @ ( initia7371379382183009595_c_d_d @ ( from_F5520467122992069737_c_d_d @ M @ Q1 ) ) @ P12 )
            & ( path_P784432154605857618_c_d_d @ ( from_F5520467122992069737_c_d_d @ M @ Q22 ) @ ( initia7371379382183009595_c_d_d @ ( from_F5520467122992069737_c_d_d @ M @ Q22 ) ) @ P22 )
            & ( ( target9056421450513965446_c_d_d @ ( initia7371379382183009595_c_d_d @ ( from_F5520467122992069737_c_d_d @ M @ Q1 ) ) @ P12 )
              = Q1 )
            & ( ( target9056421450513965446_c_d_d @ ( initia7371379382183009595_c_d_d @ ( from_F5520467122992069737_c_d_d @ M @ Q22 ) ) @ P22 )
              = Q22 )
            & ( ( map_Pr4438953839670867576od_d_d
                @ ^ [T: produc6922481318474903027_a_b_c] : ( product_Pair_d_d @ ( produc8499317692747018228_a_b_c @ ( produc4677130503389975825_a_b_c @ T ) ) @ ( produc5603417301460243854_a_b_c @ ( produc3252888468923579446_a_b_c @ ( produc4677130503389975825_a_b_c @ T ) ) ) )
                @ P12 )
              = ( map_Pr4438953839670867576od_d_d
                @ ^ [T: produc6922481318474903027_a_b_c] : ( product_Pair_d_d @ ( produc8499317692747018228_a_b_c @ ( produc4677130503389975825_a_b_c @ T ) ) @ ( produc5603417301460243854_a_b_c @ ( produc3252888468923579446_a_b_c @ ( produc4677130503389975825_a_b_c @ T ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_832_paths__for__input__def,axiom,
    ( paths_6861451799561089831_c_a_a
    = ( ^ [M2: fsm_fsm_a_b_c_a_a,Q4: fsm_a_b_c,Xs4: list_a] :
          ( collec3209657193620141994_a_b_c
          @ ^ [P4: list_P4545384517222041535_a_b_c] :
              ( ( path_fsm_a_b_c_a_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a
                  @ ( map_Pr7384117626843903474od_a_a
                    @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_833_paths__for__input__def,axiom,
    ( paths_824755907305698409od_c_a
    = ( ^ [M2: fsm_Pr306566304496394697od_c_a,Q4: produc1703568184450464039_set_a,Xs4: list_a] :
          ( collec5374521429813522028_set_a
          @ ^ [P4: list_P5750286787369789953_set_a] :
              ( ( path_P8254327248518380459od_c_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr7928508354609250845_c_a_a @ produc4637894536740104257od_c_a
                  @ ( map_Pr2884798845580615278od_c_a
                    @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_834_paths__for__input__def,axiom,
    ( paths_3189189365015657092_set_a
    = ( ^ [M2: fsm_se8204406047609318756_set_a,Q4: set_a,Xs4: list_l2826717958560394946od_c_a] :
          ( collec7846127375383488716_set_a
          @ ^ [P4: list_P3716011504951620193_set_a] :
              ( ( path_s1167107641669997510_set_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr1708894191565315447od_c_a @ produc2106691229161676581_set_a
                  @ ( map_Pr3507802118303906290_set_a
                    @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_835_paths__for__input__def,axiom,
    ( paths_4833692061487544546_set_a
    = ( ^ [M2: fsm_se2423825140862529346_set_a,Q4: set_a,Xs4: list_set_a] :
          ( collec5416562221586181674_set_a
          @ ^ [P4: list_P5960435600574695743_set_a] :
              ( ( path_s526755528433797412_set_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr8866779738092938811_set_a @ produc9088895665703139587_set_a
                  @ ( map_Pr7110208979441748082_set_a
                    @ ^ [T: produc9078272350731273017_set_a] : ( produc9088192753505129239_set_a @ ( produc38069527703528652_set_a @ ( produc7471892532163375447_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2783630374533347854_set_a @ ( produc7471892532163375447_set_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_836_paths__for__input__def,axiom,
    ( paths_1286540635615737765_c_c_a
    = ( ^ [M2: fsm_fsm_a_b_c_c_a,Q4: fsm_a_b_c,Xs4: list_c] :
          ( collec2119896848643030824_a_b_c
          @ ^ [P4: list_P3455624172244930365_a_b_c] :
              ( ( path_fsm_a_b_c_c_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a
                  @ ( map_Pr3451339525954936178od_c_a
                    @ ^ [T: produc2463281488305106231_a_b_c] : ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_837_paths__for__input__def,axiom,
    ( paths_607830118076874410_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b,Q4: product_prod_c_a,Xs4: list_a] :
          ( collec453890259647268073od_c_a
          @ ^ [P4: list_P1562850763667305982od_c_a] :
              ( ( path_P1107610407223809260_a_a_b @ M2 @ Q4 @ P4 )
              & ( ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b
                  @ ( map_Pr3411285261028437108od_a_b
                    @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_838_paths__for__input__def,axiom,
    ( paths_6942934977415288259_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q4: a,Xs4: list_b] :
          ( collec6273869032445462695od_c_a
          @ ^ [P4: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M2 @ Q4 @ P4 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_839_paths__for__io__def,axiom,
    ( paths_9186897071720686893_c_a_a
    = ( ^ [M2: fsm_fsm_a_b_c_a_a,Q4: fsm_a_b_c,Io4: list_P1396940483166286381od_a_a] :
          ( collec3209657193620141994_a_b_c
          @ ^ [P4: list_P4545384517222041535_a_b_c] :
              ( ( path_fsm_a_b_c_a_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr7384117626843903474od_a_a
                  @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                  @ P4 )
                = Io4 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_840_paths__for__io__def,axiom,
    ( paths_6507691185994526831od_c_a
    = ( ^ [M2: fsm_Pr306566304496394697od_c_a,Q4: produc1703568184450464039_set_a,Io4: list_P1175628092075131243od_c_a] :
          ( collec5374521429813522028_set_a
          @ ^ [P4: list_P5750286787369789953_set_a] :
              ( ( path_P8254327248518380459od_c_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr2884798845580615278od_c_a
                  @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                  @ P4 )
                = Io4 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_841_paths__for__io__def,axiom,
    ( paths_5979839028395814026_set_a
    = ( ^ [M2: fsm_se8204406047609318756_set_a,Q4: set_a,Io4: list_P5262903567315784527_set_a] :
          ( collec7846127375383488716_set_a
          @ ^ [P4: list_P3716011504951620193_set_a] :
              ( ( path_s1167107641669997510_set_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr3507802118303906290_set_a
                  @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                  @ P4 )
                = Io4 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_842_paths__for__io__def,axiom,
    ( paths_593845848187029424_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b,Q4: product_prod_c_a,Io4: list_P1467974523212632366od_a_b] :
          ( collec453890259647268073od_c_a
          @ ^ [P4: list_P1562850763667305982od_c_a] :
              ( ( path_P1107610407223809260_a_a_b @ M2 @ Q4 @ P4 )
              & ( ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P4 )
                = Io4 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_843_paths__for__io__def,axiom,
    ( paths_for_io_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q4: a,Io4: list_P903359562653991662od_b_c] :
          ( collec6273869032445462695od_c_a
          @ ^ [P4: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M2 @ Q4 @ P4 )
              & ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P4 )
                = Io4 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_844_le__inf__iff,axiom,
    ! [X2: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
      = ( ( ord_less_eq_set_a @ X2 @ Y )
        & ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_845_le__inf__iff,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a,Z: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ X2 @ ( inf_in980870363377917322od_c_a @ Y @ Z ) )
      = ( ( ord_le2998388488506175548od_c_a @ X2 @ Y )
        & ( ord_le2998388488506175548od_c_a @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_846_inf_Obounded__iff,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
      = ( ( ord_less_eq_set_a @ A2 @ B2 )
        & ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_847_inf_Obounded__iff,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ ( inf_in980870363377917322od_c_a @ B2 @ C ) )
      = ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
        & ( ord_le2998388488506175548od_c_a @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_848_t6,axiom,
    ! [Q: a] :
      ( ( member_a @ Q @ ( image_3273489647386116407_b_c_a @ produc2620512763296580520_a_b_c @ prs ) )
     => ( ( ord_le2998388488506175548od_c_a @ ( tps @ Q )
          @ ( collec6273869032445462695od_c_a
            @ ^ [P13: list_P6327159017948738492od_c_a] :
              ? [P23: list_P6327159017948738492od_c_a,D4: produc1703568184450464039_set_a] : ( member6493638300815972283_set_a @ ( produc6508649944902238466_set_a @ ( append3884164603879732689od_c_a @ P13 @ P23 ) @ D4 ) @ ( traver6421790936065540696_a_b_c @ m @ Q @ repetition_sets @ m2 ) ) ) )
        & ( ord_le2998388488506175548od_c_a @ ( image_865596521259543215od_c_a @ produc3242630202250842350_set_a @ ( traver6421790936065540696_a_b_c @ m @ Q @ repetition_sets @ m2 ) ) @ ( tps @ Q ) ) ) ) ).

% t6
thf(fact_849_preamble__maximal__io__paths,axiom,
    ! [P5: fsm_fsm_a_b_c_a_a,M: fsm_fsm_a_b_c_a_a,Q: fsm_a_b_c,P: list_P4545384517222041535_a_b_c] :
      ( ( state_1375479521797176539_c_a_a @ P5 @ M @ Q )
     => ( ( observ552407795947821995_c_a_a @ M )
       => ( ( path_fsm_a_b_c_a_a @ P5 @ ( initia7499468085720305490_c_a_a @ P5 ) @ P )
         => ( ( ( target_fsm_a_b_c_a_a @ ( initia7499468085720305490_c_a_a @ P5 ) @ P )
              = Q )
           => ~ ? [Io5: list_P1396940483166286381od_a_a] :
                  ( ( Io5 != nil_Product_prod_a_a )
                  & ( member6824001069763096534od_a_a
                    @ ( append5335208819046833346od_a_a
                      @ ( map_Pr7384117626843903474od_a_a
                        @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                        @ P )
                      @ Io5 )
                    @ ( lS_fsm_a_b_c_a_a @ P5 @ ( initia7499468085720305490_c_a_a @ P5 ) ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths
thf(fact_850_preamble__maximal__io__paths,axiom,
    ! [P5: fsm_Pr306566304496394697od_c_a,M: fsm_Pr306566304496394697od_c_a,Q: produc1703568184450464039_set_a,P: list_P5750286787369789953_set_a] :
      ( ( state_17727317216061981od_c_a @ P5 @ M @ Q )
     => ( ( observ7047451555672324333od_c_a @ M )
       => ( ( path_P8254327248518380459od_c_a @ P5 @ ( initia7978570708967869844od_c_a @ P5 ) @ P )
         => ( ( ( target3445005149252956639od_c_a @ ( initia7978570708967869844od_c_a @ P5 ) @ P )
              = Q )
           => ~ ? [Io5: list_P1175628092075131243od_c_a] :
                  ( ( Io5 != nil_Pr184782776552258283od_c_a )
                  & ( member4479000366354228116od_c_a
                    @ ( append2230762392551410432od_c_a
                      @ ( map_Pr2884798845580615278od_c_a
                        @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                        @ P )
                      @ Io5 )
                    @ ( lS_Pro8045283330902898057od_c_a @ P5 @ ( initia7978570708967869844od_c_a @ P5 ) ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths
thf(fact_851_preamble__maximal__io__paths,axiom,
    ! [P5: fsm_se8204406047609318756_set_a,M: fsm_se8204406047609318756_set_a,Q: set_a,P: list_P3716011504951620193_set_a] :
      ( ( state_6468868301294240824_set_a @ P5 @ M @ Q )
     => ( ( observ8827156789157454600_set_a @ M )
       => ( ( path_s1167107641669997510_set_a @ P5 @ ( initia4238633132216284847_set_a @ P5 ) @ P )
         => ( ( ( target6955975396184022522_set_a @ ( initia4238633132216284847_set_a @ P5 ) @ P )
              = Q )
           => ~ ? [Io5: list_P5262903567315784527_set_a] :
                  ( ( Io5 != nil_Pr7623349287138497231_set_a )
                  & ( member5048304341121814904_set_a
                    @ ( append1006059461745118948_set_a
                      @ ( map_Pr3507802118303906290_set_a
                        @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                        @ P )
                      @ Io5 )
                    @ ( lS_set5116714952275434404_set_a @ P5 @ ( initia4238633132216284847_set_a @ P5 ) ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths
thf(fact_852_preamble__maximal__io__paths,axiom,
    ! [P5: fsm_Pr1886043760914421514_a_a_b,M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P: list_P1562850763667305982od_c_a] :
      ( ( state_7126049871323298398_a_a_b @ P5 @ M @ Q )
     => ( ( observ2964502264774598958_a_a_b @ M )
       => ( ( path_P1107610407223809260_a_a_b @ P5 @ ( initia681974817753967189_a_a_b @ P5 ) @ P )
         => ( ( ( target6303326113586172192_a_a_b @ ( initia681974817753967189_a_a_b @ P5 ) @ P )
              = Q )
           => ~ ? [Io5: list_P1467974523212632366od_a_b] :
                  ( ( Io5 != nil_Product_prod_a_b )
                  & ( member6895035109809442519od_a_b
                    @ ( append5335208823350062147od_a_b
                      @ ( map_Pr3411285261028437108od_a_b
                        @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                        @ P )
                      @ Io5 )
                    @ ( lS_Pro5114676609533226954_a_a_b @ P5 @ ( initia681974817753967189_a_a_b @ P5 ) ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths
thf(fact_853_preamble__maximal__io__paths,axiom,
    ! [P5: fsm_a_b_c,M: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a] :
      ( ( state_3011419026027373943_a_b_c @ P5 @ M @ Q )
     => ( ( observable_a_b_c @ M )
       => ( ( path_a_b_c @ P5 @ ( initial_a_b_c @ P5 ) @ P )
         => ( ( ( target_a_b_c @ ( initial_a_b_c @ P5 ) @ P )
              = Q )
           => ~ ? [Io5: list_P903359562653991662od_b_c] :
                  ( ( Io5 != nil_Product_prod_b_c )
                  & ( member6330420149250801815od_b_c
                    @ ( append2547753245680614915od_b_c
                      @ ( map_Pr776894537227705078od_b_c
                        @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                        @ P )
                      @ Io5 )
                    @ ( lS_a_b_c @ P5 @ ( initial_a_b_c @ P5 ) ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths
thf(fact_854_preamble__maximal__io__paths__rev,axiom,
    ! [P5: fsm_fsm_a_b_c_a_a,M: fsm_fsm_a_b_c_a_a,Q: fsm_a_b_c,Io3: list_P1396940483166286381od_a_a] :
      ( ( state_1375479521797176539_c_a_a @ P5 @ M @ Q )
     => ( ( observ552407795947821995_c_a_a @ M )
       => ( ( member6824001069763096534od_a_a @ Io3 @ ( lS_fsm_a_b_c_a_a @ P5 @ ( initia7499468085720305490_c_a_a @ P5 ) ) )
         => ( ~ ? [Io6: list_P1396940483166286381od_a_a] :
                  ( ( Io6 != nil_Product_prod_a_a )
                  & ( member6824001069763096534od_a_a @ ( append5335208819046833346od_a_a @ Io3 @ Io6 ) @ ( lS_fsm_a_b_c_a_a @ P5 @ ( initia7499468085720305490_c_a_a @ P5 ) ) ) )
           => ~ ! [P3: list_P4545384517222041535_a_b_c] :
                  ( ( path_fsm_a_b_c_a_a @ P5 @ ( initia7499468085720305490_c_a_a @ P5 ) @ P3 )
                 => ( ( ( map_Pr7384117626843903474od_a_a
                        @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                        @ P3 )
                      = Io3 )
                   => ( ( target_fsm_a_b_c_a_a @ ( initia7499468085720305490_c_a_a @ P5 ) @ P3 )
                     != Q ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths_rev
thf(fact_855_preamble__maximal__io__paths__rev,axiom,
    ! [P5: fsm_Pr306566304496394697od_c_a,M: fsm_Pr306566304496394697od_c_a,Q: produc1703568184450464039_set_a,Io3: list_P1175628092075131243od_c_a] :
      ( ( state_17727317216061981od_c_a @ P5 @ M @ Q )
     => ( ( observ7047451555672324333od_c_a @ M )
       => ( ( member4479000366354228116od_c_a @ Io3 @ ( lS_Pro8045283330902898057od_c_a @ P5 @ ( initia7978570708967869844od_c_a @ P5 ) ) )
         => ( ~ ? [Io6: list_P1175628092075131243od_c_a] :
                  ( ( Io6 != nil_Pr184782776552258283od_c_a )
                  & ( member4479000366354228116od_c_a @ ( append2230762392551410432od_c_a @ Io3 @ Io6 ) @ ( lS_Pro8045283330902898057od_c_a @ P5 @ ( initia7978570708967869844od_c_a @ P5 ) ) ) )
           => ~ ! [P3: list_P5750286787369789953_set_a] :
                  ( ( path_P8254327248518380459od_c_a @ P5 @ ( initia7978570708967869844od_c_a @ P5 ) @ P3 )
                 => ( ( ( map_Pr2884798845580615278od_c_a
                        @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                        @ P3 )
                      = Io3 )
                   => ( ( target3445005149252956639od_c_a @ ( initia7978570708967869844od_c_a @ P5 ) @ P3 )
                     != Q ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths_rev
thf(fact_856_preamble__maximal__io__paths__rev,axiom,
    ! [P5: fsm_se8204406047609318756_set_a,M: fsm_se8204406047609318756_set_a,Q: set_a,Io3: list_P5262903567315784527_set_a] :
      ( ( state_6468868301294240824_set_a @ P5 @ M @ Q )
     => ( ( observ8827156789157454600_set_a @ M )
       => ( ( member5048304341121814904_set_a @ Io3 @ ( lS_set5116714952275434404_set_a @ P5 @ ( initia4238633132216284847_set_a @ P5 ) ) )
         => ( ~ ? [Io6: list_P5262903567315784527_set_a] :
                  ( ( Io6 != nil_Pr7623349287138497231_set_a )
                  & ( member5048304341121814904_set_a @ ( append1006059461745118948_set_a @ Io3 @ Io6 ) @ ( lS_set5116714952275434404_set_a @ P5 @ ( initia4238633132216284847_set_a @ P5 ) ) ) )
           => ~ ! [P3: list_P3716011504951620193_set_a] :
                  ( ( path_s1167107641669997510_set_a @ P5 @ ( initia4238633132216284847_set_a @ P5 ) @ P3 )
                 => ( ( ( map_Pr3507802118303906290_set_a
                        @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                        @ P3 )
                      = Io3 )
                   => ( ( target6955975396184022522_set_a @ ( initia4238633132216284847_set_a @ P5 ) @ P3 )
                     != Q ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths_rev
thf(fact_857_preamble__maximal__io__paths__rev,axiom,
    ! [P5: fsm_Pr1886043760914421514_a_a_b,M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,Io3: list_P1467974523212632366od_a_b] :
      ( ( state_7126049871323298398_a_a_b @ P5 @ M @ Q )
     => ( ( observ2964502264774598958_a_a_b @ M )
       => ( ( member6895035109809442519od_a_b @ Io3 @ ( lS_Pro5114676609533226954_a_a_b @ P5 @ ( initia681974817753967189_a_a_b @ P5 ) ) )
         => ( ~ ? [Io6: list_P1467974523212632366od_a_b] :
                  ( ( Io6 != nil_Product_prod_a_b )
                  & ( member6895035109809442519od_a_b @ ( append5335208823350062147od_a_b @ Io3 @ Io6 ) @ ( lS_Pro5114676609533226954_a_a_b @ P5 @ ( initia681974817753967189_a_a_b @ P5 ) ) ) )
           => ~ ! [P3: list_P1562850763667305982od_c_a] :
                  ( ( path_P1107610407223809260_a_a_b @ P5 @ ( initia681974817753967189_a_a_b @ P5 ) @ P3 )
                 => ( ( ( map_Pr3411285261028437108od_a_b
                        @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                        @ P3 )
                      = Io3 )
                   => ( ( target6303326113586172192_a_a_b @ ( initia681974817753967189_a_a_b @ P5 ) @ P3 )
                     != Q ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths_rev
thf(fact_858_preamble__maximal__io__paths__rev,axiom,
    ! [P5: fsm_a_b_c,M: fsm_a_b_c,Q: a,Io3: list_P903359562653991662od_b_c] :
      ( ( state_3011419026027373943_a_b_c @ P5 @ M @ Q )
     => ( ( observable_a_b_c @ M )
       => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ P5 @ ( initial_a_b_c @ P5 ) ) )
         => ( ~ ? [Io6: list_P903359562653991662od_b_c] :
                  ( ( Io6 != nil_Product_prod_b_c )
                  & ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ Io6 ) @ ( lS_a_b_c @ P5 @ ( initial_a_b_c @ P5 ) ) ) )
           => ~ ! [P3: list_P6327159017948738492od_c_a] :
                  ( ( path_a_b_c @ P5 @ ( initial_a_b_c @ P5 ) @ P3 )
                 => ( ( ( map_Pr776894537227705078od_b_c
                        @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                        @ P3 )
                      = Io3 )
                   => ( ( target_a_b_c @ ( initial_a_b_c @ P5 ) @ P3 )
                     != Q ) ) ) ) ) ) ) ).

% preamble_maximal_io_paths_rev
thf(fact_859_from__FSM__observable,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( observable_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) ) ).

% from_FSM_observable
thf(fact_860_submachine__observable,axiom,
    ! [S: fsm_a_b_c,M: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ S @ M )
     => ( ( observable_a_b_c @ M )
       => ( observable_a_b_c @ S ) ) ) ).

% submachine_observable
thf(fact_861_inf__set__def,axiom,
    ( inf_in7322238654750965302od_d_d
    = ( ^ [A6: set_Pr9130538228769002824od_d_d,B6: set_Pr9130538228769002824od_d_d] :
          ( collec5362705567740981075od_d_d
          @ ( inf_in3715331971855973927_d_d_o
            @ ^ [X3: produc4859908366599477352od_d_d] : ( member73601632213084689od_d_d @ X3 @ A6 )
            @ ^ [X3: produc4859908366599477352od_d_d] : ( member73601632213084689od_d_d @ X3 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_862_inf__set__def,axiom,
    ( inf_in4776242878804941978_a_b_c
    = ( ^ [A6: set_Pr2640454425625433516_a_b_c,B6: set_Pr2640454425625433516_a_b_c] :
          ( collec3867759823522760631_a_b_c
          @ ( inf_in6784492051131590851_b_c_o
            @ ^ [X3: produc8838969699183119820_a_b_c] : ( member997585926662434933_a_b_c @ X3 @ A6 )
            @ ^ [X3: produc8838969699183119820_a_b_c] : ( member997585926662434933_a_b_c @ X3 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_863_inf__set__def,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [A6: set_li6436108459499378894od_b_c,B6: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ( inf_in3324677931821484577_b_c_o
            @ ^ [X3: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X3 @ A6 )
            @ ^ [X3: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X3 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_864_inf__set__def,axiom,
    ( inf_inf_set_a
    = ( ^ [A6: set_a,B6: set_a] :
          ( collect_a
          @ ( inf_inf_a_o
            @ ^ [X3: a] : ( member_a @ X3 @ A6 )
            @ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_865_inf__set__def,axiom,
    ( inf_in980870363377917322od_c_a
    = ( ^ [A6: set_li1159382662694783132od_c_a,B6: set_li1159382662694783132od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ( inf_in4728725048587157459_c_a_o
            @ ^ [X3: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ X3 @ A6 )
            @ ^ [X3: list_P6327159017948738492od_c_a] : ( member7410604586820865893od_c_a @ X3 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_866_inf__set__def,axiom,
    ( inf_in5524221907227607964od_b_c
    = ( ^ [A6: set_se3924713247505902254od_b_c,B6: set_se3924713247505902254od_b_c] :
          ( collec1823247286932916153od_b_c
          @ ( inf_in7122225856740857921_b_c_o
            @ ^ [X3: set_li6436108459499378894od_b_c] : ( member6985331446368301687od_b_c @ X3 @ A6 )
            @ ^ [X3: set_li6436108459499378894od_b_c] : ( member6985331446368301687od_b_c @ X3 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_867_Setcompr__eq__image,axiom,
    ! [F: a > set_li6436108459499378894od_b_c,A: set_a] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [Uu: set_li6436108459499378894od_b_c] :
          ? [X3: a] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member_a @ X3 @ A ) ) )
      = ( image_6261743117854694101od_b_c @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_868_Setcompr__eq__image,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,A: set_Pr2640454425625433516_a_b_c] :
      ( ( collect_a
        @ ^ [Uu: a] :
          ? [X3: produc8838969699183119820_a_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member997585926662434933_a_b_c @ X3 @ A ) ) )
      = ( image_3273489647386116407_b_c_a @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_869_Setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [Uu: list_P903359562653991662od_b_c] :
          ? [X3: list_P903359562653991662od_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member6330420149250801815od_b_c @ X3 @ A ) ) )
      = ( image_2850395784482391045od_b_c @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_870_Setcompr__eq__image,axiom,
    ! [F: a > list_P6327159017948738492od_c_a,A: set_a] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [Uu: list_P6327159017948738492od_c_a] :
          ? [X3: a] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member_a @ X3 @ A ) ) )
      = ( image_3129299617955238851od_c_a @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_871_Setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [Uu: set_li6436108459499378894od_b_c] :
          ? [X3: list_P903359562653991662od_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member6330420149250801815od_b_c @ X3 @ A ) ) )
      = ( image_1046841848883543525od_b_c @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_872_Setcompr__eq__image,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [Uu: set_li6436108459499378894od_b_c] :
          ? [X3: produc8838969699183119820_a_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member997585926662434933_a_b_c @ X3 @ A ) ) )
      = ( image_2344789136157180295od_b_c @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_873_Setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P6327159017948738492od_c_a,A: set_li6436108459499378894od_b_c] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [Uu: list_P6327159017948738492od_c_a] :
          ? [X3: list_P903359562653991662od_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member6330420149250801815od_b_c @ X3 @ A ) ) )
      = ( image_1545733045969196755od_c_a @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_874_Setcompr__eq__image,axiom,
    ! [F: produc4859908366599477352od_d_d > set_li6436108459499378894od_b_c,A: set_Pr9130538228769002824od_d_d] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [Uu: set_li6436108459499378894od_b_c] :
          ? [X3: produc4859908366599477352od_d_d] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member73601632213084689od_d_d @ X3 @ A ) ) )
      = ( image_4140449425415760363od_b_c @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_875_Setcompr__eq__image,axiom,
    ! [F: list_P6327159017948738492od_c_a > set_li6436108459499378894od_b_c,A: set_li1159382662694783132od_c_a] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [Uu: set_li6436108459499378894od_b_c] :
          ? [X3: list_P6327159017948738492od_c_a] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member7410604586820865893od_c_a @ X3 @ A ) ) )
      = ( image_4742196336002327959od_b_c @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_876_Setcompr__eq__image,axiom,
    ! [F: produc8838969699183119820_a_b_c > list_P6327159017948738492od_c_a,A: set_Pr2640454425625433516_a_b_c] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [Uu: list_P6327159017948738492od_c_a] :
          ? [X3: produc8838969699183119820_a_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member997585926662434933_a_b_c @ X3 @ A ) ) )
      = ( image_4583360106782637685od_c_a @ F @ A ) ) ).

% Setcompr_eq_image
thf(fact_877_setcompr__eq__image,axiom,
    ! [F: produc8838969699183119820_a_b_c > a,P5: produc8838969699183119820_a_b_c > $o] :
      ( ( collect_a
        @ ^ [Uu: a] :
          ? [X3: produc8838969699183119820_a_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P5 @ X3 ) ) )
      = ( image_3273489647386116407_b_c_a @ F @ ( collec3867759823522760631_a_b_c @ P5 ) ) ) ).

% setcompr_eq_image
thf(fact_878_setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [Uu: list_P903359562653991662od_b_c] :
          ? [X3: list_P903359562653991662od_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P5 @ X3 ) ) )
      = ( image_2850395784482391045od_b_c @ F @ ( collec2280997390073109977od_b_c @ P5 ) ) ) ).

% setcompr_eq_image
thf(fact_879_setcompr__eq__image,axiom,
    ! [F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,P5: produc7049025859877086738_set_a > $o] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [Uu: list_P6327159017948738492od_c_a] :
          ? [X3: produc7049025859877086738_set_a] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P5 @ X3 ) ) )
      = ( image_865596521259543215od_c_a @ F @ ( collec5778745153847319805_set_a @ P5 ) ) ) ).

% setcompr_eq_image
thf(fact_880_setcompr__eq__image,axiom,
    ! [F: list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a,P5: list_P6327159017948738492od_c_a > $o] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [Uu: list_P6327159017948738492od_c_a] :
          ? [X3: list_P6327159017948738492od_c_a] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P5 @ X3 ) ) )
      = ( image_1061481830026363525od_c_a @ F @ ( collec6273869032445462695od_c_a @ P5 ) ) ) ).

% setcompr_eq_image
thf(fact_881_setcompr__eq__image,axiom,
    ! [F: set_li6436108459499378894od_b_c > list_P6327159017948738492od_c_a,P5: set_li6436108459499378894od_b_c > $o] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [Uu: list_P6327159017948738492od_c_a] :
          ? [X3: set_li6436108459499378894od_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P5 @ X3 ) ) )
      = ( image_154576026235826419od_c_a @ F @ ( collec1823247286932916153od_b_c @ P5 ) ) ) ).

% setcompr_eq_image
thf(fact_882_setcompr__eq__image,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,P5: produc8838969699183119820_a_b_c > $o] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [Uu: set_li6436108459499378894od_b_c] :
          ? [X3: produc8838969699183119820_a_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P5 @ X3 ) ) )
      = ( image_2344789136157180295od_b_c @ F @ ( collec3867759823522760631_a_b_c @ P5 ) ) ) ).

% setcompr_eq_image
thf(fact_883_setcompr__eq__image,axiom,
    ! [F: list_P6327159017948738492od_c_a > set_li6436108459499378894od_b_c,P5: list_P6327159017948738492od_c_a > $o] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [Uu: set_li6436108459499378894od_b_c] :
          ? [X3: list_P6327159017948738492od_c_a] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P5 @ X3 ) ) )
      = ( image_4742196336002327959od_b_c @ F @ ( collec6273869032445462695od_c_a @ P5 ) ) ) ).

% setcompr_eq_image
thf(fact_884_setcompr__eq__image,axiom,
    ! [F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,P5: set_li6436108459499378894od_b_c > $o] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [Uu: set_li6436108459499378894od_b_c] :
          ? [X3: set_li6436108459499378894od_b_c] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P5 @ X3 ) ) )
      = ( image_8181729759120431109od_b_c @ F @ ( collec1823247286932916153od_b_c @ P5 ) ) ) ).

% setcompr_eq_image
thf(fact_885_implies__completeness__for__repetition__sets__simps_I7_J,axiom,
    ! [Prs2: set_Pr2640454425625433516_a_b_c,Tps2: a > set_li1159382662694783132od_c_a,Rd_targets2: produc3220416805844688357od_c_a > set_a,Separators2: product_prod_a_a > set_Pr9130538228769002824od_d_d,M: fsm_a_b_c,M6: nat,Repetition_sets2: list_P3660316430366008877_set_a,Q: a] :
      ( ( test_i4756896328399799264_b_c_d @ ( test_t5871526445387177708_b_c_d @ Prs2 @ Tps2 @ Rd_targets2 @ Separators2 ) @ M @ M6 @ Repetition_sets2 )
     => ( ( member_a @ Q @ ( image_3273489647386116407_b_c_a @ produc2620512763296580520_a_b_c @ Prs2 ) )
       => ( ( ord_le2998388488506175548od_c_a @ ( Tps2 @ Q )
            @ ( collec6273869032445462695od_c_a
              @ ^ [P13: list_P6327159017948738492od_c_a] :
                ? [P23: list_P6327159017948738492od_c_a,D4: produc1703568184450464039_set_a] : ( member6493638300815972283_set_a @ ( produc6508649944902238466_set_a @ ( append3884164603879732689od_c_a @ P13 @ P23 ) @ D4 ) @ ( traver6421790936065540696_a_b_c @ M @ Q @ Repetition_sets2 @ M6 ) ) ) )
          & ( ord_le2998388488506175548od_c_a @ ( image_865596521259543215od_c_a @ produc3242630202250842350_set_a @ ( traver6421790936065540696_a_b_c @ M @ Q @ Repetition_sets2 @ M6 ) ) @ ( Tps2 @ Q ) ) ) ) ) ).

% implies_completeness_for_repetition_sets_simps(7)
thf(fact_886_product__target__split_I2_J,axiom,
    ! [Q1: a,Q22: a,P: list_P558552684728352636od_a_a,Q12: a,Q24: a] :
      ( ( ( target539946457023899106_a_b_c @ ( product_Pair_a_a @ Q1 @ Q22 ) @ P )
        = ( product_Pair_a_a @ Q12 @ Q24 ) )
     => ( ( target_a_b_c @ Q22
          @ ( map_Pr7629667074831319300od_c_a
            @ ^ [T: produc3308035872450592246od_a_a] : ( produc5762801557034676134od_c_a @ ( product_snd_a_a @ ( produc2892452523101172178od_a_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc806483927813978002od_a_a @ ( produc8587744923287169300od_a_a @ T ) ) @ ( product_Pair_c_a @ ( produc5237419938360121514od_a_a @ ( produc4668670682878874324od_a_a @ ( produc8587744923287169300od_a_a @ T ) ) ) @ ( product_snd_a_a @ ( produc933545150919345644od_a_a @ ( produc4668670682878874324od_a_a @ ( produc8587744923287169300od_a_a @ T ) ) ) ) ) ) )
            @ P ) )
        = Q24 ) ) ).

% product_target_split(2)
thf(fact_887_product__target__split_I2_J,axiom,
    ! [Q1: c,Q22: a,P: list_P1562850763667305982od_c_a,Q12: c,Q24: a] :
      ( ( ( target6303326113586172192_a_a_b @ ( product_Pair_c_a @ Q1 @ Q22 ) @ P )
        = ( product_Pair_c_a @ Q12 @ Q24 ) )
     => ( ( target_a_a_b @ Q22
          @ ( map_Pr834105277175768324od_b_a
            @ ^ [T: produc3205353004677322104od_c_a] : ( produc7037890857971478952od_b_a @ ( product_snd_c_a @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc6867761796305373823od_b_a @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( product_Pair_b_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
            @ P ) )
        = Q24 ) ) ).

% product_target_split(2)
thf(fact_888_product__target__split_I1_J,axiom,
    ! [Q1: a,Q22: list_P6327159017948738492od_c_a,P: list_P2673244623396751228od_c_a,Q12: a,Q24: list_P6327159017948738492od_c_a] :
      ( ( ( target1956115425879022752_a_b_c @ ( produc3580641852002036821od_c_a @ Q1 @ Q22 ) @ P )
        = ( produc3580641852002036821od_c_a @ Q12 @ Q24 ) )
     => ( ( target_a_b_c @ Q1
          @ ( map_Pr6478393753112901892od_c_a
            @ ^ [T: produc3461525747684641526od_c_a] : ( produc5762801557034676134od_c_a @ ( produc4637894536740104257od_c_a @ ( produc3992837916874053330od_c_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc7418247913919711568od_c_a @ ( produc2794907291488784404od_c_a @ T ) ) @ ( product_Pair_c_a @ ( produc3687507211660016360od_c_a @ ( produc7500877187391808914od_c_a @ ( produc2794907291488784404od_c_a @ T ) ) ) @ ( produc4637894536740104257od_c_a @ ( produc7567478381142059306od_c_a @ ( produc7500877187391808914od_c_a @ ( produc2794907291488784404od_c_a @ T ) ) ) ) ) ) )
            @ P ) )
        = Q12 ) ) ).

% product_target_split(1)
thf(fact_889_product__target__split_I1_J,axiom,
    ! [Q1: a,Q22: a,P: list_P558552684728352636od_a_a,Q12: a,Q24: a] :
      ( ( ( target539946457023899106_a_b_c @ ( product_Pair_a_a @ Q1 @ Q22 ) @ P )
        = ( product_Pair_a_a @ Q12 @ Q24 ) )
     => ( ( target_a_b_c @ Q1
          @ ( map_Pr7629667074831319300od_c_a
            @ ^ [T: produc3308035872450592246od_a_a] : ( produc5762801557034676134od_c_a @ ( product_fst_a_a @ ( produc2892452523101172178od_a_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc806483927813978002od_a_a @ ( produc8587744923287169300od_a_a @ T ) ) @ ( product_Pair_c_a @ ( produc5237419938360121514od_a_a @ ( produc4668670682878874324od_a_a @ ( produc8587744923287169300od_a_a @ T ) ) ) @ ( product_fst_a_a @ ( produc933545150919345644od_a_a @ ( produc4668670682878874324od_a_a @ ( produc8587744923287169300od_a_a @ T ) ) ) ) ) ) )
            @ P ) )
        = Q12 ) ) ).

% product_target_split(1)
thf(fact_890_product__target__split_I1_J,axiom,
    ! [Q1: a,Q22: fsm_a_b_c,P: list_P578107730873818684_a_b_c,Q12: a,Q24: fsm_a_b_c] :
      ( ( ( target5407960577604541703_c_b_c @ ( produc2795441082327197628_a_b_c @ Q1 @ Q22 ) @ P )
        = ( produc2795441082327197628_a_b_c @ Q12 @ Q24 ) )
     => ( ( target_a_b_c @ Q1
          @ ( map_Pr2496390191848272452od_c_a
            @ ^ [T: produc662032092171055542_a_b_c] : ( produc5762801557034676134od_c_a @ ( produc2620512763296580520_a_b_c @ ( produc2811850266904162578_a_b_c @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc3258500352807548343_a_b_c @ ( produc8362705341804890708_a_b_c @ T ) ) @ ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc7235443165838885369_a_b_c @ ( produc8362705341804890708_a_b_c @ T ) ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc7235443165838885369_a_b_c @ ( produc8362705341804890708_a_b_c @ T ) ) ) ) ) ) )
            @ P ) )
        = Q12 ) ) ).

% product_target_split(1)
thf(fact_891_product__target__split_I1_J,axiom,
    ! [Q1: c,Q22: a,P: list_P1562850763667305982od_c_a,Q12: c,Q24: a] :
      ( ( ( target6303326113586172192_a_a_b @ ( product_Pair_c_a @ Q1 @ Q22 ) @ P )
        = ( product_Pair_c_a @ Q12 @ Q24 ) )
     => ( ( target_c_a_b @ Q1
          @ ( map_Pr6135845499543628420od_b_c
            @ ^ [T: produc3205353004677322104od_c_a] : ( produc8300696160155326504od_b_c @ ( product_fst_c_a @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc6867761804911831425od_b_c @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
            @ P ) )
        = Q12 ) ) ).

% product_target_split(1)
thf(fact_892_inf_OcoboundedI2,axiom,
    ! [B2: set_a,C: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ C )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_893_inf_OcoboundedI2,axiom,
    ! [B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B2 @ C )
     => ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_894_inf_OcoboundedI1,axiom,
    ! [A2: set_a,C: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_895_inf_OcoboundedI1,axiom,
    ! [A2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ C )
     => ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A2 @ B2 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_896_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B5: set_a,A5: set_a] :
          ( ( inf_inf_set_a @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_897_inf_Oabsorb__iff2,axiom,
    ( ord_le2998388488506175548od_c_a
    = ( ^ [B5: set_li1159382662694783132od_c_a,A5: set_li1159382662694783132od_c_a] :
          ( ( inf_in980870363377917322od_c_a @ A5 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_898_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( ( inf_inf_set_a @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_899_inf_Oabsorb__iff1,axiom,
    ( ord_le2998388488506175548od_c_a
    = ( ^ [A5: set_li1159382662694783132od_c_a,B5: set_li1159382662694783132od_c_a] :
          ( ( inf_in980870363377917322od_c_a @ A5 @ B5 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_900_inf_Ocobounded2,axiom,
    ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_901_inf_Ocobounded2,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_902_inf_Ocobounded1,axiom,
    ! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_903_inf_Ocobounded1,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_904_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( A5
          = ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_905_inf_Oorder__iff,axiom,
    ( ord_le2998388488506175548od_c_a
    = ( ^ [A5: set_li1159382662694783132od_c_a,B5: set_li1159382662694783132od_c_a] :
          ( A5
          = ( inf_in980870363377917322od_c_a @ A5 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_906_inf__greatest,axiom,
    ! [X2: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( ord_less_eq_set_a @ X2 @ Z )
       => ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_907_inf__greatest,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a,Z: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ X2 @ Y )
     => ( ( ord_le2998388488506175548od_c_a @ X2 @ Z )
       => ( ord_le2998388488506175548od_c_a @ X2 @ ( inf_in980870363377917322od_c_a @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_908_inf_OboundedI,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( ord_less_eq_set_a @ A2 @ C )
       => ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_909_inf_OboundedI,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( ord_le2998388488506175548od_c_a @ A2 @ C )
       => ( ord_le2998388488506175548od_c_a @ A2 @ ( inf_in980870363377917322od_c_a @ B2 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_910_inf_OboundedE,axiom,
    ! [A2: set_a,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
     => ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
         => ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_911_inf_OboundedE,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ ( inf_in980870363377917322od_c_a @ B2 @ C ) )
     => ~ ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
         => ~ ( ord_le2998388488506175548od_c_a @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_912_inf__absorb2,axiom,
    ! [Y: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X2 )
     => ( ( inf_inf_set_a @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_913_inf__absorb2,axiom,
    ! [Y: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ Y @ X2 )
     => ( ( inf_in980870363377917322od_c_a @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_914_inf__absorb1,axiom,
    ! [X2: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y )
     => ( ( inf_inf_set_a @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_915_inf__absorb1,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ X2 @ Y )
     => ( ( inf_in980870363377917322od_c_a @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_916_inf_Oabsorb2,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( inf_inf_set_a @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_917_inf_Oabsorb2,axiom,
    ! [B2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B2 @ A2 )
     => ( ( inf_in980870363377917322od_c_a @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_918_inf_Oabsorb1,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( inf_inf_set_a @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_919_inf_Oabsorb1,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( inf_in980870363377917322od_c_a @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_920_le__iff__inf,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X3: set_a,Y5: set_a] :
          ( ( inf_inf_set_a @ X3 @ Y5 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_921_le__iff__inf,axiom,
    ( ord_le2998388488506175548od_c_a
    = ( ^ [X3: set_li1159382662694783132od_c_a,Y5: set_li1159382662694783132od_c_a] :
          ( ( inf_in980870363377917322od_c_a @ X3 @ Y5 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_922_inf__unique,axiom,
    ! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
      ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ X )
     => ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y3 ) @ Y3 )
       => ( ! [X: set_a,Y3: set_a,Z3: set_a] :
              ( ( ord_less_eq_set_a @ X @ Y3 )
             => ( ( ord_less_eq_set_a @ X @ Z3 )
               => ( ord_less_eq_set_a @ X @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_inf_set_a @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_923_inf__unique,axiom,
    ! [F: set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] :
      ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( F @ X @ Y3 ) @ X )
     => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( F @ X @ Y3 ) @ Y3 )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a,Z3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ X @ Y3 )
             => ( ( ord_le2998388488506175548od_c_a @ X @ Z3 )
               => ( ord_le2998388488506175548od_c_a @ X @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_in980870363377917322od_c_a @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_924_inf_OorderI,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( A2
        = ( inf_inf_set_a @ A2 @ B2 ) )
     => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_925_inf_OorderI,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( A2
        = ( inf_in980870363377917322od_c_a @ A2 @ B2 ) )
     => ( ord_le2998388488506175548od_c_a @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_926_inf_OorderE,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( A2
        = ( inf_inf_set_a @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_927_inf_OorderE,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( A2
        = ( inf_in980870363377917322od_c_a @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_928_le__infI2,axiom,
    ! [B2: set_a,X2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ X2 )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).

% le_infI2
thf(fact_929_le__infI2,axiom,
    ! [B2: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B2 @ X2 )
     => ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A2 @ B2 ) @ X2 ) ) ).

% le_infI2
thf(fact_930_le__infI1,axiom,
    ! [A2: set_a,X2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ X2 )
     => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).

% le_infI1
thf(fact_931_le__infI1,axiom,
    ! [A2: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ X2 )
     => ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A2 @ B2 ) @ X2 ) ) ).

% le_infI1
thf(fact_932_inf__mono,axiom,
    ! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C )
     => ( ( ord_less_eq_set_a @ B2 @ D2 )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_933_inf__mono,axiom,
    ! [A2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,D2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ C )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ D2 )
       => ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ A2 @ B2 ) @ ( inf_in980870363377917322od_c_a @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_934_le__infI,axiom,
    ! [X2: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ A2 )
     => ( ( ord_less_eq_set_a @ X2 @ B2 )
       => ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_935_le__infI,axiom,
    ! [X2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ X2 @ A2 )
     => ( ( ord_le2998388488506175548od_c_a @ X2 @ B2 )
       => ( ord_le2998388488506175548od_c_a @ X2 @ ( inf_in980870363377917322od_c_a @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_936_le__infE,axiom,
    ! [X2: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_set_a @ X2 @ A2 )
         => ~ ( ord_less_eq_set_a @ X2 @ B2 ) ) ) ).

% le_infE
thf(fact_937_le__infE,axiom,
    ! [X2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ X2 @ ( inf_in980870363377917322od_c_a @ A2 @ B2 ) )
     => ~ ( ( ord_le2998388488506175548od_c_a @ X2 @ A2 )
         => ~ ( ord_le2998388488506175548od_c_a @ X2 @ B2 ) ) ) ).

% le_infE
thf(fact_938_inf__le2,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_939_inf__le2,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_940_inf__le1,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_941_inf__le1,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_942_inf__sup__ord_I1_J,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_943_inf__sup__ord_I1_J,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_944_inf__sup__ord_I2_J,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_945_inf__sup__ord_I2_J,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ ( inf_in980870363377917322od_c_a @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_946_observable__path__unique,axiom,
    ! [M: fsm_fsm_a_b_c_a_a,Q: fsm_a_b_c,P: list_P4545384517222041535_a_b_c,P7: list_P4545384517222041535_a_b_c] :
      ( ( observ552407795947821995_c_a_a @ M )
     => ( ( path_fsm_a_b_c_a_a @ M @ Q @ P )
       => ( ( path_fsm_a_b_c_a_a @ M @ Q @ P7 )
         => ( ( ( map_Pr7384117626843903474od_a_a
                @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                @ P )
              = ( map_Pr7384117626843903474od_a_a
                @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                @ P7 ) )
           => ( P = P7 ) ) ) ) ) ).

% observable_path_unique
thf(fact_947_observable__path__unique,axiom,
    ! [M: fsm_Pr306566304496394697od_c_a,Q: produc1703568184450464039_set_a,P: list_P5750286787369789953_set_a,P7: list_P5750286787369789953_set_a] :
      ( ( observ7047451555672324333od_c_a @ M )
     => ( ( path_P8254327248518380459od_c_a @ M @ Q @ P )
       => ( ( path_P8254327248518380459od_c_a @ M @ Q @ P7 )
         => ( ( ( map_Pr2884798845580615278od_c_a
                @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                @ P )
              = ( map_Pr2884798845580615278od_c_a
                @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                @ P7 ) )
           => ( P = P7 ) ) ) ) ) ).

% observable_path_unique
thf(fact_948_observable__path__unique,axiom,
    ! [M: fsm_se8204406047609318756_set_a,Q: set_a,P: list_P3716011504951620193_set_a,P7: list_P3716011504951620193_set_a] :
      ( ( observ8827156789157454600_set_a @ M )
     => ( ( path_s1167107641669997510_set_a @ M @ Q @ P )
       => ( ( path_s1167107641669997510_set_a @ M @ Q @ P7 )
         => ( ( ( map_Pr3507802118303906290_set_a
                @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                @ P )
              = ( map_Pr3507802118303906290_set_a
                @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                @ P7 ) )
           => ( P = P7 ) ) ) ) ) ).

% observable_path_unique
thf(fact_949_observable__path__unique,axiom,
    ! [M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P: list_P1562850763667305982od_c_a,P7: list_P1562850763667305982od_c_a] :
      ( ( observ2964502264774598958_a_a_b @ M )
     => ( ( path_P1107610407223809260_a_a_b @ M @ Q @ P )
       => ( ( path_P1107610407223809260_a_a_b @ M @ Q @ P7 )
         => ( ( ( map_Pr3411285261028437108od_a_b
                @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                @ P )
              = ( map_Pr3411285261028437108od_a_b
                @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                @ P7 ) )
           => ( P = P7 ) ) ) ) ) ).

% observable_path_unique
thf(fact_950_observable__path__unique,axiom,
    ! [M: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a,P7: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( path_a_b_c @ M @ Q @ P )
       => ( ( path_a_b_c @ M @ Q @ P7 )
         => ( ( ( map_Pr776894537227705078od_b_c
                @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                @ P )
              = ( map_Pr776894537227705078od_b_c
                @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                @ P7 ) )
           => ( P = P7 ) ) ) ) ) ).

% observable_path_unique
thf(fact_951_LS_Oelims,axiom,
    ! [X2: fsm_fsm_a_b_c_a_a,Xa: fsm_a_b_c,Y: set_li8827807065578854541od_a_a] :
      ( ( ( lS_fsm_a_b_c_a_a @ X2 @ Xa )
        = Y )
     => ( Y
        = ( collec2774578310585404696od_a_a
          @ ^ [Uu: list_P1396940483166286381od_a_a] :
            ? [P4: list_P4545384517222041535_a_b_c] :
              ( ( Uu
                = ( map_Pr7384117626843903474od_a_a
                  @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_a_a @ X2 @ Xa @ P4 ) ) ) ) ) ).

% LS.elims
thf(fact_952_LS_Oelims,axiom,
    ! [X2: fsm_Pr306566304496394697od_c_a,Xa: produc1703568184450464039_set_a,Y: set_li2252564302641075019od_c_a] :
      ( ( ( lS_Pro8045283330902898057od_c_a @ X2 @ Xa )
        = Y )
     => ( Y
        = ( collec5490115594482549206od_c_a
          @ ^ [Uu: list_P1175628092075131243od_c_a] :
            ? [P4: list_P5750286787369789953_set_a] :
              ( ( Uu
                = ( map_Pr2884798845580615278od_c_a
                  @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P8254327248518380459od_c_a @ X2 @ Xa @ P4 ) ) ) ) ) ).

% LS.elims
thf(fact_953_LS_Oelims,axiom,
    ! [X2: fsm_se8204406047609318756_set_a,Xa: set_a,Y: set_li8632829941866831663_set_a] :
      ( ( ( lS_set5116714952275434404_set_a @ X2 @ Xa )
        = Y )
     => ( Y
        = ( collec8283632072997365690_set_a
          @ ^ [Uu: list_P5262903567315784527_set_a] :
            ? [P4: list_P3716011504951620193_set_a] :
              ( ( Uu
                = ( map_Pr3507802118303906290_set_a
                  @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s1167107641669997510_set_a @ X2 @ Xa @ P4 ) ) ) ) ) ).

% LS.elims
thf(fact_954_LS_Oelims,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b,Xa: product_prod_c_a,Y: set_li6306622547289818894od_a_b] :
      ( ( ( lS_Pro5114676609533226954_a_a_b @ X2 @ Xa )
        = Y )
     => ( Y
        = ( collec2845612350631750681od_a_b
          @ ^ [Uu: list_P1467974523212632366od_a_b] :
            ? [P4: list_P1562850763667305982od_c_a] :
              ( ( Uu
                = ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P1107610407223809260_a_a_b @ X2 @ Xa @ P4 ) ) ) ) ) ).

% LS.elims
thf(fact_955_LS_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Y: set_li6436108459499378894od_b_c] :
      ( ( ( lS_a_b_c @ X2 @ Xa )
        = Y )
     => ( Y
        = ( collec2280997390073109977od_b_c
          @ ^ [Uu: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ X2 @ Xa @ P4 ) ) ) ) ) ).

% LS.elims
thf(fact_956_LS_Osimps,axiom,
    ( lS_fsm_a_b_c_a_a
    = ( ^ [M2: fsm_fsm_a_b_c_a_a,Q4: fsm_a_b_c] :
          ( collec2774578310585404696od_a_a
          @ ^ [Uu: list_P1396940483166286381od_a_a] :
            ? [P4: list_P4545384517222041535_a_b_c] :
              ( ( Uu
                = ( map_Pr7384117626843903474od_a_a
                  @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_a_a @ M2 @ Q4 @ P4 ) ) ) ) ) ).

% LS.simps
thf(fact_957_LS_Osimps,axiom,
    ( lS_Pro8045283330902898057od_c_a
    = ( ^ [M2: fsm_Pr306566304496394697od_c_a,Q4: produc1703568184450464039_set_a] :
          ( collec5490115594482549206od_c_a
          @ ^ [Uu: list_P1175628092075131243od_c_a] :
            ? [P4: list_P5750286787369789953_set_a] :
              ( ( Uu
                = ( map_Pr2884798845580615278od_c_a
                  @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P8254327248518380459od_c_a @ M2 @ Q4 @ P4 ) ) ) ) ) ).

% LS.simps
thf(fact_958_LS_Osimps,axiom,
    ( lS_set5116714952275434404_set_a
    = ( ^ [M2: fsm_se8204406047609318756_set_a,Q4: set_a] :
          ( collec8283632072997365690_set_a
          @ ^ [Uu: list_P5262903567315784527_set_a] :
            ? [P4: list_P3716011504951620193_set_a] :
              ( ( Uu
                = ( map_Pr3507802118303906290_set_a
                  @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s1167107641669997510_set_a @ M2 @ Q4 @ P4 ) ) ) ) ) ).

% LS.simps
thf(fact_959_LS_Osimps,axiom,
    ( lS_Pro5114676609533226954_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b,Q4: product_prod_c_a] :
          ( collec2845612350631750681od_a_b
          @ ^ [Uu: list_P1467974523212632366od_a_b] :
            ? [P4: list_P1562850763667305982od_c_a] :
              ( ( Uu
                = ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P1107610407223809260_a_a_b @ M2 @ Q4 @ P4 ) ) ) ) ) ).

% LS.simps
thf(fact_960_LS_Osimps,axiom,
    ( lS_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q4: a] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ M2 @ Q4 @ P4 ) ) ) ) ) ).

% LS.simps
thf(fact_961_path__map__target,axiom,
    ! [F4: set_a > a,Q: set_a,F1: set_a > a,F2: list_P6327159017948738492od_c_a > b,F3: set_a > c,P: list_P3716011504951620193_set_a] :
      ( ( target_a_b_c @ ( F4 @ Q )
        @ ( map_Pr5191547833749763551od_c_a
          @ ^ [T: produc3127082738048029659_set_a] : ( produc5762801557034676134od_c_a @ ( F1 @ ( produc7682742971909621559_set_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( F2 @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) @ ( product_Pair_c_a @ ( F3 @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) ) @ ( F4 @ ( produc1983107199584856133_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) ) ) ) )
          @ P ) )
      = ( F4 @ ( target6955975396184022522_set_a @ Q @ P ) ) ) ).

% path_map_target
thf(fact_962_path__map__target,axiom,
    ! [F4: product_prod_c_a > a,Q: product_prod_c_a,F1: product_prod_c_a > a,F2: a > b,F3: b > c,P: list_P1562850763667305982od_c_a] :
      ( ( target_a_b_c @ ( F4 @ Q )
        @ ( map_Pr2148725087721669634od_c_a
          @ ^ [T: produc3205353004677322104od_c_a] : ( produc5762801557034676134od_c_a @ ( F1 @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( F2 @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( product_Pair_c_a @ ( F3 @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) @ ( F4 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
          @ P ) )
      = ( F4 @ ( target6303326113586172192_a_a_b @ Q @ P ) ) ) ).

% path_map_target
thf(fact_963_path__map__target,axiom,
    ! [F4: a > a,Q: a,F1: a > a,F2: b > b,F3: c > c,P: list_P6327159017948738492od_c_a] :
      ( ( target_a_b_c @ ( F4 @ Q )
        @ ( map_Pr6705208461955284164od_c_a
          @ ^ [T: produc2669932201312201782od_c_a] : ( produc5762801557034676134od_c_a @ ( F1 @ ( produc4225943634317630866od_c_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( F2 @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( product_Pair_c_a @ ( F3 @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ ( F4 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) )
          @ P ) )
      = ( F4 @ ( target_a_b_c @ Q @ P ) ) ) ).

% path_map_target
thf(fact_964_observable__path__suffix,axiom,
    ! [P: list_P4545384517222041535_a_b_c,Io3: list_P1396940483166286381od_a_a,M: fsm_fsm_a_b_c_a_a,Q: fsm_a_b_c] :
      ( ( member6824001069763096534od_a_a
        @ ( append5335208819046833346od_a_a
          @ ( map_Pr7384117626843903474od_a_a
            @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
            @ P )
          @ Io3 )
        @ ( lS_fsm_a_b_c_a_a @ M @ Q ) )
     => ( ( path_fsm_a_b_c_a_a @ M @ Q @ P )
       => ( ( observ552407795947821995_c_a_a @ M )
         => ~ ! [P8: list_P4545384517222041535_a_b_c] :
                ( ( path_fsm_a_b_c_a_a @ M @ ( target_fsm_a_b_c_a_a @ Q @ P ) @ P8 )
               => ( ( map_Pr7384117626843903474od_a_a
                    @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                    @ P8 )
                 != Io3 ) ) ) ) ) ).

% observable_path_suffix
thf(fact_965_observable__path__suffix,axiom,
    ! [P: list_P5750286787369789953_set_a,Io3: list_P1175628092075131243od_c_a,M: fsm_Pr306566304496394697od_c_a,Q: produc1703568184450464039_set_a] :
      ( ( member4479000366354228116od_c_a
        @ ( append2230762392551410432od_c_a
          @ ( map_Pr2884798845580615278od_c_a
            @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
            @ P )
          @ Io3 )
        @ ( lS_Pro8045283330902898057od_c_a @ M @ Q ) )
     => ( ( path_P8254327248518380459od_c_a @ M @ Q @ P )
       => ( ( observ7047451555672324333od_c_a @ M )
         => ~ ! [P8: list_P5750286787369789953_set_a] :
                ( ( path_P8254327248518380459od_c_a @ M @ ( target3445005149252956639od_c_a @ Q @ P ) @ P8 )
               => ( ( map_Pr2884798845580615278od_c_a
                    @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                    @ P8 )
                 != Io3 ) ) ) ) ) ).

% observable_path_suffix
thf(fact_966_observable__path__suffix,axiom,
    ! [P: list_P3716011504951620193_set_a,Io3: list_P5262903567315784527_set_a,M: fsm_se8204406047609318756_set_a,Q: set_a] :
      ( ( member5048304341121814904_set_a
        @ ( append1006059461745118948_set_a
          @ ( map_Pr3507802118303906290_set_a
            @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
            @ P )
          @ Io3 )
        @ ( lS_set5116714952275434404_set_a @ M @ Q ) )
     => ( ( path_s1167107641669997510_set_a @ M @ Q @ P )
       => ( ( observ8827156789157454600_set_a @ M )
         => ~ ! [P8: list_P3716011504951620193_set_a] :
                ( ( path_s1167107641669997510_set_a @ M @ ( target6955975396184022522_set_a @ Q @ P ) @ P8 )
               => ( ( map_Pr3507802118303906290_set_a
                    @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                    @ P8 )
                 != Io3 ) ) ) ) ) ).

% observable_path_suffix
thf(fact_967_observable__path__suffix,axiom,
    ! [P: list_P1562850763667305982od_c_a,Io3: list_P1467974523212632366od_a_b,M: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a] :
      ( ( member6895035109809442519od_a_b
        @ ( append5335208823350062147od_a_b
          @ ( map_Pr3411285261028437108od_a_b
            @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
            @ P )
          @ Io3 )
        @ ( lS_Pro5114676609533226954_a_a_b @ M @ Q ) )
     => ( ( path_P1107610407223809260_a_a_b @ M @ Q @ P )
       => ( ( observ2964502264774598958_a_a_b @ M )
         => ~ ! [P8: list_P1562850763667305982od_c_a] :
                ( ( path_P1107610407223809260_a_a_b @ M @ ( target6303326113586172192_a_a_b @ Q @ P ) @ P8 )
               => ( ( map_Pr3411285261028437108od_a_b
                    @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                    @ P8 )
                 != Io3 ) ) ) ) ) ).

% observable_path_suffix
thf(fact_968_observable__path__suffix,axiom,
    ! [P: list_P6327159017948738492od_c_a,Io3: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr776894537227705078od_b_c
            @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
            @ P )
          @ Io3 )
        @ ( lS_a_b_c @ M @ Q ) )
     => ( ( path_a_b_c @ M @ Q @ P )
       => ( ( observable_a_b_c @ M )
         => ~ ! [P8: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M @ ( target_a_b_c @ Q @ P ) @ P8 )
               => ( ( map_Pr776894537227705078od_b_c
                    @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                    @ P8 )
                 != Io3 ) ) ) ) ) ).

% observable_path_suffix
thf(fact_969_language__state__for__input_Oelims,axiom,
    ! [X2: fsm_se2423825140862529346_set_a,Xa: set_a,Xb: list_set_a,Y: set_li1839383814603465869_set_a] :
      ( ( ( langua3963165854880322359_set_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec3402100140313454872_set_a
          @ ^ [Uu: list_P3660316430366008877_set_a] :
            ? [P4: list_P5960435600574695743_set_a] :
              ( ( Uu
                = ( map_Pr7110208979441748082_set_a
                  @ ^ [T: produc9078272350731273017_set_a] : ( produc9088192753505129239_set_a @ ( produc38069527703528652_set_a @ ( produc7471892532163375447_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2783630374533347854_set_a @ ( produc7471892532163375447_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s526755528433797412_set_a @ X2 @ Xa @ P4 )
              & ( ( map_Pr8866779738092938811_set_a @ produc9088895665703139587_set_a
                  @ ( map_Pr7110208979441748082_set_a
                    @ ^ [T: produc9078272350731273017_set_a] : ( produc9088192753505129239_set_a @ ( produc38069527703528652_set_a @ ( produc7471892532163375447_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2783630374533347854_set_a @ ( produc7471892532163375447_set_a @ T ) ) ) )
                    @ P4 ) )
                = Xb ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_970_language__state__for__input_Oelims,axiom,
    ! [X2: fsm_fsm_a_b_c_a_a,Xa: fsm_a_b_c,Xb: list_a,Y: set_li8827807065578854541od_a_a] :
      ( ( ( langua7447059062660578300_c_a_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec2774578310585404696od_a_a
          @ ^ [Uu: list_P1396940483166286381od_a_a] :
            ? [P4: list_P4545384517222041535_a_b_c] :
              ( ( Uu
                = ( map_Pr7384117626843903474od_a_a
                  @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_a_a @ X2 @ Xa @ P4 )
              & ( ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a
                  @ ( map_Pr7384117626843903474od_a_a
                    @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                    @ P4 ) )
                = Xb ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_971_language__state__for__input_Oelims,axiom,
    ! [X2: fsm_fsm_a_b_c_c_a,Xa: fsm_a_b_c,Xb: list_c,Y: set_li4905775889721270027od_c_a] :
      ( ( ( langua1872147898715226234_c_c_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec1503280309375431318od_c_a
          @ ^ [Uu: list_P125642481956313003od_c_a] :
            ? [P4: list_P3455624172244930365_a_b_c] :
              ( ( Uu
                = ( map_Pr3451339525954936178od_c_a
                  @ ^ [T: produc2463281488305106231_a_b_c] : ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_c_a @ X2 @ Xa @ P4 )
              & ( ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a
                  @ ( map_Pr3451339525954936178od_c_a
                    @ ^ [T: produc2463281488305106231_a_b_c] : ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) ) )
                    @ P4 ) )
                = Xb ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_972_language__state__for__input_Oelims,axiom,
    ! [X2: fsm_Pr306566304496394697od_c_a,Xa: produc1703568184450464039_set_a,Xb: list_a,Y: set_li2252564302641075019od_c_a] :
      ( ( ( langua1147745209429016638od_c_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec5490115594482549206od_c_a
          @ ^ [Uu: list_P1175628092075131243od_c_a] :
            ? [P4: list_P5750286787369789953_set_a] :
              ( ( Uu
                = ( map_Pr2884798845580615278od_c_a
                  @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P8254327248518380459od_c_a @ X2 @ Xa @ P4 )
              & ( ( map_Pr7928508354609250845_c_a_a @ produc4637894536740104257od_c_a
                  @ ( map_Pr2884798845580615278od_c_a
                    @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                    @ P4 ) )
                = Xb ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_973_language__state__for__input_Oelims,axiom,
    ! [X2: fsm_se8204406047609318756_set_a,Xa: set_a,Xb: list_l2826717958560394946od_c_a,Y: set_li8632829941866831663_set_a] :
      ( ( ( langua2081964783094375257_set_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec8283632072997365690_set_a
          @ ^ [Uu: list_P5262903567315784527_set_a] :
            ? [P4: list_P3716011504951620193_set_a] :
              ( ( Uu
                = ( map_Pr3507802118303906290_set_a
                  @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s1167107641669997510_set_a @ X2 @ Xa @ P4 )
              & ( ( map_Pr1708894191565315447od_c_a @ produc2106691229161676581_set_a
                  @ ( map_Pr3507802118303906290_set_a
                    @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                    @ P4 ) )
                = Xb ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_974_language__state__for__input_Oelims,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b,Xa: product_prod_c_a,Xb: list_a,Y: set_li6306622547289818894od_a_b] :
      ( ( ( langua6105361646171781119_a_a_b @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec2845612350631750681od_a_b
          @ ^ [Uu: list_P1467974523212632366od_a_b] :
            ? [P4: list_P1562850763667305982od_c_a] :
              ( ( Uu
                = ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P1107610407223809260_a_a_b @ X2 @ Xa @ P4 )
              & ( ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b
                  @ ( map_Pr3411285261028437108od_a_b
                    @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xb ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_975_language__state__for__input_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: list_b,Y: set_li6436108459499378894od_b_c] :
      ( ( ( langua3305995968491465752_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec2280997390073109977od_b_c
          @ ^ [Uu: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ X2 @ Xa @ P4 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xb ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_976_language__state__for__input_Osimps,axiom,
    ( langua3963165854880322359_set_a
    = ( ^ [M2: fsm_se2423825140862529346_set_a,Q4: set_a,Xs4: list_set_a] :
          ( collec3402100140313454872_set_a
          @ ^ [Uu: list_P3660316430366008877_set_a] :
            ? [P4: list_P5960435600574695743_set_a] :
              ( ( Uu
                = ( map_Pr7110208979441748082_set_a
                  @ ^ [T: produc9078272350731273017_set_a] : ( produc9088192753505129239_set_a @ ( produc38069527703528652_set_a @ ( produc7471892532163375447_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2783630374533347854_set_a @ ( produc7471892532163375447_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s526755528433797412_set_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr8866779738092938811_set_a @ produc9088895665703139587_set_a
                  @ ( map_Pr7110208979441748082_set_a
                    @ ^ [T: produc9078272350731273017_set_a] : ( produc9088192753505129239_set_a @ ( produc38069527703528652_set_a @ ( produc7471892532163375447_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2783630374533347854_set_a @ ( produc7471892532163375447_set_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_977_language__state__for__input_Osimps,axiom,
    ( langua7447059062660578300_c_a_a
    = ( ^ [M2: fsm_fsm_a_b_c_a_a,Q4: fsm_a_b_c,Xs4: list_a] :
          ( collec2774578310585404696od_a_a
          @ ^ [Uu: list_P1396940483166286381od_a_a] :
            ? [P4: list_P4545384517222041535_a_b_c] :
              ( ( Uu
                = ( map_Pr7384117626843903474od_a_a
                  @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_a_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a
                  @ ( map_Pr7384117626843903474od_a_a
                    @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_978_language__state__for__input_Osimps,axiom,
    ( langua1872147898715226234_c_c_a
    = ( ^ [M2: fsm_fsm_a_b_c_c_a,Q4: fsm_a_b_c,Xs4: list_c] :
          ( collec1503280309375431318od_c_a
          @ ^ [Uu: list_P125642481956313003od_c_a] :
            ? [P4: list_P3455624172244930365_a_b_c] :
              ( ( Uu
                = ( map_Pr3451339525954936178od_c_a
                  @ ^ [T: produc2463281488305106231_a_b_c] : ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_c_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a
                  @ ( map_Pr3451339525954936178od_c_a
                    @ ^ [T: produc2463281488305106231_a_b_c] : ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_979_language__state__for__input_Osimps,axiom,
    ( langua1147745209429016638od_c_a
    = ( ^ [M2: fsm_Pr306566304496394697od_c_a,Q4: produc1703568184450464039_set_a,Xs4: list_a] :
          ( collec5490115594482549206od_c_a
          @ ^ [Uu: list_P1175628092075131243od_c_a] :
            ? [P4: list_P5750286787369789953_set_a] :
              ( ( Uu
                = ( map_Pr2884798845580615278od_c_a
                  @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P8254327248518380459od_c_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr7928508354609250845_c_a_a @ produc4637894536740104257od_c_a
                  @ ( map_Pr2884798845580615278od_c_a
                    @ ^ [T: produc3559677970590150011_set_a] : ( produc3580641852002036821od_c_a @ ( produc2636661781674612887_set_a @ ( produc6888891369799471001_set_a @ T ) ) @ ( produc3242630202250842350_set_a @ ( produc5085042947020727513_set_a @ ( produc6888891369799471001_set_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_980_language__state__for__input_Osimps,axiom,
    ( langua2081964783094375257_set_a
    = ( ^ [M2: fsm_se8204406047609318756_set_a,Q4: set_a,Xs4: list_l2826717958560394946od_c_a] :
          ( collec8283632072997365690_set_a
          @ ^ [Uu: list_P5262903567315784527_set_a] :
            ? [P4: list_P3716011504951620193_set_a] :
              ( ( Uu
                = ( map_Pr3507802118303906290_set_a
                  @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s1167107641669997510_set_a @ M2 @ Q4 @ P4 )
              & ( ( map_Pr1708894191565315447od_c_a @ produc2106691229161676581_set_a
                  @ ( map_Pr3507802118303906290_set_a
                    @ ^ [T: produc3127082738048029659_set_a] : ( produc1702885982572811065_set_a @ ( produc3242630202250842350_set_a @ ( produc8262290704850586489_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2451121746714373936_set_a @ ( produc8262290704850586489_set_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_981_language__state__for__input_Osimps,axiom,
    ( langua6105361646171781119_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b,Q4: product_prod_c_a,Xs4: list_a] :
          ( collec2845612350631750681od_a_b
          @ ^ [Uu: list_P1467974523212632366od_a_b] :
            ? [P4: list_P1562850763667305982od_c_a] :
              ( ( Uu
                = ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P1107610407223809260_a_a_b @ M2 @ Q4 @ P4 )
              & ( ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b
                  @ ( map_Pr3411285261028437108od_a_b
                    @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_982_language__state__for__input_Osimps,axiom,
    ( langua3305995968491465752_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q4: a,Xs4: list_b] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ M2 @ Q4 @ P4 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xs4 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_983_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_a_b_c,Xa: a,Xb: set_list_b,Y: set_li6436108459499378894od_b_c] :
      ( ( ( lS_i_n_a_b_c @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec2280997390073109977od_b_c
          @ ^ [Uu: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ X2 @ Xa @ P4 )
              & ( member_list_b
                @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_984_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_a_1741613688081531898_b_c_c,Xa: a,Xb: set_li6436108459499378894od_b_c,Y: set_li4827757560454754725_b_c_c] :
      ( ( ( lS_i_n7802350980493425083_b_c_c @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec3825286231726394416_b_c_c
          @ ^ [Uu: list_P7813578483171497029_b_c_c] :
            ? [P4: list_P245329279379811475od_c_a] :
              ( ( Uu
                = ( map_Pr7037326443982793270_b_c_c
                  @ ^ [T: produc1001868188449977229od_c_a] : ( produc794085425710893615_b_c_c @ ( produc2042588467764379584od_c_a @ ( produc1192070262971910827od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5904775222829275906od_c_a @ ( produc1192070262971910827od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a2326023411465851356_b_c_c @ X2 @ Xa @ P4 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr2145510998460861229od_b_c @ produc6878540623311373339_b_c_c
                  @ ( map_Pr7037326443982793270_b_c_c
                    @ ^ [T: produc1001868188449977229od_c_a] : ( produc794085425710893615_b_c_c @ ( produc2042588467764379584od_c_a @ ( produc1192070262971910827od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5904775222829275906od_c_a @ ( produc1192070262971910827od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_985_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b,Xa: product_prod_c_a,Xb: set_list_a,Y: set_li6306622547289818894od_a_b] :
      ( ( ( lS_i_n6583937976251382987_a_a_b @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec2845612350631750681od_a_b
          @ ^ [Uu: list_P1467974523212632366od_a_b] :
            ? [P4: list_P1562850763667305982od_c_a] :
              ( ( Uu
                = ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P1107610407223809260_a_a_b @ X2 @ Xa @ P4 )
              & ( member_list_a
                @ ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b
                  @ ( map_Pr3411285261028437108od_a_b
                    @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_986_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_se2423825140862529346_set_a,Xa: set_a,Xb: set_list_set_a,Y: set_li1839383814603465869_set_a] :
      ( ( ( lS_i_n30645114859844611_set_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec3402100140313454872_set_a
          @ ^ [Uu: list_P3660316430366008877_set_a] :
            ? [P4: list_P5960435600574695743_set_a] :
              ( ( Uu
                = ( map_Pr7110208979441748082_set_a
                  @ ^ [T: produc9078272350731273017_set_a] : ( produc9088192753505129239_set_a @ ( produc38069527703528652_set_a @ ( produc7471892532163375447_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2783630374533347854_set_a @ ( produc7471892532163375447_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s526755528433797412_set_a @ X2 @ Xa @ P4 )
              & ( member_list_set_a
                @ ( map_Pr8866779738092938811_set_a @ produc9088895665703139587_set_a
                  @ ( map_Pr7110208979441748082_set_a
                    @ ^ [T: produc9078272350731273017_set_a] : ( produc9088192753505129239_set_a @ ( produc38069527703528652_set_a @ ( produc7471892532163375447_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2783630374533347854_set_a @ ( produc7471892532163375447_set_a @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_987_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_fsm_a_b_c_a_a,Xa: fsm_a_b_c,Xb: set_list_a,Y: set_li8827807065578854541od_a_a] :
      ( ( ( lS_i_n_fsm_a_b_c_a_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec2774578310585404696od_a_a
          @ ^ [Uu: list_P1396940483166286381od_a_a] :
            ? [P4: list_P4545384517222041535_a_b_c] :
              ( ( Uu
                = ( map_Pr7384117626843903474od_a_a
                  @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_a_a @ X2 @ Xa @ P4 )
              & ( member_list_a
                @ ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a
                  @ ( map_Pr7384117626843903474od_a_a
                    @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_988_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_fsm_a_b_c_c_a,Xa: fsm_a_b_c,Xb: set_list_c,Y: set_li4905775889721270027od_c_a] :
      ( ( ( lS_i_n_fsm_a_b_c_c_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec1503280309375431318od_c_a
          @ ^ [Uu: list_P125642481956313003od_c_a] :
            ? [P4: list_P3455624172244930365_a_b_c] :
              ( ( Uu
                = ( map_Pr3451339525954936178od_c_a
                  @ ^ [T: produc2463281488305106231_a_b_c] : ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_c_a @ X2 @ Xa @ P4 )
              & ( member_list_c
                @ ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a
                  @ ( map_Pr3451339525954936178od_c_a
                    @ ^ [T: produc2463281488305106231_a_b_c] : ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_989_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_se7838025714203150264_set_a,Xa: set_a,Xb: set_li6436108459499378894od_b_c,Y: set_li8325270880847126019_set_a] :
      ( ( ( lS_i_n3023404872028126329_set_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec3448721657321074574_set_a
          @ ^ [Uu: list_P7462397787910628259_set_a] :
            ? [P4: list_P719839838134327221_set_a] :
              ( ( Uu
                = ( map_Pr6895669950699468018_set_a
                  @ ^ [T: produc4514793722652660655_set_a] : ( produc2796512730703427469_set_a @ ( produc4998674477397503810_set_a @ ( produc166407759064387533_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc3928924271393864324_set_a @ ( produc166407759064387533_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s6052385582279609498_set_a @ X2 @ Xa @ P4 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr9028310062325878351od_b_c @ produc6257147697102537081_set_a
                  @ ( map_Pr6895669950699468018_set_a
                    @ ^ [T: produc4514793722652660655_set_a] : ( produc2796512730703427469_set_a @ ( produc4998674477397503810_set_a @ ( produc166407759064387533_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc3928924271393864324_set_a @ ( produc166407759064387533_set_a @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_990_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_Pr1345674910713743136_b_c_b,Xa: product_prod_c_a,Xb: set_li6436108459499378894od_b_c,Y: set_li7348942078743790372_b_c_b] :
      ( ( ( lS_i_n5220417158231219425_b_c_b @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec3754252191680048431_b_c_b
          @ ^ [Uu: list_P7742544443125151044_b_c_b] :
            ? [P4: list_P7369010113192768532od_c_a] :
              ( ( Uu
                = ( map_Pr7927132834574546676_b_c_b
                  @ ^ [T: produc490884240983871886od_c_a] : ( produc794085425710893614_b_c_b @ ( produc8235939149734187176od_c_a @ ( produc1201159413773039276od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc5461594249200507370od_c_a @ ( produc1201159413773039276od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P1883008253792545794_b_c_b @ X2 @ Xa @ P4 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr3481561618413555566od_b_c @ produc6878540623311373338_b_c_b
                  @ ( map_Pr7927132834574546676_b_c_b
                    @ ^ [T: produc490884240983871886od_c_a] : ( produc794085425710893614_b_c_b @ ( produc8235939149734187176od_c_a @ ( produc1201159413773039276od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc5461594249200507370od_c_a @ ( produc1201159413773039276od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_991_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_fs4299613126950108317_b_c_a,Xa: fsm_a_b_c,Xb: set_li6436108459499378894od_b_c,Y: set_li646754560178050211_b_c_a] :
      ( ( ( lS_i_n2411721835518878942_b_c_a @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec3683218151633702446_b_c_a
          @ ^ [Uu: list_P7671510403078805059_b_c_a] :
            ? [P4: list_P7714905414082524373_a_b_c] :
              ( ( Uu
                = ( map_Pr4008010590358927986_b_c_a
                  @ ^ [T: produc9049374856634298831_a_b_c] : ( produc794085425710893613_b_c_a @ ( produc774877329438122343_a_b_c @ ( produc7197384099478624109_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc4751820142469459369_a_b_c @ ( produc7197384099478624109_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_f5704687797818160511_b_c_a @ X2 @ Xa @ P4 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr4817612238366249903od_b_c @ produc6878540623311373337_b_c_a
                  @ ( map_Pr4008010590358927986_b_c_a
                    @ ^ [T: produc9049374856634298831_a_b_c] : ( produc794085425710893613_b_c_a @ ( produc774877329438122343_a_b_c @ ( produc7197384099478624109_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc4751820142469459369_a_b_c @ ( produc7197384099478624109_a_b_c @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_992_LS_092_060_094sub_062i_092_060_094sub_062n_Oelims,axiom,
    ! [X2: fsm_a_9051689448454524332_c_a_c,Xa: a,Xb: set_li1159382662694783132od_c_a,Y: set_li9151619642237734743_c_a_c] :
      ( ( ( lS_i_n7240973559057048429_c_a_c @ X2 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( collec230972978439657186_c_a_c
          @ ^ [Uu: list_P4571182638172300535_c_a_c] :
            ? [P4: list_P936865259254933061od_c_a] :
              ( ( Uu
                = ( map_Pr4742243551537815990_c_a_c
                  @ ^ [T: produc8503114081040878655od_c_a] : ( produc174534808117382113_c_a_c @ ( produc6921293161622054770od_c_a @ ( produc7974962593261140573od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc3393213524953276084od_c_a @ ( produc7974962593261140573od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a9139061043061041550_c_a_c @ X2 @ Xa @ P4 )
              & ( member7410604586820865893od_c_a
                @ ( map_Pr2161286011372329929od_c_a @ produc8128955733147053517_c_a_c
                  @ ( map_Pr4742243551537815990_c_a_c
                    @ ^ [T: produc8503114081040878655od_c_a] : ( produc174534808117382113_c_a_c @ ( produc6921293161622054770od_c_a @ ( produc7974962593261140573od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc3393213524953276084od_c_a @ ( produc7974962593261140573od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xb ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.elims
thf(fact_993_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q4: a,Xss: set_list_b] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu: list_P903359562653991662od_b_c] :
            ? [P4: list_P6327159017948738492od_c_a] :
              ( ( Uu
                = ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a_b_c @ M2 @ Q4 @ P4 )
              & ( member_list_b
                @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_994_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n7802350980493425083_b_c_c
    = ( ^ [M2: fsm_a_1741613688081531898_b_c_c,Q4: a,Xss: set_li6436108459499378894od_b_c] :
          ( collec3825286231726394416_b_c_c
          @ ^ [Uu: list_P7813578483171497029_b_c_c] :
            ? [P4: list_P245329279379811475od_c_a] :
              ( ( Uu
                = ( map_Pr7037326443982793270_b_c_c
                  @ ^ [T: produc1001868188449977229od_c_a] : ( produc794085425710893615_b_c_c @ ( produc2042588467764379584od_c_a @ ( produc1192070262971910827od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5904775222829275906od_c_a @ ( produc1192070262971910827od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a2326023411465851356_b_c_c @ M2 @ Q4 @ P4 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr2145510998460861229od_b_c @ produc6878540623311373339_b_c_c
                  @ ( map_Pr7037326443982793270_b_c_c
                    @ ^ [T: produc1001868188449977229od_c_a] : ( produc794085425710893615_b_c_c @ ( produc2042588467764379584od_c_a @ ( produc1192070262971910827od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5904775222829275906od_c_a @ ( produc1192070262971910827od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_995_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n6583937976251382987_a_a_b
    = ( ^ [M2: fsm_Pr1886043760914421514_a_a_b,Q4: product_prod_c_a,Xss: set_list_a] :
          ( collec2845612350631750681od_a_b
          @ ^ [Uu: list_P1467974523212632366od_a_b] :
            ? [P4: list_P1562850763667305982od_c_a] :
              ( ( Uu
                = ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P1107610407223809260_a_a_b @ M2 @ Q4 @ P4 )
              & ( member_list_a
                @ ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b
                  @ ( map_Pr3411285261028437108od_a_b
                    @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_996_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n30645114859844611_set_a
    = ( ^ [M2: fsm_se2423825140862529346_set_a,Q4: set_a,Xss: set_list_set_a] :
          ( collec3402100140313454872_set_a
          @ ^ [Uu: list_P3660316430366008877_set_a] :
            ? [P4: list_P5960435600574695743_set_a] :
              ( ( Uu
                = ( map_Pr7110208979441748082_set_a
                  @ ^ [T: produc9078272350731273017_set_a] : ( produc9088192753505129239_set_a @ ( produc38069527703528652_set_a @ ( produc7471892532163375447_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2783630374533347854_set_a @ ( produc7471892532163375447_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s526755528433797412_set_a @ M2 @ Q4 @ P4 )
              & ( member_list_set_a
                @ ( map_Pr8866779738092938811_set_a @ produc9088895665703139587_set_a
                  @ ( map_Pr7110208979441748082_set_a
                    @ ^ [T: produc9078272350731273017_set_a] : ( produc9088192753505129239_set_a @ ( produc38069527703528652_set_a @ ( produc7471892532163375447_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc2783630374533347854_set_a @ ( produc7471892532163375447_set_a @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_997_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n_fsm_a_b_c_a_a
    = ( ^ [M2: fsm_fsm_a_b_c_a_a,Q4: fsm_a_b_c,Xss: set_list_a] :
          ( collec2774578310585404696od_a_a
          @ ^ [Uu: list_P1396940483166286381od_a_a] :
            ? [P4: list_P4545384517222041535_a_b_c] :
              ( ( Uu
                = ( map_Pr7384117626843903474od_a_a
                  @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_a_a @ M2 @ Q4 @ P4 )
              & ( member_list_a
                @ ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a
                  @ ( map_Pr7384117626843903474od_a_a
                    @ ^ [T: produc8723758207947523769_a_b_c] : ( product_Pair_a_a @ ( produc6661699919570267473_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc1528364020863863699_a_b_c @ ( produc6213890764698333783_a_b_c @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_998_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n_fsm_a_b_c_c_a
    = ( ^ [M2: fsm_fsm_a_b_c_c_a,Q4: fsm_a_b_c,Xss: set_list_c] :
          ( collec1503280309375431318od_c_a
          @ ^ [Uu: list_P125642481956313003od_c_a] :
            ? [P4: list_P3455624172244930365_a_b_c] :
              ( ( Uu
                = ( map_Pr3451339525954936178od_c_a
                  @ ^ [T: produc2463281488305106231_a_b_c] : ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_fsm_a_b_c_c_a @ M2 @ Q4 @ P4 )
              & ( member_list_c
                @ ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a
                  @ ( map_Pr3451339525954936178od_c_a
                    @ ^ [T: produc2463281488305106231_a_b_c] : ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc8582826364909909973_a_b_c @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_999_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n3023404872028126329_set_a
    = ( ^ [M2: fsm_se7838025714203150264_set_a,Q4: set_a,Xss: set_li6436108459499378894od_b_c] :
          ( collec3448721657321074574_set_a
          @ ^ [Uu: list_P7462397787910628259_set_a] :
            ? [P4: list_P719839838134327221_set_a] :
              ( ( Uu
                = ( map_Pr6895669950699468018_set_a
                  @ ^ [T: produc4514793722652660655_set_a] : ( produc2796512730703427469_set_a @ ( produc4998674477397503810_set_a @ ( produc166407759064387533_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc3928924271393864324_set_a @ ( produc166407759064387533_set_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_s6052385582279609498_set_a @ M2 @ Q4 @ P4 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr9028310062325878351od_b_c @ produc6257147697102537081_set_a
                  @ ( map_Pr6895669950699468018_set_a
                    @ ^ [T: produc4514793722652660655_set_a] : ( produc2796512730703427469_set_a @ ( produc4998674477397503810_set_a @ ( produc166407759064387533_set_a @ T ) ) @ ( produc9088895665703139587_set_a @ ( produc3928924271393864324_set_a @ ( produc166407759064387533_set_a @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_1000_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n5220417158231219425_b_c_b
    = ( ^ [M2: fsm_Pr1345674910713743136_b_c_b,Q4: product_prod_c_a,Xss: set_li6436108459499378894od_b_c] :
          ( collec3754252191680048431_b_c_b
          @ ^ [Uu: list_P7742544443125151044_b_c_b] :
            ? [P4: list_P7369010113192768532od_c_a] :
              ( ( Uu
                = ( map_Pr7927132834574546676_b_c_b
                  @ ^ [T: produc490884240983871886od_c_a] : ( produc794085425710893614_b_c_b @ ( produc8235939149734187176od_c_a @ ( produc1201159413773039276od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc5461594249200507370od_c_a @ ( produc1201159413773039276od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_P1883008253792545794_b_c_b @ M2 @ Q4 @ P4 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr3481561618413555566od_b_c @ produc6878540623311373338_b_c_b
                  @ ( map_Pr7927132834574546676_b_c_b
                    @ ^ [T: produc490884240983871886od_c_a] : ( produc794085425710893614_b_c_b @ ( produc8235939149734187176od_c_a @ ( produc1201159413773039276od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc5461594249200507370od_c_a @ ( produc1201159413773039276od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_1001_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n2411721835518878942_b_c_a
    = ( ^ [M2: fsm_fs4299613126950108317_b_c_a,Q4: fsm_a_b_c,Xss: set_li6436108459499378894od_b_c] :
          ( collec3683218151633702446_b_c_a
          @ ^ [Uu: list_P7671510403078805059_b_c_a] :
            ? [P4: list_P7714905414082524373_a_b_c] :
              ( ( Uu
                = ( map_Pr4008010590358927986_b_c_a
                  @ ^ [T: produc9049374856634298831_a_b_c] : ( produc794085425710893613_b_c_a @ ( produc774877329438122343_a_b_c @ ( produc7197384099478624109_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc4751820142469459369_a_b_c @ ( produc7197384099478624109_a_b_c @ T ) ) ) )
                  @ P4 ) )
              & ( path_f5704687797818160511_b_c_a @ M2 @ Q4 @ P4 )
              & ( member6330420149250801815od_b_c
                @ ( map_Pr4817612238366249903od_b_c @ produc6878540623311373337_b_c_a
                  @ ( map_Pr4008010590358927986_b_c_a
                    @ ^ [T: produc9049374856634298831_a_b_c] : ( produc794085425710893613_b_c_a @ ( produc774877329438122343_a_b_c @ ( produc7197384099478624109_a_b_c @ T ) ) @ ( produc2620512763296580520_a_b_c @ ( produc4751820142469459369_a_b_c @ ( produc7197384099478624109_a_b_c @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_1002_LS_092_060_094sub_062i_092_060_094sub_062n_Osimps,axiom,
    ( lS_i_n7240973559057048429_c_a_c
    = ( ^ [M2: fsm_a_9051689448454524332_c_a_c,Q4: a,Xss: set_li1159382662694783132od_c_a] :
          ( collec230972978439657186_c_a_c
          @ ^ [Uu: list_P4571182638172300535_c_a_c] :
            ? [P4: list_P936865259254933061od_c_a] :
              ( ( Uu
                = ( map_Pr4742243551537815990_c_a_c
                  @ ^ [T: produc8503114081040878655od_c_a] : ( produc174534808117382113_c_a_c @ ( produc6921293161622054770od_c_a @ ( produc7974962593261140573od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc3393213524953276084od_c_a @ ( produc7974962593261140573od_c_a @ T ) ) ) )
                  @ P4 ) )
              & ( path_a9139061043061041550_c_a_c @ M2 @ Q4 @ P4 )
              & ( member7410604586820865893od_c_a
                @ ( map_Pr2161286011372329929od_c_a @ produc8128955733147053517_c_a_c
                  @ ( map_Pr4742243551537815990_c_a_c
                    @ ^ [T: produc8503114081040878655od_c_a] : ( produc174534808117382113_c_a_c @ ( produc6921293161622054770od_c_a @ ( produc7974962593261140573od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc3393213524953276084od_c_a @ ( produc7974962593261140573od_c_a @ T ) ) ) )
                    @ P4 ) )
                @ Xss ) ) ) ) ) ).

% LS\<^sub>i\<^sub>n.simps
thf(fact_1003_separator__path__targets_I3_J,axiom,
    ! [M: fsm_a_b_c,Q1: a,Q22: a,A: fsm_a_b_c,T12: a,T23: a,P: list_P6327159017948738492od_c_a] :
      ( ( state_8910485216943051354_b_c_a @ M @ Q1 @ Q22 @ A @ T12 @ T23 )
     => ( ( path_a_b_c @ A @ ( initial_a_b_c @ A ) @ P )
       => ( ( member6330420149250801815od_b_c
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
              @ P )
            @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M @ Q1 ) @ ( lS_a_b_c @ M @ Q22 ) ) )
         => ( ( ( target_a_b_c @ ( initial_a_b_c @ A ) @ P )
             != T12 )
            & ( ( target_a_b_c @ ( initial_a_b_c @ A ) @ P )
             != T23 ) ) ) ) ) ).

% separator_path_targets(3)
thf(fact_1004_c,axiom,
    ( member6330420149250801815od_b_c @ io
    @ ( comple1250193257546725051od_b_c
      @ ( collec1823247286932916153od_b_c
        @ ^ [Uu: set_li6436108459499378894od_b_c] :
          ? [P4: list_P6327159017948738492od_c_a,Pt2: list_P6327159017948738492od_c_a,Q4: a,A6: fsm_d_b_c] :
            ( ( Uu
              = ( image_2850395784482391045od_b_c
                @ ^ [Io_atc: list_P903359562653991662od_b_c] :
                    ( append2547753245680614915od_b_c
                    @ ( map_Pr776894537227705078od_b_c
                      @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                      @ P4 )
                    @ ( append2547753245680614915od_b_c
                      @ ( map_Pr776894537227705078od_b_c
                        @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                        @ Pt2 )
                      @ Io_atc ) )
                @ ( adapti7464086540814349463_b_c_d @ ( from_FSM_a_b_c @ m @ ( target_a_b_c @ Q4 @ Pt2 ) ) @ A6 ) ) )
            & ? [P9: fsm_a_b_c,Q7: a,T13: d,T24: d] :
                ( ( member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ Q4 @ P9 ) @ prs )
                & ( path_a_b_c @ P9 @ ( initial_a_b_c @ P9 ) @ P4 )
                & ( ( target_a_b_c @ ( initial_a_b_c @ P9 ) @ P4 )
                  = Q4 )
                & ( member7410604586820865893od_c_a @ Pt2 @ ( tps @ Q4 ) )
                & ( member_a @ Q7 @ ( rd_targets @ ( produc3580641852002036821od_c_a @ Q4 @ Pt2 ) ) )
                & ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ A6 @ ( product_Pair_d_d @ T13 @ T24 ) ) @ ( atcs @ ( product_Pair_a_a @ ( target_a_b_c @ Q4 @ Pt2 ) @ Q7 ) ) ) ) ) ) ) ) ).

% c
thf(fact_1005_implies__completeness__for__repetition__sets__simps_I3_J,axiom,
    ! [Prs2: set_Pr3057116837617337708_d_b_c,Tps2: d > set_li2602923542944125148od_c_d,Rd_targets2: produc4330201775697025890od_c_d > set_d,Separators2: product_prod_d_d > set_Pr9130538228769002824od_d_d,M: fsm_d_b_c,M6: nat,Repetition_sets2: list_P7949437782354667373_set_d,A: fsm_d_b_c,D1: d,D22: d,Q1: d,Q22: d] :
      ( ( test_i9220782440284194845_b_c_d @ ( test_t1112040520416797481_b_c_d @ Prs2 @ Tps2 @ Rd_targets2 @ Separators2 ) @ M @ M6 @ Repetition_sets2 )
     => ( ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ A @ ( product_Pair_d_d @ D1 @ D22 ) ) @ ( Separators2 @ ( product_Pair_d_d @ Q1 @ Q22 ) ) )
       => ( ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ A @ ( product_Pair_d_d @ D22 @ D1 ) ) @ ( Separators2 @ ( product_Pair_d_d @ Q22 @ Q1 ) ) )
          & ( state_4150999291972671130_b_c_d @ M @ Q1 @ Q22 @ A @ D1 @ D22 ) ) ) ) ).

% implies_completeness_for_repetition_sets_simps(3)
thf(fact_1006_implies__completeness__for__repetition__sets__simps_I3_J,axiom,
    ! [Prs2: set_Pr2640454425625433516_a_b_c,Tps2: a > set_li1159382662694783132od_c_a,Rd_targets2: produc3220416805844688357od_c_a > set_a,Separators2: product_prod_a_a > set_Pr9130538228769002824od_d_d,M: fsm_a_b_c,M6: nat,Repetition_sets2: list_P3660316430366008877_set_a,A: fsm_d_b_c,D1: d,D22: d,Q1: a,Q22: a] :
      ( ( test_i4756896328399799264_b_c_d @ ( test_t5871526445387177708_b_c_d @ Prs2 @ Tps2 @ Rd_targets2 @ Separators2 ) @ M @ M6 @ Repetition_sets2 )
     => ( ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ A @ ( product_Pair_d_d @ D1 @ D22 ) ) @ ( Separators2 @ ( product_Pair_a_a @ Q1 @ Q22 ) ) )
       => ( ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ A @ ( product_Pair_d_d @ D22 @ D1 ) ) @ ( Separators2 @ ( product_Pair_a_a @ Q22 @ Q1 ) ) )
          & ( state_8910485216943051357_b_c_d @ M @ Q1 @ Q22 @ A @ D1 @ D22 ) ) ) ) ).

% implies_completeness_for_repetition_sets_simps(3)
thf(fact_1007_is__separator__simps_I13_J,axiom,
    ! [M: fsm_a_b_c,Q1: a,Q22: a,A: fsm_a_b_c,T12: a,T23: a,P: list_P6327159017948738492od_c_a] :
      ( ( state_8910485216943051354_b_c_a @ M @ Q1 @ Q22 @ A @ T12 @ T23 )
     => ( ( path_a_b_c @ A @ ( initial_a_b_c @ A ) @ P )
       => ( ( ( target_a_b_c @ ( initial_a_b_c @ A ) @ P )
           != T12 )
         => ( ( ( target_a_b_c @ ( initial_a_b_c @ A ) @ P )
             != T23 )
           => ( member6330420149250801815od_b_c
              @ ( map_Pr776894537227705078od_b_c
                @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                @ P )
              @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M @ Q1 ) @ ( lS_a_b_c @ M @ Q22 ) ) ) ) ) ) ) ).

% is_separator_simps(13)
thf(fact_1008__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062io_A_092_060in_062_A_I_092_060Union_062_Iq_M_AP_J_092_060in_062prs_O_AL_AP_J_A_092_060Longrightarrow_062_Athesis_059_Aio_A_092_060in_062_A_092_060Union_062_A_123_I_064_J_A_Ip__io_Ap_J_A_096_Aset_A_Iprefixes_A_Ip__io_Apt_J_J_A_124p_Apt_O_A_092_060exists_062q_AP_O_A_Iq_M_AP_J_A_092_060in_062_Aprs_A_092_060and_062_Apath_AP_A_IFSM_Oinitial_AP_J_Ap_A_092_060and_062_Atarget_A_IFSM_Oinitial_AP_J_Ap_A_061_Aq_A_092_060and_062_Apt_A_092_060in_062_Atps_Aq_125_A_092_060Longrightarrow_062_Athesis_059_Aio_A_092_060in_062_A_092_060Union_062_A_123_I_092_060lambda_062io__atc_O_Ap__io_Ap_A_064_Ap__io_Apt_A_064_Aio__atc_J_A_096_Aatc__to__io__set_A_Ifrom__FSM_AM_A_Itarget_Aq_Apt_J_J_AA_A_124_Ap_Apt_Aq_AA_O_A_092_060exists_062P_Aq_H_At1_At2_O_A_Iq_M_AP_J_A_092_060in_062_Aprs_A_092_060and_062_Apath_AP_A_IFSM_Oinitial_AP_J_Ap_A_092_060and_062_Atarget_A_IFSM_Oinitial_AP_J_Ap_A_061_Aq_A_092_060and_062_Apt_A_092_060in_062_Atps_Aq_A_092_060and_062_Aq_H_A_092_060in_062_Ard__targets_A_Iq_M_Apt_J_A_092_060and_062_A_IA_M_At1_M_At2_J_A_092_060in_062_Aatcs_A_Itarget_Aq_Apt_M_Aq_H_J_125_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ( ~ ( member6330420149250801815od_b_c @ io
        @ ( comple1250193257546725051od_b_c
          @ ( image_2344789136157180295od_b_c
            @ ( produc6808532261061394109od_b_c
              @ ^ [Q4: a,P9: fsm_a_b_c] : ( lS_a_b_c @ P9 @ ( initial_a_b_c @ P9 ) ) )
            @ prs ) ) )
   => ( ~ ( member6330420149250801815od_b_c @ io
          @ ( comple1250193257546725051od_b_c
            @ ( collec1823247286932916153od_b_c
              @ ^ [Uu: set_li6436108459499378894od_b_c] :
                ? [P4: list_P6327159017948738492od_c_a,Pt2: list_P6327159017948738492od_c_a] :
                  ( ( Uu
                    = ( image_2850395784482391045od_b_c
                      @ ( append2547753245680614915od_b_c
                        @ ( map_Pr776894537227705078od_b_c
                          @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                          @ P4 ) )
                      @ ( set_li4480668622519654659od_b_c
                        @ ( prefix1131979855692807669od_b_c
                          @ ( map_Pr776894537227705078od_b_c
                            @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                            @ Pt2 ) ) ) ) )
                  & ? [P9: fsm_a_b_c] :
                      ( ( member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ ( target_a_b_c @ ( initial_a_b_c @ P9 ) @ P4 ) @ P9 ) @ prs )
                      & ( path_a_b_c @ P9 @ ( initial_a_b_c @ P9 ) @ P4 )
                      & ( member7410604586820865893od_c_a @ Pt2 @ ( tps @ ( target_a_b_c @ ( initial_a_b_c @ P9 ) @ P4 ) ) ) ) ) ) ) )
     => ( member6330420149250801815od_b_c @ io
        @ ( comple1250193257546725051od_b_c
          @ ( collec1823247286932916153od_b_c
            @ ^ [Uu: set_li6436108459499378894od_b_c] :
              ? [P4: list_P6327159017948738492od_c_a,Pt2: list_P6327159017948738492od_c_a,Q4: a,A6: fsm_d_b_c] :
                ( ( Uu
                  = ( image_2850395784482391045od_b_c
                    @ ^ [Io_atc: list_P903359562653991662od_b_c] :
                        ( append2547753245680614915od_b_c
                        @ ( map_Pr776894537227705078od_b_c
                          @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                          @ P4 )
                        @ ( append2547753245680614915od_b_c
                          @ ( map_Pr776894537227705078od_b_c
                            @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                            @ Pt2 )
                          @ Io_atc ) )
                    @ ( adapti7464086540814349463_b_c_d @ ( from_FSM_a_b_c @ m @ ( target_a_b_c @ Q4 @ Pt2 ) ) @ A6 ) ) )
                & ? [P9: fsm_a_b_c] :
                    ( ( member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ Q4 @ P9 ) @ prs )
                    & ( path_a_b_c @ P9 @ ( initial_a_b_c @ P9 ) @ P4 )
                    & ( ( target_a_b_c @ ( initial_a_b_c @ P9 ) @ P4 )
                      = Q4 )
                    & ( member7410604586820865893od_c_a @ Pt2 @ ( tps @ Q4 ) )
                    & ? [Q7: a] :
                        ( ( member_a @ Q7 @ ( rd_targets @ ( produc3580641852002036821od_c_a @ Q4 @ Pt2 ) ) )
                        & ? [T13: d,T24: d] : ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ A6 @ ( product_Pair_d_d @ T13 @ T24 ) ) @ ( atcs @ ( product_Pair_a_a @ ( target_a_b_c @ Q4 @ Pt2 ) @ Q7 ) ) ) ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. \<lbrakk>io \<in> (\<Union>(q, P)\<in>prs. L P) \<Longrightarrow> thesis; io \<in> \<Union> {(@) (p_io p) ` set (prefixes (p_io pt)) |p pt. \<exists>q P. (q, P) \<in> prs \<and> path P (FSM.initial P) p \<and> target (FSM.initial P) p = q \<and> pt \<in> tps q} \<Longrightarrow> thesis; io \<in> \<Union> {(\<lambda>io_atc. p_io p @ p_io pt @ io_atc) ` atc_to_io_set (from_FSM M (target q pt)) A | p pt q A. \<exists>P q' t1 t2. (q, P) \<in> prs \<and> path P (FSM.initial P) p \<and> target (FSM.initial P) p = q \<and> pt \<in> tps q \<and> q' \<in> rd_targets (q, pt) \<and> (A, t1, t2) \<in> atcs (target q pt, q')} \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_1009_paths__from__product__path_I3_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_fsm_a_b_c_b_c,P: list_P578107730873818684_a_b_c] :
      ( ( path_P6359343318551209683_c_b_c @ ( produc7620483080144554102_a_b_c @ A @ B ) @ ( initia3722918509273585852_c_b_c @ ( produc7620483080144554102_a_b_c @ A @ B ) ) @ P )
     => ( ( target_a_b_c @ ( initial_a_b_c @ A )
          @ ( map_Pr2496390191848272452od_c_a
            @ ^ [T: produc662032092171055542_a_b_c] : ( produc5762801557034676134od_c_a @ ( produc2620512763296580520_a_b_c @ ( produc2811850266904162578_a_b_c @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc3258500352807548343_a_b_c @ ( produc8362705341804890708_a_b_c @ T ) ) @ ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc7235443165838885369_a_b_c @ ( produc8362705341804890708_a_b_c @ T ) ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc7235443165838885369_a_b_c @ ( produc8362705341804890708_a_b_c @ T ) ) ) ) ) ) )
            @ P ) )
        = ( produc2620512763296580520_a_b_c @ ( target5407960577604541703_c_b_c @ ( initia3722918509273585852_c_b_c @ ( produc7620483080144554102_a_b_c @ A @ B ) ) @ P ) ) ) ) ).

% paths_from_product_path(3)
thf(fact_1010_paths__from__product__path_I3_J,axiom,
    ! [A: fsm_c_a_b,B: fsm_a_a_b,P: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) @ ( initia681974817753967189_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) ) @ P )
     => ( ( target_c_a_b @ ( initial_c_a_b @ A )
          @ ( map_Pr6135845499543628420od_b_c
            @ ^ [T: produc3205353004677322104od_c_a] : ( produc8300696160155326504od_b_c @ ( product_fst_c_a @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc6867761804911831425od_b_c @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
            @ P ) )
        = ( product_fst_c_a @ ( target6303326113586172192_a_a_b @ ( initia681974817753967189_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) ) @ P ) ) ) ) ).

% paths_from_product_path(3)
thf(fact_1011_paths__from__product__path_I4_J,axiom,
    ! [A: fsm_c_a_b,B: fsm_a_a_b,P: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) @ ( initia681974817753967189_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) ) @ P )
     => ( ( target_a_a_b @ ( initial_a_a_b @ B )
          @ ( map_Pr834105277175768324od_b_a
            @ ^ [T: produc3205353004677322104od_c_a] : ( produc7037890857971478952od_b_a @ ( product_snd_c_a @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc6867761796305373823od_b_a @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( product_Pair_b_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
            @ P ) )
        = ( product_snd_c_a @ ( target6303326113586172192_a_a_b @ ( initia681974817753967189_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) ) @ P ) ) ) ) ).

% paths_from_product_path(4)
thf(fact_1012_paths__from__product__path_I1_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_fsm_a_b_c_b_c,P: list_P578107730873818684_a_b_c] :
      ( ( path_P6359343318551209683_c_b_c @ ( produc7620483080144554102_a_b_c @ A @ B ) @ ( initia3722918509273585852_c_b_c @ ( produc7620483080144554102_a_b_c @ A @ B ) ) @ P )
     => ( path_a_b_c @ A @ ( initial_a_b_c @ A )
        @ ( map_Pr2496390191848272452od_c_a
          @ ^ [T: produc662032092171055542_a_b_c] : ( produc5762801557034676134od_c_a @ ( produc2620512763296580520_a_b_c @ ( produc2811850266904162578_a_b_c @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc3258500352807548343_a_b_c @ ( produc8362705341804890708_a_b_c @ T ) ) @ ( product_Pair_c_a @ ( produc9030635519781843663_a_b_c @ ( produc7235443165838885369_a_b_c @ ( produc8362705341804890708_a_b_c @ T ) ) ) @ ( produc2620512763296580520_a_b_c @ ( produc3897299621075439889_a_b_c @ ( produc7235443165838885369_a_b_c @ ( produc8362705341804890708_a_b_c @ T ) ) ) ) ) ) )
          @ P ) ) ) ).

% paths_from_product_path(1)
thf(fact_1013_paths__from__product__path_I1_J,axiom,
    ! [A: fsm_c_a_b,B: fsm_a_a_b,P: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) @ ( initia681974817753967189_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) ) @ P )
     => ( path_c_a_b @ A @ ( initial_c_a_b @ A )
        @ ( map_Pr6135845499543628420od_b_c
          @ ^ [T: produc3205353004677322104od_c_a] : ( produc8300696160155326504od_b_c @ ( product_fst_c_a @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc6867761804911831425od_b_c @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
          @ P ) ) ) ).

% paths_from_product_path(1)
thf(fact_1014_case__prod__conv,axiom,
    ! [F: a > fsm_a_b_c > set_li6436108459499378894od_b_c,A2: a,B2: fsm_a_b_c] :
      ( ( produc6808532261061394109od_b_c @ F @ ( produc2795441082327197628_a_b_c @ A2 @ B2 ) )
      = ( F @ A2 @ B2 ) ) ).

% case_prod_conv
thf(fact_1015_mem__case__prodI2,axiom,
    ! [P: product_prod_d_d,Z: a,C: d > d > set_a] :
      ( ! [A4: d,B4: d] :
          ( ( P
            = ( product_Pair_d_d @ A4 @ B4 ) )
         => ( member_a @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_a @ Z @ ( produc7855159438439358834_set_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1016_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z: a,C: b > c > set_a] :
      ( ! [A4: b,B4: c] :
          ( ( P
            = ( product_Pair_b_c @ A4 @ B4 ) )
         => ( member_a @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_a @ Z @ ( produc3958908706888076337_set_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1017_mem__case__prodI2,axiom,
    ! [P: product_prod_a_a,Z: a,C: a > a > set_a] :
      ( ! [A4: a,B4: a] :
          ( ( P
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( member_a @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_a @ Z @ ( produc9217457822752978994_set_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1018_mem__case__prodI2,axiom,
    ! [P: product_prod_d_d,Z: list_P903359562653991662od_b_c,C: d > d > set_li6436108459499378894od_b_c] :
      ( ! [A4: d,B4: d] :
          ( ( P
            = ( product_Pair_d_d @ A4 @ B4 ) )
         => ( member6330420149250801815od_b_c @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member6330420149250801815od_b_c @ Z @ ( produc7257933256910036962od_b_c @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1019_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z: list_P903359562653991662od_b_c,C: b > c > set_li6436108459499378894od_b_c] :
      ( ! [A4: b,B4: c] :
          ( ( P
            = ( product_Pair_b_c @ A4 @ B4 ) )
         => ( member6330420149250801815od_b_c @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member6330420149250801815od_b_c @ Z @ ( produc2955163486331930657od_b_c @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1020_mem__case__prodI2,axiom,
    ! [P: produc8838969699183119820_a_b_c,Z: a,C: a > fsm_a_b_c > set_a] :
      ( ! [A4: a,B4: fsm_a_b_c] :
          ( ( P
            = ( produc2795441082327197628_a_b_c @ A4 @ B4 ) )
         => ( member_a @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member_a @ Z @ ( produc2268483704214043853_set_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1021_mem__case__prodI2,axiom,
    ! [P: product_prod_a_a,Z: list_P903359562653991662od_b_c,C: a > a > set_li6436108459499378894od_b_c] :
      ( ! [A4: a,B4: a] :
          ( ( P
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( member6330420149250801815od_b_c @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member6330420149250801815od_b_c @ Z @ ( produc6232784878203098786od_b_c @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1022_mem__case__prodI2,axiom,
    ! [P: product_prod_d_d,Z: produc8838969699183119820_a_b_c,C: d > d > set_Pr2640454425625433516_a_b_c] :
      ( ! [A4: d,B4: d] :
          ( ( P
            = ( product_Pair_d_d @ A4 @ B4 ) )
         => ( member997585926662434933_a_b_c @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member997585926662434933_a_b_c @ Z @ ( produc1681794554106568896_a_b_c @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1023_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z: produc8838969699183119820_a_b_c,C: b > c > set_Pr2640454425625433516_a_b_c] :
      ( ! [A4: b,B4: c] :
          ( ( P
            = ( product_Pair_b_c @ A4 @ B4 ) )
         => ( member997585926662434933_a_b_c @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member997585926662434933_a_b_c @ Z @ ( produc8075048271621630975_a_b_c @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1024_mem__case__prodI2,axiom,
    ! [P: product_prod_a_a,Z: produc8838969699183119820_a_b_c,C: a > a > set_Pr2640454425625433516_a_b_c] :
      ( ! [A4: a,B4: a] :
          ( ( P
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( member997585926662434933_a_b_c @ Z @ ( C @ A4 @ B4 ) ) )
     => ( member997585926662434933_a_b_c @ Z @ ( produc4963144222241986432_a_b_c @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1025_mem__case__prodI,axiom,
    ! [Z: a,C: d > d > set_a,A2: d,B2: d] :
      ( ( member_a @ Z @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z @ ( produc7855159438439358834_set_a @ C @ ( product_Pair_d_d @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1026_mem__case__prodI,axiom,
    ! [Z: a,C: b > c > set_a,A2: b,B2: c] :
      ( ( member_a @ Z @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z @ ( produc3958908706888076337_set_a @ C @ ( product_Pair_b_c @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1027_mem__case__prodI,axiom,
    ! [Z: a,C: a > a > set_a,A2: a,B2: a] :
      ( ( member_a @ Z @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z @ ( produc9217457822752978994_set_a @ C @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1028_mem__case__prodI,axiom,
    ! [Z: list_P903359562653991662od_b_c,C: d > d > set_li6436108459499378894od_b_c,A2: d,B2: d] :
      ( ( member6330420149250801815od_b_c @ Z @ ( C @ A2 @ B2 ) )
     => ( member6330420149250801815od_b_c @ Z @ ( produc7257933256910036962od_b_c @ C @ ( product_Pair_d_d @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1029_mem__case__prodI,axiom,
    ! [Z: list_P903359562653991662od_b_c,C: b > c > set_li6436108459499378894od_b_c,A2: b,B2: c] :
      ( ( member6330420149250801815od_b_c @ Z @ ( C @ A2 @ B2 ) )
     => ( member6330420149250801815od_b_c @ Z @ ( produc2955163486331930657od_b_c @ C @ ( product_Pair_b_c @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1030_mem__case__prodI,axiom,
    ! [Z: a,C: a > fsm_a_b_c > set_a,A2: a,B2: fsm_a_b_c] :
      ( ( member_a @ Z @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z @ ( produc2268483704214043853_set_a @ C @ ( produc2795441082327197628_a_b_c @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1031_mem__case__prodI,axiom,
    ! [Z: list_P903359562653991662od_b_c,C: a > a > set_li6436108459499378894od_b_c,A2: a,B2: a] :
      ( ( member6330420149250801815od_b_c @ Z @ ( C @ A2 @ B2 ) )
     => ( member6330420149250801815od_b_c @ Z @ ( produc6232784878203098786od_b_c @ C @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1032_mem__case__prodI,axiom,
    ! [Z: produc8838969699183119820_a_b_c,C: d > d > set_Pr2640454425625433516_a_b_c,A2: d,B2: d] :
      ( ( member997585926662434933_a_b_c @ Z @ ( C @ A2 @ B2 ) )
     => ( member997585926662434933_a_b_c @ Z @ ( produc1681794554106568896_a_b_c @ C @ ( product_Pair_d_d @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1033_mem__case__prodI,axiom,
    ! [Z: produc8838969699183119820_a_b_c,C: b > c > set_Pr2640454425625433516_a_b_c,A2: b,B2: c] :
      ( ( member997585926662434933_a_b_c @ Z @ ( C @ A2 @ B2 ) )
     => ( member997585926662434933_a_b_c @ Z @ ( produc8075048271621630975_a_b_c @ C @ ( product_Pair_b_c @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1034_mem__case__prodI,axiom,
    ! [Z: produc8838969699183119820_a_b_c,C: a > a > set_Pr2640454425625433516_a_b_c,A2: a,B2: a] :
      ( ( member997585926662434933_a_b_c @ Z @ ( C @ A2 @ B2 ) )
     => ( member997585926662434933_a_b_c @ Z @ ( produc4963144222241986432_a_b_c @ C @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1035_pair__imageI,axiom,
    ! [A2: d,B2: d,A: set_Product_prod_d_d,F: d > d > a] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_9096282059794114140_d_d_a @ ( produc8018803990352283666_d_d_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1036_pair__imageI,axiom,
    ! [A2: b,B2: c,A: set_Product_prod_b_c,F: b > c > a] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_7762865866877615259_b_c_a @ ( produc8837194378803474129_b_c_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1037_pair__imageI,axiom,
    ! [A2: a,B2: a,A: set_Product_prod_a_a,F: a > a > a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_3437945252899457948_a_a_a @ ( produc8815886927560695506_a_a_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1038_pair__imageI,axiom,
    ! [A2: d,B2: d,A: set_Product_prod_d_d,F: d > d > list_P903359562653991662od_b_c] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ A2 @ B2 ) @ A )
     => ( member6330420149250801815od_b_c @ ( F @ A2 @ B2 ) @ ( image_5194547870870335820od_b_c @ ( produc4949400892932413954od_b_c @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1039_pair__imageI,axiom,
    ! [A2: b,B2: c,A: set_Product_prod_b_c,F: b > c > list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ A2 @ B2 ) @ A )
     => ( member6330420149250801815od_b_c @ ( F @ A2 @ B2 ) @ ( image_5872333176446027019od_b_c @ ( produc2916551642269384257od_b_c @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1040_pair__imageI,axiom,
    ! [A2: a,B2: fsm_a_b_c,A: set_Pr2640454425625433516_a_b_c,F: a > fsm_a_b_c > a] :
      ( ( member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_3273489647386116407_b_c_a @ ( produc7563927974110973805_b_c_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1041_pair__imageI,axiom,
    ! [A2: a,B2: a,A: set_Product_prod_a_a,F: a > a > list_P903359562653991662od_b_c] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ A )
     => ( member6330420149250801815od_b_c @ ( F @ A2 @ B2 ) @ ( image_4717466449392219788od_b_c @ ( produc5563787839592911042od_b_c @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1042_pair__imageI,axiom,
    ! [A2: fsm_d_b_c,B2: product_prod_d_d,A: set_Pr9130538228769002824od_d_d,F: fsm_d_b_c > product_prod_d_d > a] :
      ( ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_3229965221763006363_d_d_a @ ( produc8306943490236241617_d_d_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1043_pair__imageI,axiom,
    ! [A2: d,B2: d,A: set_Product_prod_d_d,F: d > d > produc8838969699183119820_a_b_c] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ A2 @ B2 ) @ A )
     => ( member997585926662434933_a_b_c @ ( F @ A2 @ B2 ) @ ( image_2733709216643310122_a_b_c @ ( produc4866288027762747616_a_b_c @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1044_pair__imageI,axiom,
    ! [A2: b,B2: c,A: set_Product_prod_b_c,F: b > c > produc8838969699183119820_a_b_c] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ A2 @ B2 ) @ A )
     => ( member997585926662434933_a_b_c @ ( F @ A2 @ B2 ) @ ( image_1679953693920893161_a_b_c @ ( produc4109512363540721695_a_b_c @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1045_product__simps_I1_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_li6382094382827256929_a_b_c] :
      ( ( initia8976879869237502037_a_b_c @ ( produc2528431742188625167od_c_a @ A @ B ) )
      = ( produc3580641852002036821od_c_a @ ( initial_a_b_c @ A ) @ ( initia2228646373960377772_a_b_c @ B ) ) ) ).

% product_simps(1)
thf(fact_1046_product__simps_I1_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_fsm_a_b_c_b_c] :
      ( ( initia3722918509273585852_c_b_c @ ( produc7620483080144554102_a_b_c @ A @ B ) )
      = ( produc2795441082327197628_a_b_c @ ( initial_a_b_c @ A ) @ ( initia4712012503747629459_c_b_c @ B ) ) ) ).

% product_simps(1)
thf(fact_1047_product__simps_I1_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( initia4141967198046469911_a_b_c @ ( produc2510185919789642449_b_c_a @ A @ B ) )
      = ( product_Pair_a_a @ ( initial_a_b_c @ A ) @ ( initial_a_b_c @ B ) ) ) ).

% product_simps(1)
thf(fact_1048_prefixes__prefixes,axiom,
    ! [Ys: list_P6327159017948738492od_c_a,Xs: list_P6327159017948738492od_c_a,Zs: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ Ys @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) ) )
     => ( ( member7410604586820865893od_c_a @ Zs @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) ) )
       => ( ( member7410604586820865893od_c_a @ Ys @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Zs ) ) )
          | ( member7410604586820865893od_c_a @ Zs @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Ys ) ) ) ) ) ) ).

% prefixes_prefixes
thf(fact_1049_prefixes__prefixes,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Ys @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) )
     => ( ( member6330420149250801815od_b_c @ Zs @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) )
       => ( ( member6330420149250801815od_b_c @ Ys @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Zs ) ) )
          | ( member6330420149250801815od_b_c @ Zs @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Ys ) ) ) ) ) ) ).

% prefixes_prefixes
thf(fact_1050_prod_Ocase__distrib,axiom,
    ! [H: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,F: a > fsm_a_b_c > set_li6436108459499378894od_b_c,Prod: produc8838969699183119820_a_b_c] :
      ( ( H @ ( produc6808532261061394109od_b_c @ F @ Prod ) )
      = ( produc6808532261061394109od_b_c
        @ ^ [X13: a,X24: fsm_a_b_c] : ( H @ ( F @ X13 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_1051_mem__case__prodE,axiom,
    ! [Z: a,C: d > d > set_a,P: product_prod_d_d] :
      ( ( member_a @ Z @ ( produc7855159438439358834_set_a @ C @ P ) )
     => ~ ! [X: d,Y3: d] :
            ( ( P
              = ( product_Pair_d_d @ X @ Y3 ) )
           => ~ ( member_a @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1052_mem__case__prodE,axiom,
    ! [Z: a,C: b > c > set_a,P: product_prod_b_c] :
      ( ( member_a @ Z @ ( produc3958908706888076337_set_a @ C @ P ) )
     => ~ ! [X: b,Y3: c] :
            ( ( P
              = ( product_Pair_b_c @ X @ Y3 ) )
           => ~ ( member_a @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1053_mem__case__prodE,axiom,
    ! [Z: a,C: a > a > set_a,P: product_prod_a_a] :
      ( ( member_a @ Z @ ( produc9217457822752978994_set_a @ C @ P ) )
     => ~ ! [X: a,Y3: a] :
            ( ( P
              = ( product_Pair_a_a @ X @ Y3 ) )
           => ~ ( member_a @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1054_mem__case__prodE,axiom,
    ! [Z: list_P903359562653991662od_b_c,C: d > d > set_li6436108459499378894od_b_c,P: product_prod_d_d] :
      ( ( member6330420149250801815od_b_c @ Z @ ( produc7257933256910036962od_b_c @ C @ P ) )
     => ~ ! [X: d,Y3: d] :
            ( ( P
              = ( product_Pair_d_d @ X @ Y3 ) )
           => ~ ( member6330420149250801815od_b_c @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1055_mem__case__prodE,axiom,
    ! [Z: list_P903359562653991662od_b_c,C: b > c > set_li6436108459499378894od_b_c,P: product_prod_b_c] :
      ( ( member6330420149250801815od_b_c @ Z @ ( produc2955163486331930657od_b_c @ C @ P ) )
     => ~ ! [X: b,Y3: c] :
            ( ( P
              = ( product_Pair_b_c @ X @ Y3 ) )
           => ~ ( member6330420149250801815od_b_c @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1056_mem__case__prodE,axiom,
    ! [Z: a,C: a > fsm_a_b_c > set_a,P: produc8838969699183119820_a_b_c] :
      ( ( member_a @ Z @ ( produc2268483704214043853_set_a @ C @ P ) )
     => ~ ! [X: a,Y3: fsm_a_b_c] :
            ( ( P
              = ( produc2795441082327197628_a_b_c @ X @ Y3 ) )
           => ~ ( member_a @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1057_mem__case__prodE,axiom,
    ! [Z: list_P903359562653991662od_b_c,C: a > a > set_li6436108459499378894od_b_c,P: product_prod_a_a] :
      ( ( member6330420149250801815od_b_c @ Z @ ( produc6232784878203098786od_b_c @ C @ P ) )
     => ~ ! [X: a,Y3: a] :
            ( ( P
              = ( product_Pair_a_a @ X @ Y3 ) )
           => ~ ( member6330420149250801815od_b_c @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1058_mem__case__prodE,axiom,
    ! [Z: produc8838969699183119820_a_b_c,C: d > d > set_Pr2640454425625433516_a_b_c,P: product_prod_d_d] :
      ( ( member997585926662434933_a_b_c @ Z @ ( produc1681794554106568896_a_b_c @ C @ P ) )
     => ~ ! [X: d,Y3: d] :
            ( ( P
              = ( product_Pair_d_d @ X @ Y3 ) )
           => ~ ( member997585926662434933_a_b_c @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1059_mem__case__prodE,axiom,
    ! [Z: produc8838969699183119820_a_b_c,C: b > c > set_Pr2640454425625433516_a_b_c,P: product_prod_b_c] :
      ( ( member997585926662434933_a_b_c @ Z @ ( produc8075048271621630975_a_b_c @ C @ P ) )
     => ~ ! [X: b,Y3: c] :
            ( ( P
              = ( product_Pair_b_c @ X @ Y3 ) )
           => ~ ( member997585926662434933_a_b_c @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1060_mem__case__prodE,axiom,
    ! [Z: produc8838969699183119820_a_b_c,C: a > a > set_Pr2640454425625433516_a_b_c,P: product_prod_a_a] :
      ( ( member997585926662434933_a_b_c @ Z @ ( produc4963144222241986432_a_b_c @ C @ P ) )
     => ~ ! [X: a,Y3: a] :
            ( ( P
              = ( product_Pair_a_a @ X @ Y3 ) )
           => ~ ( member997585926662434933_a_b_c @ Z @ ( C @ X @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_1061_map2__map__map,axiom,
    ! [H: product_prod_b_c > product_prod_b_c > product_prod_b_c,F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a,G: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( map_Pr4341431729436552389od_b_c @ ( produc1457153799263121084od_b_c @ H ) @ ( zip_Pr9032920399426446775od_b_c @ ( map_Pr776894537227705078od_b_c @ F @ Xs ) @ ( map_Pr776894537227705078od_b_c @ G @ Xs ) ) )
      = ( map_Pr776894537227705078od_b_c
        @ ^ [X3: produc2669932201312201782od_c_a] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1062_map2__map__map,axiom,
    ! [H: a > produc2133359284458116493od_c_a > product_prod_b_c,F: produc2669932201312201782od_c_a > a,Xs: list_P6327159017948738492od_c_a,G: produc2669932201312201782od_c_a > produc2133359284458116493od_c_a] :
      ( ( map_Pr776894537227705078od_b_c @ ( produc8782340964261885549od_b_c @ H ) @ ( zip_a_8428819975400564422od_c_a @ ( map_Pr5179659585931931020_c_a_a @ F @ Xs ) @ ( map_Pr3684547939035685915od_c_a @ G @ Xs ) ) )
      = ( map_Pr776894537227705078od_b_c
        @ ^ [X3: produc2669932201312201782od_c_a] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1063_old_Oprod_Ocase,axiom,
    ! [F: a > fsm_a_b_c > set_li6436108459499378894od_b_c,X1: a,X22: fsm_a_b_c] :
      ( ( produc6808532261061394109od_b_c @ F @ ( produc2795441082327197628_a_b_c @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_1064_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_1065_prefixes__set__Nil,axiom,
    ! [Xs: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) ) ).

% prefixes_set_Nil
thf(fact_1066_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_1067_prefixes__set__ob,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Xss2: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xss2 ) ) )
     => ~ ! [Xs5: list_P903359562653991662od_b_c] :
            ( Xss2
           != ( append2547753245680614915od_b_c @ Xs @ Xs5 ) ) ) ).

% prefixes_set_ob
thf(fact_1068_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_1069_prefixes__prepend,axiom,
    ! [Xs6: list_P903359562653991662od_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Xs6 @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) )
     => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Ys @ Xs6 ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ ( append2547753245680614915od_b_c @ Ys @ Xs ) ) ) ) ) ).

% prefixes_prepend
thf(fact_1070_prefixes__prefix__subset,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Xs @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Ys ) ) )
     => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Ys ) ) ) ) ).

% prefixes_prefix_subset
thf(fact_1071_prefixes__prefix__subset,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ Xs @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Ys ) ) )
     => ( ord_le2998388488506175548od_c_a @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) ) @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Ys ) ) ) ) ).

% prefixes_prefix_subset
thf(fact_1072_cond__case__prod__eta,axiom,
    ! [F: a > fsm_a_b_c > set_li6436108459499378894od_b_c,G: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c] :
      ( ! [X: a,Y3: fsm_a_b_c] :
          ( ( F @ X @ Y3 )
          = ( G @ ( produc2795441082327197628_a_b_c @ X @ Y3 ) ) )
     => ( ( produc6808532261061394109od_b_c @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_1073_case__prod__eta,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c] :
      ( ( produc6808532261061394109od_b_c
        @ ^ [X3: a,Y5: fsm_a_b_c] : ( F @ ( produc2795441082327197628_a_b_c @ X3 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_1074_case__prodE2,axiom,
    ! [Q5: set_li6436108459499378894od_b_c > $o,P5: a > fsm_a_b_c > set_li6436108459499378894od_b_c,Z: produc8838969699183119820_a_b_c] :
      ( ( Q5 @ ( produc6808532261061394109od_b_c @ P5 @ Z ) )
     => ~ ! [X: a,Y3: fsm_a_b_c] :
            ( ( Z
              = ( produc2795441082327197628_a_b_c @ X @ Y3 ) )
           => ~ ( Q5 @ ( P5 @ X @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_1075_snd__def,axiom,
    ( produc1983107199584856133_set_a
    = ( produc4474087116711199794_set_a
      @ ^ [X13: set_a,X24: set_a] : X24 ) ) ).

% snd_def
thf(fact_1076_snd__def,axiom,
    ( produc8088130389382527188od_c_a
    = ( produc3172194304981725330od_c_a
      @ ^ [X13: a,X24: produc2133359284458116493od_c_a] : X24 ) ) ).

% snd_def
thf(fact_1077_snd__def,axiom,
    ( produc5221446324160585515od_c_a
    = ( produc5632948794012708883od_c_a
      @ ^ [X13: b,X24: product_prod_c_a] : X24 ) ) ).

% snd_def
thf(fact_1078_fst__def,axiom,
    ( produc9088895665703139587_set_a
    = ( produc4474087116711199794_set_a
      @ ^ [X13: set_a,X24: set_a] : X13 ) ) ).

% fst_def
thf(fact_1079_fst__def,axiom,
    ( produc2620512763296580520_a_b_c
    = ( produc7563927974110973805_b_c_a
      @ ^ [X13: a,X24: fsm_a_b_c] : X13 ) ) ).

% fst_def
thf(fact_1080_fst__def,axiom,
    ( produc301949074746585577od_c_a
    = ( produc2354600891577745261_c_a_b
      @ ^ [X13: b,X24: product_prod_c_a] : X13 ) ) ).

% fst_def
thf(fact_1081_fst__def,axiom,
    ( product_fst_c_a
    = ( produc1561580084227405270_c_a_c
      @ ^ [X13: c,X24: a] : X13 ) ) ).

% fst_def
thf(fact_1082_fst__def,axiom,
    ( produc3242630202250842350_set_a
    = ( produc717347551696018661od_c_a
      @ ^ [X13: list_P6327159017948738492od_c_a,X24: produc1703568184450464039_set_a] : X13 ) ) ).

% fst_def
thf(fact_1083_map__zip__map2,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_a,G: produc2133359284458116493od_c_a > produc2133359284458116493od_c_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( map_Pr776894537227705078od_b_c @ F @ ( zip_a_8428819975400564422od_c_a @ Xs @ ( map_Pr526699465062943108od_c_a @ G @ Ys ) ) )
      = ( map_Pr776894537227705078od_b_c
        @ ( produc8782340964261885549od_b_c
          @ ^ [X3: a,Y5: produc2133359284458116493od_c_a] : ( F @ ( produc5762801557034676134od_c_a @ X3 @ ( G @ Y5 ) ) ) )
        @ ( zip_a_8428819975400564422od_c_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_1084_map__zip__map2,axiom,
    ! [F: produc3220416805844688357od_c_a > product_prod_b_c,Xs: list_a,G: produc2133359284458116493od_c_a > list_P6327159017948738492od_c_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( map_Pr5832279806847121031od_b_c @ F @ ( zip_a_4713777678317597557od_c_a @ Xs @ ( map_Pr6217181894889484595od_c_a @ G @ Ys ) ) )
      = ( map_Pr776894537227705078od_b_c
        @ ( produc8782340964261885549od_b_c
          @ ^ [X3: a,Y5: produc2133359284458116493od_c_a] : ( F @ ( produc3580641852002036821od_c_a @ X3 @ ( G @ Y5 ) ) ) )
        @ ( zip_a_8428819975400564422od_c_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_1085_map__zip__map2,axiom,
    ! [F: produc8838969699183119820_a_b_c > product_prod_b_c,Xs: list_a,G: produc2133359284458116493od_c_a > fsm_a_b_c,Ys: list_P5767727266746070163od_c_a] :
      ( ( map_Pr5727606280510735712od_b_c @ F @ ( zip_a_fsm_a_b_c @ Xs @ ( map_Pr605483782884719770_a_b_c @ G @ Ys ) ) )
      = ( map_Pr776894537227705078od_b_c
        @ ( produc8782340964261885549od_b_c
          @ ^ [X3: a,Y5: produc2133359284458116493od_c_a] : ( F @ ( produc2795441082327197628_a_b_c @ X3 @ ( G @ Y5 ) ) ) )
        @ ( zip_a_8428819975400564422od_c_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_1086_map__zip__map2,axiom,
    ! [F: product_prod_a_a > product_prod_b_c,Xs: list_a,G: produc2133359284458116493od_c_a > a,Ys: list_P5767727266746070163od_c_a] :
      ( ( map_Pr5116787512092568389od_b_c @ F @ ( zip_a_a @ Xs @ ( map_Pr3016883280456414453_c_a_a @ G @ Ys ) ) )
      = ( map_Pr776894537227705078od_b_c
        @ ( produc8782340964261885549od_b_c
          @ ^ [X3: a,Y5: produc2133359284458116493od_c_a] : ( F @ ( product_Pair_a_a @ X3 @ ( G @ Y5 ) ) ) )
        @ ( zip_a_8428819975400564422od_c_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_1087_map__zip__map2,axiom,
    ! [F: produc3220416805844688357od_c_a > set_li6436108459499378894od_b_c,Xs: list_a,G: fsm_a_b_c > list_P6327159017948738492od_c_a,Ys: list_fsm_a_b_c] :
      ( ( map_Pr8995122597089061229od_b_c @ F @ ( zip_a_4713777678317597557od_c_a @ Xs @ ( map_fs536842135048473245od_c_a @ G @ Ys ) ) )
      = ( map_Pr4360419603926651846od_b_c
        @ ( produc6808532261061394109od_b_c
          @ ^ [X3: a,Y5: fsm_a_b_c] : ( F @ ( produc3580641852002036821od_c_a @ X3 @ ( G @ Y5 ) ) ) )
        @ ( zip_a_fsm_a_b_c @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_1088_map__zip__map2,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,Xs: list_a,G: fsm_a_b_c > fsm_a_b_c,Ys: list_fsm_a_b_c] :
      ( ( map_Pr4360419603926651846od_b_c @ F @ ( zip_a_fsm_a_b_c @ Xs @ ( map_fs6831279280335390212_a_b_c @ G @ Ys ) ) )
      = ( map_Pr4360419603926651846od_b_c
        @ ( produc6808532261061394109od_b_c
          @ ^ [X3: a,Y5: fsm_a_b_c] : ( F @ ( produc2795441082327197628_a_b_c @ X3 @ ( G @ Y5 ) ) ) )
        @ ( zip_a_fsm_a_b_c @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_1089_map__zip__map2,axiom,
    ! [F: product_prod_a_a > set_li6436108459499378894od_b_c,Xs: list_a,G: fsm_a_b_c > a,Ys: list_fsm_a_b_c] :
      ( ( map_Pr3648903439221452587od_b_c @ F @ ( zip_a_a @ Xs @ ( map_fsm_a_b_c_a @ G @ Ys ) ) )
      = ( map_Pr4360419603926651846od_b_c
        @ ( produc6808532261061394109od_b_c
          @ ^ [X3: a,Y5: fsm_a_b_c] : ( F @ ( product_Pair_a_a @ X3 @ ( G @ Y5 ) ) ) )
        @ ( zip_a_fsm_a_b_c @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_1090_map__zip__map,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,G: a > a,Xs: list_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( map_Pr776894537227705078od_b_c @ F @ ( zip_a_8428819975400564422od_c_a @ ( map_a_a @ G @ Xs ) @ Ys ) )
      = ( map_Pr776894537227705078od_b_c
        @ ( produc8782340964261885549od_b_c
          @ ^ [X3: a,Y5: produc2133359284458116493od_c_a] : ( F @ ( produc5762801557034676134od_c_a @ ( G @ X3 ) @ Y5 ) ) )
        @ ( zip_a_8428819975400564422od_c_a @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_1091_map__zip__map,axiom,
    ! [F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,G: a > a,Xs: list_a,Ys: list_fsm_a_b_c] :
      ( ( map_Pr4360419603926651846od_b_c @ F @ ( zip_a_fsm_a_b_c @ ( map_a_a @ G @ Xs ) @ Ys ) )
      = ( map_Pr4360419603926651846od_b_c
        @ ( produc6808532261061394109od_b_c
          @ ^ [X3: a,Y5: fsm_a_b_c] : ( F @ ( produc2795441082327197628_a_b_c @ ( G @ X3 ) @ Y5 ) ) )
        @ ( zip_a_fsm_a_b_c @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_1092_split__beta,axiom,
    ( produc6808532261061394109od_b_c
    = ( ^ [F5: a > fsm_a_b_c > set_li6436108459499378894od_b_c,Prod3: produc8838969699183119820_a_b_c] : ( F5 @ ( produc2620512763296580520_a_b_c @ Prod3 ) @ ( produc4728557313310798826_a_b_c @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_1093_case__prod__beta,axiom,
    ( produc6808532261061394109od_b_c
    = ( ^ [F5: a > fsm_a_b_c > set_li6436108459499378894od_b_c,P4: produc8838969699183119820_a_b_c] : ( F5 @ ( produc2620512763296580520_a_b_c @ P4 ) @ ( produc4728557313310798826_a_b_c @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_1094_prefixes__prefix__suffix__ob,axiom,
    ! [A2: list_P6327159017948738492od_c_a,B2: list_P6327159017948738492od_c_a,C: list_P6327159017948738492od_c_a] :
      ( ( member7410604586820865893od_c_a @ A2 @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ ( append3884164603879732689od_c_a @ B2 @ C ) ) ) )
     => ( ~ ( member7410604586820865893od_c_a @ A2 @ ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ B2 ) ) )
       => ~ ! [C4: list_P6327159017948738492od_c_a] :
              ( ? [C5: list_P6327159017948738492od_c_a] :
                  ( C
                  = ( append3884164603879732689od_c_a @ C4 @ C5 ) )
             => ( ( A2
                  = ( append3884164603879732689od_c_a @ B2 @ C4 ) )
               => ( C4 = nil_Pr1342775757158464060od_c_a ) ) ) ) ) ).

% prefixes_prefix_suffix_ob
thf(fact_1095_prefixes__prefix__suffix__ob,axiom,
    ! [A2: list_P903359562653991662od_b_c,B2: list_P903359562653991662od_b_c,C: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ A2 @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ ( append2547753245680614915od_b_c @ B2 @ C ) ) ) )
     => ( ~ ( member6330420149250801815od_b_c @ A2 @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ B2 ) ) )
       => ~ ! [C4: list_P903359562653991662od_b_c] :
              ( ? [C5: list_P903359562653991662od_b_c] :
                  ( C
                  = ( append2547753245680614915od_b_c @ C4 @ C5 ) )
             => ( ( A2
                  = ( append2547753245680614915od_b_c @ B2 @ C4 ) )
               => ( C4 = nil_Product_prod_b_c ) ) ) ) ) ).

% prefixes_prefix_suffix_ob
thf(fact_1096_zip__same__conv__map,axiom,
    ! [Xs: list_d] :
      ( ( zip_d_d @ Xs @ Xs )
      = ( map_d_6802736565751565034od_d_d
        @ ^ [X3: d] : ( product_Pair_d_d @ X3 @ X3 )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_1097_zip__same__conv__map,axiom,
    ! [Xs: list_a] :
      ( ( zip_a_a @ Xs @ Xs )
      = ( map_a_7860052162900579309od_a_a
        @ ^ [X3: a] : ( product_Pair_a_a @ X3 @ X3 )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_1098_prefixes__set__subset,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] : ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) ) @ ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ ( append2547753245680614915od_b_c @ Xs @ Ys ) ) ) ) ).

% prefixes_set_subset
thf(fact_1099_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_1100_case__prod__beta_H,axiom,
    ( produc6808532261061394109od_b_c
    = ( ^ [F5: a > fsm_a_b_c > set_li6436108459499378894od_b_c,X3: produc8838969699183119820_a_b_c] : ( F5 @ ( produc2620512763296580520_a_b_c @ X3 ) @ ( produc4728557313310798826_a_b_c @ X3 ) ) ) ) ).

% case_prod_beta'
thf(fact_1101_case__prod__unfold,axiom,
    ( produc6808532261061394109od_b_c
    = ( ^ [C6: a > fsm_a_b_c > set_li6436108459499378894od_b_c,P4: produc8838969699183119820_a_b_c] : ( C6 @ ( produc2620512763296580520_a_b_c @ P4 ) @ ( produc4728557313310798826_a_b_c @ P4 ) ) ) ) ).

% case_prod_unfold
thf(fact_1102_exE__realizer,axiom,
    ! [P5: fsm_a_b_c > a > $o,P: produc8838969699183119820_a_b_c,Q5: set_li6436108459499378894od_b_c > $o,F: a > fsm_a_b_c > set_li6436108459499378894od_b_c] :
      ( ( P5 @ ( produc4728557313310798826_a_b_c @ P ) @ ( produc2620512763296580520_a_b_c @ P ) )
     => ( ! [X: a,Y3: fsm_a_b_c] :
            ( ( P5 @ Y3 @ X )
           => ( Q5 @ ( F @ X @ Y3 ) ) )
       => ( Q5 @ ( produc6808532261061394109od_b_c @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_1103_prefixes__set,axiom,
    ! [Xs: list_P6327159017948738492od_c_a] :
      ( ( set_li8663184538779258833od_c_a @ ( prefix8878233912999784899od_c_a @ Xs ) )
      = ( collec6273869032445462695od_c_a
        @ ^ [Xs7: list_P6327159017948738492od_c_a] :
          ? [Xs8: list_P6327159017948738492od_c_a] :
            ( ( append3884164603879732689od_c_a @ Xs7 @ Xs8 )
            = Xs ) ) ) ).

% prefixes_set
thf(fact_1104_prefixes__set,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( prefix1131979855692807669od_b_c @ Xs ) )
      = ( collec2280997390073109977od_b_c
        @ ^ [Xs7: list_P903359562653991662od_b_c] :
          ? [Xs8: list_P903359562653991662od_b_c] :
            ( ( append2547753245680614915od_b_c @ Xs7 @ Xs8 )
            = Xs ) ) ) ).

% prefixes_set
thf(fact_1105_zip__map__fst__snd,axiom,
    ! [Zs: list_P5219093185649440210_a_b_c] :
      ( ( zip_a_fsm_a_b_c @ ( map_Pr811584189524267510_b_c_a @ produc2620512763296580520_a_b_c @ Zs ) @ ( map_Pr6813647754772709915_a_b_c @ produc4728557313310798826_a_b_c @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1106_zip__map__fst__snd,axiom,
    ! [Zs: list_P125642481956313003od_c_a] :
      ( ( zip_c_a @ ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Zs ) @ ( map_Pr5866436826731527133_c_a_a @ product_snd_c_a @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1107_zip__map__fst__snd,axiom,
    ! [Zs: list_P4908235088432371736_set_a] :
      ( ( zip_li7800925431565461538_set_a @ ( map_Pr6055023778478330862od_c_a @ produc3242630202250842350_set_a @ Zs ) @ ( map_Pr2257100352023006809_set_a @ produc2451121746714373936_set_a @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1108_zip__map__fst__snd,axiom,
    ! [Zs: list_P3660316430366008877_set_a] :
      ( ( zip_set_a_set_a @ ( map_Pr8866779738092938811_set_a @ produc9088895665703139587_set_a @ Zs ) @ ( map_Pr8866779738092938811_set_a @ produc1983107199584856133_set_a @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1109_zip__map__fst__snd,axiom,
    ! [Zs: list_P6327159017948738492od_c_a] :
      ( ( zip_a_8428819975400564422od_c_a @ ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Zs ) @ ( map_Pr3684547939035685915od_c_a @ produc8088130389382527188od_c_a @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1110_zip__map__fst__snd,axiom,
    ! [Zs: list_P5767727266746070163od_c_a] :
      ( ( zip_b_6857355019916857117od_c_a @ ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Zs ) @ ( map_Pr6684165712643126428od_c_a @ produc5221446324160585515od_c_a @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1111_set__zip__rightD,axiom,
    ! [X2: fsm_d_b_c,Y: product_prod_d_d,Xs: list_fsm_d_b_c,Ys: list_P8926467638345140077od_d_d] :
      ( ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ X2 @ Y ) @ ( set_Pr5977834532684968573od_d_d @ ( zip_fs8624885554530663032od_d_d @ Xs @ Ys ) ) )
     => ( member2287536781371869328od_d_d @ Y @ ( set_Product_prod_d_d2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1112_set__zip__rightD,axiom,
    ! [X2: set_a,Y: set_a,Xs: list_set_a,Ys: list_set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( zip_set_a_set_a @ Xs @ Ys ) ) )
     => ( member_set_a @ Y @ ( set_set_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1113_set__zip__rightD,axiom,
    ! [X2: d,Y: d,Xs: list_d,Ys: list_d] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( zip_d_d @ Xs @ Ys ) ) )
     => ( member_d @ Y @ ( set_d2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1114_set__zip__rightD,axiom,
    ! [X2: b,Y: c,Xs: list_b,Ys: list_c] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ ( set_Product_prod_b_c2 @ ( zip_b_c @ Xs @ Ys ) ) )
     => ( member_c @ Y @ ( set_c2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1115_set__zip__rightD,axiom,
    ! [X2: a,Y: list_P6327159017948738492od_c_a,Xs: list_a,Ys: list_l2826717958560394946od_c_a] :
      ( ( member3484049051949443214od_c_a @ ( produc3580641852002036821od_c_a @ X2 @ Y ) @ ( set_Pr5079464637073054970od_c_a @ ( zip_a_4713777678317597557od_c_a @ Xs @ Ys ) ) )
     => ( member7410604586820865893od_c_a @ Y @ ( set_li8663184538779258833od_c_a @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1116_set__zip__rightD,axiom,
    ! [X2: a,Y: fsm_a_b_c,Xs: list_a,Ys: list_fsm_a_b_c] :
      ( ( member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ X2 @ Y ) @ ( set_Pr3403252632855053537_a_b_c @ ( zip_a_fsm_a_b_c @ Xs @ Ys ) ) )
     => ( member_fsm_a_b_c @ Y @ ( set_fsm_a_b_c2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1117_set__zip__rightD,axiom,
    ! [X2: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1118_set__zip__leftD,axiom,
    ! [X2: fsm_d_b_c,Y: product_prod_d_d,Xs: list_fsm_d_b_c,Ys: list_P8926467638345140077od_d_d] :
      ( ( member73601632213084689od_d_d @ ( produc3676634857637483352od_d_d @ X2 @ Y ) @ ( set_Pr5977834532684968573od_d_d @ ( zip_fs8624885554530663032od_d_d @ Xs @ Ys ) ) )
     => ( member_fsm_d_b_c @ X2 @ ( set_fsm_d_b_c2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1119_set__zip__leftD,axiom,
    ! [X2: set_a,Y: set_a,Xs: list_set_a,Ys: list_set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( zip_set_a_set_a @ Xs @ Ys ) ) )
     => ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1120_set__zip__leftD,axiom,
    ! [X2: d,Y: d,Xs: list_d,Ys: list_d] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( zip_d_d @ Xs @ Ys ) ) )
     => ( member_d @ X2 @ ( set_d2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1121_set__zip__leftD,axiom,
    ! [X2: b,Y: c,Xs: list_b,Ys: list_c] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ ( set_Product_prod_b_c2 @ ( zip_b_c @ Xs @ Ys ) ) )
     => ( member_b @ X2 @ ( set_b2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1122_set__zip__leftD,axiom,
    ! [X2: a,Y: list_P6327159017948738492od_c_a,Xs: list_a,Ys: list_l2826717958560394946od_c_a] :
      ( ( member3484049051949443214od_c_a @ ( produc3580641852002036821od_c_a @ X2 @ Y ) @ ( set_Pr5079464637073054970od_c_a @ ( zip_a_4713777678317597557od_c_a @ Xs @ Ys ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1123_set__zip__leftD,axiom,
    ! [X2: a,Y: fsm_a_b_c,Xs: list_a,Ys: list_fsm_a_b_c] :
      ( ( member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ X2 @ Y ) @ ( set_Pr3403252632855053537_a_b_c @ ( zip_a_fsm_a_b_c @ Xs @ Ys ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1124_set__zip__leftD,axiom,
    ! [X2: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1125_in__set__zipE,axiom,
    ! [X2: d,Y: d,Xs: list_d,Ys: list_d] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ X2 @ Y ) @ ( set_Product_prod_d_d2 @ ( zip_d_d @ Xs @ Ys ) ) )
     => ~ ( ( member_d @ X2 @ ( set_d2 @ Xs ) )
         => ~ ( member_d @ Y @ ( set_d2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1126_in__set__zipE,axiom,
    ! [X2: b,Y: c,Xs: list_b,Ys: list_c] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X2 @ Y ) @ ( set_Product_prod_b_c2 @ ( zip_b_c @ Xs @ Ys ) ) )
     => ~ ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
         => ~ ( member_c @ Y @ ( set_c2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1127_in__set__zipE,axiom,
    ! [X2: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1128_in__set__zipE,axiom,
    ! [X2: set_a,Y: set_a,Xs: list_set_a,Ys: list_set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X2 @ Y ) @ ( set_Pr7878197726666307004_set_a @ ( zip_set_a_set_a @ Xs @ Ys ) ) )
     => ~ ( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
         => ~ ( member_set_a @ Y @ ( set_set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1129_in__set__zipE,axiom,
    ! [X2: a,Y: list_P903359562653991662od_b_c,Xs: list_a,Ys: list_l8907847357763382004od_b_c] :
      ( ( member8874627675880996032od_b_c @ ( produc6078649392988585351od_b_c @ X2 @ Y ) @ ( set_Pr316496599637395756od_b_c @ ( zip_a_6435887832565384359od_b_c @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1130_in__set__zipE,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: a,Xs: list_l8907847357763382004od_b_c,Ys: list_a] :
      ( ( member8373667307600438368_b_c_a @ ( produc595631027891595047_b_c_a @ X2 @ Y ) @ ( set_Pr9038908268211613900_b_c_a @ ( zip_li952869467468394055_b_c_a @ Xs @ Ys ) ) )
     => ~ ( ( member6330420149250801815od_b_c @ X2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1131_in__set__zipE,axiom,
    ! [X2: a,Y: fsm_a_b_c,Xs: list_a,Ys: list_fsm_a_b_c] :
      ( ( member997585926662434933_a_b_c @ ( produc2795441082327197628_a_b_c @ X2 @ Y ) @ ( set_Pr3403252632855053537_a_b_c @ ( zip_a_fsm_a_b_c @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member_fsm_a_b_c @ Y @ ( set_fsm_a_b_c2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1132_in__set__zipE,axiom,
    ! [X2: a,Y: produc1703568184450464039_set_a,Xs: list_a,Ys: list_P3660316430366008877_set_a] :
      ( ( member7961485381496024569_set_a @ ( produc7473614524774861888_set_a @ X2 @ Y ) @ ( set_Pr8441785722565869669_set_a @ ( zip_a_4589243773223691104_set_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member7983343339038529360_set_a @ Y @ ( set_Pr7878197726666307004_set_a @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1133_in__set__zipE,axiom,
    ! [X2: produc1703568184450464039_set_a,Y: a,Xs: list_P3660316430366008877_set_a,Ys: list_a] :
      ( ( member8769086359091386215et_a_a @ ( produc2109488108518772014et_a_a @ X2 @ Y ) @ ( set_Pr26014663306455507et_a_a @ ( zip_Pr8448489393822377038et_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member7983343339038529360_set_a @ X2 @ ( set_Pr7878197726666307004_set_a @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1134_in__set__zipE,axiom,
    ! [X2: produc8838969699183119820_a_b_c,Y: a,Xs: list_P5219093185649440210_a_b_c,Ys: list_a] :
      ( ( member3104622976533456642_b_c_a @ ( produc431331022791976905_b_c_a @ X2 @ Y ) @ ( set_Pr9008855877005340526_b_c_a @ ( zip_Pr5379581719685156585_b_c_a @ Xs @ Ys ) ) )
     => ~ ( ( member997585926662434933_a_b_c @ X2 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1135_zip__same,axiom,
    ! [A2: produc4859908366599477352od_d_d,B2: produc4859908366599477352od_d_d,Xs: list_P4069430723938770798od_d_d] :
      ( ( member6383636634632210000od_d_d @ ( produc1795809570569509655od_d_d @ A2 @ B2 ) @ ( set_Pr5589546020064100540od_d_d @ ( zip_Pr2050640324219679287od_d_d @ Xs @ Xs ) ) )
      = ( ( member73601632213084689od_d_d @ A2 @ ( set_Pr5977834532684968573od_d_d @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1136_zip__same,axiom,
    ! [A2: produc8838969699183119820_a_b_c,B2: produc8838969699183119820_a_b_c,Xs: list_P5219093185649440210_a_b_c] :
      ( ( member289795379306782288_a_b_c @ ( produc7246914118821965079_a_b_c @ A2 @ B2 ) @ ( set_Pr4521536175723917500_a_b_c @ ( zip_Pr701755647690507319_a_b_c @ Xs @ Xs ) ) )
      = ( ( member997585926662434933_a_b_c @ A2 @ ( set_Pr3403252632855053537_a_b_c @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1137_zip__same,axiom,
    ! [A2: list_P6327159017948738492od_c_a,B2: list_P6327159017948738492od_c_a,Xs: list_l2826717958560394946od_c_a] :
      ( ( member4483596368559830352od_c_a @ ( produc8447801329164731415od_c_a @ A2 @ B2 ) @ ( set_Pr1402144133550743484od_c_a @ ( zip_li3537494752898587447od_c_a @ Xs @ Xs ) ) )
      = ( ( member7410604586820865893od_c_a @ A2 @ ( set_li8663184538779258833od_c_a @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1138_zip__same,axiom,
    ! [A2: produc1703568184450464039_set_a,B2: produc1703568184450464039_set_a,Xs: list_P3660316430366008877_set_a] :
      ( ( member1214319639984524048_set_a @ ( produc4984519302771176279_set_a @ A2 @ B2 ) @ ( set_Pr6500651403171443068_set_a @ ( zip_Pr5333784274856831607_set_a @ Xs @ Xs ) ) )
      = ( ( member7983343339038529360_set_a @ A2 @ ( set_Pr7878197726666307004_set_a @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1139_zip__same,axiom,
    ! [A2: list_P903359562653991662od_b_c,B2: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
      ( ( member1533839430763132368od_b_c @ ( produc3076307724246486423od_b_c @ A2 @ B2 ) @ ( set_Pr292865031892111420od_b_c @ ( zip_li8331637159781295287od_b_c @ Xs @ Xs ) ) )
      = ( ( member6330420149250801815od_b_c @ A2 @ ( set_li4480668622519654659od_b_c @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1140_zip__same,axiom,
    ! [A2: set_a,B2: set_a,Xs: list_set_a] :
      ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A2 @ B2 ) @ ( set_Pr7878197726666307004_set_a @ ( zip_set_a_set_a @ Xs @ Xs ) ) )
      = ( ( member_set_a @ A2 @ ( set_set_a2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1141_zip__same,axiom,
    ! [A2: d,B2: d,Xs: list_d] :
      ( ( member2287536781371869328od_d_d @ ( product_Pair_d_d @ A2 @ B2 ) @ ( set_Product_prod_d_d2 @ ( zip_d_d @ Xs @ Xs ) ) )
      = ( ( member_d @ A2 @ ( set_d2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1142_zip__same,axiom,
    ! [A2: a,B2: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
      = ( ( member_a @ A2 @ ( set_a2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1143_from__FSM__product__initial,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q22: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
       => ( ( initia4141967198046469911_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q22 ) ) )
          = ( product_Pair_a_a @ Q1 @ Q22 ) ) ) ) ).

% from_FSM_product_initial
thf(fact_1144_product__from__path,axiom,
    ! [Q12: a,Q24: a,M: fsm_a_b_c,Q1: a,Q22: a,P: list_P558552684728352636od_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Q12 @ Q24 ) @ ( states3827391290251485425_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q22 ) ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
         => ( ( path_P4567602787516311982_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q12 ) @ ( from_FSM_a_b_c @ M @ Q24 ) ) @ ( product_Pair_a_a @ Q12 @ Q24 ) @ P )
           => ( path_P4567602787516311982_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q22 ) ) @ ( product_Pair_a_a @ Q12 @ Q24 ) @ P ) ) ) ) ) ).

% product_from_path
thf(fact_1145_product__from__from,axiom,
    ! [Q12: a,Q24: a,M: fsm_a_b_c,Q1: a,Q22: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Q12 @ Q24 ) @ ( states3827391290251485425_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q22 ) ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
         => ( ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q12 ) @ ( from_FSM_a_b_c @ M @ Q24 ) )
            = ( from_F964489425370879813_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q22 ) ) @ ( product_Pair_a_a @ Q12 @ Q24 ) ) ) ) ) ) ).

% product_from_from
thf(fact_1146_product__from,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q22: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
       => ( ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q22 ) )
          = ( from_F964489425370879813_a_b_c @ ( produc2510185919789642449_b_c_a @ M @ M ) @ ( product_Pair_a_a @ Q1 @ Q22 ) ) ) ) ) ).

% product_from
thf(fact_1147_product__language__state,axiom,
    ! [A: fsm_a_b_c,B: fsm_li6382094382827256929_a_b_c,Q1: a,Q22: list_P6327159017948738492od_c_a] :
      ( ( lS_Pro3096241305776238666_a_b_c @ ( produc2528431742188625167od_c_a @ A @ B ) @ ( produc3580641852002036821od_c_a @ Q1 @ Q22 ) )
      = ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ A @ Q1 ) @ ( lS_lis1262993304779633313_a_b_c @ B @ Q22 ) ) ) ).

% product_language_state
thf(fact_1148_product__language__state,axiom,
    ! [A: fsm_a_b_c,B: fsm_fsm_a_b_c_b_c,Q1: a,Q22: fsm_a_b_c] :
      ( ( lS_Pro5946834958860820657_c_b_c @ ( produc7620483080144554102_a_b_c @ A @ B ) @ ( produc2795441082327197628_a_b_c @ Q1 @ Q22 ) )
      = ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ A @ Q1 ) @ ( lS_fsm_a_b_c_b_c @ B @ Q22 ) ) ) ).

% product_language_state
thf(fact_1149_product__language__state,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c,Q1: a,Q22: a] :
      ( ( lS_Pro8574668989825729676_a_b_c @ ( produc2510185919789642449_b_c_a @ A @ B ) @ ( product_Pair_a_a @ Q1 @ Q22 ) )
      = ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ A @ Q1 ) @ ( lS_a_b_c @ B @ Q22 ) ) ) ).

% product_language_state
thf(fact_1150_product__language,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( lS_Pro8574668989825729676_a_b_c @ ( produc2510185919789642449_b_c_a @ A @ B ) @ ( initia4141967198046469911_a_b_c @ ( produc2510185919789642449_b_c_a @ A @ B ) ) )
      = ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ A @ ( initial_a_b_c @ A ) ) @ ( lS_a_b_c @ B @ ( initial_a_b_c @ B ) ) ) ) ).

% product_language
thf(fact_1151_prod_Osplit__sel__asm,axiom,
    ! [P5: set_li6436108459499378894od_b_c > $o,F: a > fsm_a_b_c > set_li6436108459499378894od_b_c,Prod: produc8838969699183119820_a_b_c] :
      ( ( P5 @ ( produc6808532261061394109od_b_c @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc2795441082327197628_a_b_c @ ( produc2620512763296580520_a_b_c @ Prod ) @ ( produc4728557313310798826_a_b_c @ Prod ) ) )
            & ~ ( P5 @ ( F @ ( produc2620512763296580520_a_b_c @ Prod ) @ ( produc4728557313310798826_a_b_c @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_1152_prod_Osplit__sel,axiom,
    ! [P5: set_li6436108459499378894od_b_c > $o,F: a > fsm_a_b_c > set_li6436108459499378894od_b_c,Prod: produc8838969699183119820_a_b_c] :
      ( ( P5 @ ( produc6808532261061394109od_b_c @ F @ Prod ) )
      = ( ( Prod
          = ( produc2795441082327197628_a_b_c @ ( produc2620512763296580520_a_b_c @ Prod ) @ ( produc4728557313310798826_a_b_c @ Prod ) ) )
       => ( P5 @ ( F @ ( produc2620512763296580520_a_b_c @ Prod ) @ ( produc4728557313310798826_a_b_c @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_1153_paths__from__product__path_I2_J,axiom,
    ! [A: fsm_c_a_b,B: fsm_a_a_b,P: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) @ ( initia681974817753967189_a_a_b @ ( produc2677346961163784847_a_b_a @ A @ B ) ) @ P )
     => ( path_a_a_b @ B @ ( initial_a_a_b @ B )
        @ ( map_Pr834105277175768324od_b_a
          @ ^ [T: produc3205353004677322104od_c_a] : ( produc7037890857971478952od_b_a @ ( product_snd_c_a @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc6867761796305373823od_b_a @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( product_Pair_b_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
          @ P ) ) ) ).

% paths_from_product_path(2)
thf(fact_1154_UN__iff,axiom,
    ! [B2: list_P903359562653991662od_b_c,B: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( member6330420149250801815od_b_c @ B2 @ ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ B @ A ) ) )
      = ( ? [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
            & ( member6330420149250801815od_b_c @ B2 @ ( B @ X3 ) ) ) ) ) ).

% UN_iff
thf(fact_1155_UN__I,axiom,
    ! [A2: a,A: set_a,B2: a,B: a > set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member_a @ B2 @ ( B @ A2 ) )
       => ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1156_UN__I,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B2: a,B: list_P903359562653991662od_b_c > set_a] :
      ( ( member6330420149250801815od_b_c @ A2 @ A )
     => ( ( member_a @ B2 @ ( B @ A2 ) )
       => ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_6303648431024068597_set_a @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1157_UN__I,axiom,
    ! [A2: a,A: set_a,B2: list_P903359562653991662od_b_c,B: a > set_li6436108459499378894od_b_c] :
      ( ( member_a @ A2 @ A )
     => ( ( member6330420149250801815od_b_c @ B2 @ ( B @ A2 ) )
       => ( member6330420149250801815od_b_c @ B2 @ ( comple1250193257546725051od_b_c @ ( image_6261743117854694101od_b_c @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1158_UN__I,axiom,
    ! [A2: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B2: a,B: produc8838969699183119820_a_b_c > set_a] :
      ( ( member997585926662434933_a_b_c @ A2 @ A )
     => ( ( member_a @ B2 @ ( B @ A2 ) )
       => ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_3806487585888175767_set_a @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1159_UN__I,axiom,
    ! [A2: a,A: set_a,B2: produc8838969699183119820_a_b_c,B: a > set_Pr2640454425625433516_a_b_c] :
      ( ( member_a @ A2 @ A )
     => ( ( member997585926662434933_a_b_c @ B2 @ ( B @ A2 ) )
       => ( member997585926662434933_a_b_c @ B2 @ ( comple8630136486246698137_a_b_c @ ( image_4303748172491449011_a_b_c @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1160_UN__I,axiom,
    ! [A2: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B2: list_P903359562653991662od_b_c,B: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A2 @ A )
     => ( ( member6330420149250801815od_b_c @ B2 @ ( B @ A2 ) )
       => ( member6330420149250801815od_b_c @ B2 @ ( comple1250193257546725051od_b_c @ ( image_1046841848883543525od_b_c @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1161_UN__I,axiom,
    ! [A2: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B2: a,B: produc4859908366599477352od_d_d > set_a] :
      ( ( member73601632213084689od_d_d @ A2 @ A )
     => ( ( member_a @ B2 @ ( B @ A2 ) )
       => ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_5843299077104563451_set_a @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1162_UN__I,axiom,
    ! [A2: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B2: a,B: list_P6327159017948738492od_c_a > set_a] :
      ( ( member7410604586820865893od_c_a @ A2 @ A )
     => ( ( member_a @ B2 @ ( B @ A2 ) )
       => ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_6475989651528893095_set_a @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1163_UN__I,axiom,
    ! [A2: a,A: set_a,B2: produc4859908366599477352od_d_d,B: a > set_Pr9130538228769002824od_d_d] :
      ( ( member_a @ A2 @ A )
     => ( ( member73601632213084689od_d_d @ B2 @ ( B @ A2 ) )
       => ( member73601632213084689od_d_d @ B2 @ ( comple1406480280551970613od_d_d @ ( image_2285191578929382223od_d_d @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1164_UN__I,axiom,
    ! [A2: a,A: set_a,B2: list_P6327159017948738492od_c_a,B: a > set_li1159382662694783132od_c_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member7410604586820865893od_c_a @ B2 @ ( B @ A2 ) )
       => ( member7410604586820865893od_c_a @ B2 @ ( comple2269964827201748105od_c_a @ ( image_5319686587417496739od_c_a @ B @ A ) ) ) ) ) ).

% UN_I
thf(fact_1165_Un__iff,axiom,
    ! [C: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ( member73601632213084689od_d_d @ C @ ( sup_su3647555221267236124od_d_d @ A @ B ) )
      = ( ( member73601632213084689od_d_d @ C @ A )
        | ( member73601632213084689od_d_d @ C @ B ) ) ) ).

% Un_iff
thf(fact_1166_Un__iff,axiom,
    ! [C: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( member997585926662434933_a_b_c @ C @ ( sup_su7650654951151286656_a_b_c @ A @ B ) )
      = ( ( member997585926662434933_a_b_c @ C @ A )
        | ( member997585926662434933_a_b_c @ C @ B ) ) ) ).

% Un_iff
thf(fact_1167_Un__iff,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) )
      = ( ( member6330420149250801815od_b_c @ C @ A )
        | ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% Un_iff
thf(fact_1168_Un__iff,axiom,
    ! [C: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A @ B ) )
      = ( ( member7410604586820865893od_c_a @ C @ A )
        | ( member7410604586820865893od_c_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_1169_Un__iff,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
      = ( ( member_a @ C @ A )
        | ( member_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_1170_UnCI,axiom,
    ! [C: produc4859908366599477352od_d_d,B: set_Pr9130538228769002824od_d_d,A: set_Pr9130538228769002824od_d_d] :
      ( ( ~ ( member73601632213084689od_d_d @ C @ B )
       => ( member73601632213084689od_d_d @ C @ A ) )
     => ( member73601632213084689od_d_d @ C @ ( sup_su3647555221267236124od_d_d @ A @ B ) ) ) ).

% UnCI
thf(fact_1171_UnCI,axiom,
    ! [C: produc8838969699183119820_a_b_c,B: set_Pr2640454425625433516_a_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( ~ ( member997585926662434933_a_b_c @ C @ B )
       => ( member997585926662434933_a_b_c @ C @ A ) )
     => ( member997585926662434933_a_b_c @ C @ ( sup_su7650654951151286656_a_b_c @ A @ B ) ) ) ).

% UnCI
thf(fact_1172_UnCI,axiom,
    ! [C: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ~ ( member6330420149250801815od_b_c @ C @ B )
       => ( member6330420149250801815od_b_c @ C @ A ) )
     => ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) ) ) ).

% UnCI
thf(fact_1173_UnCI,axiom,
    ! [C: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a,A: set_li1159382662694783132od_c_a] :
      ( ( ~ ( member7410604586820865893od_c_a @ C @ B )
       => ( member7410604586820865893od_c_a @ C @ A ) )
     => ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A @ B ) ) ) ).

% UnCI
thf(fact_1174_UnCI,axiom,
    ! [C: a,B: set_a,A: set_a] :
      ( ( ~ ( member_a @ C @ B )
       => ( member_a @ C @ A ) )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnCI
thf(fact_1175_case__prodI,axiom,
    ! [F: d > d > $o,A2: d,B2: d] :
      ( ( F @ A2 @ B2 )
     => ( produc1445293847429941932_d_d_o @ F @ ( product_Pair_d_d @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1176_case__prodI,axiom,
    ! [F: b > c > $o,A2: b,B2: c] :
      ( ( F @ A2 @ B2 )
     => ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1177_case__prodI,axiom,
    ! [F: a > list_P6327159017948738492od_c_a > $o,A2: a,B2: list_P6327159017948738492od_c_a] :
      ( ( F @ A2 @ B2 )
     => ( produc3030812177477723822_c_a_o @ F @ ( produc3580641852002036821od_c_a @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1178_case__prodI,axiom,
    ! [F: a > fsm_a_b_c > $o,A2: a,B2: fsm_a_b_c] :
      ( ( F @ A2 @ B2 )
     => ( produc9024115774385606535_b_c_o @ F @ ( produc2795441082327197628_a_b_c @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1179_case__prodI,axiom,
    ! [F: a > a > $o,A2: a,B2: a] :
      ( ( F @ A2 @ B2 )
     => ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1180_case__prodI2,axiom,
    ! [P: product_prod_d_d,C: d > d > $o] :
      ( ! [A4: d,B4: d] :
          ( ( P
            = ( product_Pair_d_d @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc1445293847429941932_d_d_o @ C @ P ) ) ).

% case_prodI2
thf(fact_1181_case__prodI2,axiom,
    ! [P: product_prod_b_c,C: b > c > $o] :
      ( ! [A4: b,B4: c] :
          ( ( P
            = ( product_Pair_b_c @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc1873627588981547243_b_c_o @ C @ P ) ) ).

% case_prodI2
thf(fact_1182_case__prodI2,axiom,
    ! [P: produc3220416805844688357od_c_a,C: a > list_P6327159017948738492od_c_a > $o] :
      ( ! [A4: a,B4: list_P6327159017948738492od_c_a] :
          ( ( P
            = ( produc3580641852002036821od_c_a @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc3030812177477723822_c_a_o @ C @ P ) ) ).

% case_prodI2
thf(fact_1183_case__prodI2,axiom,
    ! [P: produc8838969699183119820_a_b_c,C: a > fsm_a_b_c > $o] :
      ( ! [A4: a,B4: fsm_a_b_c] :
          ( ( P
            = ( produc2795441082327197628_a_b_c @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc9024115774385606535_b_c_o @ C @ P ) ) ).

% case_prodI2
thf(fact_1184_case__prodI2,axiom,
    ! [P: product_prod_a_a,C: a > a > $o] :
      ( ! [A4: a,B4: a] :
          ( ( P
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc6436628058953941356_a_a_o @ C @ P ) ) ).

% case_prodI2
thf(fact_1185_sup_Obounded__iff,axiom,
    ! [B2: set_a,C: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_set_a @ B2 @ A2 )
        & ( ord_less_eq_set_a @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1186_sup_Obounded__iff,axiom,
    ! [B2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ B2 @ C ) @ A2 )
      = ( ( ord_le2998388488506175548od_c_a @ B2 @ A2 )
        & ( ord_le2998388488506175548od_c_a @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1187_le__sup__iff,axiom,
    ! [X2: set_a,Y: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z )
      = ( ( ord_less_eq_set_a @ X2 @ Z )
        & ( ord_less_eq_set_a @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1188_le__sup__iff,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a,Z: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ X2 @ Y ) @ Z )
      = ( ( ord_le2998388488506175548od_c_a @ X2 @ Z )
        & ( ord_le2998388488506175548od_c_a @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1189_Un__subset__iff,axiom,
    ! [A: set_a,B: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C3 )
      = ( ( ord_less_eq_set_a @ A @ C3 )
        & ( ord_less_eq_set_a @ B @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_1190_Un__subset__iff,axiom,
    ! [A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a,C3: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ A @ B ) @ C3 )
      = ( ( ord_le2998388488506175548od_c_a @ A @ C3 )
        & ( ord_le2998388488506175548od_c_a @ B @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_1191_UN__ball__bex__simps_I4_J,axiom,
    ! [B: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ( ? [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ B @ A ) ) )
            & ( P5 @ X3 ) ) )
      = ( ? [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
            & ? [Y5: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Y5 @ ( B @ X3 ) )
                & ( P5 @ Y5 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_1192_UN__ball__bex__simps_I2_J,axiom,
    ! [B: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ B @ A ) ) )
           => ( P5 @ X3 ) ) )
      = ( ! [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
           => ! [Y5: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Y5 @ ( B @ X3 ) )
               => ( P5 @ Y5 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_1193_bex__UN,axiom,
    ! [B: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ( ? [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ B @ A ) ) )
            & ( P5 @ X3 ) ) )
      = ( ? [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
            & ? [Y5: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Y5 @ ( B @ X3 ) )
                & ( P5 @ Y5 ) ) ) ) ) ).

% bex_UN
thf(fact_1194_ball__UN,axiom,
    ! [B: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,P5: list_P903359562653991662od_b_c > $o] :
      ( ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ B @ A ) ) )
           => ( P5 @ X3 ) ) )
      = ( ! [X3: produc8838969699183119820_a_b_c] :
            ( ( member997585926662434933_a_b_c @ X3 @ A )
           => ! [Y5: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Y5 @ ( B @ X3 ) )
               => ( P5 @ Y5 ) ) ) ) ) ).

% ball_UN
thf(fact_1195_SUP__identity__eq,axiom,
    ! [A: set_se3924713247505902254od_b_c] :
      ( ( comple1250193257546725051od_b_c
        @ ( image_8181729759120431109od_b_c
          @ ^ [X3: set_li6436108459499378894od_b_c] : X3
          @ A ) )
      = ( comple1250193257546725051od_b_c @ A ) ) ).

% SUP_identity_eq
thf(fact_1196_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_1197_set__append,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs @ Ys ) )
      = ( sup_su2554677861087362972od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ Ys ) ) ) ).

% set_append
thf(fact_1198_set__append,axiom,
    ! [Xs: list_P3660316430366008877_set_a,Ys: list_P3660316430366008877_set_a] :
      ( ( set_Pr7878197726666307004_set_a @ ( append6456390900168921282_set_a @ Xs @ Ys ) )
      = ( sup_su5624169484884619995_set_a @ ( set_Pr7878197726666307004_set_a @ Xs ) @ ( set_Pr7878197726666307004_set_a @ Ys ) ) ) ).

% set_append
thf(fact_1199_set__append,axiom,
    ! [Xs: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c] :
      ( ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs @ Ys ) )
      = ( sup_su3823046536922626210od_b_c @ ( set_li4480668622519654659od_b_c @ Xs ) @ ( set_li4480668622519654659od_b_c @ Ys ) ) ) ).

% set_append
thf(fact_1200_if__image__distrib,axiom,
    ! [P5: produc8838969699183119820_a_b_c > $o,F: produc8838969699183119820_a_b_c > a,G: produc8838969699183119820_a_b_c > a,S: set_Pr2640454425625433516_a_b_c] :
      ( ( image_3273489647386116407_b_c_a
        @ ^ [X3: produc8838969699183119820_a_b_c] : ( if_a @ ( P5 @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) )
        @ S )
      = ( sup_sup_set_a @ ( image_3273489647386116407_b_c_a @ F @ ( inf_in4776242878804941978_a_b_c @ S @ ( collec3867759823522760631_a_b_c @ P5 ) ) )
        @ ( image_3273489647386116407_b_c_a @ G
          @ ( inf_in4776242878804941978_a_b_c @ S
            @ ( collec3867759823522760631_a_b_c
              @ ^ [X3: produc8838969699183119820_a_b_c] :
                  ~ ( P5 @ X3 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_1201_if__image__distrib,axiom,
    ! [P5: list_P903359562653991662od_b_c > $o,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,G: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,S: set_li6436108459499378894od_b_c] :
      ( ( image_2850395784482391045od_b_c
        @ ^ [X3: list_P903359562653991662od_b_c] : ( if_lis437861846949295796od_b_c @ ( P5 @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) )
        @ S )
      = ( sup_su3823046536922626210od_b_c @ ( image_2850395784482391045od_b_c @ F @ ( inf_in4978071631833541052od_b_c @ S @ ( collec2280997390073109977od_b_c @ P5 ) ) )
        @ ( image_2850395784482391045od_b_c @ G
          @ ( inf_in4978071631833541052od_b_c @ S
            @ ( collec2280997390073109977od_b_c
              @ ^ [X3: list_P903359562653991662od_b_c] :
                  ~ ( P5 @ X3 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_1202_if__image__distrib,axiom,
    ! [P5: produc7049025859877086738_set_a > $o,F: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,G: produc7049025859877086738_set_a > list_P6327159017948738492od_c_a,S: set_Pr3779000724142394866_set_a] :
      ( ( image_865596521259543215od_c_a
        @ ^ [X3: produc7049025859877086738_set_a] : ( if_lis1795051829816687490od_c_a @ ( P5 @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) )
        @ S )
      = ( sup_su500200128730103920od_c_a @ ( image_865596521259543215od_c_a @ F @ ( inf_in1791426075705210592_set_a @ S @ ( collec5778745153847319805_set_a @ P5 ) ) )
        @ ( image_865596521259543215od_c_a @ G
          @ ( inf_in1791426075705210592_set_a @ S
            @ ( collec5778745153847319805_set_a
              @ ^ [X3: produc7049025859877086738_set_a] :
                  ~ ( P5 @ X3 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_1203_if__image__distrib,axiom,
    ! [P5: produc8838969699183119820_a_b_c > $o,F: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,G: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,S: set_Pr2640454425625433516_a_b_c] :
      ( ( image_2344789136157180295od_b_c
        @ ^ [X3: produc8838969699183119820_a_b_c] : ( if_set5339988235699434644od_b_c @ ( P5 @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) )
        @ S )
      = ( sup_su8485086389179333250od_b_c @ ( image_2344789136157180295od_b_c @ F @ ( inf_in4776242878804941978_a_b_c @ S @ ( collec3867759823522760631_a_b_c @ P5 ) ) )
        @ ( image_2344789136157180295od_b_c @ G
          @ ( inf_in4776242878804941978_a_b_c @ S
            @ ( collec3867759823522760631_a_b_c
              @ ^ [X3: produc8838969699183119820_a_b_c] :
                  ~ ( P5 @ X3 ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_1204_UN__Un,axiom,
    ! [M: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ M @ ( sup_su7650654951151286656_a_b_c @ A @ B ) ) )
      = ( sup_su3823046536922626210od_b_c @ ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ M @ A ) ) @ ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ M @ B ) ) ) ) ).

% UN_Un
thf(fact_1205_reachable__states__intro,axiom,
    ! [M: fsm_a_b_c,P: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ ( initial_a_b_c @ M ) @ P )
     => ( member_a @ ( target_a_b_c @ ( initial_a_b_c @ M ) @ P ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ) ).

% reachable_states_intro
thf(fact_1206_SUP__union,axiom,
    ! [M: produc8838969699183119820_a_b_c > set_li6436108459499378894od_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ M @ ( sup_su7650654951151286656_a_b_c @ A @ B ) ) )
      = ( sup_su3823046536922626210od_b_c @ ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ M @ A ) ) @ ( comple1250193257546725051od_b_c @ ( image_2344789136157180295od_b_c @ M @ B ) ) ) ) ).

% SUP_union
thf(fact_1207_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_1208_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_1209_UnI2,axiom,
    ! [C: produc4859908366599477352od_d_d,B: set_Pr9130538228769002824od_d_d,A: set_Pr9130538228769002824od_d_d] :
      ( ( member73601632213084689od_d_d @ C @ B )
     => ( member73601632213084689od_d_d @ C @ ( sup_su3647555221267236124od_d_d @ A @ B ) ) ) ).

% UnI2
thf(fact_1210_UnI2,axiom,
    ! [C: produc8838969699183119820_a_b_c,B: set_Pr2640454425625433516_a_b_c,A: set_Pr2640454425625433516_a_b_c] :
      ( ( member997585926662434933_a_b_c @ C @ B )
     => ( member997585926662434933_a_b_c @ C @ ( sup_su7650654951151286656_a_b_c @ A @ B ) ) ) ).

% UnI2
thf(fact_1211_UnI2,axiom,
    ! [C: list_P903359562653991662od_b_c,B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ B )
     => ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) ) ) ).

% UnI2
thf(fact_1212_UnI2,axiom,
    ! [C: list_P6327159017948738492od_c_a,B: set_li1159382662694783132od_c_a,A: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ B )
     => ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A @ B ) ) ) ).

% UnI2
thf(fact_1213_UnI2,axiom,
    ! [C: a,B: set_a,A: set_a] :
      ( ( member_a @ C @ B )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI2
thf(fact_1214_UnI1,axiom,
    ! [C: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ( member73601632213084689od_d_d @ C @ A )
     => ( member73601632213084689od_d_d @ C @ ( sup_su3647555221267236124od_d_d @ A @ B ) ) ) ).

% UnI1
thf(fact_1215_UnI1,axiom,
    ! [C: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( member997585926662434933_a_b_c @ C @ A )
     => ( member997585926662434933_a_b_c @ C @ ( sup_su7650654951151286656_a_b_c @ A @ B ) ) ) ).

% UnI1
thf(fact_1216_UnI1,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ A )
     => ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) ) ) ).

% UnI1
thf(fact_1217_UnI1,axiom,
    ! [C: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ A )
     => ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A @ B ) ) ) ).

% UnI1
thf(fact_1218_UnI1,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ A )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI1
thf(fact_1219_UnE,axiom,
    ! [C: produc4859908366599477352od_d_d,A: set_Pr9130538228769002824od_d_d,B: set_Pr9130538228769002824od_d_d] :
      ( ( member73601632213084689od_d_d @ C @ ( sup_su3647555221267236124od_d_d @ A @ B ) )
     => ( ~ ( member73601632213084689od_d_d @ C @ A )
       => ( member73601632213084689od_d_d @ C @ B ) ) ) ).

% UnE
thf(fact_1220_UnE,axiom,
    ! [C: produc8838969699183119820_a_b_c,A: set_Pr2640454425625433516_a_b_c,B: set_Pr2640454425625433516_a_b_c] :
      ( ( member997585926662434933_a_b_c @ C @ ( sup_su7650654951151286656_a_b_c @ A @ B ) )
     => ( ~ ( member997585926662434933_a_b_c @ C @ A )
       => ( member997585926662434933_a_b_c @ C @ B ) ) ) ).

% UnE
thf(fact_1221_UnE,axiom,
    ! [C: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C @ ( sup_su3823046536922626210od_b_c @ A @ B ) )
     => ( ~ ( member6330420149250801815od_b_c @ C @ A )
       => ( member6330420149250801815od_b_c @ C @ B ) ) ) ).

% UnE
thf(fact_1222_UnE,axiom,
    ! [C: list_P6327159017948738492od_c_a,A: set_li1159382662694783132od_c_a,B: set_li1159382662694783132od_c_a] :
      ( ( member7410604586820865893od_c_a @ C @ ( sup_su500200128730103920od_c_a @ A @ B ) )
     => ( ~ ( member7410604586820865893od_c_a @ C @ A )
       => ( member7410604586820865893od_c_a @ C @ B ) ) ) ).

% UnE
thf(fact_1223_UnE,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
     => ( ~ ( member_a @ C @ A )
       => ( member_a @ C @ B ) ) ) ).

% UnE
thf(fact_1224_Un__def,axiom,
    ( sup_su3647555221267236124od_d_d
    = ( ^ [A6: set_Pr9130538228769002824od_d_d,B6: set_Pr9130538228769002824od_d_d] :
          ( collec5362705567740981075od_d_d
          @ ^ [X3: produc4859908366599477352od_d_d] :
              ( ( member73601632213084689od_d_d @ X3 @ A6 )
              | ( member73601632213084689od_d_d @ X3 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1225_Un__def,axiom,
    ( sup_su7650654951151286656_a_b_c
    = ( ^ [A6: set_Pr2640454425625433516_a_b_c,B6: set_Pr2640454425625433516_a_b_c] :
          ( collec3867759823522760631_a_b_c
          @ ^ [X3: produc8838969699183119820_a_b_c] :
              ( ( member997585926662434933_a_b_c @ X3 @ A6 )
              | ( member997585926662434933_a_b_c @ X3 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1226_Un__def,axiom,
    ( sup_su3823046536922626210od_b_c
    = ( ^ [A6: set_li6436108459499378894od_b_c,B6: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X3: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X3 @ A6 )
              | ( member6330420149250801815od_b_c @ X3 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1227_Un__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A6: set_a,B6: set_a] :
          ( collect_a
          @ ^ [X3: a] :
              ( ( member_a @ X3 @ A6 )
              | ( member_a @ X3 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1228_Un__def,axiom,
    ( sup_su500200128730103920od_c_a
    = ( ^ [A6: set_li1159382662694783132od_c_a,B6: set_li1159382662694783132od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [X3: list_P6327159017948738492od_c_a] :
              ( ( member7410604586820865893od_c_a @ X3 @ A6 )
              | ( member7410604586820865893od_c_a @ X3 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1229_Un__def,axiom,
    ( sup_su8485086389179333250od_b_c
    = ( ^ [A6: set_se3924713247505902254od_b_c,B6: set_se3924713247505902254od_b_c] :
          ( collec1823247286932916153od_b_c
          @ ^ [X3: set_li6436108459499378894od_b_c] :
              ( ( member6985331446368301687od_b_c @ X3 @ A6 )
              | ( member6985331446368301687od_b_c @ X3 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1230_Collect__disj__eq,axiom,
    ! [P5: list_P6327159017948738492od_c_a > $o,Q5: list_P6327159017948738492od_c_a > $o] :
      ( ( collec6273869032445462695od_c_a
        @ ^ [X3: list_P6327159017948738492od_c_a] :
            ( ( P5 @ X3 )
            | ( Q5 @ X3 ) ) )
      = ( sup_su500200128730103920od_c_a @ ( collec6273869032445462695od_c_a @ P5 ) @ ( collec6273869032445462695od_c_a @ Q5 ) ) ) ).

% Collect_disj_eq
thf(fact_1231_Collect__disj__eq,axiom,
    ! [P5: set_li6436108459499378894od_b_c > $o,Q5: set_li6436108459499378894od_b_c > $o] :
      ( ( collec1823247286932916153od_b_c
        @ ^ [X3: set_li6436108459499378894od_b_c] :
            ( ( P5 @ X3 )
            | ( Q5 @ X3 ) ) )
      = ( sup_su8485086389179333250od_b_c @ ( collec1823247286932916153od_b_c @ P5 ) @ ( collec1823247286932916153od_b_c @ Q5 ) ) ) ).

% Collect_disj_eq
thf(fact_1232_inf__sup__ord_I4_J,axiom,
    ! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1233_inf__sup__ord_I4_J,axiom,
    ! [Y: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ Y @ ( sup_su500200128730103920od_c_a @ X2 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1234_inf__sup__ord_I3_J,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1235_inf__sup__ord_I3_J,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ X2 @ ( sup_su500200128730103920od_c_a @ X2 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1236_le__supE,axiom,
    ! [A2: set_a,B2: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 )
     => ~ ( ( ord_less_eq_set_a @ A2 @ X2 )
         => ~ ( ord_less_eq_set_a @ B2 @ X2 ) ) ) ).

% le_supE
thf(fact_1237_le__supE,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ A2 @ B2 ) @ X2 )
     => ~ ( ( ord_le2998388488506175548od_c_a @ A2 @ X2 )
         => ~ ( ord_le2998388488506175548od_c_a @ B2 @ X2 ) ) ) ).

% le_supE
thf(fact_1238_le__supI,axiom,
    ! [A2: set_a,X2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ X2 )
     => ( ( ord_less_eq_set_a @ B2 @ X2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 ) ) ) ).

% le_supI
thf(fact_1239_le__supI,axiom,
    ! [A2: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ X2 )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ X2 )
       => ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ A2 @ B2 ) @ X2 ) ) ) ).

% le_supI
thf(fact_1240_sup__ge1,axiom,
    ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).

% sup_ge1
thf(fact_1241_sup__ge1,axiom,
    ! [X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ X2 @ ( sup_su500200128730103920od_c_a @ X2 @ Y ) ) ).

% sup_ge1
thf(fact_1242_sup__ge2,axiom,
    ! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).

% sup_ge2
thf(fact_1243_sup__ge2,axiom,
    ! [Y: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ Y @ ( sup_su500200128730103920od_c_a @ X2 @ Y ) ) ).

% sup_ge2
thf(fact_1244_le__supI1,axiom,
    ! [X2: set_a,A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ A2 )
     => ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_1245_le__supI1,axiom,
    ! [X2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ X2 @ A2 )
     => ( ord_le2998388488506175548od_c_a @ X2 @ ( sup_su500200128730103920od_c_a @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_1246_le__supI2,axiom,
    ! [X2: set_a,B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ B2 )
     => ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_1247_le__supI2,axiom,
    ! [X2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ X2 @ B2 )
     => ( ord_le2998388488506175548od_c_a @ X2 @ ( sup_su500200128730103920od_c_a @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_1248_sup_Omono,axiom,
    ! [C: set_a,A2: set_a,D2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ C @ A2 )
     => ( ( ord_less_eq_set_a @ D2 @ B2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_1249_sup_Omono,axiom,
    ! [C: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a,D2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ C @ A2 )
     => ( ( ord_le2998388488506175548od_c_a @ D2 @ B2 )
       => ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ C @ D2 ) @ ( sup_su500200128730103920od_c_a @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_1250_sup__mono,axiom,
    ! [A2: set_a,C: set_a,B2: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ C )
     => ( ( ord_less_eq_set_a @ B2 @ D2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1251_sup__mono,axiom,
    ! [A2: set_li1159382662694783132od_c_a,C: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a,D2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ C )
     => ( ( ord_le2998388488506175548od_c_a @ B2 @ D2 )
       => ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ A2 @ B2 ) @ ( sup_su500200128730103920od_c_a @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1252_sup__least,axiom,
    ! [Y: set_a,X2: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X2 )
     => ( ( ord_less_eq_set_a @ Z @ X2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_1253_sup__least,axiom,
    ! [Y: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a,Z: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ Y @ X2 )
     => ( ( ord_le2998388488506175548od_c_a @ Z @ X2 )
       => ( ord_le2998388488506175548od_c_a @ ( sup_su500200128730103920od_c_a @ Y @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_1254_le__iff__sup,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X3: set_a,Y5: set_a] :
          ( ( sup_sup_set_a @ X3 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_1255_le__iff__sup,axiom,
    ( ord_le2998388488506175548od_c_a
    = ( ^ [X3: set_li1159382662694783132od_c_a,Y5: set_li1159382662694783132od_c_a] :
          ( ( sup_su500200128730103920od_c_a @ X3 @ Y5 )
          = Y5 ) ) ) ).

% le_iff_sup
thf(fact_1256_sup_OorderE,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_a @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_1257_sup_OorderE,axiom,
    ! [B2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B2 @ A2 )
     => ( A2
        = ( sup_su500200128730103920od_c_a @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_1258_sup_OorderI,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( A2
        = ( sup_sup_set_a @ A2 @ B2 ) )
     => ( ord_less_eq_set_a @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_1259_sup_OorderI,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( A2
        = ( sup_su500200128730103920od_c_a @ A2 @ B2 ) )
     => ( ord_le2998388488506175548od_c_a @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_1260_sup__unique,axiom,
    ! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
      ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X @ ( F @ X @ Y3 ) )
     => ( ! [X: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X @ Y3 ) )
       => ( ! [X: set_a,Y3: set_a,Z3: set_a] :
              ( ( ord_less_eq_set_a @ Y3 @ X )
             => ( ( ord_less_eq_set_a @ Z3 @ X )
               => ( ord_less_eq_set_a @ ( F @ Y3 @ Z3 ) @ X ) ) )
         => ( ( sup_sup_set_a @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_1261_sup__unique,axiom,
    ! [F: set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a,Y: set_li1159382662694783132od_c_a] :
      ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ X @ ( F @ X @ Y3 ) )
     => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a] : ( ord_le2998388488506175548od_c_a @ Y3 @ ( F @ X @ Y3 ) )
       => ( ! [X: set_li1159382662694783132od_c_a,Y3: set_li1159382662694783132od_c_a,Z3: set_li1159382662694783132od_c_a] :
              ( ( ord_le2998388488506175548od_c_a @ Y3 @ X )
             => ( ( ord_le2998388488506175548od_c_a @ Z3 @ X )
               => ( ord_le2998388488506175548od_c_a @ ( F @ Y3 @ Z3 ) @ X ) ) )
         => ( ( sup_su500200128730103920od_c_a @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_1262_sup_Oabsorb1,axiom,
    ! [B2: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A2 )
     => ( ( sup_sup_set_a @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_1263_sup_Oabsorb1,axiom,
    ! [B2: set_li1159382662694783132od_c_a,A2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ B2 @ A2 )
     => ( ( sup_su500200128730103920od_c_a @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_1264_sup_Oabsorb2,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B2 )
     => ( ( sup_sup_set_a @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_1265_sup_Oabsorb2,axiom,
    ! [A2: set_li1159382662694783132od_c_a,B2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ A2 @ B2 )
     => ( ( sup_su500200128730103920od_c_a @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_1266_sup__absorb1,axiom,
    ! [Y: set_li1159382662694783132od_c_a,X2: set_li1159382662694783132od_c_a] :
      ( ( ord_le2998388488506175548od_c_a @ Y @ X2 )
     => ( ( sup_su500200128730103920od_c_a @ X2 @ Y )
        = X2 ) ) ).

% sup_absorb1

% Helper facts (9)
thf(help_If_2_1_If_001tf__a_T,axiom,
    ! [X2: a,Y: a] :
      ( ( if_a @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__a_T,axiom,
    ! [X2: a,Y: a] :
      ( ( if_a @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_T,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( if_lis437861846949295796od_b_c @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_T,axiom,
    ! [X2: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( if_lis437861846949295796od_b_c @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_T,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( if_set5339988235699434644od_b_c @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_T,axiom,
    ! [X2: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( if_set5339988235699434644od_b_c @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_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,
    ! [P5: $o] :
      ( ( P5 = $true )
      | ( P5 = $false ) ) ).

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,
    ! [X2: list_P6327159017948738492od_c_a,Y: list_P6327159017948738492od_c_a] :
      ( ( if_lis1795051829816687490od_c_a @ $false @ X2 @ Y )
      = Y ) ).

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,
    ! [X2: list_P6327159017948738492od_c_a,Y: list_P6327159017948738492od_c_a] :
      ( ( if_lis1795051829816687490od_c_a @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    member_a @ ( target_a_b_c @ q2 @ pt ) @ ( states_a_b_c @ m ) ).

%------------------------------------------------------------------------------