%------------------------------------------------------------------------------
% 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/0062_SPY_Framework/prob_00544_028436__20777668_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1989 ( 605 unt; 732 typ;   0 def)
%            Number of atoms       : 3843 (1624 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 15064 ( 500   ~;  22   |; 413   &;12368   @)
%                                         (   0 <=>;1761  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   7 avg)
%            Number of types       :  141 ( 140 usr)
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%            Number of symbols     :  595 ( 592 usr;  40 con; 0-11 aty)
%            Number of variables   : 4282 ( 534   ^;3522   !; 226   ?;4282   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:29:55.010
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thf(sy_c_Convergence_Opreserves__divergence_001tf__a_001tf__b_001tf__c_001tf__a,type,
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thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__c_Mtf__a_J_J,type,
    member5552703068553123156od_c_a: list_P125642481956313003od_c_a > set_li4905775889721270027od_c_a > $o ).

thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
    member_list_a: list_a > set_list_a > $o ).

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

thf(sy_c_member_001t__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    member2378525361814473399od_b_c: prefix425356687782099470od_b_c > set_Pr4395803880006080238od_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J_J,type,
    member8124909232574984543od_b_c: produc596031391177498678od_b_c > set_Pr571657135400976278od_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J_J,type,
    member943782647535249696od_b_c: produc2638276842992539639od_b_c > set_Pr3099279388799821527od_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_Itf__e_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J_J_J,type,
    member4348767779791260600od_b_c: produc387491461805067041od_b_c > set_Pr2137142054503751255od_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__e_J,type,
    member323891770073680196_b_c_e: produc4534891796109904283_b_c_e > set_Pr3237851069733663483_b_c_e > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    member5500486793640445664od_c_a: produc3729687299602777783od_c_a > set_Pr7664180988061153431od_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_Itf__a_Mt__Product____Type__Oprod_It__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mt__Product____Type__Oprod_Itf__e_Mtf__a_J_J_J,type,
    member8958573442437681166od_e_a: produc8570131207859728997od_e_a > set_Pr4321813372295601349od_e_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J_J,type,
    member9210739242824242467_nat_a: produc4554795336555609996_nat_a > set_Pr4993832276224048706_nat_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_Mtf__a_J_J_J,type,
    member83456493519966291_c_a_a: produc6003456256095645226_c_a_a > set_Pr778889097596817546_c_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mtf__a_J_J_J,type,
    member7351836221732992250_c_a_a: produc2473480641396933073_c_a_a > set_Pr7384348147666419121_c_a_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_Mt__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    member2577541133337307834od_b_c: produc3550516665487611281od_b_c > set_Pr3138385297192119153od_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
    member8088020824113751456od_a_a: produc5255912757217091959od_a_a > set_Pr8677732771921454935od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J_J,type,
    member6037585776607591613_nat_b: produc2057638696429058086_nat_b > set_Pr2215838388637617372_nat_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J_J,type,
    member7205605469418551960_nat_b: produc897840418342192897_nat_b > set_Pr7645993405536248119_nat_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_Mtf__b_J_J_J,type,
    member5468748651639462952_c_a_b: produc7163242847096752895_c_a_b > set_Pr9090245704080365023_c_a_b > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__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_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Nat__Onat_Mtf__c_J_J_J,type,
    member3840034767840562712_nat_c: produc6755641753618979457_nat_c > set_Pr459837697511470263_nat_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    member265788643176044257od_b_c: produc6657052613134160568od_b_c > set_Pr7047737848026531864od_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_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__d_J_J_J,type,
    member4231278564905875615od_c_d: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    member9073573073229046463od_c_a: set_Pr8357624050790843030od_c_a > set_se312849170327590134od_c_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_c_member_001tf__e,type,
    member_e: e > set_e > $o ).

thf(sy_v_G1____,type,
    g1: e ).

thf(sy_v_G2____,type,
    g2: e ).

thf(sy_v_G____,type,
    g: e ).

thf(sy_v_M1,type,
    m1: fsm_a_b_c ).

thf(sy_v_M2,type,
    m2: fsm_d_b_c ).

thf(sy_v_T1____,type,
    t1: prefix425356687782099470od_b_c ).

thf(sy_v_T2____,type,
    t2: prefix425356687782099470od_b_c ).

thf(sy_v_T3____,type,
    t3: prefix425356687782099470od_b_c ).

thf(sy_v_TG1____,type,
    tG1: produc4534891796109904283_b_c_e ).

thf(sy_v_TG2____,type,
    tG2: produc4534891796109904283_b_c_e ).

thf(sy_v_TGxy____,type,
    tGxy: produc4534891796109904283_b_c_e ).

thf(sy_v_T____,type,
    t: prefix425356687782099470od_b_c ).

thf(sy_v_V____,type,
    v: a > list_P903359562653991662od_b_c ).

thf(sy_v_append__io__pair,type,
    append_io_pair: fsm_a_b_c > ( a > list_P903359562653991662od_b_c ) > prefix425356687782099470od_b_c > e > ( e > list_P903359562653991662od_b_c > e ) > ( e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c ) > a > b > c > produc4534891796109904283_b_c_e ).

thf(sy_v_cg__initial,type,
    cg_initial: fsm_a_b_c > prefix425356687782099470od_b_c > e ).

thf(sy_v_cg__insert,type,
    cg_insert: e > list_P903359562653991662od_b_c > e ).

thf(sy_v_cg__lookup,type,
    cg_lookup: e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c ).

thf(sy_v_cg__merge,type,
    cg_merge: e > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > e ).

thf(sy_v_covers__unverified__transition____,type,
    covers1625163452631446971sition: produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e > $o ).

thf(sy_v_establish__convergence,type,
    establ5936990325632390625rgence: fsm_a_b_c > ( a > list_P903359562653991662od_b_c ) > prefix425356687782099470od_b_c > e > ( e > list_P903359562653991662od_b_c > e ) > ( e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c ) > nat > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e ).

thf(sy_v_get__state__cover,type,
    get_state_cover: fsm_a_b_c > a > list_P903359562653991662od_b_c ).

thf(sy_v_m,type,
    m: nat ).

thf(sy_v_n____,type,
    n: nat ).

thf(sy_v_sc__covered__transitions____,type,
    sc_cov38355202683674721itions: set_Pr8357624050790843030od_c_a ).

thf(sy_v_separate__state__cover,type,
    separate_state_cover: fsm_a_b_c > ( a > list_P903359562653991662od_b_c ) > ( fsm_a_b_c > prefix425356687782099470od_b_c > e ) > ( e > list_P903359562653991662od_b_c > e ) > ( e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c ) > produc4534891796109904283_b_c_e ).

thf(sy_v_sort__unverified__transitions,type,
    sort_u5845992193009891830itions: fsm_a_b_c > ( a > list_P903359562653991662od_b_c ) > list_P6327159017948738492od_c_a > list_P6327159017948738492od_c_a ).

thf(sy_v_t____,type,
    t4: produc2669932201312201782od_c_a ).

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

thf(sy_v_undefined__io__pairs____,type,
    undefined_io_pairs: list_P8634110456476873367od_b_c ).

thf(sy_v_unverified__transitions____,type,
    unveri8662357380954777460itions: list_P6327159017948738492od_c_a ).

thf(sy_v_verify__transition____,type,
    verify_transition: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e ).

thf(sy_v_verify__undefined__io__pair____,type,
    verify8625347136745899022o_pair: prefix425356687782099470od_b_c > produc3550516665487611281od_b_c > prefix425356687782099470od_b_c ).

% Relevant facts (1250)
thf(fact_0_assms_I3_J,axiom,
    minimal_a_b_c @ m1 ).

% assms(3)
thf(fact_1_assms_I1_J,axiom,
    observable_a_b_c @ m1 ).

% assms(1)
thf(fact_2_a3,axiom,
    member5502040268208861279od_c_a @ t4 @ ( transitions_a_b_c @ m1 ) ).

% a3
thf(fact_3_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_4_prod_Ocollapse,axiom,
    ! [Prod: produc4534891796109904283_b_c_e] :
      ( ( produc2511600812177743883_b_c_e @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_5_prod_Ocollapse,axiom,
    ! [Prod: produc2669932201312201782od_c_a] :
      ( ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ Prod ) @ ( produc8088130389382527188od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6_prod_Ocollapse,axiom,
    ! [Prod: produc2133359284458116493od_c_a] :
      ( ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ Prod ) @ ( produc5221446324160585515od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_7_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_8_V,axiom,
    ( v
    = ( get_state_cover @ m1 ) ) ).

% V
thf(fact_9__092_060open_062is__state__cover__assignment_AM1_AV_092_060close_062,axiom,
    state_7672330573471793359_a_b_c @ m1 @ v ).

% \<open>is_state_cover_assignment M1 V\<close>
thf(fact_10_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_11_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4534891796109904283_b_c_e] :
      ( Prod
      = ( produc2511600812177743883_b_c_e @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_12_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_13_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_14_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_15_exI__realizer,axiom,
    ! [P: c > b > $o,Y: c,X: b] :
      ( ( P @ Y @ X )
     => ( P @ ( product_snd_b_c @ ( product_Pair_b_c @ X @ Y ) ) @ ( product_fst_b_c @ ( product_Pair_b_c @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_16_exI__realizer,axiom,
    ! [P: e > prefix425356687782099470od_b_c > $o,Y: e,X: prefix425356687782099470od_b_c] :
      ( ( P @ Y @ X )
     => ( P @ ( produc6779600879576379961_b_c_e @ ( produc2511600812177743883_b_c_e @ X @ Y ) ) @ ( produc1447154562020694007_b_c_e @ ( produc2511600812177743883_b_c_e @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_17_exI__realizer,axiom,
    ! [P: produc2133359284458116493od_c_a > a > $o,Y: produc2133359284458116493od_c_a,X: a] :
      ( ( P @ Y @ X )
     => ( P @ ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X @ Y ) ) @ ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_18_exI__realizer,axiom,
    ! [P: product_prod_c_a > b > $o,Y: product_prod_c_a,X: b] :
      ( ( P @ Y @ X )
     => ( P @ ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X @ Y ) ) @ ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_19_exI__realizer,axiom,
    ! [P: a > c > $o,Y: a,X: c] :
      ( ( P @ Y @ X )
     => ( P @ ( product_snd_c_a @ ( product_Pair_c_a @ X @ Y ) ) @ ( product_fst_c_a @ ( product_Pair_c_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_20_conjI__realizer,axiom,
    ! [P: b > $o,P2: b,Q: c > $o,Q2: c] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_b_c @ ( product_Pair_b_c @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_b_c @ ( product_Pair_b_c @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_21_conjI__realizer,axiom,
    ! [P: prefix425356687782099470od_b_c > $o,P2: prefix425356687782099470od_b_c,Q: e > $o,Q2: e] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc1447154562020694007_b_c_e @ ( produc2511600812177743883_b_c_e @ P2 @ Q2 ) ) )
          & ( Q @ ( produc6779600879576379961_b_c_e @ ( produc2511600812177743883_b_c_e @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_22_conjI__realizer,axiom,
    ! [P: a > $o,P2: a,Q: produc2133359284458116493od_c_a > $o,Q2: produc2133359284458116493od_c_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_23_conjI__realizer,axiom,
    ! [P: b > $o,P2: b,Q: product_prod_c_a > $o,Q2: product_prod_c_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_24_conjI__realizer,axiom,
    ! [P: c > $o,P2: c,Q: a > $o,Q2: a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_c_a @ ( product_Pair_c_a @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_c_a @ ( product_Pair_c_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_25_surjective__pairing,axiom,
    ! [T: product_prod_b_c] :
      ( T
      = ( product_Pair_b_c @ ( product_fst_b_c @ T ) @ ( product_snd_b_c @ T ) ) ) ).

% surjective_pairing
thf(fact_26_surjective__pairing,axiom,
    ! [T: produc4534891796109904283_b_c_e] :
      ( T
      = ( produc2511600812177743883_b_c_e @ ( produc1447154562020694007_b_c_e @ T ) @ ( produc6779600879576379961_b_c_e @ T ) ) ) ).

% surjective_pairing
thf(fact_27_surjective__pairing,axiom,
    ! [T: produc2669932201312201782od_c_a] :
      ( T
      = ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ T ) @ ( produc8088130389382527188od_c_a @ T ) ) ) ).

% surjective_pairing
thf(fact_28_surjective__pairing,axiom,
    ! [T: produc2133359284458116493od_c_a] :
      ( T
      = ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ T ) @ ( produc5221446324160585515od_c_a @ T ) ) ) ).

% surjective_pairing
thf(fact_29_surjective__pairing,axiom,
    ! [T: product_prod_c_a] :
      ( T
      = ( product_Pair_c_a @ ( product_fst_c_a @ T ) @ ( product_snd_c_a @ T ) ) ) ).

% surjective_pairing
thf(fact_30_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: prefix425356687782099470od_b_c > e > $o,X: prefix425356687782099470od_b_c,Y: e,A: produc4534891796109904283_b_c_e] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc2511600812177743883_b_c_e @ X @ Y ) )
       => ( P @ ( produc1447154562020694007_b_c_e @ A ) @ ( produc6779600879576379961_b_c_e @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_31_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: a > produc2133359284458116493od_c_a > $o,X: a,Y: produc2133359284458116493od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc5762801557034676134od_c_a @ X @ Y ) )
       => ( P @ ( produc4225943634317630866od_c_a @ A ) @ ( produc8088130389382527188od_c_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_32_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: b > product_prod_c_a > $o,X: b,Y: product_prod_c_a,A: produc2133359284458116493od_c_a] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc3440865914000881661od_c_a @ X @ Y ) )
       => ( P @ ( produc301949074746585577od_c_a @ A ) @ ( produc5221446324160585515od_c_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_33_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: c > a > $o,X: c,Y: a,A: product_prod_c_a] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( product_Pair_c_a @ X @ Y ) )
       => ( P @ ( product_fst_c_a @ A ) @ ( product_snd_c_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_34_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: b > c > $o,X: b,Y: c,A: product_prod_b_c] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( product_Pair_b_c @ X @ Y ) )
       => ( P @ ( product_fst_b_c @ A ) @ ( product_snd_b_c @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_35_prod_Oinject,axiom,
    ! [X1: prefix425356687782099470od_b_c,X2: e,Y1: prefix425356687782099470od_b_c,Y2: e] :
      ( ( ( produc2511600812177743883_b_c_e @ X1 @ X2 )
        = ( produc2511600812177743883_b_c_e @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_36_prod_Oinject,axiom,
    ! [X1: b,X2: c,Y1: b,Y2: c] :
      ( ( ( product_Pair_b_c @ X1 @ X2 )
        = ( product_Pair_b_c @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_37_prod_Oinject,axiom,
    ! [X1: c,X2: a,Y1: c,Y2: a] :
      ( ( ( product_Pair_c_a @ X1 @ X2 )
        = ( product_Pair_c_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_38_prod_Oinject,axiom,
    ! [X1: b,X2: product_prod_c_a,Y1: b,Y2: product_prod_c_a] :
      ( ( ( produc3440865914000881661od_c_a @ X1 @ X2 )
        = ( produc3440865914000881661od_c_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_39_prod_Oinject,axiom,
    ! [X1: a,X2: produc2133359284458116493od_c_a,Y1: a,Y2: produc2133359284458116493od_c_a] :
      ( ( ( produc5762801557034676134od_c_a @ X1 @ X2 )
        = ( produc5762801557034676134od_c_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_40_old_Oprod_Oinject,axiom,
    ! [A: prefix425356687782099470od_b_c,B: e,A2: prefix425356687782099470od_b_c,B2: e] :
      ( ( ( produc2511600812177743883_b_c_e @ A @ B )
        = ( produc2511600812177743883_b_c_e @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_41_old_Oprod_Oinject,axiom,
    ! [A: b,B: c,A2: b,B2: c] :
      ( ( ( product_Pair_b_c @ A @ B )
        = ( product_Pair_b_c @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_42_old_Oprod_Oinject,axiom,
    ! [A: c,B: a,A2: c,B2: a] :
      ( ( ( product_Pair_c_a @ A @ B )
        = ( product_Pair_c_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_43_old_Oprod_Oinject,axiom,
    ! [A: b,B: product_prod_c_a,A2: b,B2: product_prod_c_a] :
      ( ( ( produc3440865914000881661od_c_a @ A @ B )
        = ( produc3440865914000881661od_c_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_44_old_Oprod_Oinject,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A2: a,B2: produc2133359284458116493od_c_a] :
      ( ( ( produc5762801557034676134od_c_a @ A @ B )
        = ( produc5762801557034676134od_c_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_45_G2,axiom,
    ( g2
    = ( produc6779600879576379961_b_c_e @ tG2 ) ) ).

% G2
thf(fact_46_assms_I12_J,axiom,
    conver2727675425255806140_c_d_e @ m1 @ m2 @ cg_lookup @ cg_insert ).

% assms(12)
thf(fact_47_assms_I9_J,axiom,
    state_7672330573471793359_a_b_c @ m1 @ ( get_state_cover @ m1 ) ).

% assms(9)
thf(fact_48_T2,axiom,
    ( t2
    = ( produc1447154562020694007_b_c_e @ tG2 ) ) ).

% T2
thf(fact_49_assms_I4_J,axiom,
    minimal_d_b_c @ m2 ).

% assms(4)
thf(fact_50_assms_I2_J,axiom,
    observable_d_b_c @ m2 ).

% assms(2)
thf(fact_51_a4,axiom,
    member_a @ ( produc4225943634317630866od_c_a @ t4 ) @ ( reacha1620305530751930115_a_b_c @ m1 ) ).

% a4
thf(fact_52_prod__induct4,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a] :
      ( ! [A3: a,B3: b,C: c,D: a] : ( P @ ( produc5762801557034676134od_c_a @ A3 @ ( produc3440865914000881661od_c_a @ B3 @ ( product_Pair_c_a @ C @ D ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_53_prod__induct3,axiom,
    ! [P: produc2133359284458116493od_c_a > $o,X: produc2133359284458116493od_c_a] :
      ( ! [A3: b,B3: c,C: a] : ( P @ ( produc3440865914000881661od_c_a @ A3 @ ( product_Pair_c_a @ B3 @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_54_prod__induct3,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a] :
      ( ! [A3: a,B3: b,C: product_prod_c_a] : ( P @ ( produc5762801557034676134od_c_a @ A3 @ ( produc3440865914000881661od_c_a @ B3 @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_55_prod__cases4,axiom,
    ! [Y: produc2669932201312201782od_c_a] :
      ~ ! [A3: a,B3: b,C: c,D: a] :
          ( Y
         != ( produc5762801557034676134od_c_a @ A3 @ ( produc3440865914000881661od_c_a @ B3 @ ( product_Pair_c_a @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_56_prod__cases3,axiom,
    ! [Y: produc2133359284458116493od_c_a] :
      ~ ! [A3: b,B3: c,C: a] :
          ( Y
         != ( produc3440865914000881661od_c_a @ A3 @ ( product_Pair_c_a @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_57_prod__cases3,axiom,
    ! [Y: produc2669932201312201782od_c_a] :
      ~ ! [A3: a,B3: b,C: product_prod_c_a] :
          ( Y
         != ( produc5762801557034676134od_c_a @ A3 @ ( produc3440865914000881661od_c_a @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_58_Pair__inject,axiom,
    ! [A: prefix425356687782099470od_b_c,B: e,A2: prefix425356687782099470od_b_c,B2: e] :
      ( ( ( produc2511600812177743883_b_c_e @ A @ B )
        = ( produc2511600812177743883_b_c_e @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_59_Pair__inject,axiom,
    ! [A: b,B: c,A2: b,B2: c] :
      ( ( ( product_Pair_b_c @ A @ B )
        = ( product_Pair_b_c @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_60_Pair__inject,axiom,
    ! [A: c,B: a,A2: c,B2: a] :
      ( ( ( product_Pair_c_a @ A @ B )
        = ( product_Pair_c_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_61_Pair__inject,axiom,
    ! [A: b,B: product_prod_c_a,A2: b,B2: product_prod_c_a] :
      ( ( ( produc3440865914000881661od_c_a @ A @ B )
        = ( produc3440865914000881661od_c_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_62_Pair__inject,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A2: a,B2: produc2133359284458116493od_c_a] :
      ( ( ( produc5762801557034676134od_c_a @ A @ B )
        = ( produc5762801557034676134od_c_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_63_prod__cases,axiom,
    ! [P: produc4534891796109904283_b_c_e > $o,P2: produc4534891796109904283_b_c_e] :
      ( ! [A3: prefix425356687782099470od_b_c,B3: e] : ( P @ ( produc2511600812177743883_b_c_e @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_64_prod__cases,axiom,
    ! [P: product_prod_b_c > $o,P2: product_prod_b_c] :
      ( ! [A3: b,B3: c] : ( P @ ( product_Pair_b_c @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_65_prod__cases,axiom,
    ! [P: product_prod_c_a > $o,P2: product_prod_c_a] :
      ( ! [A3: c,B3: a] : ( P @ ( product_Pair_c_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_66_prod__cases,axiom,
    ! [P: produc2133359284458116493od_c_a > $o,P2: produc2133359284458116493od_c_a] :
      ( ! [A3: b,B3: product_prod_c_a] : ( P @ ( produc3440865914000881661od_c_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_67_prod__cases,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,P2: produc2669932201312201782od_c_a] :
      ( ! [A3: a,B3: produc2133359284458116493od_c_a] : ( P @ ( produc5762801557034676134od_c_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_68_surj__pair,axiom,
    ! [P2: produc4534891796109904283_b_c_e] :
    ? [X3: prefix425356687782099470od_b_c,Y3: e] :
      ( P2
      = ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_69_surj__pair,axiom,
    ! [P2: product_prod_b_c] :
    ? [X3: b,Y3: c] :
      ( P2
      = ( product_Pair_b_c @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_70_surj__pair,axiom,
    ! [P2: product_prod_c_a] :
    ? [X3: c,Y3: a] :
      ( P2
      = ( product_Pair_c_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_71_surj__pair,axiom,
    ! [P2: produc2133359284458116493od_c_a] :
    ? [X3: b,Y3: product_prod_c_a] :
      ( P2
      = ( produc3440865914000881661od_c_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_72_surj__pair,axiom,
    ! [P2: produc2669932201312201782od_c_a] :
    ? [X3: a,Y3: produc2133359284458116493od_c_a] :
      ( P2
      = ( produc5762801557034676134od_c_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_73_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4534891796109904283_b_c_e] :
      ~ ! [A3: prefix425356687782099470od_b_c,B3: e] :
          ( Y
         != ( produc2511600812177743883_b_c_e @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_74_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_b_c] :
      ~ ! [A3: b,B3: c] :
          ( Y
         != ( product_Pair_b_c @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_75_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_c_a] :
      ~ ! [A3: c,B3: a] :
          ( Y
         != ( product_Pair_c_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_76_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2133359284458116493od_c_a] :
      ~ ! [A3: b,B3: product_prod_c_a] :
          ( Y
         != ( produc3440865914000881661od_c_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_77_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2669932201312201782od_c_a] :
      ~ ! [A3: a,B3: produc2133359284458116493od_c_a] :
          ( Y
         != ( produc5762801557034676134od_c_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_78_snd__conv,axiom,
    ! [X1: prefix425356687782099470od_b_c,X2: e] :
      ( ( produc6779600879576379961_b_c_e @ ( produc2511600812177743883_b_c_e @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_79_snd__conv,axiom,
    ! [X1: a,X2: produc2133359284458116493od_c_a] :
      ( ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_80_snd__conv,axiom,
    ! [X1: b,X2: product_prod_c_a] :
      ( ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_81_snd__conv,axiom,
    ! [X1: c,X2: a] :
      ( ( product_snd_c_a @ ( product_Pair_c_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_82_snd__conv,axiom,
    ! [X1: b,X2: c] :
      ( ( product_snd_b_c @ ( product_Pair_b_c @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_83_snd__eqD,axiom,
    ! [X: prefix425356687782099470od_b_c,Y: e,A: e] :
      ( ( ( produc6779600879576379961_b_c_e @ ( produc2511600812177743883_b_c_e @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_84_snd__eqD,axiom,
    ! [X: a,Y: produc2133359284458116493od_c_a,A: produc2133359284458116493od_c_a] :
      ( ( ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_85_snd__eqD,axiom,
    ! [X: b,Y: product_prod_c_a,A: product_prod_c_a] :
      ( ( ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_86_snd__eqD,axiom,
    ! [X: c,Y: a,A: a] :
      ( ( ( product_snd_c_a @ ( product_Pair_c_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_87_snd__eqD,axiom,
    ! [X: b,Y: c,A: c] :
      ( ( ( product_snd_b_c @ ( product_Pair_b_c @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_88_fst__conv,axiom,
    ! [X1: prefix425356687782099470od_b_c,X2: e] :
      ( ( produc1447154562020694007_b_c_e @ ( produc2511600812177743883_b_c_e @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_89_fst__conv,axiom,
    ! [X1: a,X2: produc2133359284458116493od_c_a] :
      ( ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_90_fst__conv,axiom,
    ! [X1: b,X2: product_prod_c_a] :
      ( ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_91_fst__conv,axiom,
    ! [X1: c,X2: a] :
      ( ( product_fst_c_a @ ( product_Pair_c_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_92_fst__conv,axiom,
    ! [X1: b,X2: c] :
      ( ( product_fst_b_c @ ( product_Pair_b_c @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_93_fst__eqD,axiom,
    ! [X: prefix425356687782099470od_b_c,Y: e,A: prefix425356687782099470od_b_c] :
      ( ( ( produc1447154562020694007_b_c_e @ ( produc2511600812177743883_b_c_e @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_94_fst__eqD,axiom,
    ! [X: a,Y: produc2133359284458116493od_c_a,A: a] :
      ( ( ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_95_fst__eqD,axiom,
    ! [X: b,Y: product_prod_c_a,A: b] :
      ( ( ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_96_fst__eqD,axiom,
    ! [X: c,Y: a,A: c] :
      ( ( ( product_fst_c_a @ ( product_Pair_c_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_97_fst__eqD,axiom,
    ! [X: b,Y: c,A: b] :
      ( ( ( product_fst_b_c @ ( product_Pair_b_c @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_98_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc4534891796109904283_b_c_e,Z: produc4534891796109904283_b_c_e] : ( Y4 = Z ) )
    = ( ^ [S: produc4534891796109904283_b_c_e,T2: produc4534891796109904283_b_c_e] :
          ( ( ( produc1447154562020694007_b_c_e @ S )
            = ( produc1447154562020694007_b_c_e @ T2 ) )
          & ( ( produc6779600879576379961_b_c_e @ S )
            = ( produc6779600879576379961_b_c_e @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_99_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc2669932201312201782od_c_a,Z: produc2669932201312201782od_c_a] : ( Y4 = Z ) )
    = ( ^ [S: produc2669932201312201782od_c_a,T2: produc2669932201312201782od_c_a] :
          ( ( ( produc4225943634317630866od_c_a @ S )
            = ( produc4225943634317630866od_c_a @ T2 ) )
          & ( ( produc8088130389382527188od_c_a @ S )
            = ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_100_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc2133359284458116493od_c_a,Z: produc2133359284458116493od_c_a] : ( Y4 = Z ) )
    = ( ^ [S: produc2133359284458116493od_c_a,T2: produc2133359284458116493od_c_a] :
          ( ( ( produc301949074746585577od_c_a @ S )
            = ( produc301949074746585577od_c_a @ T2 ) )
          & ( ( produc5221446324160585515od_c_a @ S )
            = ( produc5221446324160585515od_c_a @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_101_prod__eq__iff,axiom,
    ( ( ^ [Y4: product_prod_c_a,Z: product_prod_c_a] : ( Y4 = Z ) )
    = ( ^ [S: product_prod_c_a,T2: product_prod_c_a] :
          ( ( ( product_fst_c_a @ S )
            = ( product_fst_c_a @ T2 ) )
          & ( ( product_snd_c_a @ S )
            = ( product_snd_c_a @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_102_prod__eq__iff,axiom,
    ( ( ^ [Y4: product_prod_b_c,Z: product_prod_b_c] : ( Y4 = Z ) )
    = ( ^ [S: product_prod_b_c,T2: product_prod_b_c] :
          ( ( ( product_fst_b_c @ S )
            = ( product_fst_b_c @ T2 ) )
          & ( ( product_snd_b_c @ S )
            = ( product_snd_b_c @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_103_exE__realizer_H,axiom,
    ! [P: e > prefix425356687782099470od_b_c > $o,P2: produc4534891796109904283_b_c_e] :
      ( ( P @ ( produc6779600879576379961_b_c_e @ P2 ) @ ( produc1447154562020694007_b_c_e @ P2 ) )
     => ~ ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_104_exE__realizer_H,axiom,
    ! [P: produc2133359284458116493od_c_a > a > $o,P2: produc2669932201312201782od_c_a] :
      ( ( P @ ( produc8088130389382527188od_c_a @ P2 ) @ ( produc4225943634317630866od_c_a @ P2 ) )
     => ~ ! [X3: a,Y3: produc2133359284458116493od_c_a] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_105_exE__realizer_H,axiom,
    ! [P: product_prod_c_a > b > $o,P2: produc2133359284458116493od_c_a] :
      ( ( P @ ( produc5221446324160585515od_c_a @ P2 ) @ ( produc301949074746585577od_c_a @ P2 ) )
     => ~ ! [X3: b,Y3: product_prod_c_a] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_106_exE__realizer_H,axiom,
    ! [P: a > c > $o,P2: product_prod_c_a] :
      ( ( P @ ( product_snd_c_a @ P2 ) @ ( product_fst_c_a @ P2 ) )
     => ~ ! [X3: c,Y3: a] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_107_exE__realizer_H,axiom,
    ! [P: c > b > $o,P2: product_prod_b_c] :
      ( ( P @ ( product_snd_b_c @ P2 ) @ ( product_fst_b_c @ P2 ) )
     => ~ ! [X3: b,Y3: c] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_108_prod__eqI,axiom,
    ! [P2: produc4534891796109904283_b_c_e,Q2: produc4534891796109904283_b_c_e] :
      ( ( ( produc1447154562020694007_b_c_e @ P2 )
        = ( produc1447154562020694007_b_c_e @ Q2 ) )
     => ( ( ( produc6779600879576379961_b_c_e @ P2 )
          = ( produc6779600879576379961_b_c_e @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_109_prod__eqI,axiom,
    ! [P2: produc2669932201312201782od_c_a,Q2: produc2669932201312201782od_c_a] :
      ( ( ( produc4225943634317630866od_c_a @ P2 )
        = ( produc4225943634317630866od_c_a @ Q2 ) )
     => ( ( ( produc8088130389382527188od_c_a @ P2 )
          = ( produc8088130389382527188od_c_a @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_110_prod__eqI,axiom,
    ! [P2: produc2133359284458116493od_c_a,Q2: produc2133359284458116493od_c_a] :
      ( ( ( produc301949074746585577od_c_a @ P2 )
        = ( produc301949074746585577od_c_a @ Q2 ) )
     => ( ( ( produc5221446324160585515od_c_a @ P2 )
          = ( produc5221446324160585515od_c_a @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_111_prod__eqI,axiom,
    ! [P2: product_prod_c_a,Q2: product_prod_c_a] :
      ( ( ( product_fst_c_a @ P2 )
        = ( product_fst_c_a @ Q2 ) )
     => ( ( ( product_snd_c_a @ P2 )
          = ( product_snd_c_a @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_112_prod__eqI,axiom,
    ! [P2: product_prod_b_c,Q2: product_prod_b_c] :
      ( ( ( product_fst_b_c @ P2 )
        = ( product_fst_b_c @ Q2 ) )
     => ( ( ( product_snd_b_c @ P2 )
          = ( product_snd_b_c @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_113_prod_Oexpand,axiom,
    ! [Prod: produc4534891796109904283_b_c_e,Prod2: produc4534891796109904283_b_c_e] :
      ( ( ( ( produc1447154562020694007_b_c_e @ Prod )
          = ( produc1447154562020694007_b_c_e @ Prod2 ) )
        & ( ( produc6779600879576379961_b_c_e @ Prod )
          = ( produc6779600879576379961_b_c_e @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_114_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_115_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_116_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_117_prod_Oexpand,axiom,
    ! [Prod: product_prod_b_c,Prod2: product_prod_b_c] :
      ( ( ( ( product_fst_b_c @ Prod )
          = ( product_fst_b_c @ Prod2 ) )
        & ( ( product_snd_b_c @ Prod )
          = ( product_snd_b_c @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_118_assms_I16_J,axiom,
    sPY_ve2339642005831624696_c_e_d @ append_io_pair @ m1 @ m2 @ cg_insert @ cg_lookup ).

% assms(16)
thf(fact_119_assms_I11_J,axiom,
    conver3708542147478018715_c_d_e @ m1 @ m2 @ cg_lookup @ cg_initial ).

% assms(11)
thf(fact_120_mem__Collect__eq,axiom,
    ! [A: a,P: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_121_mem__Collect__eq,axiom,
    ! [A: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( member6330420149250801815od_b_c @ A @ ( collec2280997390073109977od_b_c @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_122_mem__Collect__eq,axiom,
    ! [A: produc2669932201312201782od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ A @ ( collec8405073525749625505od_c_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_124_Collect__mem__eq,axiom,
    ! [A4: set_a] :
      ( ( collect_a
        @ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_125_Collect__mem__eq,axiom,
    ! [A4: set_li6436108459499378894od_b_c] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_126_Collect__mem__eq,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_127_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_128_Collect__cong,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,Q: produc2669932201312201782od_c_a > $o] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec8405073525749625505od_c_a @ P )
        = ( collec8405073525749625505od_c_a @ Q ) ) ) ).

% Collect_cong
thf(fact_129_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_130_assms_I13_J,axiom,
    conver7456087639816148335_c_d_e @ m1 @ m2 @ cg_lookup @ cg_merge ).

% assms(13)
thf(fact_131_a2,axiom,
    conver7089555221631485245_c_d_e @ m1 @ m2 @ cg_lookup @ g ).

% a2
thf(fact_132__092_060open_062verifies__transition_Aestablish__convergence_AM1_AM2_AV_AT1_Acg__insert_Acg__lookup_092_060close_062,axiom,
    sPY_ve3317748999363812035_c_e_d @ establ5936990325632390625rgence @ m1 @ m2 @ v @ t1 @ cg_insert @ cg_lookup ).

% \<open>verifies_transition establish_convergence M1 M2 V T1 cg_insert cg_lookup\<close>
thf(fact_133_assms_I15_J,axiom,
    sPY_ve3317748999363812035_c_e_d @ establ5936990325632390625rgence @ m1 @ m2 @ ( get_state_cover @ m1 ) @ ( produc1447154562020694007_b_c_e @ ( separate_state_cover @ m1 @ ( get_state_cover @ m1 ) @ cg_initial @ cg_insert @ cg_lookup ) ) @ cg_insert @ cg_lookup ).

% assms(15)
thf(fact_134_observable_Osimps,axiom,
    ( observable_d_b_c
    = ( ^ [M: fsm_d_b_c] :
        ! [X4: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X4 @ ( transitions_d_b_c @ M ) )
         => ! [Y5: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ Y5 @ ( transitions_d_b_c @ M ) )
             => ( ( ( ( produc1508465569166014290od_c_d @ X4 )
                    = ( produc1508465569166014290od_c_d @ Y5 ) )
                  & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) )
                    = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ Y5 ) ) )
                  & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) ) )
                    = ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Y5 ) ) ) ) )
               => ( ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) ) )
                  = ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Y5 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_135_observable_Osimps,axiom,
    ( observ2964502264774598958_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ M ) )
         => ! [Y5: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Y5 @ ( transi2941759273464634107_a_a_b @ M ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X4 )
                    = ( produc8171714036265584980od_c_a @ Y5 ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) ) )
               => ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                  = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_136_observable_Osimps,axiom,
    ( observable_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M ) )
         => ! [Y5: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ M ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                    = ( produc4225943634317630866od_c_a @ Y5 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_137_observable_Oelims_I1_J,axiom,
    ! [X: fsm_d_b_c,Y: $o] :
      ( ( ( observable_d_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ X4 @ ( transitions_d_b_c @ X ) )
             => ! [Y5: produc1399170498009216118od_c_d] :
                  ( ( member4231278564905875615od_c_d @ Y5 @ ( transitions_d_b_c @ X ) )
                 => ( ( ( ( produc1508465569166014290od_c_d @ X4 )
                        = ( produc1508465569166014290od_c_d @ Y5 ) )
                      & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) )
                        = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ Y5 ) ) )
                      & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) ) )
                        = ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Y5 ) ) ) ) )
                   => ( ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) ) )
                      = ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Y5 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_138_observable_Oelims_I1_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b,Y: $o] :
      ( ( ( observ2964502264774598958_a_a_b @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ X ) )
             => ! [Y5: produc3205353004677322104od_c_a] :
                  ( ( member6508725278956418977od_c_a @ Y5 @ ( transi2941759273464634107_a_a_b @ X ) )
                 => ( ( ( ( produc8171714036265584980od_c_a @ X4 )
                        = ( produc8171714036265584980od_c_a @ Y5 ) )
                      & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                        = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) ) )
                   => ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                      = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_139_observable_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( observable_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X ) )
             => ! [Y5: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ X ) )
                 => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                        = ( produc4225943634317630866od_c_a @ Y5 ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) )
                      & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                        = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) ) )
                   => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                      = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_140_observable_Oelims_I2_J,axiom,
    ! [X: fsm_d_b_c] :
      ( ( observable_d_b_c @ X )
     => ! [X5: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X5 @ ( transitions_d_b_c @ X ) )
         => ! [Xa: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ Xa @ ( transitions_d_b_c @ X ) )
             => ( ( ( ( produc1508465569166014290od_c_d @ X5 )
                    = ( produc1508465569166014290od_c_d @ Xa ) )
                  & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X5 ) )
                    = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ Xa ) ) )
                  & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X5 ) ) )
                    = ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Xa ) ) ) ) )
               => ( ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X5 ) ) )
                  = ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Xa ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_141_observable_Oelims_I2_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b] :
      ( ( observ2964502264774598958_a_a_b @ X )
     => ! [X5: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X5 @ ( transi2941759273464634107_a_a_b @ X ) )
         => ! [Xa: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Xa @ ( transi2941759273464634107_a_a_b @ X ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X5 )
                    = ( produc8171714036265584980od_c_a @ Xa ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X5 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X5 ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) ) )
               => ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X5 ) ) )
                  = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_142_observable_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( observable_a_b_c @ X )
     => ! [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X ) )
         => ! [Xa: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa @ ( transitions_a_b_c @ X ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X5 )
                    = ( produc4225943634317630866od_c_a @ Xa ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_143_observable_Oelims_I3_J,axiom,
    ! [X: fsm_d_b_c] :
      ( ~ ( observable_d_b_c @ X )
     => ~ ! [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( transitions_d_b_c @ X ) )
           => ! [Xa2: produc1399170498009216118od_c_d] :
                ( ( member4231278564905875615od_c_d @ Xa2 @ ( transitions_d_b_c @ X ) )
               => ( ( ( ( produc1508465569166014290od_c_d @ X3 )
                      = ( produc1508465569166014290od_c_d @ Xa2 ) )
                    & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X3 ) )
                      = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ Xa2 ) ) )
                    & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X3 ) ) )
                      = ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Xa2 ) ) ) ) )
                 => ( ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X3 ) ) )
                    = ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_144_observable_Oelims_I3_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b] :
      ( ~ ( observ2964502264774598958_a_a_b @ X )
     => ~ ! [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ X ) )
           => ! [Xa2: produc3205353004677322104od_c_a] :
                ( ( member6508725278956418977od_c_a @ Xa2 @ ( transi2941759273464634107_a_a_b @ X ) )
               => ( ( ( ( produc8171714036265584980od_c_a @ X3 )
                      = ( produc8171714036265584980od_c_a @ Xa2 ) )
                    & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) )
                      = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) ) )
                 => ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
                    = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_145_observable_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( observable_a_b_c @ X )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X ) )
           => ! [Xa2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X ) )
               => ( ( ( ( produc4225943634317630866od_c_a @ X3 )
                      = ( produc4225943634317630866od_c_a @ Xa2 ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) )
                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                      = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) )
                 => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_146_observable__transition__unique,axiom,
    ! [M2: fsm_d_b_c,T: produc1399170498009216118od_c_d] :
      ( ( observable_d_b_c @ M2 )
     => ( ( member4231278564905875615od_c_d @ T @ ( transitions_d_b_c @ M2 ) )
       => ? [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( transitions_d_b_c @ M2 ) )
            & ( ( produc1508465569166014290od_c_d @ X3 )
              = ( produc1508465569166014290od_c_d @ T ) )
            & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X3 ) )
              = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) )
            & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X3 ) ) )
              = ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) )
            & ! [Y6: produc1399170498009216118od_c_d] :
                ( ( ( member4231278564905875615od_c_d @ Y6 @ ( transitions_d_b_c @ M2 ) )
                  & ( ( produc1508465569166014290od_c_d @ Y6 )
                    = ( produc1508465569166014290od_c_d @ T ) )
                  & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ Y6 ) )
                    = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) )
                  & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Y6 ) ) )
                    = ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_147_observable__transition__unique,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T: produc3205353004677322104od_c_a] :
      ( ( observ2964502264774598958_a_a_b @ M2 )
     => ( ( member6508725278956418977od_c_a @ T @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ? [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ M2 ) )
            & ( ( produc8171714036265584980od_c_a @ X3 )
              = ( produc8171714036265584980od_c_a @ T ) )
            & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) )
              = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
            & ! [Y6: produc3205353004677322104od_c_a] :
                ( ( ( member6508725278956418977od_c_a @ Y6 @ ( transi2941759273464634107_a_a_b @ M2 ) )
                  & ( ( produc8171714036265584980od_c_a @ Y6 )
                    = ( produc8171714036265584980od_c_a @ T ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y6 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y6 ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_148_observable__transition__unique,axiom,
    ! [M2: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M2 ) )
       => ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ M2 ) )
            & ( ( produc4225943634317630866od_c_a @ X3 )
              = ( produc4225943634317630866od_c_a @ T ) )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
            & ! [Y6: produc2669932201312201782od_c_a] :
                ( ( ( member5502040268208861279od_c_a @ Y6 @ ( transitions_a_b_c @ M2 ) )
                  & ( ( produc4225943634317630866od_c_a @ Y6 )
                    = ( produc4225943634317630866od_c_a @ T ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y6 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y6 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_149_assms_I14_J,axiom,
    h_sepa5747756078645681353_c_e_d @ separate_state_cover @ m1 @ m2 @ cg_initial @ cg_insert @ cg_lookup ).

% assms(14)
thf(fact_150_TGxy,axiom,
    ( tGxy
    = ( append_io_pair @ m1 @ v @ t @ g @ cg_insert @ cg_lookup @ ( produc4225943634317630866od_c_a @ t4 ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ t4 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ t4 ) ) ) ) ) ).

% TGxy
thf(fact_151_reachable__states__next,axiom,
    ! [Q2: product_prod_c_a,M2: fsm_Pr1886043760914421514_a_a_b,T: produc3205353004677322104od_c_a] :
      ( ( member5074992350434858958od_c_a @ Q2 @ ( reacha5035603111973943274_a_a_b @ M2 ) )
     => ( ( member6508725278956418977od_c_a @ T @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ( ( ( produc8171714036265584980od_c_a @ T )
            = Q2 )
         => ( member5074992350434858958od_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( reacha5035603111973943274_a_a_b @ M2 ) ) ) ) ) ).

% reachable_states_next
thf(fact_152_reachable__states__next,axiom,
    ! [Q2: a,M2: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( member_a @ Q2 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M2 ) )
       => ( ( ( produc4225943634317630866od_c_a @ T )
            = Q2 )
         => ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( reacha1620305530751930115_a_b_c @ M2 ) ) ) ) ) ).

% reachable_states_next
thf(fact_153_observable__alt__def,axiom,
    ( observable_d_b_c
    = ( ^ [M: fsm_d_b_c] :
        ! [Q1: d,X4: b,Y5: c,Q12: d,Q13: d] :
          ( ( ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ Q1 @ ( produc3440865926910568064od_c_d @ X4 @ ( product_Pair_c_d @ Y5 @ Q12 ) ) ) @ ( transitions_d_b_c @ M ) )
            & ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ Q1 @ ( produc3440865926910568064od_c_d @ X4 @ ( product_Pair_c_d @ Y5 @ Q13 ) ) ) @ ( transitions_d_b_c @ M ) ) )
         => ( Q12 = Q13 ) ) ) ) ).

% observable_alt_def
thf(fact_154_observable__alt__def,axiom,
    ( observ2964502264774598958_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q1: product_prod_c_a,X4: a,Y5: b,Q12: product_prod_c_a,Q13: product_prod_c_a] :
          ( ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y5 @ Q12 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) )
            & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y5 @ Q13 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) )
         => ( Q12 = Q13 ) ) ) ) ).

% observable_alt_def
thf(fact_155_observable__alt__def,axiom,
    ( observable_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [Q1: a,X4: b,Y5: c,Q12: a,Q13: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y5 @ Q12 ) ) ) @ ( transitions_a_b_c @ M ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y5 @ Q13 ) ) ) @ ( transitions_a_b_c @ M ) ) )
         => ( Q12 = Q13 ) ) ) ) ).

% observable_alt_def
thf(fact_156_TG1,axiom,
    ( tG1
    = ( separate_state_cover @ m1 @ v @ cg_initial @ cg_insert @ cg_lookup ) ) ).

% TG1
thf(fact_157_T1,axiom,
    ( t1
    = ( produc1447154562020694007_b_c_e @ tG1 ) ) ).

% T1
thf(fact_158_retains__outputs__for__states__and__inputs__def,axiom,
    ( retain6283235197096808344_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b,S2: fsm_Pr1886043760914421514_a_a_b] :
        ! [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ S2 ) )
         => ! [Y5: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Y5 @ ( transi2941759273464634107_a_a_b @ M ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X4 )
                    = ( produc8171714036265584980od_c_a @ Y5 ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) )
               => ( member6508725278956418977od_c_a @ Y5 @ ( transi2941759273464634107_a_a_b @ S2 ) ) ) ) ) ) ) ).

% retains_outputs_for_states_and_inputs_def
thf(fact_159_retains__outputs__for__states__and__inputs__def,axiom,
    ( retain2430384024318543025_a_b_c
    = ( ^ [M: fsm_a_b_c,S2: fsm_a_b_c] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ S2 ) )
         => ! [Y5: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ M ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                    = ( produc4225943634317630866od_c_a @ Y5 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) )
               => ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ S2 ) ) ) ) ) ) ) ).

% retains_outputs_for_states_and_inputs_def
thf(fact_160__092_060open_062spy__framework_AM1_Aget__state__cover_Aseparate__state__cover_Asort__unverified__transitions_Aestablish__convergence_Aappend__io__pair_Acg__initial_Acg__insert_Acg__lookup_Acg__merge_Am_A_061_AT3_092_060close_062,axiom,
    ( ( sPY_sp4487033212357905651_b_c_e @ m1 @ get_state_cover @ separate_state_cover @ sort_u5845992193009891830itions @ establ5936990325632390625rgence @ append_io_pair @ cg_initial @ cg_insert @ cg_lookup @ cg_merge @ m )
    = t3 ) ).

% \<open>spy_framework M1 get_state_cover separate_state_cover sort_unverified_transitions establish_convergence append_io_pair cg_initial cg_insert cg_lookup cg_merge m = T3\<close>
thf(fact_161_deterministic_Oelims_I3_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b] :
      ( ~ ( determ4061018006534944583_a_a_b @ X )
     => ~ ! [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ X ) )
           => ! [Xa2: produc3205353004677322104od_c_a] :
                ( ( member6508725278956418977od_c_a @ Xa2 @ ( transi2941759273464634107_a_a_b @ X ) )
               => ( ( ( ( produc8171714036265584980od_c_a @ X3 )
                      = ( produc8171714036265584980od_c_a @ Xa2 ) )
                    & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) )
                      = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) )
                 => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) )
                    & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
                      = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_162_deterministic_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( deterministic_a_b_c @ X )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X ) )
           => ! [Xa2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X ) )
               => ( ( ( ( produc4225943634317630866od_c_a @ X3 )
                      = ( produc4225943634317630866od_c_a @ Xa2 ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) )
                 => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                      = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) )
                    & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                      = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_163_deterministic_Oelims_I2_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b] :
      ( ( determ4061018006534944583_a_a_b @ X )
     => ! [X5: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X5 @ ( transi2941759273464634107_a_a_b @ X ) )
         => ! [Xa: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Xa @ ( transi2941759273464634107_a_a_b @ X ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X5 )
                    = ( produc8171714036265584980od_c_a @ Xa ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X5 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) )
               => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X5 ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) )
                  & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X5 ) ) )
                    = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_164_deterministic_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( deterministic_a_b_c @ X )
     => ! [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X ) )
         => ! [Xa: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa @ ( transitions_a_b_c @ X ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X5 )
                    = ( produc4225943634317630866od_c_a @ Xa ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) )
               => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) )
                  & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_165_deterministic_Oelims_I1_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b,Y: $o] :
      ( ( ( determ4061018006534944583_a_a_b @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ X ) )
             => ! [Y5: produc3205353004677322104od_c_a] :
                  ( ( member6508725278956418977od_c_a @ Y5 @ ( transi2941759273464634107_a_a_b @ X ) )
                 => ( ( ( ( produc8171714036265584980od_c_a @ X4 )
                        = ( produc8171714036265584980od_c_a @ Y5 ) )
                      & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                        = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) )
                   => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) )
                      & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                        = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_166_deterministic_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( deterministic_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X ) )
             => ! [Y5: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ X ) )
                 => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                        = ( produc4225943634317630866od_c_a @ Y5 ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) )
                   => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                        = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) )
                      & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                        = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_167_deterministic_Osimps,axiom,
    ( determ4061018006534944583_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ M ) )
         => ! [Y5: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Y5 @ ( transi2941759273464634107_a_a_b @ M ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X4 )
                    = ( produc8171714036265584980od_c_a @ Y5 ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) )
               => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) )
                  & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                    = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_168_deterministic_Osimps,axiom,
    ( deterministic_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M ) )
         => ! [Y5: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ M ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                    = ( produc4225943634317630866od_c_a @ Y5 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) )
               => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) )
                  & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_169_state__cover__assignment__inj,axiom,
    ! [M2: fsm_d_b_c,V: d > list_P903359562653991662od_b_c,Q14: d,Q22: d] :
      ( ( state_6014242345326633810_d_b_c @ M2 @ V )
     => ( ( observable_d_b_c @ M2 )
       => ( ( member_d @ Q14 @ ( reacha9185589339461546374_d_b_c @ M2 ) )
         => ( ( member_d @ Q22 @ ( reacha9185589339461546374_d_b_c @ M2 ) )
           => ( ( Q14 != Q22 )
             => ( ( V @ Q14 )
               != ( V @ Q22 ) ) ) ) ) ) ) ).

% state_cover_assignment_inj
thf(fact_170_state__cover__assignment__inj,axiom,
    ! [M2: fsm_a_b_c,V: a > list_P903359562653991662od_b_c,Q14: a,Q22: a] :
      ( ( state_7672330573471793359_a_b_c @ M2 @ V )
     => ( ( observable_a_b_c @ M2 )
       => ( ( member_a @ Q14 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
         => ( ( member_a @ Q22 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
           => ( ( Q14 != Q22 )
             => ( ( V @ Q14 )
               != ( V @ Q22 ) ) ) ) ) ) ) ).

% state_cover_assignment_inj
thf(fact_171_acyclic__induction,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,P: product_prod_c_a > $o] :
      ( ( acycli1378905732213473983_a_a_b @ M2 )
     => ( ! [Q3: product_prod_c_a] :
            ( ( member5074992350434858958od_c_a @ Q3 @ ( reacha5035603111973943274_a_a_b @ M2 ) )
           => ( ! [T3: produc3205353004677322104od_c_a] :
                  ( ( member6508725278956418977od_c_a @ T3 @ ( transi2941759273464634107_a_a_b @ M2 ) )
                 => ( ( ( produc8171714036265584980od_c_a @ T3 )
                      = Q3 )
                   => ( P @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T3 ) ) ) ) ) )
             => ( P @ Q3 ) ) )
       => ! [X5: product_prod_c_a] :
            ( ( member5074992350434858958od_c_a @ X5 @ ( reacha5035603111973943274_a_a_b @ M2 ) )
           => ( P @ X5 ) ) ) ) ).

% acyclic_induction
thf(fact_172_acyclic__induction,axiom,
    ! [M2: fsm_a_b_c,P: a > $o] :
      ( ( acyclic_a_b_c @ M2 )
     => ( ! [Q3: a] :
            ( ( member_a @ Q3 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
           => ( ! [T3: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ T3 @ ( transitions_a_b_c @ M2 ) )
                 => ( ( ( produc4225943634317630866od_c_a @ T3 )
                      = Q3 )
                   => ( P @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) ) ) )
             => ( P @ Q3 ) ) )
       => ! [X5: a] :
            ( ( member_a @ X5 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
           => ( P @ X5 ) ) ) ) ).

% acyclic_induction
thf(fact_173_observable__transition__target__language__eq,axiom,
    ! [M2: fsm_d_b_c,T1: produc1399170498009216118od_c_d,T22: produc1399170498009216118od_c_d] :
      ( ( ( lS_d_b_c @ M2 @ ( produc1508465569166014290od_c_d @ T1 ) )
        = ( lS_d_b_c @ M2 @ ( produc1508465569166014290od_c_d @ T22 ) ) )
     => ( ( member4231278564905875615od_c_d @ T1 @ ( transitions_d_b_c @ M2 ) )
       => ( ( member4231278564905875615od_c_d @ T22 @ ( transitions_d_b_c @ M2 ) )
         => ( ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T1 ) )
              = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T22 ) ) )
           => ( ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T1 ) ) )
                = ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T22 ) ) ) )
             => ( ( observable_d_b_c @ M2 )
               => ( ( lS_d_b_c @ M2 @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T1 ) ) ) )
                  = ( lS_d_b_c @ M2 @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_174_observable__transition__target__language__eq,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T1: produc3205353004677322104od_c_a,T22: produc3205353004677322104od_c_a] :
      ( ( ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T1 ) )
        = ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T22 ) ) )
     => ( ( member6508725278956418977od_c_a @ T1 @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ( ( member6508725278956418977od_c_a @ T22 @ ( transi2941759273464634107_a_a_b @ M2 ) )
         => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T1 ) )
              = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T22 ) ) )
           => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T1 ) ) )
                = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T22 ) ) ) )
             => ( ( observ2964502264774598958_a_a_b @ M2 )
               => ( ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T1 ) ) ) )
                  = ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_175_observable__transition__target__language__eq,axiom,
    ! [M2: fsm_a_b_c,T1: produc2669932201312201782od_c_a,T22: produc2669932201312201782od_c_a] :
      ( ( ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T1 ) )
        = ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T22 ) ) )
     => ( ( member5502040268208861279od_c_a @ T1 @ ( transitions_a_b_c @ M2 ) )
       => ( ( member5502040268208861279od_c_a @ T22 @ ( transitions_a_b_c @ M2 ) )
         => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) )
           => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) )
                = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) )
             => ( ( observable_a_b_c @ M2 )
               => ( ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) ) )
                  = ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_176_G1,axiom,
    ( g1
    = ( produc6779600879576379961_b_c_e @ tG1 ) ) ).

% G1
thf(fact_177__092_060open_062TG1_A_061_A_IT1_M_AG1_J_092_060close_062,axiom,
    ( tG1
    = ( produc2511600812177743883_b_c_e @ t1 @ g1 ) ) ).

% \<open>TG1 = (T1, G1)\<close>
thf(fact_178_acyclic__no__self__loop,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,Q2: product_prod_c_a] :
      ( ( acycli1378905732213473983_a_a_b @ M2 )
     => ( ( member5074992350434858958od_c_a @ Q2 @ ( reacha5035603111973943274_a_a_b @ M2 ) )
       => ~ ? [X5: a,Y6: b] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q2 @ ( produc5762801557034676134od_c_a @ X5 @ ( produc3440865914000881661od_c_a @ Y6 @ Q2 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ) ).

% acyclic_no_self_loop
thf(fact_179_acyclic__no__self__loop,axiom,
    ! [M2: fsm_a_b_c,Q2: a] :
      ( ( acyclic_a_b_c @ M2 )
     => ( ( member_a @ Q2 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
       => ~ ? [X5: b,Y6: c] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q2 @ ( produc3440865914000881661od_c_a @ X5 @ ( product_Pair_c_a @ Y6 @ Q2 ) ) ) @ ( transitions_a_b_c @ M2 ) ) ) ) ).

% acyclic_no_self_loop
thf(fact_180_deterministic__alt__def,axiom,
    ( determ4061018006534944583_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q1: product_prod_c_a,X4: a,Y7: b,Y8: b,Q12: product_prod_c_a,Q13: product_prod_c_a] :
          ( ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y7 @ Q12 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) )
            & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y8 @ Q13 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) )
         => ( ( Y7 = Y8 )
            & ( Q12 = Q13 ) ) ) ) ) ).

% deterministic_alt_def
thf(fact_181_deterministic__alt__def,axiom,
    ( deterministic_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [Q1: a,X4: b,Y7: c,Y8: c,Q12: a,Q13: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y7 @ Q12 ) ) ) @ ( transitions_a_b_c @ M ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y8 @ Q13 ) ) ) @ ( transitions_a_b_c @ M ) ) )
         => ( ( Y7 = Y8 )
            & ( Q12 = Q13 ) ) ) ) ) ).

% deterministic_alt_def
thf(fact_182_assms_I10_J,axiom,
    ( set_Pr5664641634098845387od_c_a
    = ( ^ [Xs: list_P6327159017948738492od_c_a] : ( set_Pr5664641634098845387od_c_a @ ( sort_u5845992193009891830itions @ m1 @ ( get_state_cover @ m1 ) @ Xs ) ) ) ) ).

% assms(10)
thf(fact_183_observable__language__transition__target,axiom,
    ! [M2: fsm_d_b_c,T: produc1399170498009216118od_c_d,Io: list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( member4231278564905875615od_c_d @ T @ ( transitions_d_b_c @ M2 ) )
       => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) @ ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) ) @ Io ) @ ( lS_d_b_c @ M2 @ ( produc1508465569166014290od_c_d @ T ) ) )
         => ( member6330420149250801815od_b_c @ Io @ ( lS_d_b_c @ M2 @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_184_observable__language__transition__target,axiom,
    ! [M2: fsm_Pr7461828476839194351_a_c_a,T: produc3729687299602777783od_c_a,Io: list_P125642481956313003od_c_a] :
      ( ( observ3736821838051356051_a_c_a @ M2 )
     => ( ( member5500486793640445664od_c_a @ T @ ( transi5448983560151880288_a_c_a @ M2 ) )
       => ( ( member5552703068553123156od_c_a @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ ( produc7173261134821384633od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) ) ) @ Io ) @ ( lS_Pro1436182445161660463_a_c_a @ M2 @ ( produc1784202036732648467od_c_a @ T ) ) )
         => ( member5552703068553123156od_c_a @ Io @ ( lS_Pro1436182445161660463_a_c_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_185_observable__language__transition__target,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T: produc3205353004677322104od_c_a,Io: list_P1467974523212632366od_a_b] :
      ( ( observ2964502264774598958_a_a_b @ M2 )
     => ( ( member6508725278956418977od_c_a @ T @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) @ Io ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T ) ) )
         => ( member6895035109809442519od_a_b @ Io @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_186_observable__language__transition__target,axiom,
    ! [M2: fsm_a_b_c,T: produc2669932201312201782od_c_a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M2 ) )
       => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ Io ) @ ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T ) ) )
         => ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_187_observable__transition__target__language__subset,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T1: produc3205353004677322104od_c_a,T22: produc3205353004677322104od_c_a] :
      ( ( ord_le153002609085230766od_a_b @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T1 ) ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T22 ) ) )
     => ( ( member6508725278956418977od_c_a @ T1 @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ( ( member6508725278956418977od_c_a @ T22 @ ( transi2941759273464634107_a_a_b @ M2 ) )
         => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T1 ) )
              = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T22 ) ) )
           => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T1 ) ) )
                = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T22 ) ) ) )
             => ( ( observ2964502264774598958_a_a_b @ M2 )
               => ( ord_le153002609085230766od_a_b @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T1 ) ) ) ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_subset
thf(fact_188_observable__transition__target__language__subset,axiom,
    ! [M2: fsm_d_b_c,T1: produc1399170498009216118od_c_d,T22: produc1399170498009216118od_c_d] :
      ( ( ord_le282488521294790766od_b_c @ ( lS_d_b_c @ M2 @ ( produc1508465569166014290od_c_d @ T1 ) ) @ ( lS_d_b_c @ M2 @ ( produc1508465569166014290od_c_d @ T22 ) ) )
     => ( ( member4231278564905875615od_c_d @ T1 @ ( transitions_d_b_c @ M2 ) )
       => ( ( member4231278564905875615od_c_d @ T22 @ ( transitions_d_b_c @ M2 ) )
         => ( ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T1 ) )
              = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T22 ) ) )
           => ( ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T1 ) ) )
                = ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T22 ) ) ) )
             => ( ( observable_d_b_c @ M2 )
               => ( ord_le282488521294790766od_b_c @ ( lS_d_b_c @ M2 @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T1 ) ) ) ) @ ( lS_d_b_c @ M2 @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_subset
thf(fact_189_observable__transition__target__language__subset,axiom,
    ! [M2: fsm_a_b_c,T1: produc2669932201312201782od_c_a,T22: produc2669932201312201782od_c_a] :
      ( ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T1 ) ) @ ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T22 ) ) )
     => ( ( member5502040268208861279od_c_a @ T1 @ ( transitions_a_b_c @ M2 ) )
       => ( ( member5502040268208861279od_c_a @ T22 @ ( transitions_a_b_c @ M2 ) )
         => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) )
           => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) )
                = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) )
             => ( ( observable_a_b_c @ M2 )
               => ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) ) ) @ ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_subset
thf(fact_190_language__next__transition__ob,axiom,
    ! [X: b,Y: c,Ios: list_P903359562653991662od_b_c,M2: fsm_d_b_c,Q2: d] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ Ios ) @ ( lS_d_b_c @ M2 @ Q2 ) )
     => ~ ! [T4: produc1399170498009216118od_c_d] :
            ( ( ( produc1508465569166014290od_c_d @ T4 )
              = Q2 )
           => ( ( member4231278564905875615od_c_d @ T4 @ ( transitions_d_b_c @ M2 ) )
             => ( ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T4 ) )
                  = X )
               => ( ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T4 ) ) )
                    = Y )
                 => ~ ( member6330420149250801815od_b_c @ Ios @ ( lS_d_b_c @ M2 @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T4 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_191_language__next__transition__ob,axiom,
    ! [X: c,Y: a,Ios: list_P125642481956313003od_c_a,M2: fsm_Pr7461828476839194351_a_c_a,Q2: produc2133359284458116493od_c_a] :
      ( ( member5552703068553123156od_c_a @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X @ Y ) @ Ios ) @ ( lS_Pro1436182445161660463_a_c_a @ M2 @ Q2 ) )
     => ~ ! [T4: produc3729687299602777783od_c_a] :
            ( ( ( produc1784202036732648467od_c_a @ T4 )
              = Q2 )
           => ( ( member5500486793640445664od_c_a @ T4 @ ( transi5448983560151880288_a_c_a @ M2 ) )
             => ( ( ( produc7173261134821384633od_c_a @ ( produc2799690148840255317od_c_a @ T4 ) )
                  = X )
               => ( ( ( produc4225943634317630866od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T4 ) ) )
                    = Y )
                 => ~ ( member5552703068553123156od_c_a @ Ios @ ( lS_Pro1436182445161660463_a_c_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T4 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_192_language__next__transition__ob,axiom,
    ! [X: a,Y: b,Ios: list_P1467974523212632366od_a_b,M2: fsm_Pr1886043760914421514_a_a_b,Q2: product_prod_c_a] :
      ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ X @ Y ) @ Ios ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ Q2 ) )
     => ~ ! [T4: produc3205353004677322104od_c_a] :
            ( ( ( produc8171714036265584980od_c_a @ T4 )
              = Q2 )
           => ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
             => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) )
                  = X )
               => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) )
                    = Y )
                 => ~ ( member6895035109809442519od_a_b @ Ios @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_193_language__next__transition__ob,axiom,
    ! [X: b,Y: c,Ios: list_P903359562653991662od_b_c,M2: fsm_a_b_c,Q2: a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ Ios ) @ ( lS_a_b_c @ M2 @ Q2 ) )
     => ~ ! [T4: produc2669932201312201782od_c_a] :
            ( ( ( produc4225943634317630866od_c_a @ T4 )
              = Q2 )
           => ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
             => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) )
                  = X )
               => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) )
                    = Y )
                 => ~ ( member6330420149250801815od_b_c @ Ios @ ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_194_LS__prepend__transition,axiom,
    ! [T: produc1399170498009216118od_c_d,M2: fsm_d_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( member4231278564905875615od_c_d @ T @ ( transitions_d_b_c @ M2 ) )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_d_b_c @ M2 @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) ) )
       => ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) @ ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) ) @ Io ) @ ( lS_d_b_c @ M2 @ ( produc1508465569166014290od_c_d @ T ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_195_LS__prepend__transition,axiom,
    ! [T: produc3729687299602777783od_c_a,M2: fsm_Pr7461828476839194351_a_c_a,Io: list_P125642481956313003od_c_a] :
      ( ( member5500486793640445664od_c_a @ T @ ( transi5448983560151880288_a_c_a @ M2 ) )
     => ( ( member5552703068553123156od_c_a @ Io @ ( lS_Pro1436182445161660463_a_c_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) ) ) )
       => ( member5552703068553123156od_c_a @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ ( produc7173261134821384633od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) ) ) @ Io ) @ ( lS_Pro1436182445161660463_a_c_a @ M2 @ ( produc1784202036732648467od_c_a @ T ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_196_LS__prepend__transition,axiom,
    ! [T: produc3205353004677322104od_c_a,M2: fsm_Pr1886043760914421514_a_a_b,Io: list_P1467974523212632366od_a_b] :
      ( ( member6508725278956418977od_c_a @ T @ ( transi2941759273464634107_a_a_b @ M2 ) )
     => ( ( member6895035109809442519od_a_b @ Io @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) )
       => ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) @ Io ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_197_LS__prepend__transition,axiom,
    ! [T: produc2669932201312201782od_c_a,M2: fsm_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M2 ) )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) )
       => ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ Io ) @ ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_198__092_060open_062L_AM1_A_092_060inter_062_APrefix__Tree_Oset_A_Ispy__framework_AM1_Aget__state__cover_Aseparate__state__cover_Asort__unverified__transitions_Aestablish__convergence_Aappend__io__pair_Acg__initial_Acg__insert_Acg__lookup_Acg__merge_Am_J_A_061_AL_AM2_A_092_060inter_062_APrefix__Tree_Oset_A_Ispy__framework_AM1_Aget__state__cover_Aseparate__state__cover_Asort__unverified__transitions_Aestablish__convergence_Aappend__io__pair_Acg__initial_Acg__insert_Acg__lookup_Acg__merge_Am_J_A_092_060Longrightarrow_062_AL_AM1_A_092_060inter_062_APrefix__Tree_Oset_AT3_A_061_AL_AM2_A_092_060inter_062_APrefix__Tree_Oset_AT3_092_060close_062,axiom,
    ( ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ m1 @ ( initial_a_b_c @ m1 ) ) @ ( prefix9085958534787526401od_b_c @ ( sPY_sp4487033212357905651_b_c_e @ m1 @ get_state_cover @ separate_state_cover @ sort_u5845992193009891830itions @ establ5936990325632390625rgence @ append_io_pair @ cg_initial @ cg_insert @ cg_lookup @ cg_merge @ m ) ) )
      = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ m2 @ ( initial_d_b_c @ m2 ) ) @ ( prefix9085958534787526401od_b_c @ ( sPY_sp4487033212357905651_b_c_e @ m1 @ get_state_cover @ separate_state_cover @ sort_u5845992193009891830itions @ establ5936990325632390625rgence @ append_io_pair @ cg_initial @ cg_insert @ cg_lookup @ cg_merge @ m ) ) ) )
   => ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ m1 @ ( initial_a_b_c @ m1 ) ) @ ( prefix9085958534787526401od_b_c @ t3 ) )
      = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ m2 @ ( initial_d_b_c @ m2 ) ) @ ( prefix9085958534787526401od_b_c @ t3 ) ) ) ) ).

% \<open>L M1 \<inter> Prefix_Tree.set (spy_framework M1 get_state_cover separate_state_cover sort_unverified_transitions establish_convergence append_io_pair cg_initial cg_insert cg_lookup cg_merge m) = L M2 \<inter> Prefix_Tree.set (spy_framework M1 get_state_cover separate_state_cover sort_unverified_transitions establish_convergence append_io_pair cg_initial cg_insert cg_lookup cg_merge m) \<Longrightarrow> L M1 \<inter> Prefix_Tree.set T3 = L M2 \<inter> Prefix_Tree.set T3\<close>
thf(fact_199__092_060open_062L_AM1_A_061_AL_AM2_A_092_060Longrightarrow_062_AL_AM1_A_092_060inter_062_APrefix__Tree_Oset_A_Ispy__framework_AM1_Aget__state__cover_Aseparate__state__cover_Asort__unverified__transitions_Aestablish__convergence_Aappend__io__pair_Acg__initial_Acg__insert_Acg__lookup_Acg__merge_Am_J_A_061_AL_AM2_A_092_060inter_062_APrefix__Tree_Oset_A_Ispy__framework_AM1_Aget__state__cover_Aseparate__state__cover_Asort__unverified__transitions_Aestablish__convergence_Aappend__io__pair_Acg__initial_Acg__insert_Acg__lookup_Acg__merge_Am_J_092_060close_062,axiom,
    ( ( ( lS_a_b_c @ m1 @ ( initial_a_b_c @ m1 ) )
      = ( lS_d_b_c @ m2 @ ( initial_d_b_c @ m2 ) ) )
   => ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ m1 @ ( initial_a_b_c @ m1 ) ) @ ( prefix9085958534787526401od_b_c @ ( sPY_sp4487033212357905651_b_c_e @ m1 @ get_state_cover @ separate_state_cover @ sort_u5845992193009891830itions @ establ5936990325632390625rgence @ append_io_pair @ cg_initial @ cg_insert @ cg_lookup @ cg_merge @ m ) ) )
      = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ m2 @ ( initial_d_b_c @ m2 ) ) @ ( prefix9085958534787526401od_b_c @ ( sPY_sp4487033212357905651_b_c_e @ m1 @ get_state_cover @ separate_state_cover @ sort_u5845992193009891830itions @ establ5936990325632390625rgence @ append_io_pair @ cg_initial @ cg_insert @ cg_lookup @ cg_merge @ m ) ) ) ) ) ).

% \<open>L M1 = L M2 \<Longrightarrow> L M1 \<inter> Prefix_Tree.set (spy_framework M1 get_state_cover separate_state_cover sort_unverified_transitions establish_convergence append_io_pair cg_initial cg_insert cg_lookup cg_merge m) = L M2 \<inter> Prefix_Tree.set (spy_framework M1 get_state_cover separate_state_cover sort_unverified_transitions establish_convergence append_io_pair cg_initial cg_insert cg_lookup cg_merge m)\<close>
thf(fact_200_single__input_Oelims_I3_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b] :
      ( ~ ( single999786218259502647_a_a_b @ X )
     => ~ ! [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ X ) )
           => ! [Xa2: produc3205353004677322104od_c_a] :
                ( ( member6508725278956418977od_c_a @ Xa2 @ ( transi2941759273464634107_a_a_b @ X ) )
               => ( ( ( produc8171714036265584980od_c_a @ X3 )
                    = ( produc8171714036265584980od_c_a @ Xa2 ) )
                 => ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_201_single__input_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( single_input_a_b_c @ X )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X ) )
           => ! [Xa2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X ) )
               => ( ( ( produc4225943634317630866od_c_a @ X3 )
                    = ( produc4225943634317630866od_c_a @ Xa2 ) )
                 => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_202_single__input_Oelims_I2_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b] :
      ( ( single999786218259502647_a_a_b @ X )
     => ! [X5: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X5 @ ( transi2941759273464634107_a_a_b @ X ) )
         => ! [Xa: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Xa @ ( transi2941759273464634107_a_a_b @ X ) )
             => ( ( ( produc8171714036265584980od_c_a @ X5 )
                  = ( produc8171714036265584980od_c_a @ Xa ) )
               => ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X5 ) )
                  = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_203_single__input_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( single_input_a_b_c @ X )
     => ! [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X ) )
         => ! [Xa: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa @ ( transitions_a_b_c @ X ) )
             => ( ( ( produc4225943634317630866od_c_a @ X5 )
                  = ( produc4225943634317630866od_c_a @ Xa ) )
               => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                  = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_204_single__input_Oelims_I1_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b,Y: $o] :
      ( ( ( single999786218259502647_a_a_b @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ X ) )
             => ! [Y5: produc3205353004677322104od_c_a] :
                  ( ( member6508725278956418977od_c_a @ Y5 @ ( transi2941759273464634107_a_a_b @ X ) )
                 => ( ( ( produc8171714036265584980od_c_a @ X4 )
                      = ( produc8171714036265584980od_c_a @ Y5 ) )
                   => ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                      = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_205_single__input_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( single_input_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X ) )
             => ! [Y5: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ X ) )
                 => ( ( ( produc4225943634317630866od_c_a @ X4 )
                      = ( produc4225943634317630866od_c_a @ Y5 ) )
                   => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_206_a5,axiom,
    ord_le282488521294790766od_b_c @ ( prefix9085958534787526401od_b_c @ t1 ) @ ( prefix9085958534787526401od_b_c @ t ) ).

% a5
thf(fact_207_cg__insert__invar,axiom,
    ! [Gamma: list_P903359562653991662od_b_c,G: e] :
      ( ( member6330420149250801815od_b_c @ Gamma @ ( lS_a_b_c @ m1 @ ( initial_a_b_c @ m1 ) ) )
     => ( ( member6330420149250801815od_b_c @ Gamma @ ( lS_d_b_c @ m2 @ ( initial_d_b_c @ m2 ) ) )
       => ( ( conver7089555221631485245_c_d_e @ m1 @ m2 @ cg_lookup @ G )
         => ( conver7089555221631485245_c_d_e @ m1 @ m2 @ cg_lookup @ ( cg_insert @ G @ Gamma ) ) ) ) ) ).

% cg_insert_invar
thf(fact_208_G1__invar,axiom,
    ( ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ m1 @ ( initial_a_b_c @ m1 ) ) @ ( prefix9085958534787526401od_b_c @ t1 ) )
      = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ m2 @ ( initial_d_b_c @ m2 ) ) @ ( prefix9085958534787526401od_b_c @ t1 ) ) )
   => ( conver7089555221631485245_c_d_e @ m1 @ m2 @ cg_lookup @ g1 ) ) ).

% G1_invar
thf(fact_209_a1,axiom,
    ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ m1 @ ( initial_a_b_c @ m1 ) ) @ ( prefix9085958534787526401od_b_c @ ( produc1447154562020694007_b_c_e @ ( verify_transition @ ( produc2511600812177743883_b_c_e @ t @ g ) @ t4 ) ) ) )
    = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ m2 @ ( initial_d_b_c @ m2 ) ) @ ( prefix9085958534787526401od_b_c @ ( produc1447154562020694007_b_c_e @ ( verify_transition @ ( produc2511600812177743883_b_c_e @ t @ g ) @ t4 ) ) ) ) ) ).

% a1
thf(fact_210_reachable__states__initial,axiom,
    ! [M2: fsm_d_b_c] : ( member_d @ ( initial_d_b_c @ M2 ) @ ( reacha9185589339461546374_d_b_c @ M2 ) ) ).

% reachable_states_initial
thf(fact_211_reachable__states__initial,axiom,
    ! [M2: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M2 ) @ ( reacha1620305530751930115_a_b_c @ M2 ) ) ).

% reachable_states_initial
thf(fact_212_is__state__cover__assignment__language,axiom,
    ! [M2: fsm_Pr6425204305901329755_a_b_c,V: produc2669932201312201782od_c_a > list_P903359562653991662od_b_c,Q2: produc2669932201312201782od_c_a] :
      ( ( state_1238012481214864903_a_b_c @ M2 @ V )
     => ( ( member5502040268208861279od_c_a @ Q2 @ ( reacha1389639598965321787_a_b_c @ M2 ) )
       => ( member6330420149250801815od_b_c @ ( V @ Q2 ) @ ( lS_Pro3731892891411965979_a_b_c @ M2 @ ( initia3341036835946388646_a_b_c @ M2 ) ) ) ) ) ).

% is_state_cover_assignment_language
thf(fact_213_is__state__cover__assignment__language,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,V: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Q2: list_P903359562653991662od_b_c] :
      ( ( state_2362830510734323903_c_b_c @ M2 @ V )
     => ( ( member6330420149250801815od_b_c @ Q2 @ ( reacha8116992257183400179_c_b_c @ M2 ) )
       => ( member6330420149250801815od_b_c @ ( V @ Q2 ) @ ( lS_lis2930931384350476499_c_b_c @ M2 @ ( initia3567573336347591134_c_b_c @ M2 ) ) ) ) ) ).

% is_state_cover_assignment_language
thf(fact_214_is__state__cover__assignment__language,axiom,
    ! [M2: fsm_d_b_c,V: d > list_P903359562653991662od_b_c,Q2: d] :
      ( ( state_6014242345326633810_d_b_c @ M2 @ V )
     => ( ( member_d @ Q2 @ ( reacha9185589339461546374_d_b_c @ M2 ) )
       => ( member6330420149250801815od_b_c @ ( V @ Q2 ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) ) ) ) ).

% is_state_cover_assignment_language
thf(fact_215_is__state__cover__assignment__language,axiom,
    ! [M2: fsm_a_b_c,V: a > list_P903359562653991662od_b_c,Q2: a] :
      ( ( state_7672330573471793359_a_b_c @ M2 @ V )
     => ( ( member_a @ Q2 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
       => ( member6330420149250801815od_b_c @ ( V @ Q2 ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ) ) ).

% is_state_cover_assignment_language
thf(fact_216_state__cover__assignment__after_I1_J,axiom,
    ! [M2: fsm_Pr6425204305901329755_a_b_c,V: produc2669932201312201782od_c_a > list_P903359562653991662od_b_c,Q2: produc2669932201312201782od_c_a] :
      ( ( observ6303369326153519487_a_b_c @ M2 )
     => ( ( state_1238012481214864903_a_b_c @ M2 @ V )
       => ( ( member5502040268208861279od_c_a @ Q2 @ ( reacha1389639598965321787_a_b_c @ M2 ) )
         => ( member6330420149250801815od_b_c @ ( V @ Q2 ) @ ( lS_Pro3731892891411965979_a_b_c @ M2 @ ( initia3341036835946388646_a_b_c @ M2 ) ) ) ) ) ) ).

% state_cover_assignment_after(1)
thf(fact_217_state__cover__assignment__after_I1_J,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,V: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,Q2: list_P903359562653991662od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M2 )
     => ( ( state_2362830510734323903_c_b_c @ M2 @ V )
       => ( ( member6330420149250801815od_b_c @ Q2 @ ( reacha8116992257183400179_c_b_c @ M2 ) )
         => ( member6330420149250801815od_b_c @ ( V @ Q2 ) @ ( lS_lis2930931384350476499_c_b_c @ M2 @ ( initia3567573336347591134_c_b_c @ M2 ) ) ) ) ) ) ).

% state_cover_assignment_after(1)
thf(fact_218_state__cover__assignment__after_I1_J,axiom,
    ! [M2: fsm_d_b_c,V: d > list_P903359562653991662od_b_c,Q2: d] :
      ( ( observable_d_b_c @ M2 )
     => ( ( state_6014242345326633810_d_b_c @ M2 @ V )
       => ( ( member_d @ Q2 @ ( reacha9185589339461546374_d_b_c @ M2 ) )
         => ( member6330420149250801815od_b_c @ ( V @ Q2 ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) ) ) ) ) ).

% state_cover_assignment_after(1)
thf(fact_219_state__cover__assignment__after_I1_J,axiom,
    ! [M2: fsm_a_b_c,V: a > list_P903359562653991662od_b_c,Q2: a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( state_7672330573471793359_a_b_c @ M2 @ V )
       => ( ( member_a @ Q2 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
         => ( member6330420149250801815od_b_c @ ( V @ Q2 ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ) ) ) ).

% state_cover_assignment_after(1)
thf(fact_220_reachable__states__initial__or__target,axiom,
    ! [Q2: d,M2: fsm_d_b_c] :
      ( ( member_d @ Q2 @ ( reacha9185589339461546374_d_b_c @ M2 ) )
     => ( ( Q2
          = ( initial_d_b_c @ M2 ) )
        | ? [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( transitions_d_b_c @ M2 ) )
            & ( member_d @ ( produc1508465569166014290od_c_d @ X3 ) @ ( reacha9185589339461546374_d_b_c @ M2 ) )
            & ( ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X3 ) ) )
              = Q2 ) ) ) ) ).

% reachable_states_initial_or_target
thf(fact_221_reachable__states__initial__or__target,axiom,
    ! [Q2: product_prod_c_a,M2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( member5074992350434858958od_c_a @ Q2 @ ( reacha5035603111973943274_a_a_b @ M2 ) )
     => ( ( Q2
          = ( initia681974817753967189_a_a_b @ M2 ) )
        | ? [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ M2 ) )
            & ( member5074992350434858958od_c_a @ ( produc8171714036265584980od_c_a @ X3 ) @ ( reacha5035603111973943274_a_a_b @ M2 ) )
            & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
              = Q2 ) ) ) ) ).

% reachable_states_initial_or_target
thf(fact_222_reachable__states__initial__or__target,axiom,
    ! [Q2: a,M2: fsm_a_b_c] :
      ( ( member_a @ Q2 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
     => ( ( Q2
          = ( initial_a_b_c @ M2 ) )
        | ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ M2 ) )
            & ( member_a @ ( produc4225943634317630866od_c_a @ X3 ) @ ( reacha1620305530751930115_a_b_c @ M2 ) )
            & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
              = Q2 ) ) ) ) ).

% reachable_states_initial_or_target
thf(fact_223_reachable__states__induct,axiom,
    ! [Q2: d,M2: fsm_d_b_c,P: d > $o] :
      ( ( member_d @ Q2 @ ( reacha9185589339461546374_d_b_c @ M2 ) )
     => ( ( P @ ( initial_d_b_c @ M2 ) )
       => ( ! [T4: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ T4 @ ( transitions_d_b_c @ M2 ) )
             => ( ( member_d @ ( produc1508465569166014290od_c_d @ T4 ) @ ( reacha9185589339461546374_d_b_c @ M2 ) )
               => ( ( P @ ( produc1508465569166014290od_c_d @ T4 ) )
                 => ( P @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T4 ) ) ) ) ) ) )
         => ( P @ Q2 ) ) ) ) ).

% reachable_states_induct
thf(fact_224_reachable__states__induct,axiom,
    ! [Q2: product_prod_c_a,M2: fsm_Pr1886043760914421514_a_a_b,P: product_prod_c_a > $o] :
      ( ( member5074992350434858958od_c_a @ Q2 @ ( reacha5035603111973943274_a_a_b @ M2 ) )
     => ( ( P @ ( initia681974817753967189_a_a_b @ M2 ) )
       => ( ! [T4: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
             => ( ( member5074992350434858958od_c_a @ ( produc8171714036265584980od_c_a @ T4 ) @ ( reacha5035603111973943274_a_a_b @ M2 ) )
               => ( ( P @ ( produc8171714036265584980od_c_a @ T4 ) )
                 => ( P @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) ) ) )
         => ( P @ Q2 ) ) ) ) ).

% reachable_states_induct
thf(fact_225_reachable__states__induct,axiom,
    ! [Q2: a,M2: fsm_a_b_c,P: a > $o] :
      ( ( member_a @ Q2 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
     => ( ( P @ ( initial_a_b_c @ M2 ) )
       => ( ! [T4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
             => ( ( member_a @ ( produc4225943634317630866od_c_a @ T4 ) @ ( reacha1620305530751930115_a_b_c @ M2 ) )
               => ( ( P @ ( produc4225943634317630866od_c_a @ T4 ) )
                 => ( P @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) ) ) )
         => ( P @ Q2 ) ) ) ) ).

% reachable_states_induct
thf(fact_226_reachable__states__cases,axiom,
    ! [Q2: d,M2: fsm_d_b_c,P: d > $o] :
      ( ( member_d @ Q2 @ ( reacha9185589339461546374_d_b_c @ M2 ) )
     => ( ( P @ ( initial_d_b_c @ M2 ) )
       => ( ! [T4: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ T4 @ ( transitions_d_b_c @ M2 ) )
             => ( ( member_d @ ( produc1508465569166014290od_c_d @ T4 ) @ ( reacha9185589339461546374_d_b_c @ M2 ) )
               => ( P @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T4 ) ) ) ) ) )
         => ( P @ Q2 ) ) ) ) ).

% reachable_states_cases
thf(fact_227_reachable__states__cases,axiom,
    ! [Q2: product_prod_c_a,M2: fsm_Pr1886043760914421514_a_a_b,P: product_prod_c_a > $o] :
      ( ( member5074992350434858958od_c_a @ Q2 @ ( reacha5035603111973943274_a_a_b @ M2 ) )
     => ( ( P @ ( initia681974817753967189_a_a_b @ M2 ) )
       => ( ! [T4: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
             => ( ( member5074992350434858958od_c_a @ ( produc8171714036265584980od_c_a @ T4 ) @ ( reacha5035603111973943274_a_a_b @ M2 ) )
               => ( P @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) ) )
         => ( P @ Q2 ) ) ) ) ).

% reachable_states_cases
thf(fact_228_reachable__states__cases,axiom,
    ! [Q2: a,M2: fsm_a_b_c,P: a > $o] :
      ( ( member_a @ Q2 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
     => ( ( P @ ( initial_a_b_c @ M2 ) )
       => ( ! [T4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
             => ( ( member_a @ ( produc4225943634317630866od_c_a @ T4 ) @ ( reacha1620305530751930115_a_b_c @ M2 ) )
               => ( P @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) ) )
         => ( P @ Q2 ) ) ) ) ).

% reachable_states_cases
thf(fact_229_single__input__alt__def,axiom,
    ( single999786218259502647_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q1: product_prod_c_a,X4: a,X6: a,Y5: b,Y7: b,Q12: product_prod_c_a,Q13: product_prod_c_a] :
          ( ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y5 @ Q12 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) )
            & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X6 @ ( produc3440865914000881661od_c_a @ Y7 @ Q13 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) )
         => ( X4 = X6 ) ) ) ) ).

% single_input_alt_def
thf(fact_230_single__input__alt__def,axiom,
    ( single_input_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [Q1: a,X4: b,X6: b,Y5: c,Y7: c,Q12: a,Q13: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y5 @ Q12 ) ) ) @ ( transitions_a_b_c @ M ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X6 @ ( product_Pair_c_a @ Y7 @ Q13 ) ) ) @ ( transitions_a_b_c @ M ) ) )
         => ( X4 = X6 ) ) ) ) ).

% single_input_alt_def
thf(fact_231_single__input_Osimps,axiom,
    ( single999786218259502647_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ M ) )
         => ! [Y5: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Y5 @ ( transi2941759273464634107_a_a_b @ M ) )
             => ( ( ( produc8171714036265584980od_c_a @ X4 )
                  = ( produc8171714036265584980od_c_a @ Y5 ) )
               => ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                  = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_232_single__input_Osimps,axiom,
    ( single_input_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M ) )
         => ! [Y5: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ M ) )
             => ( ( ( produc4225943634317630866od_c_a @ X4 )
                  = ( produc4225943634317630866od_c_a @ Y5 ) )
               => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                  = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_233_verify__transition__retains__testsuite,axiom,
    ! [T5: prefix425356687782099470od_b_c,G: e,T: produc2669932201312201782od_c_a] : ( ord_le282488521294790766od_b_c @ ( prefix9085958534787526401od_b_c @ T5 ) @ ( prefix9085958534787526401od_b_c @ ( produc1447154562020694007_b_c_e @ ( verify_transition @ ( produc2511600812177743883_b_c_e @ T5 @ G ) @ T ) ) ) ) ).

% verify_transition_retains_testsuite
thf(fact_234_T1__state__cover,axiom,
    ord_le282488521294790766od_b_c @ ( image_8656872703560917237od_b_c @ v @ ( reacha1620305530751930115_a_b_c @ m1 ) ) @ ( prefix9085958534787526401od_b_c @ t1 ) ).

% T1_state_cover
thf(fact_235_Int__subset__iff,axiom,
    ! [C2: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) )
      = ( ( ord_le282488521294790766od_b_c @ C2 @ A4 )
        & ( ord_le282488521294790766od_b_c @ C2 @ B4 ) ) ) ).

% Int_subset_iff
thf(fact_236_inf_Obounded__iff,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B @ C3 ) )
      = ( ( ord_le282488521294790766od_b_c @ A @ B )
        & ( ord_le282488521294790766od_b_c @ A @ C3 ) ) ) ).

% inf.bounded_iff
thf(fact_237_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C3 ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C3 ) ) ) ).

% inf.bounded_iff
thf(fact_238_le__inf__iff,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ Y @ Z2 ) )
      = ( ( ord_le282488521294790766od_b_c @ X @ Y )
        & ( ord_le282488521294790766od_b_c @ X @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_239_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( ord_less_eq_nat @ X @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_240_convergence__graph__insert__invar__def,axiom,
    ( conver2727675425255806140_c_d_e
    = ( ^ [M1: fsm_a_b_c,M22: fsm_d_b_c,Cg_lookup: e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,Cg_insert: e > list_P903359562653991662od_b_c > e] :
        ! [G2: e,Gamma2: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ Gamma2 @ ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) ) )
         => ( ( member6330420149250801815od_b_c @ Gamma2 @ ( lS_d_b_c @ M22 @ ( initial_d_b_c @ M22 ) ) )
           => ( ( conver7089555221631485245_c_d_e @ M1 @ M22 @ Cg_lookup @ G2 )
             => ( conver7089555221631485245_c_d_e @ M1 @ M22 @ Cg_lookup @ ( Cg_insert @ G2 @ Gamma2 ) ) ) ) ) ) ) ).

% convergence_graph_insert_invar_def
thf(fact_241_observable__language__next,axiom,
    ! [Io: product_prod_c_a,Ios: list_P125642481956313003od_c_a,M2: fsm_Pr7461828476839194351_a_c_a,T: produc3729687299602777783od_c_a] :
      ( ( member5552703068553123156od_c_a @ ( cons_P1742027962761213787od_c_a @ Io @ Ios ) @ ( lS_Pro1436182445161660463_a_c_a @ M2 @ ( produc1784202036732648467od_c_a @ T ) ) )
     => ( ( observ3736821838051356051_a_c_a @ M2 )
       => ( ( member5500486793640445664od_c_a @ T @ ( transi5448983560151880288_a_c_a @ M2 ) )
         => ( ( ( produc7173261134821384633od_c_a @ ( produc2799690148840255317od_c_a @ T ) )
              = ( product_fst_c_a @ Io ) )
           => ( ( ( produc4225943634317630866od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) )
                = ( product_snd_c_a @ Io ) )
             => ( member5552703068553123156od_c_a @ Ios @ ( lS_Pro1436182445161660463_a_c_a @ ( from_F8501008907474727400_a_c_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) ) ) @ ( initia6168580702928107450_a_c_a @ ( from_F8501008907474727400_a_c_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_242_observable__language__next,axiom,
    ! [Io: product_prod_b_c,Ios: list_P903359562653991662od_b_c,M2: fsm_d_b_c,T: produc1399170498009216118od_c_d] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ Io @ Ios ) @ ( lS_d_b_c @ M2 @ ( produc1508465569166014290od_c_d @ T ) ) )
     => ( ( observable_d_b_c @ M2 )
       => ( ( member4231278564905875615od_c_d @ T @ ( transitions_d_b_c @ M2 ) )
         => ( ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T ) )
              = ( product_fst_b_c @ Io ) )
           => ( ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) )
                = ( product_snd_b_c @ Io ) )
             => ( member6330420149250801815od_b_c @ Ios @ ( lS_d_b_c @ ( from_FSM_d_b_c @ M2 @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) ) @ ( initial_d_b_c @ ( from_FSM_d_b_c @ M2 @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_243_observable__language__next,axiom,
    ! [Io: product_prod_a_b,Ios: list_P1467974523212632366od_a_b,M2: fsm_Pr1886043760914421514_a_a_b,T: produc3205353004677322104od_c_a] :
      ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ Io @ Ios ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T ) ) )
     => ( ( observ2964502264774598958_a_a_b @ M2 )
       => ( ( member6508725278956418977od_c_a @ T @ ( transi2941759273464634107_a_a_b @ M2 ) )
         => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) )
              = ( product_fst_a_b @ Io ) )
           => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) )
                = ( product_snd_a_b @ Io ) )
             => ( member6895035109809442519od_a_b @ Ios @ ( lS_Pro5114676609533226954_a_a_b @ ( from_F6727869081933152899_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) @ ( initia681974817753967189_a_a_b @ ( from_F6727869081933152899_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_244_observable__language__next,axiom,
    ! [Io: product_prod_b_c,Ios: list_P903359562653991662od_b_c,M2: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ Io @ Ios ) @ ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T ) ) )
     => ( ( observable_a_b_c @ M2 )
       => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M2 ) )
         => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) )
              = ( product_fst_b_c @ Io ) )
           => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
                = ( product_snd_b_c @ Io ) )
             => ( member6330420149250801815od_b_c @ Ios @ ( lS_a_b_c @ ( from_FSM_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_245_a6,axiom,
    ( ( append2547753245680614915od_b_c @ ( v @ ( produc4225943634317630866od_c_a @ t4 ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ t4 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ t4 ) ) ) ) @ nil_Product_prod_b_c ) )
   != ( v @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ t4 ) ) ) ) ) ).

% a6
thf(fact_246_T1__V__div,axiom,
    ( ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ m1 @ ( initial_a_b_c @ m1 ) ) @ ( prefix9085958534787526401od_b_c @ t1 ) )
      = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ m2 @ ( initial_d_b_c @ m2 ) ) @ ( prefix9085958534787526401od_b_c @ t1 ) ) )
   => ( preser261684820601417592_b_c_d @ m1 @ m2 @ ( image_8656872703560917237od_b_c @ v @ ( reacha1620305530751930115_a_b_c @ m1 ) ) ) ) ).

% T1_V_div
thf(fact_247_LS__from__LS__acyclic,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( acyclic_a_b_c @ M2 )
     => ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
        = ( lS_acyclic_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ) ).

% LS_from_LS_acyclic
thf(fact_248_LS__from__LS__acyclic,axiom,
    ! [M2: fsm_d_b_c] :
      ( ( acyclic_d_b_c @ M2 )
     => ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
        = ( lS_acyclic_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) ) ) ).

% LS_from_LS_acyclic
thf(fact_249_image__eqI,axiom,
    ! [B: produc2669932201312201782od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5502040268208861279od_c_a @ X @ A4 )
       => ( member5502040268208861279od_c_a @ B @ ( image_2169818278710580869od_c_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_250_image__eqI,axiom,
    ! [B: a,F: produc2669932201312201782od_c_a > a,X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5502040268208861279od_c_a @ X @ A4 )
       => ( member_a @ B @ ( image_9171994905208062541_c_a_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_251_image__eqI,axiom,
    ! [B: list_P903359562653991662od_b_c,F: produc2669932201312201782od_c_a > list_P903359562653991662od_b_c,X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5502040268208861279od_c_a @ X @ A4 )
       => ( member6330420149250801815od_b_c @ B @ ( image_826490394676381629od_b_c @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_252_image__eqI,axiom,
    ! [B: set_Pr8357624050790843030od_c_a,F: a > set_Pr8357624050790843030od_c_a,X: a,A4: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A4 )
       => ( member9073573073229046463od_c_a @ B @ ( image_2946797484331234845od_c_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_253_image__eqI,axiom,
    ! [B: produc2669932201312201782od_c_a,F: a > produc2669932201312201782od_c_a,X: a,A4: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A4 )
       => ( member5502040268208861279od_c_a @ B @ ( image_4184489845059386301od_c_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_254_image__eqI,axiom,
    ! [B: a,F: a > a,X: a,A4: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A4 )
       => ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_255_image__eqI,axiom,
    ! [B: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c,X: a,A4: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A4 )
       => ( member6330420149250801815od_b_c @ B @ ( image_8656872703560917237od_b_c @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_256_image__eqI,axiom,
    ! [B: produc2669932201312201782od_c_a,F: list_P903359562653991662od_b_c > produc2669932201312201782od_c_a,X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( B
        = ( F @ X ) )
     => ( ( member6330420149250801815od_b_c @ X @ A4 )
       => ( member5502040268208861279od_c_a @ B @ ( image_7601443023479506637od_c_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_257_image__eqI,axiom,
    ! [B: a,F: list_P903359562653991662od_b_c > a,X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( B
        = ( F @ X ) )
     => ( ( member6330420149250801815od_b_c @ X @ A4 )
       => ( member_a @ B @ ( image_3173854338463926933_b_c_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_258_image__eqI,axiom,
    ! [B: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( B
        = ( F @ X ) )
     => ( ( member6330420149250801815od_b_c @ X @ A4 )
       => ( member6330420149250801815od_b_c @ B @ ( image_2850395784482391045od_b_c @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_259_subsetI,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X3 @ A4 )
         => ( member5502040268208861279od_c_a @ X3 @ B4 ) )
     => ( ord_le3226338165915859510od_c_a @ A4 @ B4 ) ) ).

% subsetI
thf(fact_260_subsetI,axiom,
    ! [A4: set_a,B4: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A4 )
         => ( member_a @ X3 @ B4 ) )
     => ( ord_less_eq_set_a @ A4 @ B4 ) ) ).

% subsetI
thf(fact_261_subsetI,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A4 )
         => ( member6330420149250801815od_b_c @ X3 @ B4 ) )
     => ( ord_le282488521294790766od_b_c @ A4 @ B4 ) ) ).

% subsetI
thf(fact_262_subset__antisym,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
     => ( ( ord_le282488521294790766od_b_c @ B4 @ A4 )
       => ( A4 = B4 ) ) ) ).

% subset_antisym
thf(fact_263_inf_Oidem,axiom,
    ! [A: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ A @ A )
      = A ) ).

% inf.idem
thf(fact_264_inf__idem,axiom,
    ! [X: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ X @ X )
      = X ) ).

% inf_idem
thf(fact_265_inf_Oleft__idem,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
      = ( inf_in4978071631833541052od_b_c @ A @ B ) ) ).

% inf.left_idem
thf(fact_266_inf__left__idem,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ X @ Y ) )
      = ( inf_in4978071631833541052od_b_c @ X @ Y ) ) ).

% inf_left_idem
thf(fact_267_inf_Oright__idem,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ B )
      = ( inf_in4978071631833541052od_b_c @ A @ B ) ) ).

% inf.right_idem
thf(fact_268_inf__right__idem,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ Y )
      = ( inf_in4978071631833541052od_b_c @ X @ Y ) ) ).

% inf_right_idem
thf(fact_269_IntI,axiom,
    ! [C3: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C3 @ A4 )
     => ( ( member5502040268208861279od_c_a @ C3 @ B4 )
       => ( member5502040268208861279od_c_a @ C3 @ ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) ) ) ) ).

% IntI
thf(fact_270_IntI,axiom,
    ! [C3: a,A4: set_a,B4: set_a] :
      ( ( member_a @ C3 @ A4 )
     => ( ( member_a @ C3 @ B4 )
       => ( member_a @ C3 @ ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).

% IntI
thf(fact_271_IntI,axiom,
    ! [C3: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C3 @ A4 )
     => ( ( member6330420149250801815od_b_c @ C3 @ B4 )
       => ( member6330420149250801815od_b_c @ C3 @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) ) ).

% IntI
thf(fact_272_Int__iff,axiom,
    ! [C3: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C3 @ ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) )
      = ( ( member5502040268208861279od_c_a @ C3 @ A4 )
        & ( member5502040268208861279od_c_a @ C3 @ B4 ) ) ) ).

% Int_iff
thf(fact_273_Int__iff,axiom,
    ! [C3: a,A4: set_a,B4: set_a] :
      ( ( member_a @ C3 @ ( inf_inf_set_a @ A4 @ B4 ) )
      = ( ( member_a @ C3 @ A4 )
        & ( member_a @ C3 @ B4 ) ) ) ).

% Int_iff
thf(fact_274_Int__iff,axiom,
    ! [C3: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C3 @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) )
      = ( ( member6330420149250801815od_b_c @ C3 @ A4 )
        & ( member6330420149250801815od_b_c @ C3 @ B4 ) ) ) ).

% Int_iff
thf(fact_275_assms_I6_J,axiom,
    ord_less_eq_nat @ ( size_d_b_c @ m2 ) @ m ).

% assms(6)
thf(fact_276_assms_I5_J,axiom,
    ord_less_eq_nat @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ m1 ) ) @ m ).

% assms(5)
thf(fact_277_imageI,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ A4 )
     => ( member5502040268208861279od_c_a @ ( F @ X ) @ ( image_2169818278710580869od_c_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_278_imageI,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > a] :
      ( ( member5502040268208861279od_c_a @ X @ A4 )
     => ( member_a @ ( F @ X ) @ ( image_9171994905208062541_c_a_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_279_imageI,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > list_P903359562653991662od_b_c] :
      ( ( member5502040268208861279od_c_a @ X @ A4 )
     => ( member6330420149250801815od_b_c @ ( F @ X ) @ ( image_826490394676381629od_b_c @ F @ A4 ) ) ) ).

% imageI
thf(fact_280_imageI,axiom,
    ! [X: a,A4: set_a,F: a > set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ X @ A4 )
     => ( member9073573073229046463od_c_a @ ( F @ X ) @ ( image_2946797484331234845od_c_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_281_imageI,axiom,
    ! [X: a,A4: set_a,F: a > produc2669932201312201782od_c_a] :
      ( ( member_a @ X @ A4 )
     => ( member5502040268208861279od_c_a @ ( F @ X ) @ ( image_4184489845059386301od_c_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_282_imageI,axiom,
    ! [X: a,A4: set_a,F: a > a] :
      ( ( member_a @ X @ A4 )
     => ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_283_imageI,axiom,
    ! [X: a,A4: set_a,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X @ A4 )
     => ( member6330420149250801815od_b_c @ ( F @ X ) @ ( image_8656872703560917237od_b_c @ F @ A4 ) ) ) ).

% imageI
thf(fact_284_imageI,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > produc2669932201312201782od_c_a] :
      ( ( member6330420149250801815od_b_c @ X @ A4 )
     => ( member5502040268208861279od_c_a @ ( F @ X ) @ ( image_7601443023479506637od_c_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_285_imageI,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > a] :
      ( ( member6330420149250801815od_b_c @ X @ A4 )
     => ( member_a @ ( F @ X ) @ ( image_3173854338463926933_b_c_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_286_imageI,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ A4 )
     => ( member6330420149250801815od_b_c @ ( F @ X ) @ ( image_2850395784482391045od_b_c @ F @ A4 ) ) ) ).

% imageI
thf(fact_287_image__iff,axiom,
    ! [Z2: set_Pr8357624050790843030od_c_a,F: a > set_Pr8357624050790843030od_c_a,A4: set_a] :
      ( ( member9073573073229046463od_c_a @ Z2 @ ( image_2946797484331234845od_c_a @ F @ A4 ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A4 )
            & ( Z2
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_288_image__iff,axiom,
    ! [Z2: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c,A4: set_a] :
      ( ( member6330420149250801815od_b_c @ Z2 @ ( image_8656872703560917237od_b_c @ F @ A4 ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A4 )
            & ( Z2
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_289_bex__imageD,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A4: set_a,P: list_P903359562653991662od_b_c > $o] :
      ( ? [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( image_8656872703560917237od_b_c @ F @ A4 ) )
          & ( P @ X5 ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ A4 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_290_bex__imageD,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,A4: set_a,P: set_Pr8357624050790843030od_c_a > $o] :
      ( ? [X5: set_Pr8357624050790843030od_c_a] :
          ( ( member9073573073229046463od_c_a @ X5 @ ( image_2946797484331234845od_c_a @ F @ A4 ) )
          & ( P @ X5 ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ A4 )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_291_image__cong,axiom,
    ! [M2: set_a,N: set_a,F: a > list_P903359562653991662od_b_c,G3: a > list_P903359562653991662od_b_c] :
      ( ( M2 = N )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G3 @ X3 ) ) )
       => ( ( image_8656872703560917237od_b_c @ F @ M2 )
          = ( image_8656872703560917237od_b_c @ G3 @ N ) ) ) ) ).

% image_cong
thf(fact_292_image__cong,axiom,
    ! [M2: set_a,N: set_a,F: a > set_Pr8357624050790843030od_c_a,G3: a > set_Pr8357624050790843030od_c_a] :
      ( ( M2 = N )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G3 @ X3 ) ) )
       => ( ( image_2946797484331234845od_c_a @ F @ M2 )
          = ( image_2946797484331234845od_c_a @ G3 @ N ) ) ) ) ).

% image_cong
thf(fact_293_ball__imageD,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A4: set_a,P: list_P903359562653991662od_b_c > $o] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ ( image_8656872703560917237od_b_c @ F @ A4 ) )
         => ( P @ X3 ) )
     => ! [X5: a] :
          ( ( member_a @ X5 @ A4 )
         => ( P @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_294_ball__imageD,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,A4: set_a,P: set_Pr8357624050790843030od_c_a > $o] :
      ( ! [X3: set_Pr8357624050790843030od_c_a] :
          ( ( member9073573073229046463od_c_a @ X3 @ ( image_2946797484331234845od_c_a @ F @ A4 ) )
         => ( P @ X3 ) )
     => ! [X5: a] :
          ( ( member_a @ X5 @ A4 )
         => ( P @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_295_rev__image__eqI,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member5502040268208861279od_c_a @ B @ ( image_2169818278710580869od_c_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_296_rev__image__eqI,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B: a,F: produc2669932201312201782od_c_a > a] :
      ( ( member5502040268208861279od_c_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_9171994905208062541_c_a_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_297_rev__image__eqI,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B: list_P903359562653991662od_b_c,F: produc2669932201312201782od_c_a > list_P903359562653991662od_b_c] :
      ( ( member5502040268208861279od_c_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member6330420149250801815od_b_c @ B @ ( image_826490394676381629od_b_c @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_298_rev__image__eqI,axiom,
    ! [X: a,A4: set_a,B: set_Pr8357624050790843030od_c_a,F: a > set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member9073573073229046463od_c_a @ B @ ( image_2946797484331234845od_c_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_299_rev__image__eqI,axiom,
    ! [X: a,A4: set_a,B: produc2669932201312201782od_c_a,F: a > produc2669932201312201782od_c_a] :
      ( ( member_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member5502040268208861279od_c_a @ B @ ( image_4184489845059386301od_c_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_300_rev__image__eqI,axiom,
    ! [X: a,A4: set_a,B: a,F: a > a] :
      ( ( member_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_301_rev__image__eqI,axiom,
    ! [X: a,A4: set_a,B: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member6330420149250801815od_b_c @ B @ ( image_8656872703560917237od_b_c @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_302_rev__image__eqI,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B: produc2669932201312201782od_c_a,F: list_P903359562653991662od_b_c > produc2669932201312201782od_c_a] :
      ( ( member6330420149250801815od_b_c @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member5502040268208861279od_c_a @ B @ ( image_7601443023479506637od_c_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_303_rev__image__eqI,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B: a,F: list_P903359562653991662od_b_c > a] :
      ( ( member6330420149250801815od_b_c @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_3173854338463926933_b_c_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_304_rev__image__eqI,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member6330420149250801815od_b_c @ B @ ( image_2850395784482391045od_b_c @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_305_image__mono,axiom,
    ! [A4: set_a,B4: set_a,F: a > set_Pr8357624050790843030od_c_a] :
      ( ( ord_less_eq_set_a @ A4 @ B4 )
     => ( ord_le7969726145505873046od_c_a @ ( image_2946797484331234845od_c_a @ F @ A4 ) @ ( image_2946797484331234845od_c_a @ F @ B4 ) ) ) ).

% image_mono
thf(fact_306_image__mono,axiom,
    ! [A4: set_a,B4: set_a,F: a > list_P903359562653991662od_b_c] :
      ( ( ord_less_eq_set_a @ A4 @ B4 )
     => ( ord_le282488521294790766od_b_c @ ( image_8656872703560917237od_b_c @ F @ A4 ) @ ( image_8656872703560917237od_b_c @ F @ B4 ) ) ) ).

% image_mono
thf(fact_307_image__mono,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
     => ( ord_le282488521294790766od_b_c @ ( image_2850395784482391045od_b_c @ F @ A4 ) @ ( image_2850395784482391045od_b_c @ F @ B4 ) ) ) ).

% image_mono
thf(fact_308_image__subsetI,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X3 @ A4 )
         => ( member5502040268208861279od_c_a @ ( F @ X3 ) @ B4 ) )
     => ( ord_le3226338165915859510od_c_a @ ( image_2169818278710580869od_c_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_309_image__subsetI,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > a,B4: set_a] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X3 @ A4 )
         => ( member_a @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_a @ ( image_9171994905208062541_c_a_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_310_image__subsetI,axiom,
    ! [A4: set_a,F: a > set_Pr8357624050790843030od_c_a,B4: set_se312849170327590134od_c_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A4 )
         => ( member9073573073229046463od_c_a @ ( F @ X3 ) @ B4 ) )
     => ( ord_le7969726145505873046od_c_a @ ( image_2946797484331234845od_c_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_311_image__subsetI,axiom,
    ! [A4: set_a,F: a > produc2669932201312201782od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A4 )
         => ( member5502040268208861279od_c_a @ ( F @ X3 ) @ B4 ) )
     => ( ord_le3226338165915859510od_c_a @ ( image_4184489845059386301od_c_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_312_image__subsetI,axiom,
    ! [A4: set_a,F: a > a,B4: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A4 )
         => ( member_a @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_313_image__subsetI,axiom,
    ! [A4: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > produc2669932201312201782od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A4 )
         => ( member5502040268208861279od_c_a @ ( F @ X3 ) @ B4 ) )
     => ( ord_le3226338165915859510od_c_a @ ( image_7601443023479506637od_c_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_314_image__subsetI,axiom,
    ! [A4: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > a,B4: set_a] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A4 )
         => ( member_a @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_a @ ( image_3173854338463926933_b_c_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_315_image__subsetI,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > list_P903359562653991662od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X3 @ A4 )
         => ( member6330420149250801815od_b_c @ ( F @ X3 ) @ B4 ) )
     => ( ord_le282488521294790766od_b_c @ ( image_826490394676381629od_b_c @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_316_image__subsetI,axiom,
    ! [A4: set_a,F: a > list_P903359562653991662od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A4 )
         => ( member6330420149250801815od_b_c @ ( F @ X3 ) @ B4 ) )
     => ( ord_le282488521294790766od_b_c @ ( image_8656872703560917237od_b_c @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_317_image__subsetI,axiom,
    ! [A4: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A4 )
         => ( member6330420149250801815od_b_c @ ( F @ X3 ) @ B4 ) )
     => ( ord_le282488521294790766od_b_c @ ( image_2850395784482391045od_b_c @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_318_subset__imageE,axiom,
    ! [B4: set_se312849170327590134od_c_a,F: a > set_Pr8357624050790843030od_c_a,A4: set_a] :
      ( ( ord_le7969726145505873046od_c_a @ B4 @ ( image_2946797484331234845od_c_a @ F @ A4 ) )
     => ~ ! [C4: set_a] :
            ( ( ord_less_eq_set_a @ C4 @ A4 )
           => ( B4
             != ( image_2946797484331234845od_c_a @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_319_subset__imageE,axiom,
    ! [B4: set_li6436108459499378894od_b_c,F: a > list_P903359562653991662od_b_c,A4: set_a] :
      ( ( ord_le282488521294790766od_b_c @ B4 @ ( image_8656872703560917237od_b_c @ F @ A4 ) )
     => ~ ! [C4: set_a] :
            ( ( ord_less_eq_set_a @ C4 @ A4 )
           => ( B4
             != ( image_8656872703560917237od_b_c @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_320_subset__imageE,axiom,
    ! [B4: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B4 @ ( image_2850395784482391045od_b_c @ F @ A4 ) )
     => ~ ! [C4: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ C4 @ A4 )
           => ( B4
             != ( image_2850395784482391045od_b_c @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_321_image__subset__iff,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,A4: set_a,B4: set_se312849170327590134od_c_a] :
      ( ( ord_le7969726145505873046od_c_a @ ( image_2946797484331234845od_c_a @ F @ A4 ) @ B4 )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A4 )
           => ( member9073573073229046463od_c_a @ ( F @ X4 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_322_image__subset__iff,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A4: set_a,B4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( image_8656872703560917237od_b_c @ F @ A4 ) @ B4 )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A4 )
           => ( member6330420149250801815od_b_c @ ( F @ X4 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_323_subset__image__iff,axiom,
    ! [B4: set_se312849170327590134od_c_a,F: a > set_Pr8357624050790843030od_c_a,A4: set_a] :
      ( ( ord_le7969726145505873046od_c_a @ B4 @ ( image_2946797484331234845od_c_a @ F @ A4 ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A4 )
            & ( B4
              = ( image_2946797484331234845od_c_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_324_subset__image__iff,axiom,
    ! [B4: set_li6436108459499378894od_b_c,F: a > list_P903359562653991662od_b_c,A4: set_a] :
      ( ( ord_le282488521294790766od_b_c @ B4 @ ( image_8656872703560917237od_b_c @ F @ A4 ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A4 )
            & ( B4
              = ( image_8656872703560917237od_b_c @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_325_subset__image__iff,axiom,
    ! [B4: set_li6436108459499378894od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B4 @ ( image_2850395784482391045od_b_c @ F @ A4 ) )
      = ( ? [AA: set_li6436108459499378894od_b_c] :
            ( ( ord_le282488521294790766od_b_c @ AA @ A4 )
            & ( B4
              = ( image_2850395784482391045od_b_c @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_326_language__prefix,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M2: fsm_a_b_c,Q2: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M2 @ Q2 ) )
     => ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M2 @ Q2 ) ) ) ).

% language_prefix
thf(fact_327_language__prefix,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M2: fsm_d_b_c,Q2: d] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_d_b_c @ M2 @ Q2 ) )
     => ( member6330420149250801815od_b_c @ Io1 @ ( lS_d_b_c @ M2 @ Q2 ) ) ) ).

% language_prefix
thf(fact_328_image__Int__subset,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,A4: set_a,B4: set_a] : ( ord_le7969726145505873046od_c_a @ ( image_2946797484331234845od_c_a @ F @ ( inf_inf_set_a @ A4 @ B4 ) ) @ ( inf_in6910154213981421796od_c_a @ ( image_2946797484331234845od_c_a @ F @ A4 ) @ ( image_2946797484331234845od_c_a @ F @ B4 ) ) ) ).

% image_Int_subset
thf(fact_329_image__Int__subset,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A4: set_a,B4: set_a] : ( ord_le282488521294790766od_b_c @ ( image_8656872703560917237od_b_c @ F @ ( inf_inf_set_a @ A4 @ B4 ) ) @ ( inf_in4978071631833541052od_b_c @ ( image_8656872703560917237od_b_c @ F @ A4 ) @ ( image_8656872703560917237od_b_c @ F @ B4 ) ) ) ).

% image_Int_subset
thf(fact_330_image__Int__subset,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( image_2850395784482391045od_b_c @ F @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) @ ( inf_in4978071631833541052od_b_c @ ( image_2850395784482391045od_b_c @ F @ A4 ) @ ( image_2850395784482391045od_b_c @ F @ B4 ) ) ) ).

% image_Int_subset
thf(fact_331_from__FSM__observable,axiom,
    ! [M2: fsm_a_b_c,Q2: a] :
      ( ( observable_a_b_c @ M2 )
     => ( observable_a_b_c @ ( from_FSM_a_b_c @ M2 @ Q2 ) ) ) ).

% from_FSM_observable
thf(fact_332_from__FSM__observable,axiom,
    ! [M2: fsm_d_b_c,Q2: d] :
      ( ( observable_d_b_c @ M2 )
     => ( observable_d_b_c @ ( from_FSM_d_b_c @ M2 @ Q2 ) ) ) ).

% from_FSM_observable
thf(fact_333_language__contains__empty__sequence,axiom,
    ! [M2: fsm_a_b_c] : ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ).

% language_contains_empty_sequence
thf(fact_334_language__contains__empty__sequence,axiom,
    ! [M2: fsm_d_b_c] : ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) ) ).

% language_contains_empty_sequence
thf(fact_335_from__FSM__reachable__states,axiom,
    ! [Q2: a,M2: fsm_a_b_c] :
      ( ( member_a @ Q2 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
     => ( ord_less_eq_set_a @ ( reacha1620305530751930115_a_b_c @ ( from_FSM_a_b_c @ M2 @ Q2 ) ) @ ( reacha1620305530751930115_a_b_c @ M2 ) ) ) ).

% from_FSM_reachable_states
thf(fact_336_from__FSM__acyclic,axiom,
    ! [Q2: a,M2: fsm_a_b_c] :
      ( ( member_a @ Q2 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
     => ( ( acyclic_a_b_c @ M2 )
       => ( acyclic_a_b_c @ ( from_FSM_a_b_c @ M2 @ Q2 ) ) ) ) ).

% from_FSM_acyclic
thf(fact_337_in__mono,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a,X: produc2669932201312201782od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ A4 @ B4 )
     => ( ( member5502040268208861279od_c_a @ X @ A4 )
       => ( member5502040268208861279od_c_a @ X @ B4 ) ) ) ).

% in_mono
thf(fact_338_in__mono,axiom,
    ! [A4: set_a,B4: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A4 @ B4 )
     => ( ( member_a @ X @ A4 )
       => ( member_a @ X @ B4 ) ) ) ).

% in_mono
thf(fact_339_in__mono,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
     => ( ( member6330420149250801815od_b_c @ X @ A4 )
       => ( member6330420149250801815od_b_c @ X @ B4 ) ) ) ).

% in_mono
thf(fact_340_subsetD,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a,C3: produc2669932201312201782od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ A4 @ B4 )
     => ( ( member5502040268208861279od_c_a @ C3 @ A4 )
       => ( member5502040268208861279od_c_a @ C3 @ B4 ) ) ) ).

% subsetD
thf(fact_341_subsetD,axiom,
    ! [A4: set_a,B4: set_a,C3: a] :
      ( ( ord_less_eq_set_a @ A4 @ B4 )
     => ( ( member_a @ C3 @ A4 )
       => ( member_a @ C3 @ B4 ) ) ) ).

% subsetD
thf(fact_342_subsetD,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c,C3: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
     => ( ( member6330420149250801815od_b_c @ C3 @ A4 )
       => ( member6330420149250801815od_b_c @ C3 @ B4 ) ) ) ).

% subsetD
thf(fact_343_equalityE,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( A4 = B4 )
     => ~ ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
         => ~ ( ord_le282488521294790766od_b_c @ B4 @ A4 ) ) ) ).

% equalityE
thf(fact_344_subset__eq,axiom,
    ( ord_le3226338165915859510od_c_a
    = ( ^ [A5: set_Pr8357624050790843030od_c_a,B5: set_Pr8357624050790843030od_c_a] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ A5 )
         => ( member5502040268208861279od_c_a @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_345_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A5 )
         => ( member_a @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_346_subset__eq,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A5: set_li6436108459499378894od_b_c,B5: set_li6436108459499378894od_b_c] :
        ! [X4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X4 @ A5 )
         => ( member6330420149250801815od_b_c @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_347_equalityD1,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( A4 = B4 )
     => ( ord_le282488521294790766od_b_c @ A4 @ B4 ) ) ).

% equalityD1
thf(fact_348_equalityD2,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( A4 = B4 )
     => ( ord_le282488521294790766od_b_c @ B4 @ A4 ) ) ).

% equalityD2
thf(fact_349_subset__iff,axiom,
    ( ord_le3226338165915859510od_c_a
    = ( ^ [A5: set_Pr8357624050790843030od_c_a,B5: set_Pr8357624050790843030od_c_a] :
        ! [T2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ T2 @ A5 )
         => ( member5502040268208861279od_c_a @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_350_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
        ! [T2: a] :
          ( ( member_a @ T2 @ A5 )
         => ( member_a @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_351_subset__iff,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A5: set_li6436108459499378894od_b_c,B5: set_li6436108459499378894od_b_c] :
        ! [T2: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ T2 @ A5 )
         => ( member6330420149250801815od_b_c @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_352_subset__refl,axiom,
    ! [A4: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ A4 @ A4 ) ).

% subset_refl
thf(fact_353_Collect__mono,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,Q: produc2669932201312201782od_c_a > $o] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le3226338165915859510od_c_a @ ( collec8405073525749625505od_c_a @ P ) @ ( collec8405073525749625505od_c_a @ Q ) ) ) ).

% Collect_mono
thf(fact_354_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_355_Collect__mono,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q: list_P903359562653991662od_b_c > $o] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le282488521294790766od_b_c @ ( collec2280997390073109977od_b_c @ P ) @ ( collec2280997390073109977od_b_c @ Q ) ) ) ).

% Collect_mono
thf(fact_356_subset__trans,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
     => ( ( ord_le282488521294790766od_b_c @ B4 @ C2 )
       => ( ord_le282488521294790766od_b_c @ A4 @ C2 ) ) ) ).

% subset_trans
thf(fact_357_set__eq__subset,axiom,
    ( ( ^ [Y4: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] : ( Y4 = Z ) )
    = ( ^ [A5: set_li6436108459499378894od_b_c,B5: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ A5 @ B5 )
          & ( ord_le282488521294790766od_b_c @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_358_Collect__mono__iff,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,Q: produc2669932201312201782od_c_a > $o] :
      ( ( ord_le3226338165915859510od_c_a @ ( collec8405073525749625505od_c_a @ P ) @ ( collec8405073525749625505od_c_a @ Q ) )
      = ( ! [X4: produc2669932201312201782od_c_a] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_359_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_360_Collect__mono__iff,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ ( collec2280997390073109977od_b_c @ P ) @ ( collec2280997390073109977od_b_c @ Q ) )
      = ( ! [X4: list_P903359562653991662od_b_c] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_361_inf__sup__aci_I4_J,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ X @ Y ) )
      = ( inf_in4978071631833541052od_b_c @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_362_inf__sup__aci_I3_J,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ Y @ Z2 ) )
      = ( inf_in4978071631833541052od_b_c @ Y @ ( inf_in4978071631833541052od_b_c @ X @ Z2 ) ) ) ).

% inf_sup_aci(3)
thf(fact_363_inf__sup__aci_I2_J,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ Z2 )
      = ( inf_in4978071631833541052od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ Y @ Z2 ) ) ) ).

% inf_sup_aci(2)
thf(fact_364_inf__sup__aci_I1_J,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [X4: set_li6436108459499378894od_b_c,Y5: set_li6436108459499378894od_b_c] : ( inf_in4978071631833541052od_b_c @ Y5 @ X4 ) ) ) ).

% inf_sup_aci(1)
thf(fact_365_inf_Oassoc,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C3: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ C3 )
      = ( inf_in4978071631833541052od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B @ C3 ) ) ) ).

% inf.assoc
thf(fact_366_inf__assoc,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ Z2 )
      = ( inf_in4978071631833541052od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ Y @ Z2 ) ) ) ).

% inf_assoc
thf(fact_367_inf_Ocommute,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [A6: set_li6436108459499378894od_b_c,B6: set_li6436108459499378894od_b_c] : ( inf_in4978071631833541052od_b_c @ B6 @ A6 ) ) ) ).

% inf.commute
thf(fact_368_inf__commute,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [X4: set_li6436108459499378894od_b_c,Y5: set_li6436108459499378894od_b_c] : ( inf_in4978071631833541052od_b_c @ Y5 @ X4 ) ) ) ).

% inf_commute
thf(fact_369_inf_Oleft__commute,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,C3: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ B @ ( inf_in4978071631833541052od_b_c @ A @ C3 ) )
      = ( inf_in4978071631833541052od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B @ C3 ) ) ) ).

% inf.left_commute
thf(fact_370_inf__left__commute,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ Y @ Z2 ) )
      = ( inf_in4978071631833541052od_b_c @ Y @ ( inf_in4978071631833541052od_b_c @ X @ Z2 ) ) ) ).

% inf_left_commute
thf(fact_371_IntE,axiom,
    ! [C3: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C3 @ ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) )
     => ~ ( ( member5502040268208861279od_c_a @ C3 @ A4 )
         => ~ ( member5502040268208861279od_c_a @ C3 @ B4 ) ) ) ).

% IntE
thf(fact_372_IntE,axiom,
    ! [C3: a,A4: set_a,B4: set_a] :
      ( ( member_a @ C3 @ ( inf_inf_set_a @ A4 @ B4 ) )
     => ~ ( ( member_a @ C3 @ A4 )
         => ~ ( member_a @ C3 @ B4 ) ) ) ).

% IntE
thf(fact_373_IntE,axiom,
    ! [C3: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C3 @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) )
     => ~ ( ( member6330420149250801815od_b_c @ C3 @ A4 )
         => ~ ( member6330420149250801815od_b_c @ C3 @ B4 ) ) ) ).

% IntE
thf(fact_374_IntD1,axiom,
    ! [C3: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C3 @ ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) )
     => ( member5502040268208861279od_c_a @ C3 @ A4 ) ) ).

% IntD1
thf(fact_375_IntD1,axiom,
    ! [C3: a,A4: set_a,B4: set_a] :
      ( ( member_a @ C3 @ ( inf_inf_set_a @ A4 @ B4 ) )
     => ( member_a @ C3 @ A4 ) ) ).

% IntD1
thf(fact_376_IntD1,axiom,
    ! [C3: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C3 @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) )
     => ( member6330420149250801815od_b_c @ C3 @ A4 ) ) ).

% IntD1
thf(fact_377_IntD2,axiom,
    ! [C3: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C3 @ ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) )
     => ( member5502040268208861279od_c_a @ C3 @ B4 ) ) ).

% IntD2
thf(fact_378_IntD2,axiom,
    ! [C3: a,A4: set_a,B4: set_a] :
      ( ( member_a @ C3 @ ( inf_inf_set_a @ A4 @ B4 ) )
     => ( member_a @ C3 @ B4 ) ) ).

% IntD2
thf(fact_379_IntD2,axiom,
    ! [C3: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ C3 @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) )
     => ( member6330420149250801815od_b_c @ C3 @ B4 ) ) ).

% IntD2
thf(fact_380_Int__assoc,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) @ C2 )
      = ( inf_in4978071631833541052od_b_c @ A4 @ ( inf_in4978071631833541052od_b_c @ B4 @ C2 ) ) ) ).

% Int_assoc
thf(fact_381_Int__absorb,axiom,
    ! [A4: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ A4 @ A4 )
      = A4 ) ).

% Int_absorb
thf(fact_382_Int__commute,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [A5: set_li6436108459499378894od_b_c,B5: set_li6436108459499378894od_b_c] : ( inf_in4978071631833541052od_b_c @ B5 @ A5 ) ) ) ).

% Int_commute
thf(fact_383_Int__left__absorb,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ A4 @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) )
      = ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ).

% Int_left_absorb
thf(fact_384_Int__left__commute,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ A4 @ ( inf_in4978071631833541052od_b_c @ B4 @ C2 ) )
      = ( inf_in4978071631833541052od_b_c @ B4 @ ( inf_in4978071631833541052od_b_c @ A4 @ C2 ) ) ) ).

% Int_left_commute
thf(fact_385_state__cover__assignment__language,axiom,
    ! [M2: fsm_d_b_c,V: d > list_P903359562653991662od_b_c] :
      ( ( state_6014242345326633810_d_b_c @ M2 @ V )
     => ( ord_le282488521294790766od_b_c @ ( image_1329551058250833970od_b_c @ V @ ( reacha9185589339461546374_d_b_c @ M2 ) ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) ) ) ).

% state_cover_assignment_language
thf(fact_386_state__cover__assignment__language,axiom,
    ! [M2: fsm_a_b_c,V: a > list_P903359562653991662od_b_c] :
      ( ( state_7672330573471793359_a_b_c @ M2 @ V )
     => ( ord_le282488521294790766od_b_c @ ( image_8656872703560917237od_b_c @ V @ ( reacha1620305530751930115_a_b_c @ M2 ) ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ) ).

% state_cover_assignment_language
thf(fact_387_inf__sup__ord_I2_J,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_388_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_389_inf__sup__ord_I1_J,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_390_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_391_inf__le1,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_392_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_393_inf__le2,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_394_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_395_le__infE,axiom,
    ! [X: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ A @ B ) )
     => ~ ( ( ord_le282488521294790766od_b_c @ X @ A )
         => ~ ( ord_le282488521294790766od_b_c @ X @ B ) ) ) ).

% le_infE
thf(fact_396_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X @ A )
         => ~ ( ord_less_eq_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_397_le__infI,axiom,
    ! [X: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ A )
     => ( ( ord_le282488521294790766od_b_c @ X @ B )
       => ( ord_le282488521294790766od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% le_infI
thf(fact_398_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ( ord_less_eq_nat @ X @ B )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_399_inf__mono,axiom,
    ! [A: set_li6436108459499378894od_b_c,C3: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,D2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ C3 )
     => ( ( ord_le282488521294790766od_b_c @ B @ D2 )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ ( inf_in4978071631833541052od_b_c @ C3 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_400_inf__mono,axiom,
    ! [A: nat,C3: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C3 )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C3 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_401_le__infI1,axiom,
    ! [A: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ X )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_402_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_403_le__infI2,axiom,
    ! [B: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ X )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_404_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_405_inf_OorderE,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( A
        = ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ).

% inf.orderE
thf(fact_406_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_407_inf_OorderI,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( A
        = ( inf_in4978071631833541052od_b_c @ A @ B ) )
     => ( ord_le282488521294790766od_b_c @ A @ B ) ) ).

% inf.orderI
thf(fact_408_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_409_inf__unique,axiom,
    ! [F: set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_li6436108459499378894od_b_c,Y3: set_li6436108459499378894od_b_c,Z3: set_li6436108459499378894od_b_c] :
              ( ( ord_le282488521294790766od_b_c @ X3 @ Y3 )
             => ( ( ord_le282488521294790766od_b_c @ X3 @ Z3 )
               => ( ord_le282488521294790766od_b_c @ X3 @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_in4978071631833541052od_b_c @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_410_inf__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ( ord_less_eq_nat @ X3 @ Z3 )
               => ( ord_less_eq_nat @ X3 @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_inf_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_411_le__iff__inf,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [X4: set_li6436108459499378894od_b_c,Y5: set_li6436108459499378894od_b_c] :
          ( ( inf_in4978071631833541052od_b_c @ X4 @ Y5 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_412_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( inf_inf_nat @ X4 @ Y5 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_413_inf_Oabsorb1,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( inf_in4978071631833541052od_b_c @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_414_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_415_inf_Oabsorb2,axiom,
    ! [B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ A )
     => ( ( inf_in4978071631833541052od_b_c @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_416_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_417_inf__absorb1,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( inf_in4978071631833541052od_b_c @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_418_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_419_inf__absorb2,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ Y @ X )
     => ( ( inf_in4978071631833541052od_b_c @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_420_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_421_inf_OboundedE,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B @ C3 ) )
     => ~ ( ( ord_le282488521294790766od_b_c @ A @ B )
         => ~ ( ord_le282488521294790766od_b_c @ A @ C3 ) ) ) ).

% inf.boundedE
thf(fact_422_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C3 ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C3 ) ) ) ).

% inf.boundedE
thf(fact_423_inf_OboundedI,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,C3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( ord_le282488521294790766od_b_c @ A @ C3 )
       => ( ord_le282488521294790766od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B @ C3 ) ) ) ) ).

% inf.boundedI
thf(fact_424_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C3 )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C3 ) ) ) ) ).

% inf.boundedI
thf(fact_425_inf__greatest,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( ord_le282488521294790766od_b_c @ X @ Z2 )
       => ( ord_le282488521294790766od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_426_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Z2 )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_427_inf_Oorder__iff,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A6: set_li6436108459499378894od_b_c,B6: set_li6436108459499378894od_b_c] :
          ( A6
          = ( inf_in4978071631833541052od_b_c @ A6 @ B6 ) ) ) ) ).

% inf.order_iff
thf(fact_428_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B6: nat] :
          ( A6
          = ( inf_inf_nat @ A6 @ B6 ) ) ) ) ).

% inf.order_iff
thf(fact_429_inf_Ocobounded1,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_430_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_431_inf_Ocobounded2,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_432_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_433_inf_Oabsorb__iff1,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A6: set_li6436108459499378894od_b_c,B6: set_li6436108459499378894od_b_c] :
          ( ( inf_in4978071631833541052od_b_c @ A6 @ B6 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_434_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B6: nat] :
          ( ( inf_inf_nat @ A6 @ B6 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_435_inf_Oabsorb__iff2,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [B6: set_li6436108459499378894od_b_c,A6: set_li6436108459499378894od_b_c] :
          ( ( inf_in4978071631833541052od_b_c @ A6 @ B6 )
          = B6 ) ) ) ).

% inf.absorb_iff2
thf(fact_436_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B6: nat,A6: nat] :
          ( ( inf_inf_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% inf.absorb_iff2
thf(fact_437_inf_OcoboundedI1,axiom,
    ! [A: set_li6436108459499378894od_b_c,C3: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ C3 )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ C3 ) ) ).

% inf.coboundedI1
thf(fact_438_inf_OcoboundedI1,axiom,
    ! [A: nat,C3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C3 ) ) ).

% inf.coboundedI1
thf(fact_439_inf_OcoboundedI2,axiom,
    ! [B: set_li6436108459499378894od_b_c,C3: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B @ C3 )
     => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A @ B ) @ C3 ) ) ).

% inf.coboundedI2
thf(fact_440_inf_OcoboundedI2,axiom,
    ! [B: nat,C3: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C3 ) ) ).

% inf.coboundedI2
thf(fact_441_Int__mono,axiom,
    ! [A4: set_li6436108459499378894od_b_c,C2: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c,D3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ C2 )
     => ( ( ord_le282488521294790766od_b_c @ B4 @ D3 )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) @ ( inf_in4978071631833541052od_b_c @ C2 @ D3 ) ) ) ) ).

% Int_mono
thf(fact_442_Int__lower1,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) @ A4 ) ).

% Int_lower1
thf(fact_443_Int__lower2,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) @ B4 ) ).

% Int_lower2
thf(fact_444_Int__absorb1,axiom,
    ! [B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B4 @ A4 )
     => ( ( inf_in4978071631833541052od_b_c @ A4 @ B4 )
        = B4 ) ) ).

% Int_absorb1
thf(fact_445_Int__absorb2,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
     => ( ( inf_in4978071631833541052od_b_c @ A4 @ B4 )
        = A4 ) ) ).

% Int_absorb2
thf(fact_446_Int__greatest,axiom,
    ! [C2: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ A4 )
     => ( ( ord_le282488521294790766od_b_c @ C2 @ B4 )
       => ( ord_le282488521294790766od_b_c @ C2 @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) ) ).

% Int_greatest
thf(fact_447_Int__Collect__mono,axiom,
    ! [A4: set_a,B4: set_a,P: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ A4 @ B4 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ A4 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B4 @ ( collect_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_448_Int__Collect__mono,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a,P: produc2669932201312201782od_c_a > $o,Q: produc2669932201312201782od_c_a > $o] :
      ( ( ord_le3226338165915859510od_c_a @ A4 @ B4 )
     => ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A4 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le3226338165915859510od_c_a @ ( inf_in3815988127369930372od_c_a @ A4 @ ( collec8405073525749625505od_c_a @ P ) ) @ ( inf_in3815988127369930372od_c_a @ B4 @ ( collec8405073525749625505od_c_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_449_Int__Collect__mono,axiom,
    ! [A4: set_nat,B4: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B4 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_450_Int__Collect__mono,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o,Q: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
     => ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ A4 )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le282488521294790766od_b_c @ ( inf_in4978071631833541052od_b_c @ A4 @ ( collec2280997390073109977od_b_c @ P ) ) @ ( inf_in4978071631833541052od_b_c @ B4 @ ( collec2280997390073109977od_b_c @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_451_LS__single__transition,axiom,
    ! [X: b,Y: c,M2: fsm_d_b_c,Q2: d] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) @ ( lS_d_b_c @ M2 @ Q2 ) )
      = ( ? [X4: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X4 @ ( transitions_d_b_c @ M2 ) )
            & ( ( produc1508465569166014290od_c_d @ X4 )
              = Q2 )
            & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) )
              = X )
            & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) ) )
              = Y ) ) ) ) ).

% LS_single_transition
thf(fact_452_LS__single__transition,axiom,
    ! [X: c,Y: a,M2: fsm_Pr7461828476839194351_a_c_a,Q2: produc2133359284458116493od_c_a] :
      ( ( member5552703068553123156od_c_a @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ X @ Y ) @ nil_Product_prod_c_a ) @ ( lS_Pro1436182445161660463_a_c_a @ M2 @ Q2 ) )
      = ( ? [X4: produc3729687299602777783od_c_a] :
            ( ( member5500486793640445664od_c_a @ X4 @ ( transi5448983560151880288_a_c_a @ M2 ) )
            & ( ( produc1784202036732648467od_c_a @ X4 )
              = Q2 )
            & ( ( produc7173261134821384633od_c_a @ ( produc2799690148840255317od_c_a @ X4 ) )
              = X )
            & ( ( produc4225943634317630866od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ X4 ) ) )
              = Y ) ) ) ) ).

% LS_single_transition
thf(fact_453_LS__single__transition,axiom,
    ! [X: a,Y: b,M2: fsm_Pr1886043760914421514_a_a_b,Q2: product_prod_c_a] :
      ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ X @ Y ) @ nil_Product_prod_a_b ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ Q2 ) )
      = ( ? [X4: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
            & ( ( produc8171714036265584980od_c_a @ X4 )
              = Q2 )
            & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
              = X )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
              = Y ) ) ) ) ).

% LS_single_transition
thf(fact_454_LS__single__transition,axiom,
    ! [X: b,Y: c,M2: fsm_a_b_c,Q2: a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) @ ( lS_a_b_c @ M2 @ Q2 ) )
      = ( ? [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M2 ) )
            & ( ( produc4225943634317630866od_c_a @ X4 )
              = Q2 )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
              = X )
            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
              = Y ) ) ) ) ).

% LS_single_transition
thf(fact_455_T1__covered__transitions__conv,axiom,
    ! [T: produc2669932201312201782od_c_a] :
      ( ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ m1 @ ( initial_a_b_c @ m1 ) ) @ ( prefix9085958534787526401od_b_c @ t1 ) )
        = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ m2 @ ( initial_d_b_c @ m2 ) ) @ ( prefix9085958534787526401od_b_c @ t1 ) ) )
     => ( ( member5502040268208861279od_c_a @ T @ sc_cov38355202683674721itions )
       => ( converge_d_b_c @ m2 @ ( v @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ ( append2547753245680614915od_b_c @ ( v @ ( produc4225943634317630866od_c_a @ T ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ).

% T1_covered_transitions_conv
thf(fact_456_append1__eq__conv,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,X: product_prod_b_c,Ys: list_P903359562653991662od_b_c,Y: product_prod_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs2 @ ( cons_P4529483553340347422od_b_c @ X @ nil_Product_prod_b_c ) )
        = ( append2547753245680614915od_b_c @ Ys @ ( cons_P4529483553340347422od_b_c @ Y @ nil_Product_prod_b_c ) ) )
      = ( ( Xs2 = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_457_pass__io__set__maximal__def,axiom,
    ( pass_i5868551379204665570_a_b_c
    = ( ^ [M: fsm_a_b_c,Ios2: set_li6436108459499378894od_b_c] :
        ! [Io3: list_P903359562653991662od_b_c,X4: b,Y5: c,Io4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ nil_Product_prod_b_c ) @ Io4 ) ) @ Ios2 )
         => ! [Y7: c] :
              ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y7 ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
             => ? [Io5: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y7 ) @ nil_Product_prod_b_c ) @ Io5 ) ) @ Ios2 ) ) ) ) ) ).

% pass_io_set_maximal_def
thf(fact_458_pass__io__set__maximal__def,axiom,
    ( pass_i4210463151059506021_d_b_c
    = ( ^ [M: fsm_d_b_c,Ios2: set_li6436108459499378894od_b_c] :
        ! [Io3: list_P903359562653991662od_b_c,X4: b,Y5: c,Io4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ nil_Product_prod_b_c ) @ Io4 ) ) @ Ios2 )
         => ! [Y7: c] :
              ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y7 ) @ nil_Product_prod_b_c ) ) @ ( lS_d_b_c @ M @ ( initial_d_b_c @ M ) ) )
             => ? [Io5: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y7 ) @ nil_Product_prod_b_c ) @ Io5 ) ) @ Ios2 ) ) ) ) ) ).

% pass_io_set_maximal_def
thf(fact_459_pass__io__set__def,axiom,
    ( pass_io_set_a_b_c
    = ( ^ [M: fsm_a_b_c,Ios2: set_li6436108459499378894od_b_c] :
        ! [Io3: list_P903359562653991662od_b_c,X4: b,Y5: c] :
          ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ nil_Product_prod_b_c ) ) @ Ios2 )
         => ! [Y7: c] :
              ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y7 ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
             => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y7 ) @ nil_Product_prod_b_c ) ) @ Ios2 ) ) ) ) ) ).

% pass_io_set_def
thf(fact_460_pass__io__set__def,axiom,
    ( pass_io_set_d_b_c
    = ( ^ [M: fsm_d_b_c,Ios2: set_li6436108459499378894od_b_c] :
        ! [Io3: list_P903359562653991662od_b_c,X4: b,Y5: c] :
          ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ nil_Product_prod_b_c ) ) @ Ios2 )
         => ! [Y7: c] :
              ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y7 ) @ nil_Product_prod_b_c ) ) @ ( lS_d_b_c @ M @ ( initial_d_b_c @ M ) ) )
             => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Y7 ) @ nil_Product_prod_b_c ) ) @ Ios2 ) ) ) ) ) ).

% pass_io_set_def
thf(fact_461_append_Oright__neutral,axiom,
    ! [A: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ A @ nil_Product_prod_b_c )
      = A ) ).

% append.right_neutral
thf(fact_462_append__Nil2,axiom,
    ! [Xs2: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ Xs2 @ nil_Product_prod_b_c )
      = Xs2 ) ).

% append_Nil2
thf(fact_463_append__self__conv,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs2 @ Ys )
        = Xs2 )
      = ( Ys = nil_Product_prod_b_c ) ) ).

% append_self_conv
thf(fact_464_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_465_append__self__conv2,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs2 @ Ys )
        = Ys )
      = ( Xs2 = nil_Product_prod_b_c ) ) ).

% append_self_conv2
thf(fact_466_list_Oinject,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c,Y21: product_prod_b_c,Y22: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X21 @ X22 )
        = ( cons_P4529483553340347422od_b_c @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_467_same__append__eq,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs2 @ Ys )
        = ( append2547753245680614915od_b_c @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_468_append__same__eq,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Xs2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Ys @ Xs2 )
        = ( append2547753245680614915od_b_c @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_469_append__assoc,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ ( append2547753245680614915od_b_c @ Xs2 @ Ys ) @ Zs )
      = ( append2547753245680614915od_b_c @ Xs2 @ ( append2547753245680614915od_b_c @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_470_append_Oassoc,axiom,
    ! [A: list_P903359562653991662od_b_c,B: list_P903359562653991662od_b_c,C3: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ ( append2547753245680614915od_b_c @ A @ B ) @ C3 )
      = ( append2547753245680614915od_b_c @ A @ ( append2547753245680614915od_b_c @ B @ C3 ) ) ) ).

% append.assoc
thf(fact_471_append__is__Nil__conv,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs2 @ Ys )
        = nil_Product_prod_b_c )
      = ( ( Xs2 = nil_Product_prod_b_c )
        & ( Ys = nil_Product_prod_b_c ) ) ) ).

% append_is_Nil_conv
thf(fact_472_Nil__is__append__conv,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( nil_Product_prod_b_c
        = ( append2547753245680614915od_b_c @ Xs2 @ Ys ) )
      = ( ( Xs2 = nil_Product_prod_b_c )
        & ( Ys = nil_Product_prod_b_c ) ) ) ).

% Nil_is_append_conv
thf(fact_473_self__append__conv2,axiom,
    ! [Y: list_P903359562653991662od_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( Y
        = ( append2547753245680614915od_b_c @ Xs2 @ Y ) )
      = ( Xs2 = nil_Product_prod_b_c ) ) ).

% self_append_conv2
thf(fact_474_n,axiom,
    ( n
    = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ m1 ) ) ) ).

% n
thf(fact_475_state__cover__assignment__card,axiom,
    ! [M2: fsm_d_b_c,V: d > list_P903359562653991662od_b_c] :
      ( ( state_6014242345326633810_d_b_c @ M2 @ V )
     => ( ( observable_d_b_c @ M2 )
       => ( ( finite5583770498833199894od_b_c @ ( image_1329551058250833970od_b_c @ V @ ( reacha9185589339461546374_d_b_c @ M2 ) ) )
          = ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M2 ) ) ) ) ) ).

% state_cover_assignment_card
thf(fact_476_state__cover__assignment__card,axiom,
    ! [M2: fsm_a_b_c,V: a > list_P903359562653991662od_b_c] :
      ( ( state_7672330573471793359_a_b_c @ M2 @ V )
     => ( ( observable_a_b_c @ M2 )
       => ( ( finite5583770498833199894od_b_c @ ( image_8656872703560917237od_b_c @ V @ ( reacha1620305530751930115_a_b_c @ M2 ) ) )
          = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M2 ) ) ) ) ) ).

% state_cover_assignment_card
thf(fact_477_convergence__graph__merge__invar__def,axiom,
    ( conver7456087639816148335_c_d_e
    = ( ^ [M1: fsm_a_b_c,M22: fsm_d_b_c,Cg_lookup: e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,Cg_merge: e > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > e] :
        ! [G2: e,Gamma2: list_P903359562653991662od_b_c,Gamma3: list_P903359562653991662od_b_c] :
          ( ( converge_a_b_c @ M1 @ Gamma2 @ Gamma3 )
         => ( ( converge_d_b_c @ M22 @ Gamma2 @ Gamma3 )
           => ( ( conver7089555221631485245_c_d_e @ M1 @ M22 @ Cg_lookup @ G2 )
             => ( conver7089555221631485245_c_d_e @ M1 @ M22 @ Cg_lookup @ ( Cg_merge @ G2 @ Gamma2 @ Gamma3 ) ) ) ) ) ) ) ).

% convergence_graph_merge_invar_def
thf(fact_478_not__Cons__self2,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( cons_P4529483553340347422od_b_c @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_479_append__eq__append__conv2,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,Ts: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Xs2 @ Ys )
        = ( append2547753245680614915od_b_c @ Zs @ Ts ) )
      = ( ? [Us: list_P903359562653991662od_b_c] :
            ( ( ( Xs2
                = ( append2547753245680614915od_b_c @ Zs @ Us ) )
              & ( ( append2547753245680614915od_b_c @ Us @ Ys )
                = Ts ) )
            | ( ( ( append2547753245680614915od_b_c @ Xs2 @ Us )
                = Zs )
              & ( Ys
                = ( append2547753245680614915od_b_c @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_480_append__eq__appendI,axiom,
    ! [Xs2: 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 @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append2547753245680614915od_b_c @ Xs1 @ Us2 ) )
       => ( ( append2547753245680614915od_b_c @ Xs2 @ Ys )
          = ( append2547753245680614915od_b_c @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_481_minimal__equivalence__size__r,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_d_b_c] :
      ( ( minimal_d_b_c @ M12 )
     => ( ( minimal_d_b_c @ M23 )
       => ( ( observable_d_b_c @ M12 )
         => ( ( observable_d_b_c @ M23 )
           => ( ( ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) )
                = ( lS_d_b_c @ M23 @ ( initial_d_b_c @ M23 ) ) )
             => ( ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M12 ) )
                = ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M23 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_482_minimal__equivalence__size__r,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_nat_b_c] :
      ( ( minimal_d_b_c @ M12 )
     => ( ( minimal_nat_b_c @ M23 )
       => ( ( observable_d_b_c @ M12 )
         => ( ( observable_nat_b_c @ M23 )
           => ( ( ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) )
                = ( lS_nat_b_c @ M23 @ ( initial_nat_b_c @ M23 ) ) )
             => ( ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M12 ) )
                = ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M23 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_483_minimal__equivalence__size__r,axiom,
    ! [M12: fsm_nat_b_c,M23: fsm_d_b_c] :
      ( ( minimal_nat_b_c @ M12 )
     => ( ( minimal_d_b_c @ M23 )
       => ( ( observable_nat_b_c @ M12 )
         => ( ( observable_d_b_c @ M23 )
           => ( ( ( lS_nat_b_c @ M12 @ ( initial_nat_b_c @ M12 ) )
                = ( lS_d_b_c @ M23 @ ( initial_d_b_c @ M23 ) ) )
             => ( ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M12 ) )
                = ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M23 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_484_minimal__equivalence__size__r,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_a_b_c] :
      ( ( minimal_d_b_c @ M12 )
     => ( ( minimal_a_b_c @ M23 )
       => ( ( observable_d_b_c @ M12 )
         => ( ( observable_a_b_c @ M23 )
           => ( ( ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) )
                = ( lS_a_b_c @ M23 @ ( initial_a_b_c @ M23 ) ) )
             => ( ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M12 ) )
                = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M23 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_485_minimal__equivalence__size__r,axiom,
    ! [M12: fsm_nat_b_c,M23: fsm_a_b_c] :
      ( ( minimal_nat_b_c @ M12 )
     => ( ( minimal_a_b_c @ M23 )
       => ( ( observable_nat_b_c @ M12 )
         => ( ( observable_a_b_c @ M23 )
           => ( ( ( lS_nat_b_c @ M12 @ ( initial_nat_b_c @ M12 ) )
                = ( lS_a_b_c @ M23 @ ( initial_a_b_c @ M23 ) ) )
             => ( ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M12 ) )
                = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M23 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_486_minimal__equivalence__size__r,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c] :
      ( ( minimal_a_b_c @ M12 )
     => ( ( minimal_d_b_c @ M23 )
       => ( ( observable_a_b_c @ M12 )
         => ( ( observable_d_b_c @ M23 )
           => ( ( ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) )
                = ( lS_d_b_c @ M23 @ ( initial_d_b_c @ M23 ) ) )
             => ( ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M12 ) )
                = ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M23 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_487_minimal__equivalence__size__r,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_nat_b_c] :
      ( ( minimal_a_b_c @ M12 )
     => ( ( minimal_nat_b_c @ M23 )
       => ( ( observable_a_b_c @ M12 )
         => ( ( observable_nat_b_c @ M23 )
           => ( ( ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) )
                = ( lS_nat_b_c @ M23 @ ( initial_nat_b_c @ M23 ) ) )
             => ( ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M12 ) )
                = ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M23 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_488_minimal__equivalence__size__r,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_a_b_c] :
      ( ( minimal_a_b_c @ M12 )
     => ( ( minimal_a_b_c @ M23 )
       => ( ( observable_a_b_c @ M12 )
         => ( ( observable_a_b_c @ M23 )
           => ( ( ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) )
                = ( lS_a_b_c @ M23 @ ( initial_a_b_c @ M23 ) ) )
             => ( ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M12 ) )
                = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M23 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_489_splice_Ocases,axiom,
    ! [X: produc5878855887281848999od_b_c] :
      ( ! [Ys2: list_P903359562653991662od_b_c] :
          ( X
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys2 ) )
     => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Ys2: list_P903359562653991662od_b_c] :
            ( X
           != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_490_shuffles_Ocases,axiom,
    ! [X: produc5878855887281848999od_b_c] :
      ( ! [Ys2: list_P903359562653991662od_b_c] :
          ( X
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys2 ) )
     => ( ! [Xs3: list_P903359562653991662od_b_c] :
            ( X
           != ( produc3076307724246486423od_b_c @ Xs3 @ nil_Product_prod_b_c ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
              ( X
             != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_491_sorted__wrt_Ocases,axiom,
    ! [X: produc2416979464461413415od_b_c] :
      ( ! [P3: product_prod_b_c > product_prod_b_c > $o] :
          ( X
         != ( produc6092892548594537111od_b_c @ P3 @ nil_Product_prod_b_c ) )
     => ~ ! [P3: product_prod_b_c > product_prod_b_c > $o,X3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
            ( X
           != ( produc6092892548594537111od_b_c @ P3 @ ( cons_P4529483553340347422od_b_c @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_492_successively_Ocases,axiom,
    ! [X: produc2416979464461413415od_b_c] :
      ( ! [P3: product_prod_b_c > product_prod_b_c > $o] :
          ( X
         != ( produc6092892548594537111od_b_c @ P3 @ nil_Product_prod_b_c ) )
     => ( ! [P3: product_prod_b_c > product_prod_b_c > $o,X3: product_prod_b_c] :
            ( X
           != ( produc6092892548594537111od_b_c @ P3 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) )
       => ~ ! [P3: product_prod_b_c > product_prod_b_c > $o,X3: product_prod_b_c,Y3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( X
             != ( produc6092892548594537111od_b_c @ P3 @ ( cons_P4529483553340347422od_b_c @ X3 @ ( cons_P4529483553340347422od_b_c @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_493_list__nonempty__induct,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( Xs2 != nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
       => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( ( Xs3 != nil_Product_prod_b_c )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_494_list__induct2_H,axiom,
    ! [P: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( P @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ nil_Product_prod_b_c )
       => ( ! [Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] : ( P @ nil_Product_prod_b_c @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) )
         => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
                ( ( P @ Xs3 @ Ys2 )
               => ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) )
           => ( P @ Xs2 @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_495_neq__Nil__conv,axiom,
    ! [Xs2: list_P903359562653991662od_b_c] :
      ( ( Xs2 != nil_Product_prod_b_c )
      = ( ? [Y5: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] :
            ( Xs2
            = ( cons_P4529483553340347422od_b_c @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_496_remdups__adj_Ocases,axiom,
    ! [X: list_P903359562653991662od_b_c] :
      ( ( X != nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c] :
            ( X
           != ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
       => ~ ! [X3: product_prod_b_c,Y3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( X
             != ( cons_P4529483553340347422od_b_c @ X3 @ ( cons_P4529483553340347422od_b_c @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_497_min__list_Ocases,axiom,
    ! [X: list_P903359562653991662od_b_c] :
      ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
          ( X
         != ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) )
     => ( X = nil_Product_prod_b_c ) ) ).

% min_list.cases
thf(fact_498_list_Oexhaust,axiom,
    ! [Y: list_P903359562653991662od_b_c] :
      ( ( Y != nil_Product_prod_b_c )
     => ~ ! [X212: product_prod_b_c,X222: list_P903359562653991662od_b_c] :
            ( Y
           != ( cons_P4529483553340347422od_b_c @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_499_list_OdiscI,axiom,
    ! [List: list_P903359562653991662od_b_c,X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( ( List
        = ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) )
     => ( List != nil_Product_prod_b_c ) ) ).

% list.discI
thf(fact_500_list_Odistinct_I1_J,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( nil_Product_prod_b_c
     != ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_501_transpose_Ocases,axiom,
    ! [X: list_l8907847357763382004od_b_c] :
      ( ( X != nil_li8071162985807626740od_b_c )
     => ( ! [Xss: list_l8907847357763382004od_b_c] :
            ( X
           != ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Xss: list_l8907847357763382004od_b_c] :
              ( X
             != ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_502_set__ConsD,axiom,
    ! [Y: a,X: a,Xs2: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_a @ Y @ ( set_a2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_503_set__ConsD,axiom,
    ! [Y: list_P903359562653991662od_b_c,X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( cons_l342771267123639716od_b_c @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_504_set__ConsD,axiom,
    ! [Y: produc2669932201312201782od_c_a,X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ Y @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member5502040268208861279od_c_a @ Y @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_505_set__ConsD,axiom,
    ! [Y: product_prod_b_c,X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_506_list_Oset__cases,axiom,
    ! [E: a,A: list_a] :
      ( ( member_a @ E @ ( set_a2 @ A ) )
     => ( ! [Z22: list_a] :
            ( A
           != ( cons_a @ E @ Z22 ) )
       => ~ ! [Z1: a,Z22: list_a] :
              ( ( A
                = ( cons_a @ Z1 @ Z22 ) )
             => ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_507_list_Oset__cases,axiom,
    ! [E: list_P903359562653991662od_b_c,A: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ E @ ( set_li4480668622519654659od_b_c @ A ) )
     => ( ! [Z22: list_l8907847357763382004od_b_c] :
            ( A
           != ( cons_l342771267123639716od_b_c @ E @ Z22 ) )
       => ~ ! [Z1: list_P903359562653991662od_b_c,Z22: list_l8907847357763382004od_b_c] :
              ( ( A
                = ( cons_l342771267123639716od_b_c @ Z1 @ Z22 ) )
             => ~ ( member6330420149250801815od_b_c @ E @ ( set_li4480668622519654659od_b_c @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_508_list_Oset__cases,axiom,
    ! [E: produc2669932201312201782od_c_a,A: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ E @ ( set_Pr5664641634098845387od_c_a @ A ) )
     => ( ! [Z22: list_P6327159017948738492od_c_a] :
            ( A
           != ( cons_P8659537020926305772od_c_a @ E @ Z22 ) )
       => ~ ! [Z1: produc2669932201312201782od_c_a,Z22: list_P6327159017948738492od_c_a] :
              ( ( A
                = ( cons_P8659537020926305772od_c_a @ Z1 @ Z22 ) )
             => ~ ( member5502040268208861279od_c_a @ E @ ( set_Pr5664641634098845387od_c_a @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_509_list_Oset__cases,axiom,
    ! [E: product_prod_b_c,A: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ E @ ( set_Product_prod_b_c2 @ A ) )
     => ( ! [Z22: list_P903359562653991662od_b_c] :
            ( A
           != ( cons_P4529483553340347422od_b_c @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_b_c,Z22: list_P903359562653991662od_b_c] :
              ( ( A
                = ( cons_P4529483553340347422od_b_c @ Z1 @ Z22 ) )
             => ~ ( member7862447941013992593od_b_c @ E @ ( set_Product_prod_b_c2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_510_list_Oset__intros_I1_J,axiom,
    ! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_511_list_Oset__intros_I1_J,axiom,
    ! [X21: list_P903359562653991662od_b_c,X22: list_l8907847357763382004od_b_c] : ( member6330420149250801815od_b_c @ X21 @ ( set_li4480668622519654659od_b_c @ ( cons_l342771267123639716od_b_c @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_512_list_Oset__intros_I1_J,axiom,
    ! [X21: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a] : ( member5502040268208861279od_c_a @ X21 @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_513_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] : ( member7862447941013992593od_b_c @ X21 @ ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_514_list_Oset__intros_I2_J,axiom,
    ! [Y: a,X22: list_a,X21: a] :
      ( ( member_a @ Y @ ( set_a2 @ X22 ) )
     => ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_515_list_Oset__intros_I2_J,axiom,
    ! [Y: list_P903359562653991662od_b_c,X22: list_l8907847357763382004od_b_c,X21: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ X22 ) )
     => ( member6330420149250801815od_b_c @ Y @ ( set_li4480668622519654659od_b_c @ ( cons_l342771267123639716od_b_c @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_516_list_Oset__intros_I2_J,axiom,
    ! [Y: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a,X21: produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ Y @ ( set_Pr5664641634098845387od_c_a @ X22 ) )
     => ( member5502040268208861279od_c_a @ Y @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_517_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_b_c,X22: list_P903359562653991662od_b_c,X21: product_prod_b_c] :
      ( ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ X22 ) )
     => ( member7862447941013992593od_b_c @ Y @ ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_518_Cons__eq__appendI,axiom,
    ! [X: product_prod_b_c,Xs1: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Xs2: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append2547753245680614915od_b_c @ Xs1 @ Zs ) )
       => ( ( cons_P4529483553340347422od_b_c @ X @ Xs2 )
          = ( append2547753245680614915od_b_c @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_519_append__Cons,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) @ Ys )
      = ( cons_P4529483553340347422od_b_c @ X @ ( append2547753245680614915od_b_c @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_520_eq__Nil__appendI,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( Xs2 = Ys )
     => ( Xs2
        = ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_521_append_Oleft__neutral,axiom,
    ! [A: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ A )
      = A ) ).

% append.left_neutral
thf(fact_522_append__Nil,axiom,
    ! [Ys: list_P903359562653991662od_b_c] :
      ( ( append2547753245680614915od_b_c @ nil_Product_prod_b_c @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_523_subset__code_I1_J,axiom,
    ! [Xs2: list_a,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B4 )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
           => ( member_a @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_524_subset__code_I1_J,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) @ B4 )
      = ( ! [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
           => ( member5502040268208861279od_c_a @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_525_subset__code_I1_J,axiom,
    ! [Xs2: list_l8907847357763382004od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs2 ) @ B4 )
      = ( ! [X4: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X4 @ ( set_li4480668622519654659od_b_c @ Xs2 ) )
           => ( member6330420149250801815od_b_c @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_526_rev__induct,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Xs2: list_P903359562653991662od_b_c] :
      ( ( P @ nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
            ( ( P @ Xs3 )
           => ( P @ ( append2547753245680614915od_b_c @ Xs3 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) ) )
       => ( P @ Xs2 ) ) ) ).

% rev_induct
thf(fact_527_rev__exhaust,axiom,
    ! [Xs2: list_P903359562653991662od_b_c] :
      ( ( Xs2 != nil_Product_prod_b_c )
     => ~ ! [Ys2: list_P903359562653991662od_b_c,Y3: product_prod_b_c] :
            ( Xs2
           != ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ Y3 @ nil_Product_prod_b_c ) ) ) ) ).

% rev_exhaust
thf(fact_528_Cons__eq__append__conv,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X @ Xs2 )
        = ( append2547753245680614915od_b_c @ Ys @ Zs ) )
      = ( ( ( Ys = nil_Product_prod_b_c )
          & ( ( cons_P4529483553340347422od_b_c @ X @ Xs2 )
            = Zs ) )
        | ? [Ys4: list_P903359562653991662od_b_c] :
            ( ( ( cons_P4529483553340347422od_b_c @ X @ Ys4 )
              = Ys )
            & ( Xs2
              = ( append2547753245680614915od_b_c @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_529_append__eq__Cons__conv,axiom,
    ! [Ys: list_P903359562653991662od_b_c,Zs: list_P903359562653991662od_b_c,X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( ( append2547753245680614915od_b_c @ Ys @ Zs )
        = ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) )
      = ( ( ( Ys = nil_Product_prod_b_c )
          & ( Zs
            = ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) ) )
        | ? [Ys4: list_P903359562653991662od_b_c] :
            ( ( Ys
              = ( cons_P4529483553340347422od_b_c @ X @ Ys4 ) )
            & ( ( append2547753245680614915od_b_c @ Ys4 @ Zs )
              = Xs2 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_530_rev__nonempty__induct,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( Xs2 != nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c] : ( P @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
       => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
              ( ( Xs3 != nil_Product_prod_b_c )
             => ( ( P @ Xs3 )
               => ( P @ ( append2547753245680614915od_b_c @ Xs3 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_531_set__subset__Cons,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,X: produc2669932201312201782od_c_a] : ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) @ ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_532_set__subset__Cons,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,X: product_prod_b_c] : ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs2 ) @ ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_533_set__subset__Cons,axiom,
    ! [Xs2: list_l8907847357763382004od_b_c,X: list_P903359562653991662od_b_c] : ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs2 ) @ ( set_li4480668622519654659od_b_c @ ( cons_l342771267123639716od_b_c @ X @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_534_split__list__first__prop__iff,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( ? [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,X4: produc2669932201312201782od_c_a] :
            ( ? [Zs2: list_P6327159017948738492od_c_a] :
                ( Xs2
                = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y5 @ ( set_Pr5664641634098845387od_c_a @ Ys3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_535_split__list__first__prop__iff,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ( ? [X4: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X4 @ ( set_Product_prod_b_c2 @ Xs2 ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,X4: product_prod_b_c] :
            ( ? [Zs2: list_P903359562653991662od_b_c] :
                ( Xs2
                = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: product_prod_b_c] :
                ( ( member7862447941013992593od_b_c @ Y5 @ ( set_Product_prod_b_c2 @ Ys3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_536_split__list__last__prop__iff,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( ? [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,X4: produc2669932201312201782od_c_a,Zs2: list_P6327159017948738492od_c_a] :
            ( ( Xs2
              = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y5 @ ( set_Pr5664641634098845387od_c_a @ Zs2 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_537_split__list__last__prop__iff,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ( ? [X4: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X4 @ ( set_Product_prod_b_c2 @ Xs2 ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,X4: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs2
              = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: product_prod_b_c] :
                ( ( member7862447941013992593od_b_c @ Y5 @ ( set_Product_prod_b_c2 @ Zs2 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_538_in__set__conv__decomp__first,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
      = ( ? [Ys3: list_a,Zs2: list_a] :
            ( ( Xs2
              = ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_539_in__set__conv__decomp__first,axiom,
    ! [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs2 ) )
      = ( ? [Ys3: list_l8907847357763382004od_b_c,Zs2: list_l8907847357763382004od_b_c] :
            ( ( Xs2
              = ( append6992831555715836169od_b_c @ Ys3 @ ( cons_l342771267123639716od_b_c @ X @ Zs2 ) ) )
            & ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_540_in__set__conv__decomp__first,axiom,
    ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,Zs2: list_P6327159017948738492od_c_a] :
            ( ( Xs2
              = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X @ Zs2 ) ) )
            & ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_541_in__set__conv__decomp__first,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs2 ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs2
              = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X @ Zs2 ) ) )
            & ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_542_in__set__conv__decomp__last,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
      = ( ? [Ys3: list_a,Zs2: list_a] :
            ( ( Xs2
              = ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_543_in__set__conv__decomp__last,axiom,
    ! [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs2 ) )
      = ( ? [Ys3: list_l8907847357763382004od_b_c,Zs2: list_l8907847357763382004od_b_c] :
            ( ( Xs2
              = ( append6992831555715836169od_b_c @ Ys3 @ ( cons_l342771267123639716od_b_c @ X @ Zs2 ) ) )
            & ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_544_in__set__conv__decomp__last,axiom,
    ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,Zs2: list_P6327159017948738492od_c_a] :
            ( ( Xs2
              = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X @ Zs2 ) ) )
            & ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_545_in__set__conv__decomp__last,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs2 ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs2
              = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X @ Zs2 ) ) )
            & ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_546_split__list__first__propE,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ? [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
            ( ? [Zs3: list_P6327159017948738492od_c_a] :
                ( Xs2
                = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa @ ( set_Pr5664641634098845387od_c_a @ Ys2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_547_split__list__first__propE,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c] :
            ( ? [Zs3: list_P903359562653991662od_b_c] :
                ( Xs2
                = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa: product_prod_b_c] :
                    ( ( member7862447941013992593od_b_c @ Xa @ ( set_Product_prod_b_c2 @ Ys2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_548_split__list__last__propE,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ? [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a,Zs3: list_P6327159017948738492od_c_a] :
            ( ( Xs2
              = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa @ ( set_Pr5664641634098845387od_c_a @ Zs3 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_549_split__list__last__propE,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c,Zs3: list_P903359562653991662od_b_c] :
            ( ( Xs2
              = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa: product_prod_b_c] :
                    ( ( member7862447941013992593od_b_c @ Xa @ ( set_Product_prod_b_c2 @ Zs3 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_550_split__list__first__prop,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ? [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
          ( ? [Zs3: list_P6327159017948738492od_c_a] :
              ( Xs2
              = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa @ ( set_Pr5664641634098845387od_c_a @ Ys2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_551_split__list__first__prop,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c] :
          ( ? [Zs3: list_P903359562653991662od_b_c] :
              ( Xs2
              = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa: product_prod_b_c] :
              ( ( member7862447941013992593od_b_c @ Xa @ ( set_Product_prod_b_c2 @ Ys2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_552_split__list__last__prop,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ? [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a,Zs3: list_P6327159017948738492od_c_a] :
          ( ( Xs2
            = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa @ ( set_Pr5664641634098845387od_c_a @ Zs3 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_553_split__list__last__prop,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c,Zs3: list_P903359562653991662od_b_c] :
          ( ( Xs2
            = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa: product_prod_b_c] :
              ( ( member7862447941013992593od_b_c @ Xa @ ( set_Product_prod_b_c2 @ Zs3 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_554_in__set__conv__decomp,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
      = ( ? [Ys3: list_a,Zs2: list_a] :
            ( Xs2
            = ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_555_in__set__conv__decomp,axiom,
    ! [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs2 ) )
      = ( ? [Ys3: list_l8907847357763382004od_b_c,Zs2: list_l8907847357763382004od_b_c] :
            ( Xs2
            = ( append6992831555715836169od_b_c @ Ys3 @ ( cons_l342771267123639716od_b_c @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_556_in__set__conv__decomp,axiom,
    ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
      = ( ? [Ys3: list_P6327159017948738492od_c_a,Zs2: list_P6327159017948738492od_c_a] :
            ( Xs2
            = ( append3884164603879732689od_c_a @ Ys3 @ ( cons_P8659537020926305772od_c_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_557_in__set__conv__decomp,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs2 ) )
      = ( ? [Ys3: list_P903359562653991662od_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( Xs2
            = ( append2547753245680614915od_b_c @ Ys3 @ ( cons_P4529483553340347422od_b_c @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_558_append__Cons__eq__iff,axiom,
    ! [X: a,Xs2: list_a,Ys: list_a,Xs4: list_a,Ys5: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
       => ( ( ( append_a @ Xs2 @ ( cons_a @ X @ Ys ) )
            = ( append_a @ Xs4 @ ( cons_a @ X @ Ys5 ) ) )
          = ( ( Xs2 = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_559_append__Cons__eq__iff,axiom,
    ! [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c,Xs4: list_l8907847357763382004od_b_c,Ys5: list_l8907847357763382004od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs2 ) )
     => ( ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Ys ) )
       => ( ( ( append6992831555715836169od_b_c @ Xs2 @ ( cons_l342771267123639716od_b_c @ X @ Ys ) )
            = ( append6992831555715836169od_b_c @ Xs4 @ ( cons_l342771267123639716od_b_c @ X @ Ys5 ) ) )
          = ( ( Xs2 = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_560_append__Cons__eq__iff,axiom,
    ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a,Xs4: list_P6327159017948738492od_c_a,Ys5: list_P6327159017948738492od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
     => ( ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ( ( ( append3884164603879732689od_c_a @ Xs2 @ ( cons_P8659537020926305772od_c_a @ X @ Ys ) )
            = ( append3884164603879732689od_c_a @ Xs4 @ ( cons_P8659537020926305772od_c_a @ X @ Ys5 ) ) )
          = ( ( Xs2 = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_561_append__Cons__eq__iff,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,Xs4: list_P903359562653991662od_b_c,Ys5: list_P903359562653991662od_b_c] :
      ( ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs2 ) )
     => ( ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Ys ) )
       => ( ( ( append2547753245680614915od_b_c @ Xs2 @ ( cons_P4529483553340347422od_b_c @ X @ Ys ) )
            = ( append2547753245680614915od_b_c @ Xs4 @ ( cons_P4529483553340347422od_b_c @ X @ Ys5 ) ) )
          = ( ( Xs2 = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_562_split__list__propE,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ? [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
            ( ? [Zs3: list_P6327159017948738492od_c_a] :
                ( Xs2
                = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs3 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_563_split__list__propE,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c] :
            ( ? [Zs3: list_P903359562653991662od_b_c] :
                ( Xs2
                = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs3 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_564_split__list__first,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ? [Ys2: list_a,Zs3: list_a] :
          ( ( Xs2
            = ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_565_split__list__first,axiom,
    ! [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs2 ) )
     => ? [Ys2: list_l8907847357763382004od_b_c,Zs3: list_l8907847357763382004od_b_c] :
          ( ( Xs2
            = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X @ Zs3 ) ) )
          & ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_566_split__list__first,axiom,
    ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,Zs3: list_P6327159017948738492od_c_a] :
          ( ( Xs2
            = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X @ Zs3 ) ) )
          & ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_567_split__list__first,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs2 ) )
     => ? [Ys2: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c] :
          ( ( Xs2
            = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X @ Zs3 ) ) )
          & ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_568_split__list__prop,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ? [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,X3: produc2669932201312201782od_c_a] :
          ( ? [Zs3: list_P6327159017948738492od_c_a] :
              ( Xs2
              = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_569_split__list__prop,axiom,
    ! [Xs2: list_P903359562653991662od_b_c,P: product_prod_b_c > $o] :
      ( ? [X5: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X5 @ ( set_Product_prod_b_c2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_P903359562653991662od_b_c,X3: product_prod_b_c] :
          ( ? [Zs3: list_P903359562653991662od_b_c] :
              ( Xs2
              = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Zs3 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_570_split__list__last,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ? [Ys2: list_a,Zs3: list_a] :
          ( ( Xs2
            = ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_571_split__list__last,axiom,
    ! [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs2 ) )
     => ? [Ys2: list_l8907847357763382004od_b_c,Zs3: list_l8907847357763382004od_b_c] :
          ( ( Xs2
            = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X @ Zs3 ) ) )
          & ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_572_split__list__last,axiom,
    ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,Zs3: list_P6327159017948738492od_c_a] :
          ( ( Xs2
            = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X @ Zs3 ) ) )
          & ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_573_split__list__last,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs2 ) )
     => ? [Ys2: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c] :
          ( ( Xs2
            = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X @ Zs3 ) ) )
          & ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_574_split__list,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ? [Ys2: list_a,Zs3: list_a] :
          ( Xs2
          = ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_575_split__list,axiom,
    ! [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs2 ) )
     => ? [Ys2: list_l8907847357763382004od_b_c,Zs3: list_l8907847357763382004od_b_c] :
          ( Xs2
          = ( append6992831555715836169od_b_c @ Ys2 @ ( cons_l342771267123639716od_b_c @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_576_split__list,axiom,
    ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
     => ? [Ys2: list_P6327159017948738492od_c_a,Zs3: list_P6327159017948738492od_c_a] :
          ( Xs2
          = ( append3884164603879732689od_c_a @ Ys2 @ ( cons_P8659537020926305772od_c_a @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_577_split__list,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs2 ) )
     => ? [Ys2: list_P903359562653991662od_b_c,Zs3: list_P903359562653991662od_b_c] :
          ( Xs2
          = ( append2547753245680614915od_b_c @ Ys2 @ ( cons_P4529483553340347422od_b_c @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_578_state__cover__transition__converges,axiom,
    ! [M2: fsm_a_c_a,V: a > list_P125642481956313003od_c_a,T: produc5255912757217091959od_a_a] :
      ( ( observable_a_c_a @ M2 )
     => ( ( state_4884874991499117324_a_c_a @ M2 @ V )
       => ( ( member8088020824113751456od_a_a @ T @ ( transitions_a_c_a @ M2 ) )
         => ( ( member_a @ ( produc4857801335195788115od_a_a @ T ) @ ( reacha8056221985634029888_a_c_a @ M2 ) )
           => ( converge_a_c_a @ M2 @ ( append8983669691956257088od_c_a @ ( V @ ( produc4857801335195788115od_a_a @ T ) ) @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ ( produc5237419938360121514od_a_a @ ( produc8719988090260684437od_a_a @ T ) ) @ ( product_fst_a_a @ ( produc933545150919345644od_a_a @ ( produc8719988090260684437od_a_a @ T ) ) ) ) @ nil_Product_prod_c_a ) ) @ ( V @ ( product_snd_a_a @ ( produc933545150919345644od_a_a @ ( produc8719988090260684437od_a_a @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_579_state__cover__transition__converges,axiom,
    ! [M2: fsm_d_b_c,V: d > list_P903359562653991662od_b_c,T: produc1399170498009216118od_c_d] :
      ( ( observable_d_b_c @ M2 )
     => ( ( state_6014242345326633810_d_b_c @ M2 @ V )
       => ( ( member4231278564905875615od_c_d @ T @ ( transitions_d_b_c @ M2 ) )
         => ( ( member_d @ ( produc1508465569166014290od_c_d @ T ) @ ( reacha9185589339461546374_d_b_c @ M2 ) )
           => ( converge_d_b_c @ M2 @ ( append2547753245680614915od_b_c @ ( V @ ( produc1508465569166014290od_c_d @ T ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) @ ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) ) @ nil_Product_prod_b_c ) ) @ ( V @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_580_state__cover__transition__converges,axiom,
    ! [M2: fsm_a_b_c,V: a > list_P903359562653991662od_b_c,T: produc2669932201312201782od_c_a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( state_7672330573471793359_a_b_c @ M2 @ V )
       => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M2 ) )
         => ( ( member_a @ ( produc4225943634317630866od_c_a @ T ) @ ( reacha1620305530751930115_a_b_c @ M2 ) )
           => ( converge_a_b_c @ M2 @ ( append2547753245680614915od_b_c @ ( V @ ( produc4225943634317630866od_c_a @ T ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ nil_Product_prod_b_c ) ) @ ( V @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_581_state__cover__transition__converges,axiom,
    ! [M2: fsm_a_4970532336856918856od_c_a,V: a > list_P5767727266746070163od_c_a,T: produc2473480641396933073_c_a_a] :
      ( ( observ6829695757306629100od_c_a @ M2 )
     => ( ( state_1890649935170971764od_c_a @ M2 @ V )
       => ( ( member7351836221732992250_c_a_a @ T @ ( transi6806952765996664249od_c_a @ M2 ) )
         => ( ( member_a @ ( produc2137243269946864045_c_a_a @ T ) @ ( reacha8900796604505973416od_c_a @ M2 ) )
           => ( conver5434926665748456575od_c_a @ M2 @ ( append2814445079200870952od_c_a @ ( V @ ( produc2137243269946864045_c_a_a @ T ) ) @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ ( produc3813224185952080260_c_a_a @ ( produc8586270406267960047_c_a_a @ T ) ) @ ( produc4522685202854701596_c_a_a @ ( produc7675410941016976582_c_a_a @ ( produc8586270406267960047_c_a_a @ T ) ) ) ) @ nil_Pr5380409136740620691od_c_a ) ) @ ( V @ ( produc218810415413925726_c_a_a @ ( produc7675410941016976582_c_a_a @ ( produc8586270406267960047_c_a_a @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_582_state__cover__transition__converges,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,V: product_prod_c_a > list_P1467974523212632366od_a_b,T: produc3205353004677322104od_c_a] :
      ( ( observ2964502264774598958_a_a_b @ M2 )
     => ( ( state_7248828479493717430_a_a_b @ M2 @ V )
       => ( ( member6508725278956418977od_c_a @ T @ ( transi2941759273464634107_a_a_b @ M2 ) )
         => ( ( member5074992350434858958od_c_a @ ( produc8171714036265584980od_c_a @ T ) @ ( reacha5035603111973943274_a_a_b @ M2 ) )
           => ( conver1569733173216426433_a_a_b @ M2 @ ( append5335208823350062147od_a_b @ ( V @ ( produc8171714036265584980od_c_a @ T ) ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) @ nil_Product_prod_a_b ) ) @ ( V @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_583_state__cover__transition__converges,axiom,
    ! [M2: fsm_a_8518373703003225814_b_c_e,V: a > list_P268689088907702561_b_c_e,T: produc8570131207859728997od_e_a] :
      ( ( observ7286853435796284794_b_c_e @ M2 )
     => ( ( state_8320733454665479682_b_c_e @ M2 @ V )
       => ( ( member8958573442437681166od_e_a @ T @ ( transi8235709246594610119_b_c_e @ M2 ) )
         => ( ( member_a @ ( produc141068547207231937od_e_a @ T ) @ ( reacha4051624011855102518_b_c_e @ M2 ) )
           => ( conver3658120378338990477_b_c_e @ M2 @ ( append3015380627477509558_b_c_e @ ( V @ ( produc141068547207231937od_e_a @ T ) ) @ ( cons_P1373155238876956753_b_c_e @ ( produc2511600812177743883_b_c_e @ ( produc2488811074921770520od_e_a @ ( produc7077781598552879107od_e_a @ T ) ) @ ( product_fst_e_a @ ( produc3877390521829078618od_e_a @ ( produc7077781598552879107od_e_a @ T ) ) ) ) @ nil_Pr3869772438633755809_b_c_e ) ) @ ( V @ ( product_snd_e_a @ ( produc3877390521829078618od_e_a @ ( produc7077781598552879107od_e_a @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_584_state__cover__transition__converges,axiom,
    ! [M2: fsm_li6801133765522507155_c_b_c,V: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,T: produc596031391177498678od_b_c] :
      ( ( observ6293852833591064631_c_b_c @ M2 )
     => ( ( state_2362830510734323903_c_b_c @ M2 @ V )
       => ( ( member8124909232574984543od_b_c @ T @ ( transi3161963839906981764_c_b_c @ M2 ) )
         => ( ( member6330420149250801815od_b_c @ ( produc5084291339869529746od_b_c @ T ) @ ( reacha8116992257183400179_c_b_c @ M2 ) )
           => ( conver6852411701971397450_c_b_c @ M2 @ ( append2547753245680614915od_b_c @ ( V @ ( produc5084291339869529746od_b_c @ T ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc4488652850390612441od_b_c @ ( produc829356566976367572od_b_c @ T ) ) @ ( produc5388473464130769137od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T ) ) ) ) @ nil_Product_prod_b_c ) ) @ ( V @ ( produc8475728678562477363od_b_c @ ( produc3182598933741882395od_b_c @ ( produc829356566976367572od_b_c @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_585_state__cover__transition__converges,axiom,
    ! [M2: fsm_li4013678174943373520_c_c_a,V: list_P903359562653991662od_b_c > list_P125642481956313003od_c_a,T: produc2638276842992539639od_b_c] :
      ( ( observ3506397251618388596_c_c_a @ M2 )
     => ( ( state_8798746965616423676_c_c_a @ M2 @ V )
       => ( ( member943782647535249696od_b_c @ T @ ( transi374508257934305729_c_c_a @ M2 ) )
         => ( ( member6330420149250801815od_b_c @ ( produc4656216724876455635od_b_c @ T ) @ ( reacha5329536675210724144_c_c_a @ M2 ) )
           => ( conver4064956119998721415_c_c_a @ M2 @ ( append8983669691956257088od_c_a @ ( V @ ( produc4656216724876455635od_b_c @ T ) ) @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ ( produc7862681159774872346od_b_c @ ( produc401281951983293461od_b_c @ T ) ) @ ( produc7198897215385899379od_b_c @ ( produc6556627243126142300od_b_c @ ( produc401281951983293461od_b_c @ T ) ) ) ) @ nil_Product_prod_c_a ) ) @ ( V @ ( produc1062780392962831797od_b_c @ ( produc6556627243126142300od_b_c @ ( produc401281951983293461od_b_c @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_586_state__cover__transition__converges,axiom,
    ! [M2: fsm_a_1173558889237794545od_c_a,V: a > list_P6327159017948738492od_c_a,T: produc6003456256095645226_c_a_a] :
      ( ( observ5806242608082402453od_c_a @ M2 )
     => ( ( state_6079102352357429533od_c_a @ M2 @ V )
       => ( ( member83456493519966291_c_a_a @ T @ ( transi7518404330182926690od_c_a @ M2 ) )
         => ( ( member_a @ ( produc2976558321828830086_c_a_a @ T ) @ ( reacha8031885885039376209od_c_a @ M2 ) )
           => ( conver9013735908743793192od_c_a @ M2 @ ( append3884164603879732689od_c_a @ ( V @ ( produc2976558321828830086_c_a_a @ T ) ) @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ ( produc2803981006637266269_c_a_a @ ( produc8671850722014827208_c_a_a @ T ) ) @ ( produc1462292479987911732_c_a_a @ ( produc29636106103586463_c_a_a @ ( produc8671850722014827208_c_a_a @ T ) ) ) ) @ nil_Pr1342775757158464060od_c_a ) ) @ ( V @ ( produc5324479235052808054_c_a_a @ ( produc29636106103586463_c_a_a @ ( produc8671850722014827208_c_a_a @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_587_state__cover__transition__converges,axiom,
    ! [M2: fsm_Pr7461828476839194351_a_c_a,V: produc2133359284458116493od_c_a > list_P125642481956313003od_c_a,T: produc3729687299602777783od_c_a] :
      ( ( observ3736821838051356051_a_c_a @ M2 )
     => ( ( state_4009681582326383131_a_c_a @ M2 @ V )
       => ( ( member5500486793640445664od_c_a @ T @ ( transi5448983560151880288_a_c_a @ M2 ) )
         => ( ( member1160383752307813046od_c_a @ ( produc1784202036732648467od_c_a @ T ) @ ( reacha5962465115008329807_a_c_a @ M2 ) )
           => ( conver6944315138712746790_a_c_a @ M2 @ ( append8983669691956257088od_c_a @ ( V @ ( produc1784202036732648467od_c_a @ T ) ) @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ ( produc7173261134821384633od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) ) ) @ nil_Product_prod_c_a ) ) @ ( V @ ( produc8088130389382527188od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) ) ) ) ) ) ) ) ).

% state_cover_transition_converges
thf(fact_588_cg__merge__invar,axiom,
    ! [G: e,Gamma: list_P903359562653991662od_b_c,Gamma4: list_P903359562653991662od_b_c] :
      ( ( conver7089555221631485245_c_d_e @ m1 @ m2 @ cg_lookup @ G )
     => ( ( converge_a_b_c @ m1 @ Gamma @ Gamma4 )
       => ( ( converge_d_b_c @ m2 @ Gamma @ Gamma4 )
         => ( conver7089555221631485245_c_d_e @ m1 @ m2 @ cg_lookup @ ( cg_merge @ G @ Gamma @ Gamma4 ) ) ) ) ) ).

% cg_merge_invar
thf(fact_589_state__cover__assignment__diverges,axiom,
    ! [M2: fsm_d_b_c,F: d > list_P903359562653991662od_b_c,Q14: d,Q22: d] :
      ( ( observable_d_b_c @ M2 )
     => ( ( minimal_d_b_c @ M2 )
       => ( ( state_6014242345326633810_d_b_c @ M2 @ F )
         => ( ( member_d @ Q14 @ ( reacha9185589339461546374_d_b_c @ M2 ) )
           => ( ( member_d @ Q22 @ ( reacha9185589339461546374_d_b_c @ M2 ) )
             => ( ( Q14 != Q22 )
               => ~ ( converge_d_b_c @ M2 @ ( F @ Q14 ) @ ( F @ Q22 ) ) ) ) ) ) ) ) ).

% state_cover_assignment_diverges
thf(fact_590_state__cover__assignment__diverges,axiom,
    ! [M2: fsm_a_b_c,F: a > list_P903359562653991662od_b_c,Q14: a,Q22: a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( minimal_a_b_c @ M2 )
       => ( ( state_7672330573471793359_a_b_c @ M2 @ F )
         => ( ( member_a @ Q14 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
           => ( ( member_a @ Q22 @ ( reacha1620305530751930115_a_b_c @ M2 ) )
             => ( ( Q14 != Q22 )
               => ~ ( converge_a_b_c @ M2 @ ( F @ Q14 ) @ ( F @ Q22 ) ) ) ) ) ) ) ) ).

% state_cover_assignment_diverges
thf(fact_591_preserves__divergence_Oelims_I3_J,axiom,
    ! [X: fsm_d_b_c,Xa3: fsm_d_b_c,Xb: set_li6436108459499378894od_b_c] :
      ( ~ ( preser4725570932485813173_b_c_d @ X @ Xa3 @ Xb )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
           => ! [Xa2: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Xa2 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
               => ( ~ ( converge_d_b_c @ X @ X3 @ Xa2 )
                 => ~ ( converge_d_b_c @ Xa3 @ X3 @ Xa2 ) ) ) ) ) ).

% preserves_divergence.elims(3)
thf(fact_592_preserves__divergence_Oelims_I3_J,axiom,
    ! [X: fsm_d_b_c,Xa3: fsm_a_b_c,Xb: set_li6436108459499378894od_b_c] :
      ( ~ ( preser4725570932485813170_b_c_a @ X @ Xa3 @ Xb )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
           => ! [Xa2: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Xa2 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
               => ( ~ ( converge_d_b_c @ X @ X3 @ Xa2 )
                 => ~ ( converge_a_b_c @ Xa3 @ X3 @ Xa2 ) ) ) ) ) ).

% preserves_divergence.elims(3)
thf(fact_593_preserves__divergence_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c,Xa3: fsm_a_b_c,Xb: set_li6436108459499378894od_b_c] :
      ( ~ ( preser261684820601417589_b_c_a @ X @ Xa3 @ Xb )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
           => ! [Xa2: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Xa2 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
               => ( ~ ( converge_a_b_c @ X @ X3 @ Xa2 )
                 => ~ ( converge_a_b_c @ Xa3 @ X3 @ Xa2 ) ) ) ) ) ).

% preserves_divergence.elims(3)
thf(fact_594_preserves__divergence_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c,Xa3: fsm_d_b_c,Xb: set_li6436108459499378894od_b_c] :
      ( ~ ( preser261684820601417592_b_c_d @ X @ Xa3 @ Xb )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
           => ! [Xa2: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Xa2 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
               => ( ~ ( converge_a_b_c @ X @ X3 @ Xa2 )
                 => ~ ( converge_d_b_c @ Xa3 @ X3 @ Xa2 ) ) ) ) ) ).

% preserves_divergence.elims(3)
thf(fact_595_preserves__divergence_Oelims_I2_J,axiom,
    ! [X: fsm_d_b_c,Xa3: fsm_d_b_c,Xb: set_li6436108459499378894od_b_c] :
      ( ( preser4725570932485813173_b_c_d @ X @ Xa3 @ Xb )
     => ! [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
         => ! [Xa: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Xa @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
             => ( ~ ( converge_d_b_c @ X @ X5 @ Xa )
               => ~ ( converge_d_b_c @ Xa3 @ X5 @ Xa ) ) ) ) ) ).

% preserves_divergence.elims(2)
thf(fact_596_preserves__divergence_Oelims_I2_J,axiom,
    ! [X: fsm_d_b_c,Xa3: fsm_a_b_c,Xb: set_li6436108459499378894od_b_c] :
      ( ( preser4725570932485813170_b_c_a @ X @ Xa3 @ Xb )
     => ! [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
         => ! [Xa: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Xa @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
             => ( ~ ( converge_d_b_c @ X @ X5 @ Xa )
               => ~ ( converge_a_b_c @ Xa3 @ X5 @ Xa ) ) ) ) ) ).

% preserves_divergence.elims(2)
thf(fact_597_preserves__divergence_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c,Xa3: fsm_a_b_c,Xb: set_li6436108459499378894od_b_c] :
      ( ( preser261684820601417589_b_c_a @ X @ Xa3 @ Xb )
     => ! [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
         => ! [Xa: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Xa @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
             => ( ~ ( converge_a_b_c @ X @ X5 @ Xa )
               => ~ ( converge_a_b_c @ Xa3 @ X5 @ Xa ) ) ) ) ) ).

% preserves_divergence.elims(2)
thf(fact_598_preserves__divergence_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c,Xa3: fsm_d_b_c,Xb: set_li6436108459499378894od_b_c] :
      ( ( preser261684820601417592_b_c_d @ X @ Xa3 @ Xb )
     => ! [X5: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X5 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
         => ! [Xa: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Xa @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
             => ( ~ ( converge_a_b_c @ X @ X5 @ Xa )
               => ~ ( converge_d_b_c @ Xa3 @ X5 @ Xa ) ) ) ) ) ).

% preserves_divergence.elims(2)
thf(fact_599_preserves__divergence_Oelims_I1_J,axiom,
    ! [X: fsm_d_b_c,Xa3: fsm_d_b_c,Xb: set_li6436108459499378894od_b_c,Y: $o] :
      ( ( ( preser4725570932485813173_b_c_d @ X @ Xa3 @ Xb )
        = Y )
     => ( Y
        = ( ! [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
             => ! [Y5: list_P903359562653991662od_b_c] :
                  ( ( member6330420149250801815od_b_c @ Y5 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
                 => ( ~ ( converge_d_b_c @ X @ X4 @ Y5 )
                   => ~ ( converge_d_b_c @ Xa3 @ X4 @ Y5 ) ) ) ) ) ) ) ).

% preserves_divergence.elims(1)
thf(fact_600_preserves__divergence_Oelims_I1_J,axiom,
    ! [X: fsm_d_b_c,Xa3: fsm_a_b_c,Xb: set_li6436108459499378894od_b_c,Y: $o] :
      ( ( ( preser4725570932485813170_b_c_a @ X @ Xa3 @ Xb )
        = Y )
     => ( Y
        = ( ! [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
             => ! [Y5: list_P903359562653991662od_b_c] :
                  ( ( member6330420149250801815od_b_c @ Y5 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ X @ ( initial_d_b_c @ X ) ) @ Xb ) )
                 => ( ~ ( converge_d_b_c @ X @ X4 @ Y5 )
                   => ~ ( converge_a_b_c @ Xa3 @ X4 @ Y5 ) ) ) ) ) ) ) ).

% preserves_divergence.elims(1)
thf(fact_601_preserves__divergence_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Xa3: fsm_a_b_c,Xb: set_li6436108459499378894od_b_c,Y: $o] :
      ( ( ( preser261684820601417589_b_c_a @ X @ Xa3 @ Xb )
        = Y )
     => ( Y
        = ( ! [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
             => ! [Y5: list_P903359562653991662od_b_c] :
                  ( ( member6330420149250801815od_b_c @ Y5 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
                 => ( ~ ( converge_a_b_c @ X @ X4 @ Y5 )
                   => ~ ( converge_a_b_c @ Xa3 @ X4 @ Y5 ) ) ) ) ) ) ) ).

% preserves_divergence.elims(1)
thf(fact_602_preserves__divergence_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Xa3: fsm_d_b_c,Xb: set_li6436108459499378894od_b_c,Y: $o] :
      ( ( ( preser261684820601417592_b_c_d @ X @ Xa3 @ Xb )
        = Y )
     => ( Y
        = ( ! [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
             => ! [Y5: list_P903359562653991662od_b_c] :
                  ( ( member6330420149250801815od_b_c @ Y5 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ X @ ( initial_a_b_c @ X ) ) @ Xb ) )
                 => ( ~ ( converge_a_b_c @ X @ X4 @ Y5 )
                   => ~ ( converge_d_b_c @ Xa3 @ X4 @ Y5 ) ) ) ) ) ) ) ).

% preserves_divergence.elims(1)
thf(fact_603_preserves__divergence_Osimps,axiom,
    ( preser4725570932485813173_b_c_d
    = ( ^ [M1: fsm_d_b_c,M22: fsm_d_b_c,A5: set_li6436108459499378894od_b_c] :
        ! [X4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X4 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M1 @ ( initial_d_b_c @ M1 ) ) @ A5 ) )
         => ! [Y5: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Y5 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M1 @ ( initial_d_b_c @ M1 ) ) @ A5 ) )
             => ( ~ ( converge_d_b_c @ M1 @ X4 @ Y5 )
               => ~ ( converge_d_b_c @ M22 @ X4 @ Y5 ) ) ) ) ) ) ).

% preserves_divergence.simps
thf(fact_604_preserves__divergence_Osimps,axiom,
    ( preser4725570932485813170_b_c_a
    = ( ^ [M1: fsm_d_b_c,M22: fsm_a_b_c,A5: set_li6436108459499378894od_b_c] :
        ! [X4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X4 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M1 @ ( initial_d_b_c @ M1 ) ) @ A5 ) )
         => ! [Y5: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Y5 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M1 @ ( initial_d_b_c @ M1 ) ) @ A5 ) )
             => ( ~ ( converge_d_b_c @ M1 @ X4 @ Y5 )
               => ~ ( converge_a_b_c @ M22 @ X4 @ Y5 ) ) ) ) ) ) ).

% preserves_divergence.simps
thf(fact_605_preserves__divergence_Osimps,axiom,
    ( preser261684820601417589_b_c_a
    = ( ^ [M1: fsm_a_b_c,M22: fsm_a_b_c,A5: set_li6436108459499378894od_b_c] :
        ! [X4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X4 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) ) @ A5 ) )
         => ! [Y5: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Y5 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) ) @ A5 ) )
             => ( ~ ( converge_a_b_c @ M1 @ X4 @ Y5 )
               => ~ ( converge_a_b_c @ M22 @ X4 @ Y5 ) ) ) ) ) ) ).

% preserves_divergence.simps
thf(fact_606_preserves__divergence_Osimps,axiom,
    ( preser261684820601417592_b_c_d
    = ( ^ [M1: fsm_a_b_c,M22: fsm_d_b_c,A5: set_li6436108459499378894od_b_c] :
        ! [X4: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X4 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) ) @ A5 ) )
         => ! [Y5: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ Y5 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) ) @ A5 ) )
             => ( ~ ( converge_a_b_c @ M1 @ X4 @ Y5 )
               => ~ ( converge_d_b_c @ M22 @ X4 @ Y5 ) ) ) ) ) ) ).

% preserves_divergence.simps
thf(fact_607_convergence__graph__lookup__invar__simp_I2_J,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,Cg_lookup2: e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,G: e,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( conver7089555221631485245_c_d_e @ M12 @ M23 @ Cg_lookup2 @ G )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) )
       => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_d_b_c @ M23 @ ( initial_d_b_c @ M23 ) ) )
         => ( ( member6330420149250801815od_b_c @ Beta @ ( set_li4480668622519654659od_b_c @ ( Cg_lookup2 @ G @ Alpha ) ) )
           => ( converge_d_b_c @ M23 @ Alpha @ Beta ) ) ) ) ) ).

% convergence_graph_lookup_invar_simp(2)
thf(fact_608_convergence__graph__lookup__invar__simp_I1_J,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,Cg_lookup2: e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,G: e,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( conver7089555221631485245_c_d_e @ M12 @ M23 @ Cg_lookup2 @ G )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) )
       => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_d_b_c @ M23 @ ( initial_d_b_c @ M23 ) ) )
         => ( ( member6330420149250801815od_b_c @ Beta @ ( set_li4480668622519654659od_b_c @ ( Cg_lookup2 @ G @ Alpha ) ) )
           => ( converge_a_b_c @ M12 @ Alpha @ Beta ) ) ) ) ) ).

% convergence_graph_lookup_invar_simp(1)
thf(fact_609_convergence__graph__lookup__invar__def,axiom,
    ( conver7089555221631485245_c_d_e
    = ( ^ [M1: fsm_a_b_c,M22: fsm_d_b_c,Cg_lookup: e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,G2: e] :
        ! [Alpha2: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ Alpha2 @ ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) ) )
         => ( ( member6330420149250801815od_b_c @ Alpha2 @ ( lS_d_b_c @ M22 @ ( initial_d_b_c @ M22 ) ) )
           => ( ( member6330420149250801815od_b_c @ Alpha2 @ ( set_li4480668622519654659od_b_c @ ( Cg_lookup @ G2 @ Alpha2 ) ) )
              & ! [Beta2: list_P903359562653991662od_b_c] :
                  ( ( member6330420149250801815od_b_c @ Beta2 @ ( set_li4480668622519654659od_b_c @ ( Cg_lookup @ G2 @ Alpha2 ) ) )
                 => ( ( converge_a_b_c @ M1 @ Alpha2 @ Beta2 )
                    & ( converge_d_b_c @ M22 @ Alpha2 @ Beta2 ) ) ) ) ) ) ) ) ).

% convergence_graph_lookup_invar_def
thf(fact_610_preserves__divergence__subset,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( preser261684820601417592_b_c_d @ M12 @ M23 @ B4 )
     => ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
       => ( preser261684820601417592_b_c_d @ M12 @ M23 @ A4 ) ) ) ).

% preserves_divergence_subset
thf(fact_611_converge__refl,axiom,
    ! [Alpha: list_P903359562653991662od_b_c,M2: fsm_d_b_c] :
      ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) )
     => ( converge_d_b_c @ M2 @ Alpha @ Alpha ) ) ).

% converge_refl
thf(fact_612_converge__refl,axiom,
    ! [Alpha: list_P903359562653991662od_b_c,M2: fsm_a_b_c] :
      ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
     => ( converge_a_b_c @ M2 @ Alpha @ Alpha ) ) ).

% converge_refl
thf(fact_613_converge__append__iff,axiom,
    ! [M2: fsm_d_b_c,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Omega: list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( converge_d_b_c @ M2 @ Alpha @ Beta )
       => ( ( converge_d_b_c @ M2 @ Gamma @ ( append2547753245680614915od_b_c @ Alpha @ Omega ) )
          = ( converge_d_b_c @ M2 @ Gamma @ ( append2547753245680614915od_b_c @ Beta @ Omega ) ) ) ) ) ).

% converge_append_iff
thf(fact_614_converge__append__iff,axiom,
    ! [M2: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Omega: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( converge_a_b_c @ M2 @ Alpha @ Beta )
       => ( ( converge_a_b_c @ M2 @ Gamma @ ( append2547753245680614915od_b_c @ Alpha @ Omega ) )
          = ( converge_a_b_c @ M2 @ Gamma @ ( append2547753245680614915od_b_c @ Beta @ Omega ) ) ) ) ) ).

% converge_append_iff
thf(fact_615_equivalence__preserves__divergence,axiom,
    ! [M2: fsm_a_b_c,I: fsm_a_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
            = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) )
         => ( preser261684820601417589_b_c_a @ M2 @ I @ A4 ) ) ) ) ).

% equivalence_preserves_divergence
thf(fact_616_equivalence__preserves__divergence,axiom,
    ! [M2: fsm_d_b_c,I: fsm_a_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
            = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) )
         => ( preser4725570932485813170_b_c_a @ M2 @ I @ A4 ) ) ) ) ).

% equivalence_preserves_divergence
thf(fact_617_equivalence__preserves__divergence,axiom,
    ! [M2: fsm_d_b_c,I: fsm_d_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
            = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) )
         => ( preser4725570932485813173_b_c_d @ M2 @ I @ A4 ) ) ) ) ).

% equivalence_preserves_divergence
thf(fact_618_equivalence__preserves__divergence,axiom,
    ! [M2: fsm_a_b_c,I: fsm_d_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
            = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) )
         => ( preser261684820601417592_b_c_d @ M2 @ I @ A4 ) ) ) ) ).

% equivalence_preserves_divergence
thf(fact_619_converge__trans__2_I3_J,axiom,
    ! [M2: fsm_d_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,W1: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( minimal_d_b_c @ M2 )
       => ( ( converge_d_b_c @ M2 @ U @ V2 )
         => ( ( converge_d_b_c @ M2 @ ( append2547753245680614915od_b_c @ U @ W1 ) @ ( append2547753245680614915od_b_c @ U @ W2 ) )
            = ( converge_d_b_c @ M2 @ ( append2547753245680614915od_b_c @ V2 @ W1 ) @ ( append2547753245680614915od_b_c @ U @ W2 ) ) ) ) ) ) ).

% converge_trans_2(3)
thf(fact_620_converge__trans__2_I3_J,axiom,
    ! [M2: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,W1: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( minimal_a_b_c @ M2 )
       => ( ( converge_a_b_c @ M2 @ U @ V2 )
         => ( ( converge_a_b_c @ M2 @ ( append2547753245680614915od_b_c @ U @ W1 ) @ ( append2547753245680614915od_b_c @ U @ W2 ) )
            = ( converge_a_b_c @ M2 @ ( append2547753245680614915od_b_c @ V2 @ W1 ) @ ( append2547753245680614915od_b_c @ U @ W2 ) ) ) ) ) ) ).

% converge_trans_2(3)
thf(fact_621_converge__trans__2_I2_J,axiom,
    ! [M2: fsm_d_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,W1: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( minimal_d_b_c @ M2 )
       => ( ( converge_d_b_c @ M2 @ U @ V2 )
         => ( ( converge_d_b_c @ M2 @ ( append2547753245680614915od_b_c @ U @ W1 ) @ ( append2547753245680614915od_b_c @ U @ W2 ) )
            = ( converge_d_b_c @ M2 @ ( append2547753245680614915od_b_c @ U @ W1 ) @ ( append2547753245680614915od_b_c @ V2 @ W2 ) ) ) ) ) ) ).

% converge_trans_2(2)
thf(fact_622_converge__trans__2_I2_J,axiom,
    ! [M2: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,W1: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( minimal_a_b_c @ M2 )
       => ( ( converge_a_b_c @ M2 @ U @ V2 )
         => ( ( converge_a_b_c @ M2 @ ( append2547753245680614915od_b_c @ U @ W1 ) @ ( append2547753245680614915od_b_c @ U @ W2 ) )
            = ( converge_a_b_c @ M2 @ ( append2547753245680614915od_b_c @ U @ W1 ) @ ( append2547753245680614915od_b_c @ V2 @ W2 ) ) ) ) ) ) ).

% converge_trans_2(2)
thf(fact_623_converge__trans__2_I1_J,axiom,
    ! [M2: fsm_d_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,W1: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( minimal_d_b_c @ M2 )
       => ( ( converge_d_b_c @ M2 @ U @ V2 )
         => ( ( converge_d_b_c @ M2 @ ( append2547753245680614915od_b_c @ U @ W1 ) @ ( append2547753245680614915od_b_c @ U @ W2 ) )
            = ( converge_d_b_c @ M2 @ ( append2547753245680614915od_b_c @ V2 @ W1 ) @ ( append2547753245680614915od_b_c @ V2 @ W2 ) ) ) ) ) ) ).

% converge_trans_2(1)
thf(fact_624_converge__trans__2_I1_J,axiom,
    ! [M2: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,W1: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( minimal_a_b_c @ M2 )
       => ( ( converge_a_b_c @ M2 @ U @ V2 )
         => ( ( converge_a_b_c @ M2 @ ( append2547753245680614915od_b_c @ U @ W1 ) @ ( append2547753245680614915od_b_c @ U @ W2 ) )
            = ( converge_a_b_c @ M2 @ ( append2547753245680614915od_b_c @ V2 @ W1 ) @ ( append2547753245680614915od_b_c @ V2 @ W2 ) ) ) ) ) ) ).

% converge_trans_2(1)
thf(fact_625_converge__append__language__iff,axiom,
    ! [M2: fsm_d_b_c,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( converge_d_b_c @ M2 @ Alpha @ Beta )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) ) ) ) ) ).

% converge_append_language_iff
thf(fact_626_converge__append__language__iff,axiom,
    ! [M2: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( converge_a_b_c @ M2 @ Alpha @ Beta )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ) ) ) ).

% converge_append_language_iff
thf(fact_627_converge__extend,axiom,
    ! [M2: fsm_d_b_c,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( converge_d_b_c @ M2 @ Alpha @ Beta )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) )
         => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) )
           => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) ) ) ) ) ) ).

% converge_extend
thf(fact_628_converge__extend,axiom,
    ! [M2: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( converge_a_b_c @ M2 @ Alpha @ Beta )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
         => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
           => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ) ) ) ) ).

% converge_extend
thf(fact_629_converge__append,axiom,
    ! [M2: fsm_d_b_c,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( converge_d_b_c @ M2 @ Alpha @ Beta )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) )
         => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) )
           => ( converge_d_b_c @ M2 @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) ) ) ) ) ) ).

% converge_append
thf(fact_630_converge__append,axiom,
    ! [M2: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( converge_a_b_c @ M2 @ Alpha @ Beta )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
         => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
           => ( converge_a_b_c @ M2 @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) ) ) ) ) ) ).

% converge_append
thf(fact_631_diverge__prefix,axiom,
    ! [M2: fsm_d_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) @ ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) ) )
         => ( ~ ( converge_d_b_c @ M2 @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) )
           => ~ ( converge_d_b_c @ M2 @ Alpha @ Beta ) ) ) ) ) ).

% diverge_prefix
thf(fact_632_diverge__prefix,axiom,
    ! [M2: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) )
         => ( ~ ( converge_a_b_c @ M2 @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( append2547753245680614915od_b_c @ Beta @ Gamma ) )
           => ~ ( converge_a_b_c @ M2 @ Alpha @ Beta ) ) ) ) ) ).

% diverge_prefix
thf(fact_633_verify__transition__retains__finiteness,axiom,
    ! [T5: prefix425356687782099470od_b_c,G: e,T: produc2669932201312201782od_c_a] :
      ( ( prefix6767660319289835678od_b_c @ T5 )
     => ( prefix6767660319289835678od_b_c @ ( produc1447154562020694007_b_c_e @ ( verify_transition @ ( produc2511600812177743883_b_c_e @ T5 @ G ) @ T ) ) ) ) ).

% verify_transition_retains_finiteness
thf(fact_634_delete__aux_Ocases,axiom,
    ! [X: produc4983050515642857530_b_c_e] :
      ( ! [K: prefix425356687782099470od_b_c] :
          ( X
         != ( produc3397272950601485866_b_c_e @ K @ nil_Pr3869772438633755809_b_c_e ) )
     => ~ ! [K: prefix425356687782099470od_b_c,K2: prefix425356687782099470od_b_c,V3: e,Xs3: list_P268689088907702561_b_c_e] :
            ( X
           != ( produc3397272950601485866_b_c_e @ K @ ( cons_P1373155238876956753_b_c_e @ ( produc2511600812177743883_b_c_e @ K2 @ V3 ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_635_delete__aux_Ocases,axiom,
    ! [X: produc6475121930738208726od_b_c] :
      ( ! [K: b] :
          ( X
         != ( produc561751498933632326od_b_c @ K @ nil_Product_prod_b_c ) )
     => ~ ! [K: b,K2: b,V3: c,Xs3: list_P903359562653991662od_b_c] :
            ( X
           != ( produc561751498933632326od_b_c @ K @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K2 @ V3 ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_636_delete__aux_Ocases,axiom,
    ! [X: produc8651263503759922642od_c_a] :
      ( ! [K: c] :
          ( X
         != ( produc3490508561035776450od_c_a @ K @ nil_Product_prod_c_a ) )
     => ~ ! [K: c,K2: c,V3: a,Xs3: list_P125642481956313003od_c_a] :
            ( X
           != ( produc3490508561035776450od_c_a @ K @ ( cons_P1742027962761213787od_c_a @ ( product_Pair_c_a @ K2 @ V3 ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_637_delete__aux_Ocases,axiom,
    ! [X: produc7270830264722373115od_c_a] :
      ( ! [K: b] :
          ( X
         != ( produc1443866078074754795od_c_a @ K @ nil_Pr5380409136740620691od_c_a ) )
     => ~ ! [K: b,K2: b,V3: product_prod_c_a,Xs3: list_P5767727266746070163od_c_a] :
            ( X
           != ( produc1443866078074754795od_c_a @ K @ ( cons_P7376644893796496707od_c_a @ ( produc3440865914000881661od_c_a @ K2 @ V3 ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_638_delete__aux_Ocases,axiom,
    ! [X: produc3220416805844688357od_c_a] :
      ( ! [K: a] :
          ( X
         != ( produc3580641852002036821od_c_a @ K @ nil_Pr1342775757158464060od_c_a ) )
     => ~ ! [K: a,K2: a,V3: produc2133359284458116493od_c_a,Xs3: list_P6327159017948738492od_c_a] :
            ( X
           != ( produc3580641852002036821od_c_a @ K @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ K2 @ V3 ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_639_sc__covered__transitions__alt__def,axiom,
    ( sc_cov38355202683674721itions
    = ( collec8405073525749625505od_c_a
      @ ^ [T2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ m1 ) )
          & ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ ( reacha1620305530751930115_a_b_c @ m1 ) )
          & ( ( v @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
            = ( append2547753245680614915od_b_c @ ( v @ ( produc4225943634317630866od_c_a @ T2 ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ) ).

% sc_covered_transitions_alt_def
thf(fact_640_TG2,axiom,
    ( tG2
    = ( foldl_2588721813272579068od_c_a @ verify_transition @ tG1 @ unveri8662357380954777460itions ) ) ).

% TG2
thf(fact_641__092_060open_062verify__transition_A_092_060equiv_062_A_092_060lambda_062_IT_M_AG_J_At_O_Alet_ATGxy_A_061_Aappend__io__pair_AM1_AV_AT_AG_Acg__insert_Acg__lookup_A_It__source_At_J_A_It__input_At_J_A_It__output_At_J_059_A_IT_H_M_AG_H_J_A_061_Aestablish__convergence_AM1_AV_A_Ifst_ATGxy_J_A_Isnd_ATGxy_J_Acg__insert_Acg__lookup_Am_At_Ain_ALet_A_Icg__merge_AG_H_A_IV_A_It__source_At_J_A_064_A_091_It__input_At_M_At__output_At_J_093_J_A_IV_A_It__target_At_J_J_J_A_IPair_AT_H_J_092_060close_062,axiom,
    ( verify_transition
    = ( produc4791951524949757546_b_c_e
      @ ^ [T6: prefix425356687782099470od_b_c,G2: e,T2: produc2669932201312201782od_c_a] :
          ( produc7591640664947284859_b_c_e
          @ ^ [T7: prefix425356687782099470od_b_c,G4: e] : ( produc2511600812177743883_b_c_e @ T7 @ ( cg_merge @ G4 @ ( append2547753245680614915od_b_c @ ( v @ ( produc4225943634317630866od_c_a @ T2 ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) @ nil_Product_prod_b_c ) ) @ ( v @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) )
          @ ( establ5936990325632390625rgence @ m1 @ v @ ( produc1447154562020694007_b_c_e @ ( append_io_pair @ m1 @ v @ T6 @ G2 @ cg_insert @ cg_lookup @ ( produc4225943634317630866od_c_a @ T2 ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) @ ( produc6779600879576379961_b_c_e @ ( append_io_pair @ m1 @ v @ T6 @ G2 @ cg_insert @ cg_lookup @ ( produc4225943634317630866od_c_a @ T2 ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) @ cg_insert @ cg_lookup @ m @ T2 ) ) ) ) ).

% \<open>verify_transition \<equiv> \<lambda>(T, G) t. let TGxy = append_io_pair M1 V T G cg_insert cg_lookup (t_source t) (t_input t) (t_output t); (T', G') = establish_convergence M1 V (fst TGxy) (snd TGxy) cg_insert cg_lookup m t in Let (cg_merge G' (V (t_source t) @ [(t_input t, t_output t)]) (V (t_target t))) (Pair T')\<close>
thf(fact_642_assms_I8_J,axiom,
    ( ( outputs_d_b_c @ m2 )
    = ( outputs_a_b_c @ m1 ) ) ).

% assms(8)
thf(fact_643_assms_I7_J,axiom,
    ( ( inputs_d_b_c @ m2 )
    = ( inputs_a_b_c @ m1 ) ) ).

% assms(7)
thf(fact_644_T1__finite,axiom,
    prefix6767660319289835678od_b_c @ t1 ).

% T1_finite
thf(fact_645_foldl__append,axiom,
    ! [F: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,A: produc4534891796109904283_b_c_e,Xs2: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( foldl_2588721813272579068od_c_a @ F @ A @ ( append3884164603879732689od_c_a @ Xs2 @ Ys ) )
      = ( foldl_2588721813272579068od_c_a @ F @ ( foldl_2588721813272579068od_c_a @ F @ A @ Xs2 ) @ Ys ) ) ).

% foldl_append
thf(fact_646_foldl__append,axiom,
    ! [F: prefix425356687782099470od_b_c > produc3550516665487611281od_b_c > prefix425356687782099470od_b_c,A: prefix425356687782099470od_b_c,Xs2: list_P8634110456476873367od_b_c,Ys: list_P8634110456476873367od_b_c] :
      ( ( foldl_6461310528730867108od_b_c @ F @ A @ ( append4231602460230365740od_b_c @ Xs2 @ Ys ) )
      = ( foldl_6461310528730867108od_b_c @ F @ ( foldl_6461310528730867108od_b_c @ F @ A @ Xs2 ) @ Ys ) ) ).

% foldl_append
thf(fact_647_pair__imageI,axiom,
    ! [A: b,B: c,A4: set_Product_prod_b_c,F: b > c > a] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ A @ B ) @ A4 )
     => ( member_a @ ( F @ A @ B ) @ ( image_7762865866877615259_b_c_a @ ( produc8837194378803474129_b_c_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_648_pair__imageI,axiom,
    ! [A: c,B: a,A4: set_Product_prod_c_a,F: c > a > a] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ A4 )
     => ( member_a @ ( F @ A @ B ) @ ( image_5407010446420943518_c_a_a @ ( produc1561580084227405268_c_a_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_649_pair__imageI,axiom,
    ! [A: b,B: product_prod_c_a,A4: set_Pr272002107461315949od_c_a,F: b > product_prod_c_a > a] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ A4 )
     => ( member_a @ ( F @ A @ B ) @ ( image_8972666397057282742_c_a_a @ ( produc2354600891577745260_c_a_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_650_pair__imageI,axiom,
    ! [A: prefix425356687782099470od_b_c,B: e,A4: set_Pr3237851069733663483_b_c_e,F: prefix425356687782099470od_b_c > e > a] :
      ( ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ A @ B ) @ A4 )
     => ( member_a @ ( F @ A @ B ) @ ( image_2493620005906216232_c_e_a @ ( produc4180796069478548702_c_e_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_651_pair__imageI,axiom,
    ! [A: b,B: c,A4: set_Product_prod_b_c,F: b > c > list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ A @ B ) @ A4 )
     => ( member6330420149250801815od_b_c @ ( F @ A @ B ) @ ( image_5872333176446027019od_b_c @ ( produc2916551642269384257od_b_c @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_652_pair__imageI,axiom,
    ! [A: c,B: a,A4: set_Product_prod_c_a,F: c > a > list_P903359562653991662od_b_c] :
      ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ A @ B ) @ A4 )
     => ( member6330420149250801815od_b_c @ ( F @ A @ B ) @ ( image_3043841557255426702od_b_c @ ( produc3890162947456117956od_b_c @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_653_pair__imageI,axiom,
    ! [A: prefix425356687782099470od_b_c,B: e,A4: set_Pr3237851069733663483_b_c_e,F: prefix425356687782099470od_b_c > e > $o] :
      ( ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ A @ B ) @ A4 )
     => ( member_o @ ( F @ A @ B ) @ ( image_6871687335040213698_c_e_o @ ( produc8093372032475639160_c_e_o @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_654_pair__imageI,axiom,
    ! [A: b,B: c,A4: set_Product_prod_b_c,F: b > c > prefix425356687782099470od_b_c] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ A @ B ) @ A4 )
     => ( member2378525361814473399od_b_c @ ( F @ A @ B ) @ ( image_7251446585419991851od_b_c @ ( produc7288989217166112353od_b_c @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_655_pair__imageI,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A4: set_Pr8357624050790843030od_c_a,F: a > produc2133359284458116493od_c_a > a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ A4 )
     => ( member_a @ ( F @ A @ B ) @ ( image_9171994905208062541_c_a_a @ ( produc4731461908963710979_c_a_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_656_pair__imageI,axiom,
    ! [A: b,B: product_prod_c_a,A4: set_Pr272002107461315949od_c_a,F: b > product_prod_c_a > list_P903359562653991662od_b_c] :
      ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ A @ B ) @ A4 )
     => ( member6330420149250801815od_b_c @ ( F @ A @ B ) @ ( image_6970894258237677222od_b_c @ ( produc3980176248775602780od_b_c @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_657_case__prod__conv,axiom,
    ! [F: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,A: prefix425356687782099470od_b_c,B: e] :
      ( ( produc4791951524949757546_b_c_e @ F @ ( produc2511600812177743883_b_c_e @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_658_case__prod__conv,axiom,
    ! [F: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,A: prefix425356687782099470od_b_c,B: e] :
      ( ( produc7591640664947284859_b_c_e @ F @ ( produc2511600812177743883_b_c_e @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_659_case__prod__conv,axiom,
    ! [F: prefix425356687782099470od_b_c > e > $o,A: prefix425356687782099470od_b_c,B: e] :
      ( ( produc8093372032475639160_c_e_o @ F @ ( produc2511600812177743883_b_c_e @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_660_case__prod__conv,axiom,
    ! [F: a > product_prod_b_c > prefix425356687782099470od_b_c,A: a,B: product_prod_b_c] :
      ( ( produc2053673206102377720od_b_c @ F @ ( produc6867761804911831425od_b_c @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_661_case__prod__conv,axiom,
    ! [F: b > c > prefix425356687782099470od_b_c,A: b,B: c] :
      ( ( produc7288989217166112353od_b_c @ F @ ( product_Pair_b_c @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_662_fsm__transition__input,axiom,
    ! [T: produc1399170498009216118od_c_d,M2: fsm_d_b_c] :
      ( ( member4231278564905875615od_c_d @ T @ ( transitions_d_b_c @ M2 ) )
     => ( member_b @ ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) @ ( inputs_d_b_c @ M2 ) ) ) ).

% fsm_transition_input
thf(fact_663_fsm__transition__input,axiom,
    ! [T: produc3205353004677322104od_c_a,M2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( member6508725278956418977od_c_a @ T @ ( transi2941759273464634107_a_a_b @ M2 ) )
     => ( member_a @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( inputs8636541066849307272_a_a_b @ M2 ) ) ) ).

% fsm_transition_input
thf(fact_664_fsm__transition__input,axiom,
    ! [T: produc2669932201312201782od_c_a,M2: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M2 ) )
     => ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M2 ) ) ) ).

% fsm_transition_input
thf(fact_665_fsm__transition__output,axiom,
    ! [T: produc1399170498009216118od_c_d,M2: fsm_d_b_c] :
      ( ( member4231278564905875615od_c_d @ T @ ( transitions_d_b_c @ M2 ) )
     => ( member_c @ ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) @ ( outputs_d_b_c @ M2 ) ) ) ).

% fsm_transition_output
thf(fact_666_fsm__transition__output,axiom,
    ! [T: produc3205353004677322104od_c_a,M2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( member6508725278956418977od_c_a @ T @ ( transi2941759273464634107_a_a_b @ M2 ) )
     => ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( output2371653660386375239_a_a_b @ M2 ) ) ) ).

% fsm_transition_output
thf(fact_667_fsm__transition__output,axiom,
    ! [T: produc2669932201312201782od_c_a,M2: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M2 ) )
     => ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M2 ) ) ) ).

% fsm_transition_output
thf(fact_668_unverified__transitions__alt__def,axiom,
    ( ( set_Pr5664641634098845387od_c_a @ unveri8662357380954777460itions )
    = ( collec8405073525749625505od_c_a
      @ ^ [T2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ m1 ) )
          & ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ ( reacha1620305530751930115_a_b_c @ m1 ) )
          & ( ( v @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
           != ( append2547753245680614915od_b_c @ ( v @ ( produc4225943634317630866od_c_a @ T2 ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ) ).

% unverified_transitions_alt_def
thf(fact_669_verify__transition,axiom,
    ( verify_transition
    = ( produc4791951524949757546_b_c_e
      @ ^ [T6: prefix425356687782099470od_b_c,G2: e,T2: produc2669932201312201782od_c_a] :
          ( produc7591640664947284859_b_c_e
          @ ^ [T7: prefix425356687782099470od_b_c,G4: e] : ( produc2511600812177743883_b_c_e @ T7 @ ( cg_merge @ G4 @ ( append2547753245680614915od_b_c @ ( v @ ( produc4225943634317630866od_c_a @ T2 ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) @ nil_Product_prod_b_c ) ) @ ( v @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) )
          @ ( establ5936990325632390625rgence @ m1 @ v @ ( produc1447154562020694007_b_c_e @ ( append_io_pair @ m1 @ v @ T6 @ G2 @ cg_insert @ cg_lookup @ ( produc4225943634317630866od_c_a @ T2 ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) @ ( produc6779600879576379961_b_c_e @ ( append_io_pair @ m1 @ v @ T6 @ G2 @ cg_insert @ cg_lookup @ ( produc4225943634317630866od_c_a @ T2 ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) @ cg_insert @ cg_lookup @ m @ T2 ) ) ) ) ).

% verify_transition
thf(fact_670_case__prodE2,axiom,
    ! [Q: ( produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e ) > $o,P: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,Z2: produc4534891796109904283_b_c_e] :
      ( ( Q @ ( produc4791951524949757546_b_c_e @ P @ Z2 ) )
     => ~ ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ( ( Z2
              = ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_671_case__prodE2,axiom,
    ! [Q: produc4534891796109904283_b_c_e > $o,P: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,Z2: produc4534891796109904283_b_c_e] :
      ( ( Q @ ( produc7591640664947284859_b_c_e @ P @ Z2 ) )
     => ~ ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ( ( Z2
              = ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_672_case__prodE2,axiom,
    ! [Q: $o > $o,P: prefix425356687782099470od_b_c > e > $o,Z2: produc4534891796109904283_b_c_e] :
      ( ( Q @ ( produc8093372032475639160_c_e_o @ P @ Z2 ) )
     => ~ ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ( ( Z2
              = ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_673_case__prodE2,axiom,
    ! [Q: prefix425356687782099470od_b_c > $o,P: a > product_prod_b_c > prefix425356687782099470od_b_c,Z2: produc3550516665487611281od_b_c] :
      ( ( Q @ ( produc2053673206102377720od_b_c @ P @ Z2 ) )
     => ~ ! [X3: a,Y3: product_prod_b_c] :
            ( ( Z2
              = ( produc6867761804911831425od_b_c @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_674_case__prodE2,axiom,
    ! [Q: prefix425356687782099470od_b_c > $o,P: b > c > prefix425356687782099470od_b_c,Z2: product_prod_b_c] :
      ( ( Q @ ( produc7288989217166112353od_b_c @ P @ Z2 ) )
     => ~ ! [X3: b,Y3: c] :
            ( ( Z2
              = ( product_Pair_b_c @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_675_case__prod__eta,axiom,
    ! [F: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e] :
      ( ( produc4791951524949757546_b_c_e
        @ ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( F @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_676_case__prod__eta,axiom,
    ! [F: produc4534891796109904283_b_c_e > produc4534891796109904283_b_c_e] :
      ( ( produc7591640664947284859_b_c_e
        @ ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( F @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_677_case__prod__eta,axiom,
    ! [F: produc4534891796109904283_b_c_e > $o] :
      ( ( produc8093372032475639160_c_e_o
        @ ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( F @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_678_case__prod__eta,axiom,
    ! [F: produc3550516665487611281od_b_c > prefix425356687782099470od_b_c] :
      ( ( produc2053673206102377720od_b_c
        @ ^ [X4: a,Y5: product_prod_b_c] : ( F @ ( produc6867761804911831425od_b_c @ X4 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_679_case__prod__eta,axiom,
    ! [F: product_prod_b_c > prefix425356687782099470od_b_c] :
      ( ( produc7288989217166112353od_b_c
        @ ^ [X4: b,Y5: c] : ( F @ ( product_Pair_b_c @ X4 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_680_cond__case__prod__eta,axiom,
    ! [F: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,G3: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e] :
      ( ! [X3: prefix425356687782099470od_b_c,Y3: e] :
          ( ( F @ X3 @ Y3 )
          = ( G3 @ ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) ) )
     => ( ( produc4791951524949757546_b_c_e @ F )
        = G3 ) ) ).

% cond_case_prod_eta
thf(fact_681_cond__case__prod__eta,axiom,
    ! [F: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,G3: produc4534891796109904283_b_c_e > produc4534891796109904283_b_c_e] :
      ( ! [X3: prefix425356687782099470od_b_c,Y3: e] :
          ( ( F @ X3 @ Y3 )
          = ( G3 @ ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) ) )
     => ( ( produc7591640664947284859_b_c_e @ F )
        = G3 ) ) ).

% cond_case_prod_eta
thf(fact_682_cond__case__prod__eta,axiom,
    ! [F: prefix425356687782099470od_b_c > e > $o,G3: produc4534891796109904283_b_c_e > $o] :
      ( ! [X3: prefix425356687782099470od_b_c,Y3: e] :
          ( ( F @ X3 @ Y3 )
          = ( G3 @ ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) ) )
     => ( ( produc8093372032475639160_c_e_o @ F )
        = G3 ) ) ).

% cond_case_prod_eta
thf(fact_683_cond__case__prod__eta,axiom,
    ! [F: a > product_prod_b_c > prefix425356687782099470od_b_c,G3: produc3550516665487611281od_b_c > prefix425356687782099470od_b_c] :
      ( ! [X3: a,Y3: product_prod_b_c] :
          ( ( F @ X3 @ Y3 )
          = ( G3 @ ( produc6867761804911831425od_b_c @ X3 @ Y3 ) ) )
     => ( ( produc2053673206102377720od_b_c @ F )
        = G3 ) ) ).

% cond_case_prod_eta
thf(fact_684_cond__case__prod__eta,axiom,
    ! [F: b > c > prefix425356687782099470od_b_c,G3: product_prod_b_c > prefix425356687782099470od_b_c] :
      ( ! [X3: b,Y3: c] :
          ( ( F @ X3 @ Y3 )
          = ( G3 @ ( product_Pair_b_c @ X3 @ Y3 ) ) )
     => ( ( produc7288989217166112353od_b_c @ F )
        = G3 ) ) ).

% cond_case_prod_eta
thf(fact_685_Collect__conj__eq,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,Q: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X4: produc2669932201312201782od_c_a] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_in3815988127369930372od_c_a @ ( collec8405073525749625505od_c_a @ P ) @ ( collec8405073525749625505od_c_a @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_686_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_687_Collect__conj__eq,axiom,
    ! [P: list_P903359562653991662od_b_c > $o,Q: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X4: list_P903359562653991662od_b_c] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_in4978071631833541052od_b_c @ ( collec2280997390073109977od_b_c @ P ) @ ( collec2280997390073109977od_b_c @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_688_inf__set__def,axiom,
    ( inf_inf_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( collect_a
          @ ( inf_inf_a_o
            @ ^ [X4: a] : ( member_a @ X4 @ A5 )
            @ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_689_inf__set__def,axiom,
    ( inf_in3815988127369930372od_c_a
    = ( ^ [A5: set_Pr8357624050790843030od_c_a,B5: set_Pr8357624050790843030od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ( inf_in5491342246039287129_c_a_o
            @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ A5 )
            @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_690_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_691_inf__set__def,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [A5: set_li6436108459499378894od_b_c,B5: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ( inf_in3324677931821484577_b_c_o
            @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ A5 )
            @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_692_Int__Collect,axiom,
    ! [X: a,A4: set_a,P: a > $o] :
      ( ( member_a @ X @ ( inf_inf_set_a @ A4 @ ( collect_a @ P ) ) )
      = ( ( member_a @ X @ A4 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_693_Int__Collect,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ X @ ( inf_in3815988127369930372od_c_a @ A4 @ ( collec8405073525749625505od_c_a @ P ) ) )
      = ( ( member5502040268208861279od_c_a @ X @ A4 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_694_Int__Collect,axiom,
    ! [X: nat,A4: set_nat,P: nat > $o] :
      ( ( member_nat @ X @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X @ A4 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_695_Int__Collect,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( member6330420149250801815od_b_c @ X @ ( inf_in4978071631833541052od_b_c @ A4 @ ( collec2280997390073109977od_b_c @ P ) ) )
      = ( ( member6330420149250801815od_b_c @ X @ A4 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_696_Int__def,axiom,
    ( inf_inf_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A5 )
              & ( member_a @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_697_Int__def,axiom,
    ( inf_in3815988127369930372od_c_a
    = ( ^ [A5: set_Pr8357624050790843030od_c_a,B5: set_Pr8357624050790843030od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ A5 )
              & ( member5502040268208861279od_c_a @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_698_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A5 )
              & ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_699_Int__def,axiom,
    ( inf_in4978071631833541052od_b_c
    = ( ^ [A5: set_li6436108459499378894od_b_c,B5: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ A5 )
              & ( member6330420149250801815od_b_c @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_700_snd__def,axiom,
    ( produc6779600879576379961_b_c_e
    = ( produc4180796069478548706_c_e_e
      @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : X23 ) ) ).

% snd_def
thf(fact_701_snd__def,axiom,
    ( produc8088130389382527188od_c_a
    = ( produc3172194304981725330od_c_a
      @ ^ [X12: a,X23: produc2133359284458116493od_c_a] : X23 ) ) ).

% snd_def
thf(fact_702_snd__def,axiom,
    ( produc5221446324160585515od_c_a
    = ( produc5632948794012708883od_c_a
      @ ^ [X12: b,X23: product_prod_c_a] : X23 ) ) ).

% snd_def
thf(fact_703_snd__def,axiom,
    ( product_snd_c_a
    = ( produc1561580084227405268_c_a_a
      @ ^ [X12: c,X23: a] : X23 ) ) ).

% snd_def
thf(fact_704_snd__def,axiom,
    ( product_snd_b_c
    = ( produc8837194378803474131_b_c_c
      @ ^ [X12: b,X23: c] : X23 ) ) ).

% snd_def
thf(fact_705_exE__realizer,axiom,
    ! [P: e > prefix425356687782099470od_b_c > $o,P2: produc4534891796109904283_b_c_e,Q: ( produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e ) > $o,F: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e] :
      ( ( P @ ( produc6779600879576379961_b_c_e @ P2 ) @ ( produc1447154562020694007_b_c_e @ P2 ) )
     => ( ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc4791951524949757546_b_c_e @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_706_exE__realizer,axiom,
    ! [P: e > prefix425356687782099470od_b_c > $o,P2: produc4534891796109904283_b_c_e,Q: produc4534891796109904283_b_c_e > $o,F: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e] :
      ( ( P @ ( produc6779600879576379961_b_c_e @ P2 ) @ ( produc1447154562020694007_b_c_e @ P2 ) )
     => ( ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc7591640664947284859_b_c_e @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_707_exE__realizer,axiom,
    ! [P: e > prefix425356687782099470od_b_c > $o,P2: produc4534891796109904283_b_c_e,Q: $o > $o,F: prefix425356687782099470od_b_c > e > $o] :
      ( ( P @ ( produc6779600879576379961_b_c_e @ P2 ) @ ( produc1447154562020694007_b_c_e @ P2 ) )
     => ( ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc8093372032475639160_c_e_o @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_708_exE__realizer,axiom,
    ! [P: product_prod_b_c > a > $o,P2: produc3550516665487611281od_b_c,Q: prefix425356687782099470od_b_c > $o,F: a > product_prod_b_c > prefix425356687782099470od_b_c] :
      ( ( P @ ( produc8648342215071535279od_b_c @ P2 ) @ ( produc3728844965657535341od_b_c @ P2 ) )
     => ( ! [X3: a,Y3: product_prod_b_c] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc2053673206102377720od_b_c @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_709_exE__realizer,axiom,
    ! [P: c > b > $o,P2: product_prod_b_c,Q: prefix425356687782099470od_b_c > $o,F: b > c > prefix425356687782099470od_b_c] :
      ( ( P @ ( product_snd_b_c @ P2 ) @ ( product_fst_b_c @ P2 ) )
     => ( ! [X3: b,Y3: c] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc7288989217166112353od_b_c @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_710_case__prod__beta_H,axiom,
    ( produc4791951524949757546_b_c_e
    = ( ^ [F2: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,X4: produc4534891796109904283_b_c_e] : ( F2 @ ( produc1447154562020694007_b_c_e @ X4 ) @ ( produc6779600879576379961_b_c_e @ X4 ) ) ) ) ).

% case_prod_beta'
thf(fact_711_case__prod__beta_H,axiom,
    ( produc7591640664947284859_b_c_e
    = ( ^ [F2: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,X4: produc4534891796109904283_b_c_e] : ( F2 @ ( produc1447154562020694007_b_c_e @ X4 ) @ ( produc6779600879576379961_b_c_e @ X4 ) ) ) ) ).

% case_prod_beta'
thf(fact_712_case__prod__beta_H,axiom,
    ( produc8093372032475639160_c_e_o
    = ( ^ [F2: prefix425356687782099470od_b_c > e > $o,X4: produc4534891796109904283_b_c_e] : ( F2 @ ( produc1447154562020694007_b_c_e @ X4 ) @ ( produc6779600879576379961_b_c_e @ X4 ) ) ) ) ).

% case_prod_beta'
thf(fact_713_case__prod__beta_H,axiom,
    ( produc2053673206102377720od_b_c
    = ( ^ [F2: a > product_prod_b_c > prefix425356687782099470od_b_c,X4: produc3550516665487611281od_b_c] : ( F2 @ ( produc3728844965657535341od_b_c @ X4 ) @ ( produc8648342215071535279od_b_c @ X4 ) ) ) ) ).

% case_prod_beta'
thf(fact_714_case__prod__beta_H,axiom,
    ( produc7288989217166112353od_b_c
    = ( ^ [F2: b > c > prefix425356687782099470od_b_c,X4: product_prod_b_c] : ( F2 @ ( product_fst_b_c @ X4 ) @ ( product_snd_b_c @ X4 ) ) ) ) ).

% case_prod_beta'
thf(fact_715_case__prod__unfold,axiom,
    ( produc4791951524949757546_b_c_e
    = ( ^ [C5: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,P4: produc4534891796109904283_b_c_e] : ( C5 @ ( produc1447154562020694007_b_c_e @ P4 ) @ ( produc6779600879576379961_b_c_e @ P4 ) ) ) ) ).

% case_prod_unfold
thf(fact_716_case__prod__unfold,axiom,
    ( produc7591640664947284859_b_c_e
    = ( ^ [C5: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,P4: produc4534891796109904283_b_c_e] : ( C5 @ ( produc1447154562020694007_b_c_e @ P4 ) @ ( produc6779600879576379961_b_c_e @ P4 ) ) ) ) ).

% case_prod_unfold
thf(fact_717_case__prod__unfold,axiom,
    ( produc8093372032475639160_c_e_o
    = ( ^ [C5: prefix425356687782099470od_b_c > e > $o,P4: produc4534891796109904283_b_c_e] : ( C5 @ ( produc1447154562020694007_b_c_e @ P4 ) @ ( produc6779600879576379961_b_c_e @ P4 ) ) ) ) ).

% case_prod_unfold
thf(fact_718_case__prod__unfold,axiom,
    ( produc2053673206102377720od_b_c
    = ( ^ [C5: a > product_prod_b_c > prefix425356687782099470od_b_c,P4: produc3550516665487611281od_b_c] : ( C5 @ ( produc3728844965657535341od_b_c @ P4 ) @ ( produc8648342215071535279od_b_c @ P4 ) ) ) ) ).

% case_prod_unfold
thf(fact_719_case__prod__unfold,axiom,
    ( produc7288989217166112353od_b_c
    = ( ^ [C5: b > c > prefix425356687782099470od_b_c,P4: product_prod_b_c] : ( C5 @ ( product_fst_b_c @ P4 ) @ ( product_snd_b_c @ P4 ) ) ) ) ).

% case_prod_unfold
thf(fact_720_fst__def,axiom,
    ( produc1447154562020694007_b_c_e
    = ( produc7263634267956358126od_b_c
      @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : X12 ) ) ).

% fst_def
thf(fact_721_fst__def,axiom,
    ( produc4225943634317630866od_c_a
    = ( produc4731461908963710979_c_a_a
      @ ^ [X12: a,X23: produc2133359284458116493od_c_a] : X12 ) ) ).

% fst_def
thf(fact_722_fst__def,axiom,
    ( produc301949074746585577od_c_a
    = ( produc2354600891577745261_c_a_b
      @ ^ [X12: b,X23: product_prod_c_a] : X12 ) ) ).

% fst_def
thf(fact_723_fst__def,axiom,
    ( product_fst_c_a
    = ( produc1561580084227405270_c_a_c
      @ ^ [X12: c,X23: a] : X12 ) ) ).

% fst_def
thf(fact_724_fst__def,axiom,
    ( product_fst_b_c
    = ( produc8837194378803474130_b_c_b
      @ ^ [X12: b,X23: c] : X12 ) ) ).

% fst_def
thf(fact_725_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: prefix425356687782099470od_b_c > e > $o,Prod: produc4534891796109904283_b_c_e] :
      ( ( H @ ( produc8093372032475639160_c_e_o @ F @ Prod ) )
      = ( produc8093372032475639160_c_e_o
        @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_726_prod_Ocase__distrib,axiom,
    ! [H: prefix425356687782099470od_b_c > prefix425356687782099470od_b_c,F: b > c > prefix425356687782099470od_b_c,Prod: product_prod_b_c] :
      ( ( H @ ( produc7288989217166112353od_b_c @ F @ Prod ) )
      = ( produc7288989217166112353od_b_c
        @ ^ [X12: b,X23: c] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_727_prod_Ocase__distrib,axiom,
    ! [H: produc4534891796109904283_b_c_e > $o,F: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,Prod: produc4534891796109904283_b_c_e] :
      ( ( H @ ( produc7591640664947284859_b_c_e @ F @ Prod ) )
      = ( produc8093372032475639160_c_e_o
        @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_728_prod_Ocase__distrib,axiom,
    ! [H: $o > produc4534891796109904283_b_c_e,F: prefix425356687782099470od_b_c > e > $o,Prod: produc4534891796109904283_b_c_e] :
      ( ( H @ ( produc8093372032475639160_c_e_o @ F @ Prod ) )
      = ( produc7591640664947284859_b_c_e
        @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_729_prod_Ocase__distrib,axiom,
    ! [H: prefix425356687782099470od_b_c > prefix425356687782099470od_b_c,F: a > product_prod_b_c > prefix425356687782099470od_b_c,Prod: produc3550516665487611281od_b_c] :
      ( ( H @ ( produc2053673206102377720od_b_c @ F @ Prod ) )
      = ( produc2053673206102377720od_b_c
        @ ^ [X12: a,X23: product_prod_b_c] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_730_prod_Ocase__distrib,axiom,
    ! [H: produc4534891796109904283_b_c_e > produc4534891796109904283_b_c_e,F: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,Prod: produc4534891796109904283_b_c_e] :
      ( ( H @ ( produc7591640664947284859_b_c_e @ F @ Prod ) )
      = ( produc7591640664947284859_b_c_e
        @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_731_prod_Ocase__distrib,axiom,
    ! [H: ( produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e ) > $o,F: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,Prod: produc4534891796109904283_b_c_e] :
      ( ( H @ ( produc4791951524949757546_b_c_e @ F @ Prod ) )
      = ( produc8093372032475639160_c_e_o
        @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_732_prod_Ocase__distrib,axiom,
    ! [H: $o > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,F: prefix425356687782099470od_b_c > e > $o,Prod: produc4534891796109904283_b_c_e] :
      ( ( H @ ( produc8093372032475639160_c_e_o @ F @ Prod ) )
      = ( produc4791951524949757546_b_c_e
        @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_733_prod_Ocase__distrib,axiom,
    ! [H: ( produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e ) > produc4534891796109904283_b_c_e,F: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,Prod: produc4534891796109904283_b_c_e] :
      ( ( H @ ( produc4791951524949757546_b_c_e @ F @ Prod ) )
      = ( produc7591640664947284859_b_c_e
        @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_734_prod_Ocase__distrib,axiom,
    ! [H: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,F: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,Prod: produc4534891796109904283_b_c_e] :
      ( ( H @ ( produc7591640664947284859_b_c_e @ F @ Prod ) )
      = ( produc4791951524949757546_b_c_e
        @ ^ [X12: prefix425356687782099470od_b_c,X23: e] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_735_Compr__image__eq,axiom,
    ! [F: a > a,A4: set_a,P: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_a_a @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_a_a @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_736_Compr__image__eq,axiom,
    ! [F: nat > a,A4: set_nat,P: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_nat_a @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_nat_a @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_737_Compr__image__eq,axiom,
    ! [F: a > nat,A4: set_a,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ ( image_a_nat @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_a_nat @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_738_Compr__image__eq,axiom,
    ! [F: nat > nat,A4: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_nat_nat @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_739_Compr__image__eq,axiom,
    ! [F: list_P903359562653991662od_b_c > a,A4: set_li6436108459499378894od_b_c,P: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_3173854338463926933_b_c_a @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_3173854338463926933_b_c_a @ F
        @ ( collec2280997390073109977od_b_c
          @ ^ [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_740_Compr__image__eq,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A4: set_a,P: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X4: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X4 @ ( image_8656872703560917237od_b_c @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_8656872703560917237od_b_c @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_741_Compr__image__eq,axiom,
    ! [F: nat > list_P903359562653991662od_b_c,A4: set_nat,P: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X4: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X4 @ ( image_2316969277338295127od_b_c @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_2316969277338295127od_b_c @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_742_Compr__image__eq,axiom,
    ! [F: list_P903359562653991662od_b_c > nat,A4: set_li6436108459499378894od_b_c,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ ( image_4545125072327328889_c_nat @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_4545125072327328889_c_nat @ F
        @ ( collec2280997390073109977od_b_c
          @ ^ [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_743_Compr__image__eq,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [X4: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X4 @ ( image_2850395784482391045od_b_c @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_2850395784482391045od_b_c @ F
        @ ( collec2280997390073109977od_b_c
          @ ^ [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_744_Compr__image__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A4: set_Pr8357624050790843030od_c_a,P: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_9171994905208062541_c_a_a @ F @ A4 ) )
            & ( P @ X4 ) ) )
      = ( image_9171994905208062541_c_a_a @ F
        @ ( collec8405073525749625505od_c_a
          @ ^ [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ A4 )
              & ( P @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_745_image__image,axiom,
    ! [F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,G3: a > list_P903359562653991662od_b_c,A4: set_a] :
      ( ( image_2850395784482391045od_b_c @ F @ ( image_8656872703560917237od_b_c @ G3 @ A4 ) )
      = ( image_8656872703560917237od_b_c
        @ ^ [X4: a] : ( F @ ( G3 @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_746_image__image,axiom,
    ! [F: list_P903359562653991662od_b_c > set_Pr8357624050790843030od_c_a,G3: a > list_P903359562653991662od_b_c,A4: set_a] :
      ( ( image_1227955527963590957od_c_a @ F @ ( image_8656872703560917237od_b_c @ G3 @ A4 ) )
      = ( image_2946797484331234845od_c_a
        @ ^ [X4: a] : ( F @ ( G3 @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_747_image__image,axiom,
    ! [F: set_Pr8357624050790843030od_c_a > list_P903359562653991662od_b_c,G3: a > set_Pr8357624050790843030od_c_a,A4: set_a] :
      ( ( image_3695866549450572125od_b_c @ F @ ( image_2946797484331234845od_c_a @ G3 @ A4 ) )
      = ( image_8656872703560917237od_b_c
        @ ^ [X4: a] : ( F @ ( G3 @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_748_image__image,axiom,
    ! [F: set_Pr8357624050790843030od_c_a > set_Pr8357624050790843030od_c_a,G3: a > set_Pr8357624050790843030od_c_a,A4: set_a] :
      ( ( image_3355043664154344069od_c_a @ F @ ( image_2946797484331234845od_c_a @ G3 @ A4 ) )
      = ( image_2946797484331234845od_c_a
        @ ^ [X4: a] : ( F @ ( G3 @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_749_image__image,axiom,
    ! [F: a > list_P903359562653991662od_b_c,G3: a > a,A4: set_a] :
      ( ( image_8656872703560917237od_b_c @ F @ ( image_a_a @ G3 @ A4 ) )
      = ( image_8656872703560917237od_b_c
        @ ^ [X4: a] : ( F @ ( G3 @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_750_image__image,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,G3: a > a,A4: set_a] :
      ( ( image_2946797484331234845od_c_a @ F @ ( image_a_a @ G3 @ A4 ) )
      = ( image_2946797484331234845od_c_a
        @ ^ [X4: a] : ( F @ ( G3 @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_751_imageE,axiom,
    ! [B: set_Pr8357624050790843030od_c_a,F: a > set_Pr8357624050790843030od_c_a,A4: set_a] :
      ( ( member9073573073229046463od_c_a @ B @ ( image_2946797484331234845od_c_a @ F @ A4 ) )
     => ~ ! [X3: a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_752_imageE,axiom,
    ! [B: produc2669932201312201782od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_2169818278710580869od_c_a @ F @ A4 ) )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member5502040268208861279od_c_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_753_imageE,axiom,
    ! [B: produc2669932201312201782od_c_a,F: a > produc2669932201312201782od_c_a,A4: set_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_4184489845059386301od_c_a @ F @ A4 ) )
     => ~ ! [X3: a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_754_imageE,axiom,
    ! [B: produc2669932201312201782od_c_a,F: list_P903359562653991662od_b_c > produc2669932201312201782od_c_a,A4: set_li6436108459499378894od_b_c] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_7601443023479506637od_c_a @ F @ A4 ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member6330420149250801815od_b_c @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_755_imageE,axiom,
    ! [B: a,F: produc2669932201312201782od_c_a > a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ B @ ( image_9171994905208062541_c_a_a @ F @ A4 ) )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member5502040268208861279od_c_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_756_imageE,axiom,
    ! [B: a,F: a > a,A4: set_a] :
      ( ( member_a @ B @ ( image_a_a @ F @ A4 ) )
     => ~ ! [X3: a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_757_imageE,axiom,
    ! [B: a,F: list_P903359562653991662od_b_c > a,A4: set_li6436108459499378894od_b_c] :
      ( ( member_a @ B @ ( image_3173854338463926933_b_c_a @ F @ A4 ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member6330420149250801815od_b_c @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_758_imageE,axiom,
    ! [B: list_P903359562653991662od_b_c,F: produc2669932201312201782od_c_a > list_P903359562653991662od_b_c,A4: set_Pr8357624050790843030od_c_a] :
      ( ( member6330420149250801815od_b_c @ B @ ( image_826490394676381629od_b_c @ F @ A4 ) )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member5502040268208861279od_c_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_759_imageE,axiom,
    ! [B: list_P903359562653991662od_b_c,F: a > list_P903359562653991662od_b_c,A4: set_a] :
      ( ( member6330420149250801815od_b_c @ B @ ( image_8656872703560917237od_b_c @ F @ A4 ) )
     => ~ ! [X3: a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_760_imageE,axiom,
    ! [B: list_P903359562653991662od_b_c,F: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ B @ ( image_2850395784482391045od_b_c @ F @ A4 ) )
     => ~ ! [X3: list_P903359562653991662od_b_c] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member6330420149250801815od_b_c @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_761_Collect__subset,axiom,
    ! [A4: set_a,P: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_762_Collect__subset,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ord_le3226338165915859510od_c_a
      @ ( collec8405073525749625505od_c_a
        @ ^ [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_763_Collect__subset,axiom,
    ! [A4: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_764_Collect__subset,axiom,
    ! [A4: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ord_le282488521294790766od_b_c
      @ ( collec2280997390073109977od_b_c
        @ ^ [X4: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_765_less__eq__set__def,axiom,
    ( ord_le3226338165915859510od_c_a
    = ( ^ [A5: set_Pr8357624050790843030od_c_a,B5: set_Pr8357624050790843030od_c_a] :
          ( ord_le753561201022589991_c_a_o
          @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ A5 )
          @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_766_less__eq__set__def,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( ord_less_eq_a_o
          @ ^ [X4: a] : ( member_a @ X4 @ A5 )
          @ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_767_less__eq__set__def,axiom,
    ( ord_le282488521294790766od_b_c
    = ( ^ [A5: set_li6436108459499378894od_b_c,B5: set_li6436108459499378894od_b_c] :
          ( ord_le1261880702084804079_b_c_o
          @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ A5 )
          @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_768_subset__Collect__iff,axiom,
    ! [B4: set_a,A4: set_a,P: a > $o] :
      ( ( ord_less_eq_set_a @ B4 @ A4 )
     => ( ( ord_less_eq_set_a @ B4
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: a] :
              ( ( member_a @ X4 @ B4 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_769_subset__Collect__iff,axiom,
    ! [B4: set_Pr8357624050790843030od_c_a,A4: set_Pr8357624050790843030od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( ord_le3226338165915859510od_c_a @ B4 @ A4 )
     => ( ( ord_le3226338165915859510od_c_a @ B4
          @ ( collec8405073525749625505od_c_a
            @ ^ [X4: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ B4 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_770_subset__Collect__iff,axiom,
    ! [B4: set_nat,A4: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( ord_less_eq_set_nat @ B4
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_771_subset__Collect__iff,axiom,
    ! [B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c,P: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ B4 @ A4 )
     => ( ( ord_le282488521294790766od_b_c @ B4
          @ ( collec2280997390073109977od_b_c
            @ ^ [X4: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ B4 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_772_subset__CollectI,axiom,
    ! [B4: set_a,A4: set_a,Q: a > $o,P: a > $o] :
      ( ( ord_less_eq_set_a @ B4 @ A4 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_a
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ B4 )
                & ( Q @ X4 ) ) )
          @ ( collect_a
            @ ^ [X4: a] :
                ( ( member_a @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_773_subset__CollectI,axiom,
    ! [B4: set_Pr8357624050790843030od_c_a,A4: set_Pr8357624050790843030od_c_a,Q: produc2669932201312201782od_c_a > $o,P: produc2669932201312201782od_c_a > $o] :
      ( ( ord_le3226338165915859510od_c_a @ B4 @ A4 )
     => ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3226338165915859510od_c_a
          @ ( collec8405073525749625505od_c_a
            @ ^ [X4: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X4 @ B4 )
                & ( Q @ X4 ) ) )
          @ ( collec8405073525749625505od_c_a
            @ ^ [X4: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_774_subset__CollectI,axiom,
    ! [B4: set_nat,A4: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ B4 )
                & ( Q @ X4 ) ) )
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_775_subset__CollectI,axiom,
    ! [B4: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c,Q: list_P903359562653991662od_b_c > $o,P: list_P903359562653991662od_b_c > $o] :
      ( ( ord_le282488521294790766od_b_c @ B4 @ A4 )
     => ( ! [X3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le282488521294790766od_b_c
          @ ( collec2280997390073109977od_b_c
            @ ^ [X4: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X4 @ B4 )
                & ( Q @ X4 ) ) )
          @ ( collec2280997390073109977od_b_c
            @ ^ [X4: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_776_old_Oprod_Ocase,axiom,
    ! [F: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,X1: prefix425356687782099470od_b_c,X2: e] :
      ( ( produc4791951524949757546_b_c_e @ F @ ( produc2511600812177743883_b_c_e @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_777_old_Oprod_Ocase,axiom,
    ! [F: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,X1: prefix425356687782099470od_b_c,X2: e] :
      ( ( produc7591640664947284859_b_c_e @ F @ ( produc2511600812177743883_b_c_e @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_778_old_Oprod_Ocase,axiom,
    ! [F: prefix425356687782099470od_b_c > e > $o,X1: prefix425356687782099470od_b_c,X2: e] :
      ( ( produc8093372032475639160_c_e_o @ F @ ( produc2511600812177743883_b_c_e @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_779_old_Oprod_Ocase,axiom,
    ! [F: a > product_prod_b_c > prefix425356687782099470od_b_c,X1: a,X2: product_prod_b_c] :
      ( ( produc2053673206102377720od_b_c @ F @ ( produc6867761804911831425od_b_c @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_780_old_Oprod_Ocase,axiom,
    ! [F: b > c > prefix425356687782099470od_b_c,X1: b,X2: c] :
      ( ( produc7288989217166112353od_b_c @ F @ ( product_Pair_b_c @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_781_foldl__Cons,axiom,
    ! [F: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,A: produc4534891796109904283_b_c_e,X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a] :
      ( ( foldl_2588721813272579068od_c_a @ F @ A @ ( cons_P8659537020926305772od_c_a @ X @ Xs2 ) )
      = ( foldl_2588721813272579068od_c_a @ F @ ( F @ A @ X ) @ Xs2 ) ) ).

% foldl_Cons
thf(fact_782_foldl__Cons,axiom,
    ! [F: prefix425356687782099470od_b_c > produc3550516665487611281od_b_c > prefix425356687782099470od_b_c,A: prefix425356687782099470od_b_c,X: produc3550516665487611281od_b_c,Xs2: list_P8634110456476873367od_b_c] :
      ( ( foldl_6461310528730867108od_b_c @ F @ A @ ( cons_P8793802274825991495od_b_c @ X @ Xs2 ) )
      = ( foldl_6461310528730867108od_b_c @ F @ ( F @ A @ X ) @ Xs2 ) ) ).

% foldl_Cons
thf(fact_783_foldl__Nil,axiom,
    ! [F: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,A: produc4534891796109904283_b_c_e] :
      ( ( foldl_2588721813272579068od_c_a @ F @ A @ nil_Pr1342775757158464060od_c_a )
      = A ) ).

% foldl_Nil
thf(fact_784_foldl__Nil,axiom,
    ! [F: prefix425356687782099470od_b_c > produc3550516665487611281od_b_c > prefix425356687782099470od_b_c,A: prefix425356687782099470od_b_c] :
      ( ( foldl_6461310528730867108od_b_c @ F @ A @ nil_Pr6797566517770115479od_b_c )
      = A ) ).

% foldl_Nil
thf(fact_785_foldl__cong,axiom,
    ! [A: prefix425356687782099470od_b_c,B: prefix425356687782099470od_b_c,L: list_P8634110456476873367od_b_c,K3: list_P8634110456476873367od_b_c,F: prefix425356687782099470od_b_c > produc3550516665487611281od_b_c > prefix425356687782099470od_b_c,G3: prefix425356687782099470od_b_c > produc3550516665487611281od_b_c > prefix425356687782099470od_b_c] :
      ( ( A = B )
     => ( ( L = K3 )
       => ( ! [A3: prefix425356687782099470od_b_c,X3: produc3550516665487611281od_b_c] :
              ( ( member2577541133337307834od_b_c @ X3 @ ( set_Pr6515338384934575398od_b_c @ L ) )
             => ( ( F @ A3 @ X3 )
                = ( G3 @ A3 @ X3 ) ) )
         => ( ( foldl_6461310528730867108od_b_c @ F @ A @ L )
            = ( foldl_6461310528730867108od_b_c @ G3 @ B @ K3 ) ) ) ) ) ).

% foldl_cong
thf(fact_786_foldl__cong,axiom,
    ! [A: produc4534891796109904283_b_c_e,B: produc4534891796109904283_b_c_e,L: list_P6327159017948738492od_c_a,K3: list_P6327159017948738492od_c_a,F: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,G3: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e] :
      ( ( A = B )
     => ( ( L = K3 )
       => ( ! [A3: produc4534891796109904283_b_c_e,X3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X3 @ ( set_Pr5664641634098845387od_c_a @ L ) )
             => ( ( F @ A3 @ X3 )
                = ( G3 @ A3 @ X3 ) ) )
         => ( ( foldl_2588721813272579068od_c_a @ F @ A @ L )
            = ( foldl_2588721813272579068od_c_a @ G3 @ B @ K3 ) ) ) ) ) ).

% foldl_cong
thf(fact_787_split__beta,axiom,
    ( produc4791951524949757546_b_c_e
    = ( ^ [F2: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,Prod3: produc4534891796109904283_b_c_e] : ( F2 @ ( produc1447154562020694007_b_c_e @ Prod3 ) @ ( produc6779600879576379961_b_c_e @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_788_split__beta,axiom,
    ( produc7591640664947284859_b_c_e
    = ( ^ [F2: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,Prod3: produc4534891796109904283_b_c_e] : ( F2 @ ( produc1447154562020694007_b_c_e @ Prod3 ) @ ( produc6779600879576379961_b_c_e @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_789_split__beta,axiom,
    ( produc8093372032475639160_c_e_o
    = ( ^ [F2: prefix425356687782099470od_b_c > e > $o,Prod3: produc4534891796109904283_b_c_e] : ( F2 @ ( produc1447154562020694007_b_c_e @ Prod3 ) @ ( produc6779600879576379961_b_c_e @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_790_split__beta,axiom,
    ( produc2053673206102377720od_b_c
    = ( ^ [F2: a > product_prod_b_c > prefix425356687782099470od_b_c,Prod3: produc3550516665487611281od_b_c] : ( F2 @ ( produc3728844965657535341od_b_c @ Prod3 ) @ ( produc8648342215071535279od_b_c @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_791_split__beta,axiom,
    ( produc7288989217166112353od_b_c
    = ( ^ [F2: b > c > prefix425356687782099470od_b_c,Prod3: product_prod_b_c] : ( F2 @ ( product_fst_b_c @ Prod3 ) @ ( product_snd_b_c @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_792_case__prod__beta,axiom,
    ( produc4791951524949757546_b_c_e
    = ( ^ [F2: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,P4: produc4534891796109904283_b_c_e] : ( F2 @ ( produc1447154562020694007_b_c_e @ P4 ) @ ( produc6779600879576379961_b_c_e @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_793_case__prod__beta,axiom,
    ( produc7591640664947284859_b_c_e
    = ( ^ [F2: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,P4: produc4534891796109904283_b_c_e] : ( F2 @ ( produc1447154562020694007_b_c_e @ P4 ) @ ( produc6779600879576379961_b_c_e @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_794_case__prod__beta,axiom,
    ( produc8093372032475639160_c_e_o
    = ( ^ [F2: prefix425356687782099470od_b_c > e > $o,P4: produc4534891796109904283_b_c_e] : ( F2 @ ( produc1447154562020694007_b_c_e @ P4 ) @ ( produc6779600879576379961_b_c_e @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_795_case__prod__beta,axiom,
    ( produc2053673206102377720od_b_c
    = ( ^ [F2: a > product_prod_b_c > prefix425356687782099470od_b_c,P4: produc3550516665487611281od_b_c] : ( F2 @ ( produc3728844965657535341od_b_c @ P4 ) @ ( produc8648342215071535279od_b_c @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_796_case__prod__beta,axiom,
    ( produc7288989217166112353od_b_c
    = ( ^ [F2: b > c > prefix425356687782099470od_b_c,P4: product_prod_b_c] : ( F2 @ ( product_fst_b_c @ P4 ) @ ( product_snd_b_c @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_797_language__io_I2_J,axiom,
    ! [Io: list_P903359562653991662od_b_c,M2: fsm_d_b_c,Q2: d,X: b,Y: c] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_d_b_c @ M2 @ Q2 ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ ( set_Product_prod_b_c2 @ Io ) )
       => ( member_c @ Y @ ( outputs_d_b_c @ M2 ) ) ) ) ).

% language_io(2)
thf(fact_798_language__io_I2_J,axiom,
    ! [Io: list_P903359562653991662od_b_c,M2: fsm_a_b_c,Q2: a,X: b,Y: c] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ Q2 ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ ( set_Product_prod_b_c2 @ Io ) )
       => ( member_c @ Y @ ( outputs_a_b_c @ M2 ) ) ) ) ).

% language_io(2)
thf(fact_799_language__io_I1_J,axiom,
    ! [Io: list_P903359562653991662od_b_c,M2: fsm_d_b_c,Q2: d,X: b,Y: c] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_d_b_c @ M2 @ Q2 ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ ( set_Product_prod_b_c2 @ Io ) )
       => ( member_b @ X @ ( inputs_d_b_c @ M2 ) ) ) ) ).

% language_io(1)
thf(fact_800_language__io_I1_J,axiom,
    ! [Io: list_P903359562653991662od_b_c,M2: fsm_a_b_c,Q2: a,X: b,Y: c] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ Q2 ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ ( set_Product_prod_b_c2 @ Io ) )
       => ( member_b @ X @ ( inputs_a_b_c @ M2 ) ) ) ) ).

% language_io(1)
thf(fact_801_prod_Osplit__sel,axiom,
    ! [P: ( produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e ) > $o,F: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,Prod: produc4534891796109904283_b_c_e] :
      ( ( P @ ( produc4791951524949757546_b_c_e @ F @ Prod ) )
      = ( ( Prod
          = ( produc2511600812177743883_b_c_e @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) )
       => ( P @ ( F @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_802_prod_Osplit__sel,axiom,
    ! [P: produc4534891796109904283_b_c_e > $o,F: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,Prod: produc4534891796109904283_b_c_e] :
      ( ( P @ ( produc7591640664947284859_b_c_e @ F @ Prod ) )
      = ( ( Prod
          = ( produc2511600812177743883_b_c_e @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) )
       => ( P @ ( F @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_803_prod_Osplit__sel,axiom,
    ! [P: $o > $o,F: prefix425356687782099470od_b_c > e > $o,Prod: produc4534891796109904283_b_c_e] :
      ( ( P @ ( produc8093372032475639160_c_e_o @ F @ Prod ) )
      = ( ( Prod
          = ( produc2511600812177743883_b_c_e @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) )
       => ( P @ ( F @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_804_prod_Osplit__sel,axiom,
    ! [P: prefix425356687782099470od_b_c > $o,F: a > product_prod_b_c > prefix425356687782099470od_b_c,Prod: produc3550516665487611281od_b_c] :
      ( ( P @ ( produc2053673206102377720od_b_c @ F @ Prod ) )
      = ( ( Prod
          = ( produc6867761804911831425od_b_c @ ( produc3728844965657535341od_b_c @ Prod ) @ ( produc8648342215071535279od_b_c @ Prod ) ) )
       => ( P @ ( F @ ( produc3728844965657535341od_b_c @ Prod ) @ ( produc8648342215071535279od_b_c @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_805_prod_Osplit__sel,axiom,
    ! [P: prefix425356687782099470od_b_c > $o,F: b > c > prefix425356687782099470od_b_c,Prod: product_prod_b_c] :
      ( ( P @ ( produc7288989217166112353od_b_c @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_806_prod_Osplit__sel__asm,axiom,
    ! [P: ( produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e ) > $o,F: prefix425356687782099470od_b_c > e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,Prod: produc4534891796109904283_b_c_e] :
      ( ( P @ ( produc4791951524949757546_b_c_e @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc2511600812177743883_b_c_e @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_807_prod_Osplit__sel__asm,axiom,
    ! [P: produc4534891796109904283_b_c_e > $o,F: prefix425356687782099470od_b_c > e > produc4534891796109904283_b_c_e,Prod: produc4534891796109904283_b_c_e] :
      ( ( P @ ( produc7591640664947284859_b_c_e @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc2511600812177743883_b_c_e @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_808_prod_Osplit__sel__asm,axiom,
    ! [P: $o > $o,F: prefix425356687782099470od_b_c > e > $o,Prod: produc4534891796109904283_b_c_e] :
      ( ( P @ ( produc8093372032475639160_c_e_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc2511600812177743883_b_c_e @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc1447154562020694007_b_c_e @ Prod ) @ ( produc6779600879576379961_b_c_e @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_809_prod_Osplit__sel__asm,axiom,
    ! [P: prefix425356687782099470od_b_c > $o,F: a > product_prod_b_c > prefix425356687782099470od_b_c,Prod: produc3550516665487611281od_b_c] :
      ( ( P @ ( produc2053673206102377720od_b_c @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc6867761804911831425od_b_c @ ( produc3728844965657535341od_b_c @ Prod ) @ ( produc8648342215071535279od_b_c @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc3728844965657535341od_b_c @ Prod ) @ ( produc8648342215071535279od_b_c @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_810_prod_Osplit__sel__asm,axiom,
    ! [P: prefix425356687782099470od_b_c > $o,F: b > c > prefix425356687782099470od_b_c,Prod: product_prod_b_c] :
      ( ( P @ ( produc7288989217166112353od_b_c @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_811_map__entry_Ocases,axiom,
    ! [X: produc6452300523622238572od_b_c] :
      ( ! [K: b,F3: c > c] :
          ( X
         != ( produc7015427828931889894od_b_c @ K @ ( produc1101678828523717360od_b_c @ F3 @ nil_Product_prod_b_c ) ) )
     => ~ ! [K: b,F3: c > c,P5: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
            ( X
           != ( produc7015427828931889894od_b_c @ K @ ( produc1101678828523717360od_b_c @ F3 @ ( cons_P4529483553340347422od_b_c @ P5 @ Ps ) ) ) ) ) ).

% map_entry.cases
thf(fact_812_map__default_Ocases,axiom,
    ! [X: produc2009451851958091931od_b_c] :
      ( ! [K: b,V3: c,F3: c > c] :
          ( X
         != ( produc1384939585547926677od_b_c @ K @ ( produc3590379597927054951od_b_c @ V3 @ ( produc1101678828523717360od_b_c @ F3 @ nil_Product_prod_b_c ) ) ) )
     => ~ ! [K: b,V3: c,F3: c > c,P5: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
            ( X
           != ( produc1384939585547926677od_b_c @ K @ ( produc3590379597927054951od_b_c @ V3 @ ( produc1101678828523717360od_b_c @ F3 @ ( cons_P4529483553340347422od_b_c @ P5 @ Ps ) ) ) ) ) ) ).

% map_default.cases
thf(fact_813_clearjunk_Ocases,axiom,
    ! [X: list_P903359562653991662od_b_c] :
      ( ( X != nil_Product_prod_b_c )
     => ~ ! [P5: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
            ( X
           != ( cons_P4529483553340347422od_b_c @ P5 @ Ps ) ) ) ).

% clearjunk.cases
thf(fact_814_convergence__graph__initial__invar__def,axiom,
    ( conver3708542147478018715_c_d_e
    = ( ^ [M1: fsm_a_b_c,M22: fsm_d_b_c,Cg_lookup: e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c,Cg_initial: fsm_a_b_c > prefix425356687782099470od_b_c > e] :
        ! [T6: prefix425356687782099470od_b_c] :
          ( ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) ) @ ( prefix9085958534787526401od_b_c @ T6 ) )
            = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M22 @ ( initial_d_b_c @ M22 ) ) @ ( prefix9085958534787526401od_b_c @ T6 ) ) )
         => ( ( prefix6767660319289835678od_b_c @ T6 )
           => ( conver7089555221631485245_c_d_e @ M1 @ M22 @ Cg_lookup @ ( Cg_initial @ M1 @ T6 ) ) ) ) ) ) ).

% convergence_graph_initial_invar_def
thf(fact_815_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_c_a_b,F: c > product_prod_c_a] :
      ( ( transi2941759273464634107_a_a_b @ ( rename5198635604026011160od_c_a @ M2 @ F ) )
      = ( image_8330237349178395077od_c_a
        @ ^ [T2: produc6657052613134160568od_b_c] : ( produc7911353075612250984od_c_a @ ( F @ ( produc6763838237438281236od_b_c @ T2 ) ) @ ( produc5762801557034676134od_c_a @ ( produc3728844965657535341od_b_c @ ( produc1402652955648401750od_b_c @ T2 ) ) @ ( produc3440865914000881661od_c_a @ ( product_fst_b_c @ ( produc8648342215071535279od_b_c @ ( produc1402652955648401750od_b_c @ T2 ) ) ) @ ( F @ ( product_snd_b_c @ ( produc8648342215071535279od_b_c @ ( produc1402652955648401750od_b_c @ T2 ) ) ) ) ) ) )
        @ ( transitions_c_a_b @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_816_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,F: product_prod_c_a > c] :
      ( ( transitions_c_a_b @ ( rename6043294581747517582_a_b_c @ M2 @ F ) )
      = ( image_3434781565917999685od_b_c
        @ ^ [T2: produc3205353004677322104od_c_a] : ( produc8300696160155326504od_b_c @ ( F @ ( produc8171714036265584980od_c_a @ T2 ) ) @ ( produc6867761804911831425od_b_c @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T2 ) ) @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T2 ) ) ) @ ( F @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T2 ) ) ) ) ) ) )
        @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_817_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,F: product_prod_c_a > product_prod_c_a] :
      ( ( transi2941759273464634107_a_a_b @ ( rename3079062586811799859od_c_a @ M2 @ F ) )
      = ( image_5098445917802613509od_c_a
        @ ^ [T2: produc3205353004677322104od_c_a] : ( produc7911353075612250984od_c_a @ ( F @ ( produc8171714036265584980od_c_a @ T2 ) ) @ ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T2 ) ) @ ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T2 ) ) ) @ ( F @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T2 ) ) ) ) ) ) )
        @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_818_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_a_b_c,F: a > a] :
      ( ( transitions_a_b_c @ ( rename4556187690751568563_b_c_a @ M2 @ F ) )
      = ( image_2169818278710580869od_c_a
        @ ^ [T2: produc2669932201312201782od_c_a] : ( produc5762801557034676134od_c_a @ ( F @ ( produc4225943634317630866od_c_a @ T2 ) ) @ ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_Pair_c_a @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( F @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) ) )
        @ ( transitions_a_b_c @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_819_abstract__h__condition__exhaustiveness,axiom,
    ! [M2: fsm_d_b_c,I: fsm_a_b_c,M3: nat,V: d > list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( minimal_d_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_a_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_a_b_c @ I )
                  = ( inputs_d_b_c @ M2 ) )
               => ( ( ( outputs_a_b_c @ I )
                    = ( outputs_d_b_c @ M2 ) )
                 => ( ( state_6014242345326633810_d_b_c @ M2 @ V )
                   => ( ( h_sati4005454913823510672_b_c_a @ M2 @ I @ V @ M3 )
                     => ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
                        = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_exhaustiveness
thf(fact_820_abstract__h__condition__exhaustiveness,axiom,
    ! [M2: fsm_d_b_c,I: fsm_d_b_c,M3: nat,V: d > list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( minimal_d_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_d_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_d_b_c @ I )
                  = ( inputs_d_b_c @ M2 ) )
               => ( ( ( outputs_d_b_c @ I )
                    = ( outputs_d_b_c @ M2 ) )
                 => ( ( state_6014242345326633810_d_b_c @ M2 @ V )
                   => ( ( h_sati4005454913823510675_b_c_d @ M2 @ I @ V @ M3 )
                     => ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
                        = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_exhaustiveness
thf(fact_821_abstract__h__condition__exhaustiveness,axiom,
    ! [M2: fsm_nat_b_c,I: fsm_a_b_c,M3: nat,V: nat > list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( minimal_nat_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_a_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_a_b_c @ I )
                  = ( inputs_nat_b_c @ M2 ) )
               => ( ( ( outputs_a_b_c @ I )
                    = ( outputs_nat_b_c @ M2 ) )
                 => ( ( state_5280168802736327169at_b_c @ M2 @ V )
                   => ( ( h_sati1514657294798470871_b_c_a @ M2 @ I @ V @ M3 )
                     => ( ( lS_nat_b_c @ M2 @ ( initial_nat_b_c @ M2 ) )
                        = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_exhaustiveness
thf(fact_822_abstract__h__condition__exhaustiveness,axiom,
    ! [M2: fsm_nat_b_c,I: fsm_d_b_c,M3: nat,V: nat > list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( minimal_nat_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_d_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_d_b_c @ I )
                  = ( inputs_nat_b_c @ M2 ) )
               => ( ( ( outputs_d_b_c @ I )
                    = ( outputs_nat_b_c @ M2 ) )
                 => ( ( state_5280168802736327169at_b_c @ M2 @ V )
                   => ( ( h_sati1514657294798470874_b_c_d @ M2 @ I @ V @ M3 )
                     => ( ( lS_nat_b_c @ M2 @ ( initial_nat_b_c @ M2 ) )
                        = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_exhaustiveness
thf(fact_823_abstract__h__condition__exhaustiveness,axiom,
    ! [M2: fsm_a_b_c,I: fsm_a_b_c,M3: nat,V: a > list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( minimal_a_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_a_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_a_b_c @ I )
                  = ( inputs_a_b_c @ M2 ) )
               => ( ( ( outputs_a_b_c @ I )
                    = ( outputs_a_b_c @ M2 ) )
                 => ( ( state_7672330573471793359_a_b_c @ M2 @ V )
                   => ( ( h_sati8764940838793890899_b_c_a @ M2 @ I @ V @ M3 )
                     => ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
                        = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_exhaustiveness
thf(fact_824_abstract__h__condition__exhaustiveness,axiom,
    ! [M2: fsm_a_b_c,I: fsm_d_b_c,M3: nat,V: a > list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( minimal_a_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_d_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_d_b_c @ I )
                  = ( inputs_a_b_c @ M2 ) )
               => ( ( ( outputs_d_b_c @ I )
                    = ( outputs_a_b_c @ M2 ) )
                 => ( ( state_7672330573471793359_a_b_c @ M2 @ V )
                   => ( ( h_sati8764940838793890902_b_c_d @ M2 @ I @ V @ M3 )
                     => ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
                        = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_exhaustiveness
thf(fact_825_abstract__h__condition__completeness,axiom,
    ! [M2: fsm_d_b_c,I: fsm_a_b_c,M3: nat,V: d > list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( minimal_d_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_a_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_a_b_c @ I )
                  = ( inputs_d_b_c @ M2 ) )
               => ( ( ( outputs_a_b_c @ I )
                    = ( outputs_d_b_c @ M2 ) )
                 => ( ( state_6014242345326633810_d_b_c @ M2 @ V )
                   => ( ( h_sati4005454913823510672_b_c_a @ M2 @ I @ V @ M3 )
                      = ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
                        = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_completeness
thf(fact_826_abstract__h__condition__completeness,axiom,
    ! [M2: fsm_d_b_c,I: fsm_d_b_c,M3: nat,V: d > list_P903359562653991662od_b_c] :
      ( ( observable_d_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( minimal_d_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_d_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_d @ ( reacha9185589339461546374_d_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_d_b_c @ I )
                  = ( inputs_d_b_c @ M2 ) )
               => ( ( ( outputs_d_b_c @ I )
                    = ( outputs_d_b_c @ M2 ) )
                 => ( ( state_6014242345326633810_d_b_c @ M2 @ V )
                   => ( ( h_sati4005454913823510675_b_c_d @ M2 @ I @ V @ M3 )
                      = ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
                        = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_completeness
thf(fact_827_abstract__h__condition__completeness,axiom,
    ! [M2: fsm_nat_b_c,I: fsm_a_b_c,M3: nat,V: nat > list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( minimal_nat_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_a_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_a_b_c @ I )
                  = ( inputs_nat_b_c @ M2 ) )
               => ( ( ( outputs_a_b_c @ I )
                    = ( outputs_nat_b_c @ M2 ) )
                 => ( ( state_5280168802736327169at_b_c @ M2 @ V )
                   => ( ( h_sati1514657294798470871_b_c_a @ M2 @ I @ V @ M3 )
                      = ( ( lS_nat_b_c @ M2 @ ( initial_nat_b_c @ M2 ) )
                        = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_completeness
thf(fact_828_abstract__h__condition__completeness,axiom,
    ! [M2: fsm_nat_b_c,I: fsm_d_b_c,M3: nat,V: nat > list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( minimal_nat_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_d_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_d_b_c @ I )
                  = ( inputs_nat_b_c @ M2 ) )
               => ( ( ( outputs_d_b_c @ I )
                    = ( outputs_nat_b_c @ M2 ) )
                 => ( ( state_5280168802736327169at_b_c @ M2 @ V )
                   => ( ( h_sati1514657294798470874_b_c_d @ M2 @ I @ V @ M3 )
                      = ( ( lS_nat_b_c @ M2 @ ( initial_nat_b_c @ M2 ) )
                        = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_completeness
thf(fact_829_abstract__h__condition__completeness,axiom,
    ! [M2: fsm_a_b_c,I: fsm_a_b_c,M3: nat,V: a > list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( minimal_a_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_a_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_a_b_c @ I )
                  = ( inputs_a_b_c @ M2 ) )
               => ( ( ( outputs_a_b_c @ I )
                    = ( outputs_a_b_c @ M2 ) )
                 => ( ( state_7672330573471793359_a_b_c @ M2 @ V )
                   => ( ( h_sati8764940838793890899_b_c_a @ M2 @ I @ V @ M3 )
                      = ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
                        = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_completeness
thf(fact_830_abstract__h__condition__completeness,axiom,
    ! [M2: fsm_a_b_c,I: fsm_d_b_c,M3: nat,V: a > list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( minimal_a_b_c @ M2 )
         => ( ( ord_less_eq_nat @ ( size_d_b_c @ I ) @ M3 )
           => ( ( ord_less_eq_nat @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M2 ) ) @ M3 )
             => ( ( ( inputs_d_b_c @ I )
                  = ( inputs_a_b_c @ M2 ) )
               => ( ( ( outputs_d_b_c @ I )
                    = ( outputs_a_b_c @ M2 ) )
                 => ( ( state_7672330573471793359_a_b_c @ M2 @ V )
                   => ( ( h_sati8764940838793890902_b_c_d @ M2 @ I @ V @ M3 )
                      = ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
                        = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) ) ) ) ) ) ) ) ) ) ) ).

% abstract_h_condition_completeness
thf(fact_831_fsm__from__list__simps_I4_J,axiom,
    ! [Q2: d,Ts: list_P7416935848109857788od_c_d] :
      ( ( outputs_d_b_c @ ( fsm_from_list_d_b_c @ Q2 @ Ts ) )
      = ( image_7388000213934049807_c_d_c
        @ ^ [A6: produc1399170498009216118od_c_d] : ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ A6 ) ) )
        @ ( set_Pr4393879930795859723od_c_d @ Ts ) ) ) ).

% fsm_from_list_simps(4)
thf(fact_832_fsm__from__list__simps_I4_J,axiom,
    ! [Q2: product_prod_c_a,Ts: list_P1562850763667305982od_c_a] :
      ( ( output2371653660386375239_a_a_b @ ( fsm_fr4203749811987807313_a_a_b @ Q2 @ Ts ) )
      = ( image_7535793878625517964_c_a_b
        @ ^ [A6: produc3205353004677322104od_c_a] : ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ A6 ) ) )
        @ ( set_Pr6987804667698513421od_c_a @ Ts ) ) ) ).

% fsm_from_list_simps(4)
thf(fact_833_fsm__from__list__simps_I4_J,axiom,
    ! [Q2: a,Ts: list_P6327159017948738492od_c_a] :
      ( ( outputs_a_b_c @ ( fsm_from_list_a_b_c @ Q2 @ Ts ) )
      = ( image_9171994905208062543_c_a_c
        @ ^ [A6: produc2669932201312201782od_c_a] : ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A6 ) ) )
        @ ( set_Pr5664641634098845387od_c_a @ Ts ) ) ) ).

% fsm_from_list_simps(4)
thf(fact_834_covers__unverified__transition,axiom,
    ( covers1625163452631446971sition
    = ( ^ [T2: produc2669932201312201782od_c_a] :
          ( produc8093372032475639160_c_e_o
          @ ^ [T7: prefix425356687782099470od_b_c,G4: e] :
              ( ? [Alpha2: list_P903359562653991662od_b_c,Beta2: list_P903359562653991662od_b_c] :
                  ( ( converge_a_b_c @ m1 @ Alpha2 @ ( v @ ( produc4225943634317630866od_c_a @ T2 ) ) )
                  & ( converge_d_b_c @ m2 @ Alpha2 @ ( v @ ( produc4225943634317630866od_c_a @ T2 ) ) )
                  & ( converge_a_b_c @ m1 @ Beta2 @ ( v @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) )
                  & ( converge_d_b_c @ m2 @ Beta2 @ ( v @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) )
                  & ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha2 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) @ nil_Product_prod_b_c ) ) @ ( prefix9085958534787526401od_b_c @ T7 ) )
                  & ( member6330420149250801815od_b_c @ Beta2 @ ( prefix9085958534787526401od_b_c @ T7 ) ) )
              & ( converge_d_b_c @ m2 @ ( append2547753245680614915od_b_c @ ( v @ ( produc4225943634317630866od_c_a @ T2 ) ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) @ nil_Product_prod_b_c ) ) @ ( v @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) )
              & ( conver7089555221631485245_c_d_e @ m1 @ m2 @ cg_lookup @ G4 ) ) ) ) ) ).

% covers_unverified_transition
thf(fact_835_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_c,Z2: a,C3: b > c > set_a] :
      ( ! [A3: b,B3: c] :
          ( ( P2
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc3958908706888076337_set_a @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_836_mem__case__prodI2,axiom,
    ! [P2: product_prod_c_a,Z2: a,C3: c > a > set_a] :
      ( ! [A3: c,B3: a] :
          ( ( P2
            = ( product_Pair_c_a @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc2545871480639997748_set_a @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_837_mem__case__prodI2,axiom,
    ! [P2: produc2133359284458116493od_c_a,Z2: a,C3: b > product_prod_c_a > set_a] :
      ( ! [A3: b,B3: product_prod_c_a] :
          ( ( P2
            = ( produc3440865914000881661od_c_a @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc711231852148568268_set_a @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_838_mem__case__prodI2,axiom,
    ! [P2: produc4534891796109904283_b_c_e,Z2: a,C3: prefix425356687782099470od_b_c > e > set_a] :
      ( ! [A3: prefix425356687782099470od_b_c,B3: e] :
          ( ( P2
            = ( produc2511600812177743883_b_c_e @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc8713890162468351550_set_a @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_839_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_c,Z2: list_P903359562653991662od_b_c,C3: b > c > set_li6436108459499378894od_b_c] :
      ( ! [A3: b,B3: c] :
          ( ( P2
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member6330420149250801815od_b_c @ Z2 @ ( produc2955163486331930657od_b_c @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_840_mem__case__prodI2,axiom,
    ! [P2: product_prod_c_a,Z2: list_P903359562653991662od_b_c,C3: c > a > set_li6436108459499378894od_b_c] :
      ( ! [A3: c,B3: a] :
          ( ( P2
            = ( product_Pair_c_a @ A3 @ B3 ) )
         => ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member6330420149250801815od_b_c @ Z2 @ ( produc1857062784414984868od_b_c @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_841_mem__case__prodI2,axiom,
    ! [P2: produc2669932201312201782od_c_a,Z2: a,C3: a > produc2133359284458116493od_c_a > set_a] :
      ( ! [A3: a,B3: produc2133359284458116493od_c_a] :
          ( ( P2
            = ( produc5762801557034676134od_c_a @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc2035080022814879075_set_a @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_842_mem__case__prodI2,axiom,
    ! [P2: produc2133359284458116493od_c_a,Z2: list_P903359562653991662od_b_c,C3: b > product_prod_c_a > set_li6436108459499378894od_b_c] :
      ( ! [A3: b,B3: product_prod_c_a] :
          ( ( P2
            = ( produc3440865914000881661od_c_a @ A3 @ B3 ) )
         => ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member6330420149250801815od_b_c @ Z2 @ ( produc3626413167509901372od_b_c @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_843_mem__case__prodI2,axiom,
    ! [P2: produc4534891796109904283_b_c_e,Z2: list_P903359562653991662od_b_c,C3: prefix425356687782099470od_b_c > e > set_li6436108459499378894od_b_c] :
      ( ! [A3: prefix425356687782099470od_b_c,B3: e] :
          ( ( P2
            = ( produc2511600812177743883_b_c_e @ A3 @ B3 ) )
         => ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member6330420149250801815od_b_c @ Z2 @ ( produc8964711679942503086od_b_c @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_844_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_c,Z2: produc2669932201312201782od_c_a,C3: b > c > set_Pr8357624050790843030od_c_a] :
      ( ! [A3: b,B3: c] :
          ( ( P2
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( member5502040268208861279od_c_a @ Z2 @ ( C3 @ A3 @ B3 ) ) )
     => ( member5502040268208861279od_c_a @ Z2 @ ( produc7297509611420049769od_c_a @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_845_mem__case__prodI,axiom,
    ! [Z2: a,C3: b > c > set_a,A: b,B: c] :
      ( ( member_a @ Z2 @ ( C3 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc3958908706888076337_set_a @ C3 @ ( product_Pair_b_c @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_846_mem__case__prodI,axiom,
    ! [Z2: a,C3: c > a > set_a,A: c,B: a] :
      ( ( member_a @ Z2 @ ( C3 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc2545871480639997748_set_a @ C3 @ ( product_Pair_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_847_mem__case__prodI,axiom,
    ! [Z2: a,C3: b > product_prod_c_a > set_a,A: b,B: product_prod_c_a] :
      ( ( member_a @ Z2 @ ( C3 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc711231852148568268_set_a @ C3 @ ( produc3440865914000881661od_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_848_mem__case__prodI,axiom,
    ! [Z2: a,C3: prefix425356687782099470od_b_c > e > set_a,A: prefix425356687782099470od_b_c,B: e] :
      ( ( member_a @ Z2 @ ( C3 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc8713890162468351550_set_a @ C3 @ ( produc2511600812177743883_b_c_e @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_849_mem__case__prodI,axiom,
    ! [Z2: list_P903359562653991662od_b_c,C3: b > c > set_li6436108459499378894od_b_c,A: b,B: c] :
      ( ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ A @ B ) )
     => ( member6330420149250801815od_b_c @ Z2 @ ( produc2955163486331930657od_b_c @ C3 @ ( product_Pair_b_c @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_850_mem__case__prodI,axiom,
    ! [Z2: list_P903359562653991662od_b_c,C3: c > a > set_li6436108459499378894od_b_c,A: c,B: a] :
      ( ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ A @ B ) )
     => ( member6330420149250801815od_b_c @ Z2 @ ( produc1857062784414984868od_b_c @ C3 @ ( product_Pair_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_851_mem__case__prodI,axiom,
    ! [Z2: a,C3: a > produc2133359284458116493od_c_a > set_a,A: a,B: produc2133359284458116493od_c_a] :
      ( ( member_a @ Z2 @ ( C3 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc2035080022814879075_set_a @ C3 @ ( produc5762801557034676134od_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_852_mem__case__prodI,axiom,
    ! [Z2: list_P903359562653991662od_b_c,C3: b > product_prod_c_a > set_li6436108459499378894od_b_c,A: b,B: product_prod_c_a] :
      ( ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ A @ B ) )
     => ( member6330420149250801815od_b_c @ Z2 @ ( produc3626413167509901372od_b_c @ C3 @ ( produc3440865914000881661od_c_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_853_mem__case__prodI,axiom,
    ! [Z2: list_P903359562653991662od_b_c,C3: prefix425356687782099470od_b_c > e > set_li6436108459499378894od_b_c,A: prefix425356687782099470od_b_c,B: e] :
      ( ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ A @ B ) )
     => ( member6330420149250801815od_b_c @ Z2 @ ( produc8964711679942503086od_b_c @ C3 @ ( produc2511600812177743883_b_c_e @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_854_mem__case__prodI,axiom,
    ! [Z2: produc2669932201312201782od_c_a,C3: b > c > set_Pr8357624050790843030od_c_a,A: b,B: c] :
      ( ( member5502040268208861279od_c_a @ Z2 @ ( C3 @ A @ B ) )
     => ( member5502040268208861279od_c_a @ Z2 @ ( produc7297509611420049769od_c_a @ C3 @ ( product_Pair_b_c @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_855_split__part,axiom,
    ! [P: $o,Q: prefix425356687782099470od_b_c > e > $o] :
      ( ( produc8093372032475639160_c_e_o
        @ ^ [A6: prefix425356687782099470od_b_c,B6: e] :
            ( P
            & ( Q @ A6 @ B6 ) ) )
      = ( ^ [Ab: produc4534891796109904283_b_c_e] :
            ( P
            & ( produc8093372032475639160_c_e_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_856_rename__states__simps_I1_J,axiom,
    ! [M2: fsm_a_b_c,F: a > a] :
      ( ( initial_a_b_c @ ( rename4556187690751568563_b_c_a @ M2 @ F ) )
      = ( F @ ( initial_a_b_c @ M2 ) ) ) ).

% rename_states_simps(1)
thf(fact_857_rename__states__simps_I1_J,axiom,
    ! [M2: fsm_d_b_c,F: d > a] :
      ( ( initial_a_b_c @ ( rename9020073802635964144_b_c_a @ M2 @ F ) )
      = ( F @ ( initial_d_b_c @ M2 ) ) ) ).

% rename_states_simps(1)
thf(fact_858_rename__states__simps_I1_J,axiom,
    ! [M2: fsm_a_b_c,F: a > d] :
      ( ( initial_d_b_c @ ( rename4556187690751568566_b_c_d @ M2 @ F ) )
      = ( F @ ( initial_a_b_c @ M2 ) ) ) ).

% rename_states_simps(1)
thf(fact_859_rename__states__simps_I1_J,axiom,
    ! [M2: fsm_d_b_c,F: d > d] :
      ( ( initial_d_b_c @ ( rename9020073802635964147_b_c_d @ M2 @ F ) )
      = ( F @ ( initial_d_b_c @ M2 ) ) ) ).

% rename_states_simps(1)
thf(fact_860_rename__states__simps_I3_J,axiom,
    ! [M2: fsm_d_b_c,F: d > d] :
      ( ( inputs_d_b_c @ ( rename9020073802635964147_b_c_d @ M2 @ F ) )
      = ( inputs_d_b_c @ M2 ) ) ).

% rename_states_simps(3)
thf(fact_861_rename__states__simps_I3_J,axiom,
    ! [M2: fsm_a_b_c,F: a > d] :
      ( ( inputs_d_b_c @ ( rename4556187690751568566_b_c_d @ M2 @ F ) )
      = ( inputs_a_b_c @ M2 ) ) ).

% rename_states_simps(3)
thf(fact_862_rename__states__simps_I3_J,axiom,
    ! [M2: fsm_d_b_c,F: d > a] :
      ( ( inputs_a_b_c @ ( rename9020073802635964144_b_c_a @ M2 @ F ) )
      = ( inputs_d_b_c @ M2 ) ) ).

% rename_states_simps(3)
thf(fact_863_rename__states__simps_I3_J,axiom,
    ! [M2: fsm_a_b_c,F: a > a] :
      ( ( inputs_a_b_c @ ( rename4556187690751568563_b_c_a @ M2 @ F ) )
      = ( inputs_a_b_c @ M2 ) ) ).

% rename_states_simps(3)
thf(fact_864_rename__states__simps_I4_J,axiom,
    ! [M2: fsm_d_b_c,F: d > d] :
      ( ( outputs_d_b_c @ ( rename9020073802635964147_b_c_d @ M2 @ F ) )
      = ( outputs_d_b_c @ M2 ) ) ).

% rename_states_simps(4)
thf(fact_865_rename__states__simps_I4_J,axiom,
    ! [M2: fsm_a_b_c,F: a > d] :
      ( ( outputs_d_b_c @ ( rename4556187690751568566_b_c_d @ M2 @ F ) )
      = ( outputs_a_b_c @ M2 ) ) ).

% rename_states_simps(4)
thf(fact_866_rename__states__simps_I4_J,axiom,
    ! [M2: fsm_d_b_c,F: d > a] :
      ( ( outputs_a_b_c @ ( rename9020073802635964144_b_c_a @ M2 @ F ) )
      = ( outputs_d_b_c @ M2 ) ) ).

% rename_states_simps(4)
thf(fact_867_rename__states__simps_I4_J,axiom,
    ! [M2: fsm_a_b_c,F: a > a] :
      ( ( outputs_a_b_c @ ( rename4556187690751568563_b_c_a @ M2 @ F ) )
      = ( outputs_a_b_c @ M2 ) ) ).

% rename_states_simps(4)
thf(fact_868_case__prodI2,axiom,
    ! [P2: product_prod_b_c,C3: b > c > $o] :
      ( ! [A3: b,B3: c] :
          ( ( P2
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( C3 @ A3 @ B3 ) )
     => ( produc1873627588981547243_b_c_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_869_case__prodI2,axiom,
    ! [P2: product_prod_c_a,C3: c > a > $o] :
      ( ! [A3: c,B3: a] :
          ( ( P2
            = ( product_Pair_c_a @ A3 @ B3 ) )
         => ( C3 @ A3 @ B3 ) )
     => ( produc8907516716866730350_c_a_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_870_case__prodI2,axiom,
    ! [P2: produc2133359284458116493od_c_a,C3: b > product_prod_c_a > $o] :
      ( ! [A3: b,B3: product_prod_c_a] :
          ( ( P2
            = ( produc3440865914000881661od_c_a @ A3 @ B3 ) )
         => ( C3 @ A3 @ B3 ) )
     => ( produc6211283585785106182_c_a_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_871_case__prodI2,axiom,
    ! [P2: produc2669932201312201782od_c_a,C3: a > produc2133359284458116493od_c_a > $o] :
      ( ! [A3: a,B3: produc2133359284458116493od_c_a] :
          ( ( P2
            = ( produc5762801557034676134od_c_a @ A3 @ B3 ) )
         => ( C3 @ A3 @ B3 ) )
     => ( produc1940289858450342685_c_a_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_872_case__prodI2,axiom,
    ! [P2: produc4534891796109904283_b_c_e,C3: prefix425356687782099470od_b_c > e > $o] :
      ( ! [A3: prefix425356687782099470od_b_c,B3: e] :
          ( ( P2
            = ( produc2511600812177743883_b_c_e @ A3 @ B3 ) )
         => ( C3 @ A3 @ B3 ) )
     => ( produc8093372032475639160_c_e_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_873_case__prodI,axiom,
    ! [F: b > c > $o,A: b,B: c] :
      ( ( F @ A @ B )
     => ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ A @ B ) ) ) ).

% case_prodI
thf(fact_874_case__prodI,axiom,
    ! [F: c > a > $o,A: c,B: a] :
      ( ( F @ A @ B )
     => ( produc8907516716866730350_c_a_o @ F @ ( product_Pair_c_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_875_case__prodI,axiom,
    ! [F: b > product_prod_c_a > $o,A: b,B: product_prod_c_a] :
      ( ( F @ A @ B )
     => ( produc6211283585785106182_c_a_o @ F @ ( produc3440865914000881661od_c_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_876_case__prodI,axiom,
    ! [F: a > produc2133359284458116493od_c_a > $o,A: a,B: produc2133359284458116493od_c_a] :
      ( ( F @ A @ B )
     => ( produc1940289858450342685_c_a_o @ F @ ( produc5762801557034676134od_c_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_877_case__prodI,axiom,
    ! [F: prefix425356687782099470od_b_c > e > $o,A: prefix425356687782099470od_b_c,B: e] :
      ( ( F @ A @ B )
     => ( produc8093372032475639160_c_e_o @ F @ ( produc2511600812177743883_b_c_e @ A @ B ) ) ) ).

% case_prodI
thf(fact_878_fsm__from__list__simps_I5_J,axiom,
    ! [Q2: a,Ts: list_P6327159017948738492od_c_a] :
      ( ( transitions_a_b_c @ ( fsm_from_list_a_b_c @ Q2 @ Ts ) )
      = ( set_Pr5664641634098845387od_c_a @ Ts ) ) ).

% fsm_from_list_simps(5)
thf(fact_879_fsm__from__list__simps_I3_J,axiom,
    ! [Q2: d,Ts: list_P7416935848109857788od_c_d] :
      ( ( inputs_d_b_c @ ( fsm_from_list_d_b_c @ Q2 @ Ts ) )
      = ( image_7388000213934049806_c_d_b
        @ ^ [A6: produc1399170498009216118od_c_d] : ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ A6 ) )
        @ ( set_Pr4393879930795859723od_c_d @ Ts ) ) ) ).

% fsm_from_list_simps(3)
thf(fact_880_fsm__from__list__simps_I3_J,axiom,
    ! [Q2: product_prod_c_a,Ts: list_P1562850763667305982od_c_a] :
      ( ( inputs8636541066849307272_a_a_b @ ( fsm_fr4203749811987807313_a_a_b @ Q2 @ Ts ) )
      = ( image_7535793878625517963_c_a_a
        @ ^ [A6: produc3205353004677322104od_c_a] : ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ A6 ) )
        @ ( set_Pr6987804667698513421od_c_a @ Ts ) ) ) ).

% fsm_from_list_simps(3)
thf(fact_881_fsm__from__list__simps_I3_J,axiom,
    ! [Q2: a,Ts: list_P6327159017948738492od_c_a] :
      ( ( inputs_a_b_c @ ( fsm_from_list_a_b_c @ Q2 @ Ts ) )
      = ( image_9171994905208062542_c_a_b
        @ ^ [A6: produc2669932201312201782od_c_a] : ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ A6 ) )
        @ ( set_Pr5664641634098845387od_c_a @ Ts ) ) ) ).

% fsm_from_list_simps(3)
thf(fact_882_verify__undefined__io__pair,axiom,
    ( verify8625347136745899022o_pair
    = ( ^ [T6: prefix425356687782099470od_b_c] :
          ( produc2053673206102377720od_b_c
          @ ^ [Q4: a] :
              ( produc7288989217166112353od_b_c
              @ ^ [X4: b,Y5: c] : ( produc1447154562020694007_b_c_e @ ( append_io_pair @ m1 @ v @ T6 @ ( produc6779600879576379961_b_c_e @ tG2 ) @ cg_insert @ cg_lookup @ Q4 @ X4 @ Y5 ) ) ) ) ) ) ).

% verify_undefined_io_pair
thf(fact_883_prod_Odisc__eq__case,axiom,
    ! [Prod: produc4534891796109904283_b_c_e] :
      ( produc8093372032475639160_c_e_o
      @ ^ [Uu: prefix425356687782099470od_b_c,Uv: e] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_884_Collect__case__prod__mono,axiom,
    ! [A4: a > produc2133359284458116493od_c_a > $o,B4: a > produc2133359284458116493od_c_a > $o] :
      ( ( ord_le3047557917845655871_c_a_o @ A4 @ B4 )
     => ( ord_le3226338165915859510od_c_a @ ( collec8405073525749625505od_c_a @ ( produc1940289858450342685_c_a_o @ A4 ) ) @ ( collec8405073525749625505od_c_a @ ( produc1940289858450342685_c_a_o @ B4 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_885_Collect__case__prod__mono,axiom,
    ! [A4: prefix425356687782099470od_b_c > e > $o,B4: prefix425356687782099470od_b_c > e > $o] :
      ( ( ord_le7893044087947676058_c_e_o @ A4 @ B4 )
     => ( ord_le1795193733426387611_b_c_e @ ( collec7881274725622991494_b_c_e @ ( produc8093372032475639160_c_e_o @ A4 ) ) @ ( collec7881274725622991494_b_c_e @ ( produc8093372032475639160_c_e_o @ B4 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_886_mem__case__prodE,axiom,
    ! [Z2: a,C3: b > c > set_a,P2: product_prod_b_c] :
      ( ( member_a @ Z2 @ ( produc3958908706888076337_set_a @ C3 @ P2 ) )
     => ~ ! [X3: b,Y3: c] :
            ( ( P2
              = ( product_Pair_b_c @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_887_mem__case__prodE,axiom,
    ! [Z2: a,C3: c > a > set_a,P2: product_prod_c_a] :
      ( ( member_a @ Z2 @ ( produc2545871480639997748_set_a @ C3 @ P2 ) )
     => ~ ! [X3: c,Y3: a] :
            ( ( P2
              = ( product_Pair_c_a @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_888_mem__case__prodE,axiom,
    ! [Z2: a,C3: b > product_prod_c_a > set_a,P2: produc2133359284458116493od_c_a] :
      ( ( member_a @ Z2 @ ( produc711231852148568268_set_a @ C3 @ P2 ) )
     => ~ ! [X3: b,Y3: product_prod_c_a] :
            ( ( P2
              = ( produc3440865914000881661od_c_a @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_889_mem__case__prodE,axiom,
    ! [Z2: a,C3: prefix425356687782099470od_b_c > e > set_a,P2: produc4534891796109904283_b_c_e] :
      ( ( member_a @ Z2 @ ( produc8713890162468351550_set_a @ C3 @ P2 ) )
     => ~ ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ( ( P2
              = ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_890_mem__case__prodE,axiom,
    ! [Z2: list_P903359562653991662od_b_c,C3: b > c > set_li6436108459499378894od_b_c,P2: product_prod_b_c] :
      ( ( member6330420149250801815od_b_c @ Z2 @ ( produc2955163486331930657od_b_c @ C3 @ P2 ) )
     => ~ ! [X3: b,Y3: c] :
            ( ( P2
              = ( product_Pair_b_c @ X3 @ Y3 ) )
           => ~ ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_891_mem__case__prodE,axiom,
    ! [Z2: list_P903359562653991662od_b_c,C3: c > a > set_li6436108459499378894od_b_c,P2: product_prod_c_a] :
      ( ( member6330420149250801815od_b_c @ Z2 @ ( produc1857062784414984868od_b_c @ C3 @ P2 ) )
     => ~ ! [X3: c,Y3: a] :
            ( ( P2
              = ( product_Pair_c_a @ X3 @ Y3 ) )
           => ~ ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_892_mem__case__prodE,axiom,
    ! [Z2: a,C3: a > produc2133359284458116493od_c_a > set_a,P2: produc2669932201312201782od_c_a] :
      ( ( member_a @ Z2 @ ( produc2035080022814879075_set_a @ C3 @ P2 ) )
     => ~ ! [X3: a,Y3: produc2133359284458116493od_c_a] :
            ( ( P2
              = ( produc5762801557034676134od_c_a @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_893_mem__case__prodE,axiom,
    ! [Z2: list_P903359562653991662od_b_c,C3: b > product_prod_c_a > set_li6436108459499378894od_b_c,P2: produc2133359284458116493od_c_a] :
      ( ( member6330420149250801815od_b_c @ Z2 @ ( produc3626413167509901372od_b_c @ C3 @ P2 ) )
     => ~ ! [X3: b,Y3: product_prod_c_a] :
            ( ( P2
              = ( produc3440865914000881661od_c_a @ X3 @ Y3 ) )
           => ~ ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_894_mem__case__prodE,axiom,
    ! [Z2: list_P903359562653991662od_b_c,C3: prefix425356687782099470od_b_c > e > set_li6436108459499378894od_b_c,P2: produc4534891796109904283_b_c_e] :
      ( ( member6330420149250801815od_b_c @ Z2 @ ( produc8964711679942503086od_b_c @ C3 @ P2 ) )
     => ~ ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ( ( P2
              = ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) )
           => ~ ( member6330420149250801815od_b_c @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_895_mem__case__prodE,axiom,
    ! [Z2: produc2669932201312201782od_c_a,C3: b > c > set_Pr8357624050790843030od_c_a,P2: product_prod_b_c] :
      ( ( member5502040268208861279od_c_a @ Z2 @ ( produc7297509611420049769od_c_a @ C3 @ P2 ) )
     => ~ ! [X3: b,Y3: c] :
            ( ( P2
              = ( product_Pair_b_c @ X3 @ Y3 ) )
           => ~ ( member5502040268208861279od_c_a @ Z2 @ ( C3 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_896_case__prodE,axiom,
    ! [C3: b > c > $o,P2: product_prod_b_c] :
      ( ( produc1873627588981547243_b_c_o @ C3 @ P2 )
     => ~ ! [X3: b,Y3: c] :
            ( ( P2
              = ( product_Pair_b_c @ X3 @ Y3 ) )
           => ~ ( C3 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_897_case__prodE,axiom,
    ! [C3: c > a > $o,P2: product_prod_c_a] :
      ( ( produc8907516716866730350_c_a_o @ C3 @ P2 )
     => ~ ! [X3: c,Y3: a] :
            ( ( P2
              = ( product_Pair_c_a @ X3 @ Y3 ) )
           => ~ ( C3 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_898_case__prodE,axiom,
    ! [C3: b > product_prod_c_a > $o,P2: produc2133359284458116493od_c_a] :
      ( ( produc6211283585785106182_c_a_o @ C3 @ P2 )
     => ~ ! [X3: b,Y3: product_prod_c_a] :
            ( ( P2
              = ( produc3440865914000881661od_c_a @ X3 @ Y3 ) )
           => ~ ( C3 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_899_case__prodE,axiom,
    ! [C3: a > produc2133359284458116493od_c_a > $o,P2: produc2669932201312201782od_c_a] :
      ( ( produc1940289858450342685_c_a_o @ C3 @ P2 )
     => ~ ! [X3: a,Y3: produc2133359284458116493od_c_a] :
            ( ( P2
              = ( produc5762801557034676134od_c_a @ X3 @ Y3 ) )
           => ~ ( C3 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_900_case__prodE,axiom,
    ! [C3: prefix425356687782099470od_b_c > e > $o,P2: produc4534891796109904283_b_c_e] :
      ( ( produc8093372032475639160_c_e_o @ C3 @ P2 )
     => ~ ! [X3: prefix425356687782099470od_b_c,Y3: e] :
            ( ( P2
              = ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) )
           => ~ ( C3 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_901_case__prodD,axiom,
    ! [F: b > c > $o,A: b,B: c] :
      ( ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_902_case__prodD,axiom,
    ! [F: c > a > $o,A: c,B: a] :
      ( ( produc8907516716866730350_c_a_o @ F @ ( product_Pair_c_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_903_case__prodD,axiom,
    ! [F: b > product_prod_c_a > $o,A: b,B: product_prod_c_a] :
      ( ( produc6211283585785106182_c_a_o @ F @ ( produc3440865914000881661od_c_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_904_case__prodD,axiom,
    ! [F: a > produc2133359284458116493od_c_a > $o,A: a,B: produc2133359284458116493od_c_a] :
      ( ( produc1940289858450342685_c_a_o @ F @ ( produc5762801557034676134od_c_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_905_case__prodD,axiom,
    ! [F: prefix425356687782099470od_b_c > e > $o,A: prefix425356687782099470od_b_c,B: e] :
      ( ( produc8093372032475639160_c_e_o @ F @ ( produc2511600812177743883_b_c_e @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_906_Setcompr__eq__image,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A4: set_a] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [Uu: list_P903359562653991662od_b_c] :
          ? [X4: a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( member_a @ X4 @ A4 ) ) )
      = ( image_8656872703560917237od_b_c @ F @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_907_Setcompr__eq__image,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,A4: set_a] :
      ( ( collec3436578395599707393od_c_a
        @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
          ? [X4: a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( member_a @ X4 @ A4 ) ) )
      = ( image_2946797484331234845od_c_a @ F @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_908_Setcompr__eq__image,axiom,
    ! [F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [Uu: produc2669932201312201782od_c_a] :
          ? [X4: produc2669932201312201782od_c_a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( member5502040268208861279od_c_a @ X4 @ A4 ) ) )
      = ( image_2169818278710580869od_c_a @ F @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_909_Setcompr__eq__image,axiom,
    ! [F: a > produc2669932201312201782od_c_a,A4: set_a] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [Uu: produc2669932201312201782od_c_a] :
          ? [X4: a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( member_a @ X4 @ A4 ) ) )
      = ( image_4184489845059386301od_c_a @ F @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_910_Setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > produc2669932201312201782od_c_a,A4: set_li6436108459499378894od_b_c] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [Uu: produc2669932201312201782od_c_a] :
          ? [X4: list_P903359562653991662od_b_c] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( member6330420149250801815od_b_c @ X4 @ A4 ) ) )
      = ( image_7601443023479506637od_c_a @ F @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_911_Setcompr__eq__image,axiom,
    ! [F: produc2669932201312201782od_c_a > nat,A4: set_Pr8357624050790843030od_c_a] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X4: produc2669932201312201782od_c_a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( member5502040268208861279od_c_a @ X4 @ A4 ) ) )
      = ( image_1800417606190878401_a_nat @ F @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_912_Setcompr__eq__image,axiom,
    ! [F: a > nat,A4: set_a] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X4: a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( member_a @ X4 @ A4 ) ) )
      = ( image_a_nat @ F @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_913_Setcompr__eq__image,axiom,
    ! [F: list_P903359562653991662od_b_c > nat,A4: set_li6436108459499378894od_b_c] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X4: list_P903359562653991662od_b_c] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( member6330420149250801815od_b_c @ X4 @ A4 ) ) )
      = ( image_4545125072327328889_c_nat @ F @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_914_setcompr__eq__image,axiom,
    ! [F: a > list_P903359562653991662od_b_c,P: a > $o] :
      ( ( collec2280997390073109977od_b_c
        @ ^ [Uu: list_P903359562653991662od_b_c] :
          ? [X4: a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( P @ X4 ) ) )
      = ( image_8656872703560917237od_b_c @ F @ ( collect_a @ P ) ) ) ).

% setcompr_eq_image
thf(fact_915_setcompr__eq__image,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,P: a > $o] :
      ( ( collec3436578395599707393od_c_a
        @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
          ? [X4: a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( P @ X4 ) ) )
      = ( image_2946797484331234845od_c_a @ F @ ( collect_a @ P ) ) ) ).

% setcompr_eq_image
thf(fact_916_setcompr__eq__image,axiom,
    ! [F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [Uu: produc2669932201312201782od_c_a] :
          ? [X4: produc2669932201312201782od_c_a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( P @ X4 ) ) )
      = ( image_2169818278710580869od_c_a @ F @ ( collec8405073525749625505od_c_a @ P ) ) ) ).

% setcompr_eq_image
thf(fact_917_setcompr__eq__image,axiom,
    ! [F: nat > produc2669932201312201782od_c_a,P: nat > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [Uu: produc2669932201312201782od_c_a] :
          ? [X4: nat] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( P @ X4 ) ) )
      = ( image_6077155709069503263od_c_a @ F @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_918_setcompr__eq__image,axiom,
    ! [F: produc2669932201312201782od_c_a > nat,P: produc2669932201312201782od_c_a > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X4: produc2669932201312201782od_c_a] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( P @ X4 ) ) )
      = ( image_1800417606190878401_a_nat @ F @ ( collec8405073525749625505od_c_a @ P ) ) ) ).

% setcompr_eq_image
thf(fact_919_setcompr__eq__image,axiom,
    ! [F: nat > nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X4: nat] :
            ( ( Uu
              = ( F @ X4 ) )
            & ( P @ X4 ) ) )
      = ( image_nat_nat @ F @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_920_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: a > produc2133359284458116493od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ X @ ( collec8405073525749625505od_c_a @ ( produc1940289858450342685_c_a_o @ A4 ) ) )
     => ( A4 @ ( produc4225943634317630866od_c_a @ X ) @ ( produc8088130389382527188od_c_a @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_921_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc2133359284458116493od_c_a,A4: b > product_prod_c_a > $o] :
      ( ( member1160383752307813046od_c_a @ X @ ( collec3135042693873630968od_c_a @ ( produc6211283585785106182_c_a_o @ A4 ) ) )
     => ( A4 @ ( produc301949074746585577od_c_a @ X ) @ ( produc5221446324160585515od_c_a @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_922_Product__Type_OCollect__case__prodD,axiom,
    ! [X: product_prod_c_a,A4: c > a > $o] :
      ( ( member5074992350434858958od_c_a @ X @ ( collec6984858670293876240od_c_a @ ( produc8907516716866730350_c_a_o @ A4 ) ) )
     => ( A4 @ ( product_fst_c_a @ X ) @ ( product_snd_c_a @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_923_Product__Type_OCollect__case__prodD,axiom,
    ! [X: product_prod_b_c,A4: b > c > $o] :
      ( ( member7862447941013992593od_b_c @ X @ ( collec548942224018234067od_b_c @ ( produc1873627588981547243_b_c_o @ A4 ) ) )
     => ( A4 @ ( product_fst_b_c @ X ) @ ( product_snd_b_c @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_924_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc4534891796109904283_b_c_e,A4: prefix425356687782099470od_b_c > e > $o] :
      ( ( member323891770073680196_b_c_e @ X @ ( collec7881274725622991494_b_c_e @ ( produc8093372032475639160_c_e_o @ A4 ) ) )
     => ( A4 @ ( produc1447154562020694007_b_c_e @ X ) @ ( produc6779600879576379961_b_c_e @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_925_Collect__split__mono__strong,axiom,
    ! [X7: set_a,A4: set_Pr8357624050790843030od_c_a,Y9: set_Pr272002107461315949od_c_a,P: a > produc2133359284458116493od_c_a > $o,Q: a > produc2133359284458116493od_c_a > $o] :
      ( ( X7
        = ( image_9171994905208062541_c_a_a @ produc4225943634317630866od_c_a @ A4 ) )
     => ( ( Y9
          = ( image_644133890040260316od_c_a @ produc8088130389382527188od_c_a @ A4 ) )
       => ( ! [X3: a] :
              ( ( member_a @ X3 @ X7 )
             => ! [Xa2: produc2133359284458116493od_c_a] :
                  ( ( member1160383752307813046od_c_a @ Xa2 @ Y9 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le3226338165915859510od_c_a @ A4 @ ( collec8405073525749625505od_c_a @ ( produc1940289858450342685_c_a_o @ P ) ) )
           => ( ord_le3226338165915859510od_c_a @ A4 @ ( collec8405073525749625505od_c_a @ ( produc1940289858450342685_c_a_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_926_Collect__split__mono__strong,axiom,
    ! [X7: set_b,A4: set_Pr272002107461315949od_c_a,Y9: set_Product_prod_c_a,P: b > product_prod_c_a > $o,Q: b > product_prod_c_a > $o] :
      ( ( X7
        = ( image_8972666397057282743_c_a_b @ produc301949074746585577od_c_a @ A4 ) )
     => ( ( Y9
          = ( image_1453128995064482141od_c_a @ produc5221446324160585515od_c_a @ A4 ) )
       => ( ! [X3: b] :
              ( ( member_b @ X3 @ X7 )
             => ! [Xa2: product_prod_c_a] :
                  ( ( member5074992350434858958od_c_a @ Xa2 @ Y9 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le4080174072498570509od_c_a @ A4 @ ( collec3135042693873630968od_c_a @ ( produc6211283585785106182_c_a_o @ P ) ) )
           => ( ord_le4080174072498570509od_c_a @ A4 @ ( collec3135042693873630968od_c_a @ ( produc6211283585785106182_c_a_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_927_Collect__split__mono__strong,axiom,
    ! [X7: set_c,A4: set_Product_prod_c_a,Y9: set_a,P: c > a > $o,Q: c > a > $o] :
      ( ( X7
        = ( image_5407010446420943520_c_a_c @ product_fst_c_a @ A4 ) )
     => ( ( Y9
          = ( image_5407010446420943518_c_a_a @ product_snd_c_a @ A4 ) )
       => ( ! [X3: c] :
              ( ( member_c @ X3 @ X7 )
             => ! [Xa2: a] :
                  ( ( member_a @ Xa2 @ Y9 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le8698776994054418981od_c_a @ A4 @ ( collec6984858670293876240od_c_a @ ( produc8907516716866730350_c_a_o @ P ) ) )
           => ( ord_le8698776994054418981od_c_a @ A4 @ ( collec6984858670293876240od_c_a @ ( produc8907516716866730350_c_a_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_928_Collect__split__mono__strong,axiom,
    ! [X7: set_b,A4: set_Product_prod_b_c,Y9: set_c,P: b > c > $o,Q: b > c > $o] :
      ( ( X7
        = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ A4 ) )
     => ( ( Y9
          = ( image_7762865866877615261_b_c_c @ product_snd_b_c @ A4 ) )
       => ( ! [X3: b] :
              ( ( member_b @ X3 @ X7 )
             => ! [Xa2: c] :
                  ( ( member_c @ Xa2 @ Y9 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le253122037897321832od_b_c @ A4 @ ( collec548942224018234067od_b_c @ ( produc1873627588981547243_b_c_o @ P ) ) )
           => ( ord_le253122037897321832od_b_c @ A4 @ ( collec548942224018234067od_b_c @ ( produc1873627588981547243_b_c_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_929_Collect__split__mono__strong,axiom,
    ! [X7: set_Pr4395803880006080238od_b_c,A4: set_Pr3237851069733663483_b_c_e,Y9: set_e,P: prefix425356687782099470od_b_c > e > $o,Q: prefix425356687782099470od_b_c > e > $o] :
      ( ( X7
        = ( image_3005832862676689720od_b_c @ produc1447154562020694007_b_c_e @ A4 ) )
     => ( ( Y9
          = ( image_2493620005906216236_c_e_e @ produc6779600879576379961_b_c_e @ A4 ) )
       => ( ! [X3: prefix425356687782099470od_b_c] :
              ( ( member2378525361814473399od_b_c @ X3 @ X7 )
             => ! [Xa2: e] :
                  ( ( member_e @ Xa2 @ Y9 )
                 => ( ( P @ X3 @ Xa2 )
                   => ( Q @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le1795193733426387611_b_c_e @ A4 @ ( collec7881274725622991494_b_c_e @ ( produc8093372032475639160_c_e_o @ P ) ) )
           => ( ord_le1795193733426387611_b_c_e @ A4 @ ( collec7881274725622991494_b_c_e @ ( produc8093372032475639160_c_e_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_930_abstract__h__condition__soundness,axiom,
    ! [M2: fsm_d_b_c,I: fsm_a_b_c,V: d > list_P903359562653991662od_b_c,M3: nat] :
      ( ( observable_d_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( state_6014242345326633810_d_b_c @ M2 @ V )
         => ( ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
              = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) )
           => ( h_sati4005454913823510672_b_c_a @ M2 @ I @ V @ M3 ) ) ) ) ) ).

% abstract_h_condition_soundness
thf(fact_931_abstract__h__condition__soundness,axiom,
    ! [M2: fsm_d_b_c,I: fsm_d_b_c,V: d > list_P903359562653991662od_b_c,M3: nat] :
      ( ( observable_d_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( state_6014242345326633810_d_b_c @ M2 @ V )
         => ( ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
              = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) )
           => ( h_sati4005454913823510675_b_c_d @ M2 @ I @ V @ M3 ) ) ) ) ) ).

% abstract_h_condition_soundness
thf(fact_932_abstract__h__condition__soundness,axiom,
    ! [M2: fsm_a_b_c,I: fsm_a_b_c,V: a > list_P903359562653991662od_b_c,M3: nat] :
      ( ( observable_a_b_c @ M2 )
     => ( ( observable_a_b_c @ I )
       => ( ( state_7672330573471793359_a_b_c @ M2 @ V )
         => ( ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
              = ( lS_a_b_c @ I @ ( initial_a_b_c @ I ) ) )
           => ( h_sati8764940838793890899_b_c_a @ M2 @ I @ V @ M3 ) ) ) ) ) ).

% abstract_h_condition_soundness
thf(fact_933_abstract__h__condition__soundness,axiom,
    ! [M2: fsm_a_b_c,I: fsm_d_b_c,V: a > list_P903359562653991662od_b_c,M3: nat] :
      ( ( observable_a_b_c @ M2 )
     => ( ( observable_d_b_c @ I )
       => ( ( state_7672330573471793359_a_b_c @ M2 @ V )
         => ( ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
              = ( lS_d_b_c @ I @ ( initial_d_b_c @ I ) ) )
           => ( h_sati8764940838793890902_b_c_d @ M2 @ I @ V @ M3 ) ) ) ) ) ).

% abstract_h_condition_soundness
thf(fact_934_T3,axiom,
    ( t3
    = ( foldl_6461310528730867108od_b_c @ verify8625347136745899022o_pair @ ( produc1447154562020694007_b_c_e @ tG2 ) @ undefined_io_pairs ) ) ).

% T3
thf(fact_935_output__complete_Oelims_I3_J,axiom,
    ! [X: fsm_d_b_c] :
      ( ~ ( output2176235794172393715_d_b_c @ X )
     => ~ ! [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( transitions_d_b_c @ X ) )
           => ! [Xa2: c] :
                ( ( member_c @ Xa2 @ ( outputs_d_b_c @ X ) )
               => ? [Xb2: produc1399170498009216118od_c_d] :
                    ( ( member4231278564905875615od_c_d @ Xb2 @ ( transitions_d_b_c @ X ) )
                    & ( ( produc1508465569166014290od_c_d @ X3 )
                      = ( produc1508465569166014290od_c_d @ Xb2 ) )
                    & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X3 ) )
                      = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ Xb2 ) ) )
                    & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Xb2 ) ) )
                      = Xa2 ) ) ) ) ) ).

% output_complete.elims(3)
thf(fact_936_output__complete_Oelims_I3_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b] :
      ( ~ ( output1640370057126496855_a_a_b @ X )
     => ~ ! [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ X ) )
           => ! [Xa2: b] :
                ( ( member_b @ Xa2 @ ( output2371653660386375239_a_a_b @ X ) )
               => ? [Xb2: produc3205353004677322104od_c_a] :
                    ( ( member6508725278956418977od_c_a @ Xb2 @ ( transi2941759273464634107_a_a_b @ X ) )
                    & ( ( produc8171714036265584980od_c_a @ X3 )
                      = ( produc8171714036265584980od_c_a @ Xb2 ) )
                    & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) )
                      = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xb2 ) ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xb2 ) ) )
                      = Xa2 ) ) ) ) ) ).

% output_complete.elims(3)
thf(fact_937_output__complete_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( output3834324022317553264_a_b_c @ X )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X ) )
           => ! [Xa2: c] :
                ( ( member_c @ Xa2 @ ( outputs_a_b_c @ X ) )
               => ? [Xb2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xb2 @ ( transitions_a_b_c @ X ) )
                    & ( ( produc4225943634317630866od_c_a @ X3 )
                      = ( produc4225943634317630866od_c_a @ Xb2 ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                      = Xa2 ) ) ) ) ) ).

% output_complete.elims(3)
thf(fact_938_output__complete_Oelims_I2_J,axiom,
    ! [X: fsm_d_b_c] :
      ( ( output2176235794172393715_d_b_c @ X )
     => ! [X5: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X5 @ ( transitions_d_b_c @ X ) )
         => ! [Xa: c] :
              ( ( member_c @ Xa @ ( outputs_d_b_c @ X ) )
             => ? [Xb3: produc1399170498009216118od_c_d] :
                  ( ( member4231278564905875615od_c_d @ Xb3 @ ( transitions_d_b_c @ X ) )
                  & ( ( produc1508465569166014290od_c_d @ X5 )
                    = ( produc1508465569166014290od_c_d @ Xb3 ) )
                  & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X5 ) )
                    = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ Xb3 ) ) )
                  & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Xb3 ) ) )
                    = Xa ) ) ) ) ) ).

% output_complete.elims(2)
thf(fact_939_output__complete_Oelims_I2_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b] :
      ( ( output1640370057126496855_a_a_b @ X )
     => ! [X5: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X5 @ ( transi2941759273464634107_a_a_b @ X ) )
         => ! [Xa: b] :
              ( ( member_b @ Xa @ ( output2371653660386375239_a_a_b @ X ) )
             => ? [Xb3: produc3205353004677322104od_c_a] :
                  ( ( member6508725278956418977od_c_a @ Xb3 @ ( transi2941759273464634107_a_a_b @ X ) )
                  & ( ( produc8171714036265584980od_c_a @ X5 )
                    = ( produc8171714036265584980od_c_a @ Xb3 ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X5 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xb3 ) ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xb3 ) ) )
                    = Xa ) ) ) ) ) ).

% output_complete.elims(2)
thf(fact_940_output__complete_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( output3834324022317553264_a_b_c @ X )
     => ! [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X ) )
         => ! [Xa: c] :
              ( ( member_c @ Xa @ ( outputs_a_b_c @ X ) )
             => ? [Xb3: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Xb3 @ ( transitions_a_b_c @ X ) )
                  & ( ( produc4225943634317630866od_c_a @ X5 )
                    = ( produc4225943634317630866od_c_a @ Xb3 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb3 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb3 ) ) )
                    = Xa ) ) ) ) ) ).

% output_complete.elims(2)
thf(fact_941_verifies__io__pair__handled,axiom,
    ! [F: fsm_a_b_c > ( a > list_P903359562653991662od_b_c ) > prefix425356687782099470od_b_c > e > ( e > list_P903359562653991662od_b_c > e ) > ( e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c ) > a > b > c > produc4534891796109904283_b_c_e,M12: fsm_a_b_c,M23: fsm_d_b_c,Cg_insert2: e > list_P903359562653991662od_b_c > e,Cg_lookup2: e > list_P903359562653991662od_b_c > list_l8907847357763382004od_b_c] :
      ( ( sPY_ve2339642005831624696_c_e_d @ F @ M12 @ M23 @ Cg_insert2 @ Cg_lookup2 )
     => ( h_hand4722993169802688916_c_e_d @ F @ M12 @ M23 @ Cg_insert2 @ Cg_lookup2 ) ) ).

% verifies_io_pair_handled
thf(fact_942_output__complete__alt__def,axiom,
    ( output2176235794172393715_d_b_c
    = ( ^ [M: fsm_d_b_c] :
        ! [Q4: d,X4: b] :
          ( ? [Y5: c,Q5: d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ Q4 @ ( produc3440865926910568064od_c_d @ X4 @ ( product_Pair_c_d @ Y5 @ Q5 ) ) ) @ ( transitions_d_b_c @ M ) )
         => ! [Y5: c] :
              ( ( member_c @ Y5 @ ( outputs_d_b_c @ M ) )
             => ? [Q5: d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ Q4 @ ( produc3440865926910568064od_c_d @ X4 @ ( product_Pair_c_d @ Y5 @ Q5 ) ) ) @ ( transitions_d_b_c @ M ) ) ) ) ) ) ).

% output_complete_alt_def
thf(fact_943_output__complete__alt__def,axiom,
    ( output1640370057126496855_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q4: product_prod_c_a,X4: a] :
          ( ? [Y5: b,Q5: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q4 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y5 @ Q5 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) )
         => ! [Y5: b] :
              ( ( member_b @ Y5 @ ( output2371653660386375239_a_a_b @ M ) )
             => ? [Q5: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q4 @ ( produc5762801557034676134od_c_a @ X4 @ ( produc3440865914000881661od_c_a @ Y5 @ Q5 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) ) ) ) ) ).

% output_complete_alt_def
thf(fact_944_output__complete__alt__def,axiom,
    ( output3834324022317553264_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [Q4: a,X4: b] :
          ( ? [Y5: c,Q5: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q4 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y5 @ Q5 ) ) ) @ ( transitions_a_b_c @ M ) )
         => ! [Y5: c] :
              ( ( member_c @ Y5 @ ( outputs_a_b_c @ M ) )
             => ? [Q5: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q4 @ ( produc3440865914000881661od_c_a @ X4 @ ( product_Pair_c_a @ Y5 @ Q5 ) ) ) @ ( transitions_a_b_c @ M ) ) ) ) ) ) ).

% output_complete_alt_def
thf(fact_945_output__complete_Osimps,axiom,
    ( output2176235794172393715_d_b_c
    = ( ^ [M: fsm_d_b_c] :
        ! [X4: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X4 @ ( transitions_d_b_c @ M ) )
         => ! [Y5: c] :
              ( ( member_c @ Y5 @ ( outputs_d_b_c @ M ) )
             => ? [Z4: produc1399170498009216118od_c_d] :
                  ( ( member4231278564905875615od_c_d @ Z4 @ ( transitions_d_b_c @ M ) )
                  & ( ( produc1508465569166014290od_c_d @ X4 )
                    = ( produc1508465569166014290od_c_d @ Z4 ) )
                  & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) )
                    = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ Z4 ) ) )
                  & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Z4 ) ) )
                    = Y5 ) ) ) ) ) ) ).

% output_complete.simps
thf(fact_946_output__complete_Osimps,axiom,
    ( output1640370057126496855_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ M ) )
         => ! [Y5: b] :
              ( ( member_b @ Y5 @ ( output2371653660386375239_a_a_b @ M ) )
             => ? [Z4: produc3205353004677322104od_c_a] :
                  ( ( member6508725278956418977od_c_a @ Z4 @ ( transi2941759273464634107_a_a_b @ M ) )
                  & ( ( produc8171714036265584980od_c_a @ X4 )
                    = ( produc8171714036265584980od_c_a @ Z4 ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Z4 ) ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Z4 ) ) )
                    = Y5 ) ) ) ) ) ) ).

% output_complete.simps
thf(fact_947_output__complete_Osimps,axiom,
    ( output3834324022317553264_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M ) )
         => ! [Y5: c] :
              ( ( member_c @ Y5 @ ( outputs_a_b_c @ M ) )
             => ? [Z4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Z4 @ ( transitions_a_b_c @ M ) )
                  & ( ( produc4225943634317630866od_c_a @ X4 )
                    = ( produc4225943634317630866od_c_a @ Z4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                    = Y5 ) ) ) ) ) ) ).

% output_complete.simps
thf(fact_948_output__complete_Oelims_I1_J,axiom,
    ! [X: fsm_d_b_c,Y: $o] :
      ( ( ( output2176235794172393715_d_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ X4 @ ( transitions_d_b_c @ X ) )
             => ! [Y5: c] :
                  ( ( member_c @ Y5 @ ( outputs_d_b_c @ X ) )
                 => ? [Z4: produc1399170498009216118od_c_d] :
                      ( ( member4231278564905875615od_c_d @ Z4 @ ( transitions_d_b_c @ X ) )
                      & ( ( produc1508465569166014290od_c_d @ X4 )
                        = ( produc1508465569166014290od_c_d @ Z4 ) )
                      & ( ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ X4 ) )
                        = ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ Z4 ) ) )
                      & ( ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ Z4 ) ) )
                        = Y5 ) ) ) ) ) ) ) ).

% output_complete.elims(1)
thf(fact_949_output__complete_Oelims_I1_J,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b,Y: $o] :
      ( ( ( output1640370057126496855_a_a_b @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ X ) )
             => ! [Y5: b] :
                  ( ( member_b @ Y5 @ ( output2371653660386375239_a_a_b @ X ) )
                 => ? [Z4: produc3205353004677322104od_c_a] :
                      ( ( member6508725278956418977od_c_a @ Z4 @ ( transi2941759273464634107_a_a_b @ X ) )
                      & ( ( produc8171714036265584980od_c_a @ X4 )
                        = ( produc8171714036265584980od_c_a @ Z4 ) )
                      & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                        = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Z4 ) ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Z4 ) ) )
                        = Y5 ) ) ) ) ) ) ) ).

% output_complete.elims(1)
thf(fact_950_output__complete_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( output3834324022317553264_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X ) )
             => ! [Y5: c] :
                  ( ( member_c @ Y5 @ ( outputs_a_b_c @ X ) )
                 => ? [Z4: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Z4 @ ( transitions_a_b_c @ X ) )
                      & ( ( produc4225943634317630866od_c_a @ X4 )
                        = ( produc4225943634317630866od_c_a @ Z4 ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                      & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                        = Y5 ) ) ) ) ) ) ) ).

% output_complete.elims(1)
thf(fact_951_h__out__alt__def,axiom,
    ! [M2: fsm_b_c_nat,Q2: b,X: c] :
      ( ( h_out_b_c_nat @ M2 @ ( product_Pair_b_c @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Uu: nat] :
          ? [T2: produc897840418342192897_nat_b] :
            ( ( Uu
              = ( product_fst_nat_b @ ( produc936460378501341983_nat_b @ ( produc6092916665544297165_nat_b @ T2 ) ) ) )
            & ( member7205605469418551960_nat_b @ T2 @ ( transitions_b_c_nat @ M2 ) )
            & ( ( produc6660495173513958031_nat_b @ T2 )
              = Q2 )
            & ( ( produc383478219494620385_nat_b @ ( produc6092916665544297165_nat_b @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_952_h__out__alt__def,axiom,
    ! [M2: fsm_c_a_nat,Q2: c,X: a] :
      ( ( h_out_c_a_nat @ M2 @ ( product_Pair_c_a @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Uu: nat] :
          ? [T2: produc6755641753618979457_nat_c] :
            ( ( Uu
              = ( product_fst_nat_c @ ( produc5169316498785520798_nat_c @ ( produc8208945774677536205_nat_c @ T2 ) ) ) )
            & ( member3840034767840562712_nat_c @ T2 @ ( transitions_c_a_nat @ M2 ) )
            & ( ( produc8776524282647197071_nat_c @ T2 )
              = Q2 )
            & ( ( produc4616334339778799200_nat_c @ ( produc8208945774677536205_nat_c @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_953_h__out__alt__def,axiom,
    ! [M2: fsm_c_a_b,Q2: c,X: a] :
      ( ( h_out_c_a_b @ M2 @ ( product_Pair_c_a @ Q2 @ X ) )
      = ( collect_b
        @ ^ [Uu: b] :
          ? [T2: produc6657052613134160568od_b_c] :
            ( ( Uu
              = ( product_fst_b_c @ ( produc8648342215071535279od_b_c @ ( produc1402652955648401750od_b_c @ T2 ) ) ) )
            & ( member265788643176044257od_b_c @ T2 @ ( transitions_c_a_b @ M2 ) )
            & ( ( produc6763838237438281236od_b_c @ T2 )
              = Q2 )
            & ( ( produc3728844965657535341od_b_c @ ( produc1402652955648401750od_b_c @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_954_h__out__alt__def,axiom,
    ! [M2: fsm_a_b_c,Q2: a,X: b] :
      ( ( h_out_a_b_c @ M2 @ ( product_Pair_a_b @ Q2 @ X ) )
      = ( collect_c
        @ ^ [Uu: c] :
          ? [T2: produc2669932201312201782od_c_a] :
            ( ( Uu
              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) )
            & ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ M2 ) )
            & ( ( produc4225943634317630866od_c_a @ T2 )
              = Q2 )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_955_h__out__alt__def,axiom,
    ! [M2: fsm_b_8881565123868762728_a_nat,Q2: b,X: product_prod_c_a] :
      ( ( h_out_7043617306241251473_a_nat @ M2 @ ( produc3440865914000881661od_c_a @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Uu: nat] :
          ? [T2: produc2057638696429058086_nat_b] :
            ( ( Uu
              = ( product_fst_nat_b @ ( produc1315313831680247492_nat_b @ ( produc6880390115534073714_nat_b @ T2 ) ) ) )
            & ( member6037585776607591613_nat_b @ T2 @ ( transi4389418769516128417_a_nat @ M2 ) )
            & ( ( produc2805774169205826868_nat_b @ T2 )
              = Q2 )
            & ( ( produc1882892339649908358_nat_b @ ( produc6880390115534073714_nat_b @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_956_h__out__alt__def,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,Q2: product_prod_c_a,X: a] :
      ( ( h_out_8774345398997575435_a_a_b @ M2 @ ( produc7661602042108997680_c_a_a @ Q2 @ X ) )
      = ( collect_b
        @ ^ [Uu: b] :
          ? [T2: produc3205353004677322104od_c_a] :
            ( ( Uu
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T2 ) ) ) )
            & ( member6508725278956418977od_c_a @ T2 @ ( transi2941759273464634107_a_a_b @ M2 ) )
            & ( ( produc8171714036265584980od_c_a @ T2 )
              = Q2 )
            & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_957_h__out__alt__def,axiom,
    ! [M2: fsm_Pr1631622435462234230_e_nat,Q2: prefix425356687782099470od_b_c,X: e] :
      ( ( h_out_7860023044450888735_e_nat @ M2 @ ( produc2511600812177743883_b_c_e @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Uu: nat] :
          ? [T2: produc387491461805067041od_b_c] :
            ( ( Uu
              = ( produc3141030035374696313od_b_c @ ( produc8955329273187884848od_b_c @ ( produc5715416556317142637od_b_c @ T2 ) ) ) )
            & ( member4348767779791260600od_b_c @ T2 @ ( transi6891536161183548463_e_nat @ M2 ) )
            & ( ( produc3078523373971156015od_b_c @ T2 )
              = Q2 )
            & ( ( produc8220628286839526386od_b_c @ ( produc5715416556317142637od_b_c @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_958_h__out__alt__def,axiom,
    ! [M2: fsm_a_1723543254342851473_a_nat,Q2: a,X: produc2133359284458116493od_c_a] :
      ( ( h_out_5121717952982053178_a_nat @ M2 @ ( produc5762801557034676134od_c_a @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Uu: nat] :
          ? [T2: produc4554795336555609996_nat_a] :
            ( ( Uu
              = ( product_fst_nat_a @ ( produc5298802324949756203_nat_a @ ( produc2406474377285217112_nat_a @ T2 ) ) ) )
            & ( member9210739242824242467_nat_a @ T2 @ ( transi8412874649895244106_a_nat @ M2 ) )
            & ( ( produc4555553616425619226_nat_a @ T2 )
              = Q2 )
            & ( ( produc1224186378621509357_nat_a @ ( produc2406474377285217112_nat_a @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_959_h__out__alt__def,axiom,
    ! [M2: fsm_a_836089231483141141_c_a_c,Q2: a,X: produc2133359284458116493od_c_a] :
      ( ( h_out_1068671350592293910_c_a_c @ M2 @ ( produc5762801557034676134od_c_a @ Q2 @ X ) )
      = ( collect_c
        @ ^ [Uu: c] :
          ? [T2: produc5283743357988929576od_c_a] :
            ( ( Uu
              = ( product_fst_c_a @ ( produc6859116959324569117od_c_a @ ( produc5173695862223815622od_c_a @ T2 ) ) ) )
            & ( member8587115632268026449od_c_a @ T2 @ ( transi4754753175853207558_c_a_c @ M2 ) )
            & ( ( produc8701775498892594308od_c_a @ T2 )
              = Q2 )
            & ( ( produc410089823003473115od_c_a @ ( produc5173695862223815622od_c_a @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_960_h__out__alt__def,axiom,
    ! [M2: fsm_b_5636252348735175577od_c_a,Q2: b,X: c] :
      ( ( h_out_165584265889796634od_c_a @ M2 @ ( product_Pair_b_c @ Q2 @ X ) )
      = ( collec8405073525749625505od_c_a
        @ ^ [Uu: produc2669932201312201782od_c_a] :
          ? [T2: produc7163242847096752895_c_a_b] :
            ( ( Uu
              = ( produc8128955733147053516_c_a_b @ ( produc7166236969271656501_c_a_b @ ( produc6841312944981846429_c_a_b @ T2 ) ) ) )
            & ( member5468748651639462952_c_a_b @ T2 @ ( transi4984929337990653450od_c_a @ M2 ) )
            & ( ( produc1872875681020232795_c_a_b @ T2 )
              = Q2 )
            & ( ( produc1470944569085659379_c_a_b @ ( produc6841312944981846429_c_a_b @ T2 ) )
              = X ) ) ) ) ).

% h_out_alt_def
thf(fact_961_filter__states__simps_I5_J,axiom,
    ! [P: d > $o,M2: fsm_d_b_c] :
      ( ( P @ ( initial_d_b_c @ M2 ) )
     => ( ( transitions_d_b_c @ ( filter_states_d_b_c @ M2 @ P ) )
        = ( collec7134311822446639841od_c_d
          @ ^ [T2: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ T2 @ ( transitions_d_b_c @ M2 ) )
              & ( P @ ( produc1508465569166014290od_c_d @ T2 ) )
              & ( P @ ( product_snd_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T2 ) ) ) ) ) ) ) ) ).

% filter_states_simps(5)
thf(fact_962_filter__states__simps_I5_J,axiom,
    ! [P: product_prod_c_a > $o,M2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( P @ ( initia681974817753967189_a_a_b @ M2 ) )
     => ( ( transi2941759273464634107_a_a_b @ ( filter1780482953621413783_a_a_b @ M2 @ P ) )
        = ( collec7519840507084740067od_c_a
          @ ^ [T2: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ T2 @ ( transi2941759273464634107_a_a_b @ M2 ) )
              & ( P @ ( produc8171714036265584980od_c_a @ T2 ) )
              & ( P @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T2 ) ) ) ) ) ) ) ) ).

% filter_states_simps(5)
thf(fact_963_filter__states__simps_I5_J,axiom,
    ! [P: a > $o,M2: fsm_a_b_c] :
      ( ( P @ ( initial_a_b_c @ M2 ) )
     => ( ( transitions_a_b_c @ ( filter_states_a_b_c @ M2 @ P ) )
        = ( collec8405073525749625505od_c_a
          @ ^ [T2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ M2 ) )
              & ( P @ ( produc4225943634317630866od_c_a @ T2 ) )
              & ( P @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) ) ) ) ).

% filter_states_simps(5)
thf(fact_964_foldl__elem__check,axiom,
    ! [Xs2: list_P8634110456476873367od_b_c,A4: set_Pr3138385297192119153od_b_c,F: prefix425356687782099470od_b_c > produc3550516665487611281od_b_c > prefix425356687782099470od_b_c,V2: prefix425356687782099470od_b_c] :
      ( ( ord_le6946557262229373713od_b_c @ ( set_Pr6515338384934575398od_b_c @ Xs2 ) @ A4 )
     => ( ( foldl_6461310528730867108od_b_c
          @ ^ [Y5: prefix425356687782099470od_b_c,X4: produc3550516665487611281od_b_c] :
              ( if_Pre4314254153497284564od_b_c
              @ ~ ( member2577541133337307834od_b_c @ X4 @ A4 )
              @ Y5
              @ ( F @ Y5 @ X4 ) )
          @ V2
          @ Xs2 )
        = ( foldl_6461310528730867108od_b_c @ F @ V2 @ Xs2 ) ) ) ).

% foldl_elem_check
thf(fact_965_foldl__elem__check,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a,A4: set_Pr8357624050790843030od_c_a,F: produc4534891796109904283_b_c_e > produc2669932201312201782od_c_a > produc4534891796109904283_b_c_e,V2: produc4534891796109904283_b_c_e] :
      ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) @ A4 )
     => ( ( foldl_2588721813272579068od_c_a
          @ ^ [Y5: produc4534891796109904283_b_c_e,X4: produc2669932201312201782od_c_a] :
              ( if_Pro509616640563734881_b_c_e
              @ ~ ( member5502040268208861279od_c_a @ X4 @ A4 )
              @ Y5
              @ ( F @ Y5 @ X4 ) )
          @ V2
          @ Xs2 )
        = ( foldl_2588721813272579068od_c_a @ F @ V2 @ Xs2 ) ) ) ).

% foldl_elem_check
thf(fact_966_pred__subset__eq2,axiom,
    ! [R: set_Pr3237851069733663483_b_c_e,S3: set_Pr3237851069733663483_b_c_e] :
      ( ( ord_le7893044087947676058_c_e_o
        @ ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) @ R )
        @ ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) @ S3 ) )
      = ( ord_le1795193733426387611_b_c_e @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_967_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_b_c,S3: set_Product_prod_b_c] :
      ( ( ord_less_eq_b_c_o
        @ ^ [X4: b,Y5: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ R )
        @ ^ [X4: b,Y5: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ S3 ) )
      = ( ord_le253122037897321832od_b_c @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_968_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_c_a,S3: set_Product_prod_c_a] :
      ( ( ord_less_eq_c_a_o
        @ ^ [X4: c,Y5: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y5 ) @ R )
        @ ^ [X4: c,Y5: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y5 ) @ S3 ) )
      = ( ord_le8698776994054418981od_c_a @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_969_pred__subset__eq2,axiom,
    ! [R: set_Pr272002107461315949od_c_a,S3: set_Pr272002107461315949od_c_a] :
      ( ( ord_le4841357954007891880_c_a_o
        @ ^ [X4: b,Y5: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y5 ) @ R )
        @ ^ [X4: b,Y5: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y5 ) @ S3 ) )
      = ( ord_le4080174072498570509od_c_a @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_970_pred__subset__eq2,axiom,
    ! [R: set_Pr8357624050790843030od_c_a,S3: set_Pr8357624050790843030od_c_a] :
      ( ( ord_le3047557917845655871_c_a_o
        @ ^ [X4: a,Y5: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y5 ) @ R )
        @ ^ [X4: a,Y5: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y5 ) @ S3 ) )
      = ( ord_le3226338165915859510od_c_a @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_971_empty__Collect__eq,axiom,
    ! [P: produc2669932201312201782od_c_a > $o] :
      ( ( bot_bo267491471446728962od_c_a
        = ( collec8405073525749625505od_c_a @ P ) )
      = ( ! [X4: produc2669932201312201782od_c_a] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_972_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_973_Collect__empty__eq,axiom,
    ! [P: produc2669932201312201782od_c_a > $o] :
      ( ( ( collec8405073525749625505od_c_a @ P )
        = bot_bo267491471446728962od_c_a )
      = ( ! [X4: produc2669932201312201782od_c_a] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_974_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_975_all__not__in__conv,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a] :
      ( ( ! [X4: produc2669932201312201782od_c_a] :
            ~ ( member5502040268208861279od_c_a @ X4 @ A4 ) )
      = ( A4 = bot_bo267491471446728962od_c_a ) ) ).

% all_not_in_conv
thf(fact_976_all__not__in__conv,axiom,
    ! [A4: set_a] :
      ( ( ! [X4: a] :
            ~ ( member_a @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_977_all__not__in__conv,axiom,
    ! [A4: set_li6436108459499378894od_b_c] :
      ( ( ! [X4: list_P903359562653991662od_b_c] :
            ~ ( member6330420149250801815od_b_c @ X4 @ A4 ) )
      = ( A4 = bot_bo4166481423041325370od_b_c ) ) ).

% all_not_in_conv
thf(fact_978_empty__iff,axiom,
    ! [C3: produc2669932201312201782od_c_a] :
      ~ ( member5502040268208861279od_c_a @ C3 @ bot_bo267491471446728962od_c_a ) ).

% empty_iff
thf(fact_979_empty__iff,axiom,
    ! [C3: a] :
      ~ ( member_a @ C3 @ bot_bot_set_a ) ).

% empty_iff
thf(fact_980_empty__iff,axiom,
    ! [C3: list_P903359562653991662od_b_c] :
      ~ ( member6330420149250801815od_b_c @ C3 @ bot_bo4166481423041325370od_b_c ) ).

% empty_iff
thf(fact_981_insert__iff,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ ( insert8105206408023001606od_c_a @ B @ A4 ) )
      = ( ( A = B )
        | ( member5502040268208861279od_c_a @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_982_insert__iff,axiom,
    ! [A: a,B: a,A4: set_a] :
      ( ( member_a @ A @ ( insert_a @ B @ A4 ) )
      = ( ( A = B )
        | ( member_a @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_983_insert__iff,axiom,
    ! [A: list_P903359562653991662od_b_c,B: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ B @ A4 ) )
      = ( ( A = B )
        | ( member6330420149250801815od_b_c @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_984_insertCI,axiom,
    ! [A: produc2669932201312201782od_c_a,B4: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a] :
      ( ( ~ ( member5502040268208861279od_c_a @ A @ B4 )
       => ( A = B ) )
     => ( member5502040268208861279od_c_a @ A @ ( insert8105206408023001606od_c_a @ B @ B4 ) ) ) ).

% insertCI
thf(fact_985_insertCI,axiom,
    ! [A: a,B4: set_a,B: a] :
      ( ( ~ ( member_a @ A @ B4 )
       => ( A = B ) )
     => ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).

% insertCI
thf(fact_986_insertCI,axiom,
    ! [A: list_P903359562653991662od_b_c,B4: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c] :
      ( ( ~ ( member6330420149250801815od_b_c @ A @ B4 )
       => ( A = B ) )
     => ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ B @ B4 ) ) ) ).

% insertCI
thf(fact_987_image__empty,axiom,
    ! [F: a > list_P903359562653991662od_b_c] :
      ( ( image_8656872703560917237od_b_c @ F @ bot_bot_set_a )
      = bot_bo4166481423041325370od_b_c ) ).

% image_empty
thf(fact_988_image__empty,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a] :
      ( ( image_2946797484331234845od_c_a @ F @ bot_bot_set_a )
      = bot_bo3605389620251476834od_c_a ) ).

% image_empty
thf(fact_989_empty__is__image,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A4: set_a] :
      ( ( bot_bo4166481423041325370od_b_c
        = ( image_8656872703560917237od_b_c @ F @ A4 ) )
      = ( A4 = bot_bot_set_a ) ) ).

% empty_is_image
thf(fact_990_empty__is__image,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,A4: set_a] :
      ( ( bot_bo3605389620251476834od_c_a
        = ( image_2946797484331234845od_c_a @ F @ A4 ) )
      = ( A4 = bot_bot_set_a ) ) ).

% empty_is_image
thf(fact_991_image__is__empty,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A4: set_a] :
      ( ( ( image_8656872703560917237od_b_c @ F @ A4 )
        = bot_bo4166481423041325370od_b_c )
      = ( A4 = bot_bot_set_a ) ) ).

% image_is_empty
thf(fact_992_image__is__empty,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,A4: set_a] :
      ( ( ( image_2946797484331234845od_c_a @ F @ A4 )
        = bot_bo3605389620251476834od_c_a )
      = ( A4 = bot_bot_set_a ) ) ).

% image_is_empty
thf(fact_993_inf__bot__left,axiom,
    ! [X: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ bot_bo4166481423041325370od_b_c @ X )
      = bot_bo4166481423041325370od_b_c ) ).

% inf_bot_left
thf(fact_994_inf__bot__right,axiom,
    ! [X: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ X @ bot_bo4166481423041325370od_b_c )
      = bot_bo4166481423041325370od_b_c ) ).

% inf_bot_right
thf(fact_995_subset__empty,axiom,
    ! [A4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ bot_bo4166481423041325370od_b_c )
      = ( A4 = bot_bo4166481423041325370od_b_c ) ) ).

% subset_empty
thf(fact_996_empty__subsetI,axiom,
    ! [A4: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ bot_bo4166481423041325370od_b_c @ A4 ) ).

% empty_subsetI
thf(fact_997_image__insert,axiom,
    ! [F: a > list_P903359562653991662od_b_c,A: a,B4: set_a] :
      ( ( image_8656872703560917237od_b_c @ F @ ( insert_a @ A @ B4 ) )
      = ( insert6227932334100060350od_b_c @ ( F @ A ) @ ( image_8656872703560917237od_b_c @ F @ B4 ) ) ) ).

% image_insert
thf(fact_998_image__insert,axiom,
    ! [F: a > set_Pr8357624050790843030od_c_a,A: a,B4: set_a] :
      ( ( image_2946797484331234845od_c_a @ F @ ( insert_a @ A @ B4 ) )
      = ( insert7672950595763708006od_c_a @ ( F @ A ) @ ( image_2946797484331234845od_c_a @ F @ B4 ) ) ) ).

% image_insert
thf(fact_999_insert__image,axiom,
    ! [X: a,A4: set_a,F: a > list_P903359562653991662od_b_c] :
      ( ( member_a @ X @ A4 )
     => ( ( insert6227932334100060350od_b_c @ ( F @ X ) @ ( image_8656872703560917237od_b_c @ F @ A4 ) )
        = ( image_8656872703560917237od_b_c @ F @ A4 ) ) ) ).

% insert_image
thf(fact_1000_insert__image,axiom,
    ! [X: a,A4: set_a,F: a > set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ X @ A4 )
     => ( ( insert7672950595763708006od_c_a @ ( F @ X ) @ ( image_2946797484331234845od_c_a @ F @ A4 ) )
        = ( image_2946797484331234845od_c_a @ F @ A4 ) ) ) ).

% insert_image
thf(fact_1001_singletonI,axiom,
    ! [A: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ A @ ( insert8105206408023001606od_c_a @ A @ bot_bo267491471446728962od_c_a ) ) ).

% singletonI
thf(fact_1002_singletonI,axiom,
    ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_1003_singletonI,axiom,
    ! [A: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) ).

% singletonI
thf(fact_1004_insert__subset,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ ( insert8105206408023001606od_c_a @ X @ A4 ) @ B4 )
      = ( ( member5502040268208861279od_c_a @ X @ B4 )
        & ( ord_le3226338165915859510od_c_a @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1005_insert__subset,axiom,
    ! [X: a,A4: set_a,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a @ X @ A4 ) @ B4 )
      = ( ( member_a @ X @ B4 )
        & ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1006_insert__subset,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( insert6227932334100060350od_b_c @ X @ A4 ) @ B4 )
      = ( ( member6330420149250801815od_b_c @ X @ B4 )
        & ( ord_le282488521294790766od_b_c @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1007_Int__insert__left__if0,axiom,
    ! [A: produc2669932201312201782od_c_a,C2: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ A @ C2 )
     => ( ( inf_in3815988127369930372od_c_a @ ( insert8105206408023001606od_c_a @ A @ B4 ) @ C2 )
        = ( inf_in3815988127369930372od_c_a @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1008_Int__insert__left__if0,axiom,
    ! [A: a,C2: set_a,B4: set_a] :
      ( ~ ( member_a @ A @ C2 )
     => ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
        = ( inf_inf_set_a @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1009_Int__insert__left__if0,axiom,
    ! [A: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ A @ C2 )
     => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ B4 ) @ C2 )
        = ( inf_in4978071631833541052od_b_c @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1010_Int__insert__left__if1,axiom,
    ! [A: produc2669932201312201782od_c_a,C2: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ C2 )
     => ( ( inf_in3815988127369930372od_c_a @ ( insert8105206408023001606od_c_a @ A @ B4 ) @ C2 )
        = ( insert8105206408023001606od_c_a @ A @ ( inf_in3815988127369930372od_c_a @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1011_Int__insert__left__if1,axiom,
    ! [A: a,C2: set_a,B4: set_a] :
      ( ( member_a @ A @ C2 )
     => ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
        = ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1012_Int__insert__left__if1,axiom,
    ! [A: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ C2 )
     => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ B4 ) @ C2 )
        = ( insert6227932334100060350od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1013_insert__inter__insert,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ A4 ) @ ( insert6227932334100060350od_b_c @ A @ B4 ) )
      = ( insert6227932334100060350od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_1014_Int__insert__right__if0,axiom,
    ! [A: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ A @ A4 )
     => ( ( inf_in3815988127369930372od_c_a @ A4 @ ( insert8105206408023001606od_c_a @ A @ B4 ) )
        = ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1015_Int__insert__right__if0,axiom,
    ! [A: a,A4: set_a,B4: set_a] :
      ( ~ ( member_a @ A @ A4 )
     => ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B4 ) )
        = ( inf_inf_set_a @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1016_Int__insert__right__if0,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ A @ A4 )
     => ( ( inf_in4978071631833541052od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ A @ B4 ) )
        = ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1017_Int__insert__right__if1,axiom,
    ! [A: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ A4 )
     => ( ( inf_in3815988127369930372od_c_a @ A4 @ ( insert8105206408023001606od_c_a @ A @ B4 ) )
        = ( insert8105206408023001606od_c_a @ A @ ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1018_Int__insert__right__if1,axiom,
    ! [A: a,A4: set_a,B4: set_a] :
      ( ( member_a @ A @ A4 )
     => ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B4 ) )
        = ( insert_a @ A @ ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1019_Int__insert__right__if1,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ A4 )
     => ( ( inf_in4978071631833541052od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ A @ B4 ) )
        = ( insert6227932334100060350od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1020_filter__states__simps_I1_J,axiom,
    ! [P: a > $o,M2: fsm_a_b_c] :
      ( ( P @ ( initial_a_b_c @ M2 ) )
     => ( ( initial_a_b_c @ ( filter_states_a_b_c @ M2 @ P ) )
        = ( initial_a_b_c @ M2 ) ) ) ).

% filter_states_simps(1)
thf(fact_1021_filter__states__simps_I1_J,axiom,
    ! [P: d > $o,M2: fsm_d_b_c] :
      ( ( P @ ( initial_d_b_c @ M2 ) )
     => ( ( initial_d_b_c @ ( filter_states_d_b_c @ M2 @ P ) )
        = ( initial_d_b_c @ M2 ) ) ) ).

% filter_states_simps(1)
thf(fact_1022_singleton__conv,axiom,
    ! [A: produc2669932201312201782od_c_a] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X4: produc2669932201312201782od_c_a] : ( X4 = A ) )
      = ( insert8105206408023001606od_c_a @ A @ bot_bo267491471446728962od_c_a ) ) ).

% singleton_conv
thf(fact_1023_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( X4 = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_1024_singleton__conv2,axiom,
    ! [A: produc2669932201312201782od_c_a] :
      ( ( collec8405073525749625505od_c_a
        @ ( ^ [Y4: produc2669932201312201782od_c_a,Z: produc2669932201312201782od_c_a] : ( Y4 = Z )
          @ A ) )
      = ( insert8105206408023001606od_c_a @ A @ bot_bo267491471446728962od_c_a ) ) ).

% singleton_conv2
thf(fact_1025_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y4: nat,Z: nat] : ( Y4 = Z )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_1026_set__empty2,axiom,
    ! [Xs2: list_P903359562653991662od_b_c] :
      ( ( bot_bo2863795366942399540od_b_c
        = ( set_Product_prod_b_c2 @ Xs2 ) )
      = ( Xs2 = nil_Product_prod_b_c ) ) ).

% set_empty2
thf(fact_1027_set__empty2,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a] :
      ( ( bot_bo267491471446728962od_c_a
        = ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
      = ( Xs2 = nil_Pr1342775757158464060od_c_a ) ) ).

% set_empty2
thf(fact_1028_List_Oset__empty,axiom,
    ! [Xs2: list_P903359562653991662od_b_c] :
      ( ( ( set_Product_prod_b_c2 @ Xs2 )
        = bot_bo2863795366942399540od_b_c )
      = ( Xs2 = nil_Product_prod_b_c ) ) ).

% List.set_empty
thf(fact_1029_List_Oset__empty,axiom,
    ! [Xs2: list_P6327159017948738492od_c_a] :
      ( ( ( set_Pr5664641634098845387od_c_a @ Xs2 )
        = bot_bo267491471446728962od_c_a )
      = ( Xs2 = nil_Pr1342775757158464060od_c_a ) ) ).

% List.set_empty
thf(fact_1030_list_Osimps_I15_J,axiom,
    ! [X21: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a] :
      ( ( set_Pr5664641634098845387od_c_a @ ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) )
      = ( insert8105206408023001606od_c_a @ X21 @ ( set_Pr5664641634098845387od_c_a @ X22 ) ) ) ).

% list.simps(15)
thf(fact_1031_list_Osimps_I15_J,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( ( set_Product_prod_b_c2 @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) )
      = ( insert1747480808674937912od_b_c @ X21 @ ( set_Product_prod_b_c2 @ X22 ) ) ) ).

% list.simps(15)
thf(fact_1032_singleton__insert__inj__eq,axiom,
    ! [B: list_P903359562653991662od_b_c,A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( ( insert6227932334100060350od_b_c @ B @ bot_bo4166481423041325370od_b_c )
        = ( insert6227932334100060350od_b_c @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_le282488521294790766od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ B @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1033_singleton__insert__inj__eq_H,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c] :
      ( ( ( insert6227932334100060350od_b_c @ A @ A4 )
        = ( insert6227932334100060350od_b_c @ B @ bot_bo4166481423041325370od_b_c ) )
      = ( ( A = B )
        & ( ord_le282488521294790766od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ B @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1034_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( bot_bo267491471446728962od_c_a
        = ( inf_in3815988127369930372od_c_a @ A4 @ ( insert8105206408023001606od_c_a @ B @ B4 ) ) )
      = ( ~ ( member5502040268208861279od_c_a @ B @ A4 )
        & ( bot_bo267491471446728962od_c_a
          = ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1035_disjoint__insert_I2_J,axiom,
    ! [A4: set_a,B: a,B4: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ A4 @ ( insert_a @ B @ B4 ) ) )
      = ( ~ ( member_a @ B @ A4 )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1036_disjoint__insert_I2_J,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( bot_bo4166481423041325370od_b_c
        = ( inf_in4978071631833541052od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ B @ B4 ) ) )
      = ( ~ ( member6330420149250801815od_b_c @ B @ A4 )
        & ( bot_bo4166481423041325370od_b_c
          = ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1037_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( ( inf_in3815988127369930372od_c_a @ B4 @ ( insert8105206408023001606od_c_a @ A @ A4 ) )
        = bot_bo267491471446728962od_c_a )
      = ( ~ ( member5502040268208861279od_c_a @ A @ B4 )
        & ( ( inf_in3815988127369930372od_c_a @ B4 @ A4 )
          = bot_bo267491471446728962od_c_a ) ) ) ).

% disjoint_insert(1)
thf(fact_1038_disjoint__insert_I1_J,axiom,
    ! [B4: set_a,A: a,A4: set_a] :
      ( ( ( inf_inf_set_a @ B4 @ ( insert_a @ A @ A4 ) )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A @ B4 )
        & ( ( inf_inf_set_a @ B4 @ A4 )
          = bot_bot_set_a ) ) ) ).

% disjoint_insert(1)
thf(fact_1039_disjoint__insert_I1_J,axiom,
    ! [B4: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ B4 @ ( insert6227932334100060350od_b_c @ A @ A4 ) )
        = bot_bo4166481423041325370od_b_c )
      = ( ~ ( member6330420149250801815od_b_c @ A @ B4 )
        & ( ( inf_in4978071631833541052od_b_c @ B4 @ A4 )
          = bot_bo4166481423041325370od_b_c ) ) ) ).

% disjoint_insert(1)
thf(fact_1040_insert__disjoint_I2_J,axiom,
    ! [A: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( bot_bo267491471446728962od_c_a
        = ( inf_in3815988127369930372od_c_a @ ( insert8105206408023001606od_c_a @ A @ A4 ) @ B4 ) )
      = ( ~ ( member5502040268208861279od_c_a @ A @ B4 )
        & ( bot_bo267491471446728962od_c_a
          = ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1041_insert__disjoint_I2_J,axiom,
    ! [A: a,A4: set_a,B4: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ ( insert_a @ A @ A4 ) @ B4 ) )
      = ( ~ ( member_a @ A @ B4 )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1042_insert__disjoint_I2_J,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( bot_bo4166481423041325370od_b_c
        = ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ A4 ) @ B4 ) )
      = ( ~ ( member6330420149250801815od_b_c @ A @ B4 )
        & ( bot_bo4166481423041325370od_b_c
          = ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1043_insert__disjoint_I1_J,axiom,
    ! [A: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( ( inf_in3815988127369930372od_c_a @ ( insert8105206408023001606od_c_a @ A @ A4 ) @ B4 )
        = bot_bo267491471446728962od_c_a )
      = ( ~ ( member5502040268208861279od_c_a @ A @ B4 )
        & ( ( inf_in3815988127369930372od_c_a @ A4 @ B4 )
          = bot_bo267491471446728962od_c_a ) ) ) ).

% insert_disjoint(1)
thf(fact_1044_insert__disjoint_I1_J,axiom,
    ! [A: a,A4: set_a,B4: set_a] :
      ( ( ( inf_inf_set_a @ ( insert_a @ A @ A4 ) @ B4 )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A @ B4 )
        & ( ( inf_inf_set_a @ A4 @ B4 )
          = bot_bot_set_a ) ) ) ).

% insert_disjoint(1)
thf(fact_1045_insert__disjoint_I1_J,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ A4 ) @ B4 )
        = bot_bo4166481423041325370od_b_c )
      = ( ~ ( member6330420149250801815od_b_c @ A @ B4 )
        & ( ( inf_in4978071631833541052od_b_c @ A4 @ B4 )
          = bot_bo4166481423041325370od_b_c ) ) ) ).

% insert_disjoint(1)
thf(fact_1046_filter__states__simps_I3_J,axiom,
    ! [P: a > $o,M2: fsm_a_b_c] :
      ( ( P @ ( initial_a_b_c @ M2 ) )
     => ( ( inputs_a_b_c @ ( filter_states_a_b_c @ M2 @ P ) )
        = ( inputs_a_b_c @ M2 ) ) ) ).

% filter_states_simps(3)
thf(fact_1047_filter__states__simps_I3_J,axiom,
    ! [P: d > $o,M2: fsm_d_b_c] :
      ( ( P @ ( initial_d_b_c @ M2 ) )
     => ( ( inputs_d_b_c @ ( filter_states_d_b_c @ M2 @ P ) )
        = ( inputs_d_b_c @ M2 ) ) ) ).

% filter_states_simps(3)
thf(fact_1048_filter__states__simps_I4_J,axiom,
    ! [P: d > $o,M2: fsm_d_b_c] :
      ( ( P @ ( initial_d_b_c @ M2 ) )
     => ( ( outputs_d_b_c @ ( filter_states_d_b_c @ M2 @ P ) )
        = ( outputs_d_b_c @ M2 ) ) ) ).

% filter_states_simps(4)
thf(fact_1049_filter__states__simps_I4_J,axiom,
    ! [P: a > $o,M2: fsm_a_b_c] :
      ( ( P @ ( initial_a_b_c @ M2 ) )
     => ( ( outputs_a_b_c @ ( filter_states_a_b_c @ M2 @ P ) )
        = ( outputs_a_b_c @ M2 ) ) ) ).

% filter_states_simps(4)
thf(fact_1050_subset__singleton__iff,axiom,
    ! [X7: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X7 @ ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) )
      = ( ( X7 = bot_bo4166481423041325370od_b_c )
        | ( X7
          = ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% subset_singleton_iff
thf(fact_1051_subset__singletonD,axiom,
    ! [A4: set_li6436108459499378894od_b_c,X: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ X @ bot_bo4166481423041325370od_b_c ) )
     => ( ( A4 = bot_bo4166481423041325370od_b_c )
        | ( A4
          = ( insert6227932334100060350od_b_c @ X @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% subset_singletonD
thf(fact_1052_subset__insertI2,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
     => ( ord_le282488521294790766od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_1053_subset__insertI,axiom,
    ! [B4: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c] : ( ord_le282488521294790766od_b_c @ B4 @ ( insert6227932334100060350od_b_c @ A @ B4 ) ) ).

% subset_insertI
thf(fact_1054_subset__insert,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X @ A4 )
     => ( ( ord_le3226338165915859510od_c_a @ A4 @ ( insert8105206408023001606od_c_a @ X @ B4 ) )
        = ( ord_le3226338165915859510od_c_a @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1055_subset__insert,axiom,
    ! [X: a,A4: set_a,B4: set_a] :
      ( ~ ( member_a @ X @ A4 )
     => ( ( ord_less_eq_set_a @ A4 @ ( insert_a @ X @ B4 ) )
        = ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1056_subset__insert,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X @ A4 )
     => ( ( ord_le282488521294790766od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ X @ B4 ) )
        = ( ord_le282488521294790766od_b_c @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1057_insert__mono,axiom,
    ! [C2: set_li6436108459499378894od_b_c,D3: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ C2 @ D3 )
     => ( ord_le282488521294790766od_b_c @ ( insert6227932334100060350od_b_c @ A @ C2 ) @ ( insert6227932334100060350od_b_c @ A @ D3 ) ) ) ).

% insert_mono
thf(fact_1058_Int__insert__right,axiom,
    ! [A: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( ( member5502040268208861279od_c_a @ A @ A4 )
       => ( ( inf_in3815988127369930372od_c_a @ A4 @ ( insert8105206408023001606od_c_a @ A @ B4 ) )
          = ( insert8105206408023001606od_c_a @ A @ ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) ) ) )
      & ( ~ ( member5502040268208861279od_c_a @ A @ A4 )
       => ( ( inf_in3815988127369930372od_c_a @ A4 @ ( insert8105206408023001606od_c_a @ A @ B4 ) )
          = ( inf_in3815988127369930372od_c_a @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_1059_Int__insert__right,axiom,
    ! [A: a,A4: set_a,B4: set_a] :
      ( ( ( member_a @ A @ A4 )
       => ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B4 ) )
          = ( insert_a @ A @ ( inf_inf_set_a @ A4 @ B4 ) ) ) )
      & ( ~ ( member_a @ A @ A4 )
       => ( ( inf_inf_set_a @ A4 @ ( insert_a @ A @ B4 ) )
          = ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_1060_Int__insert__right,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ( member6330420149250801815od_b_c @ A @ A4 )
       => ( ( inf_in4978071631833541052od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ A @ B4 ) )
          = ( insert6227932334100060350od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) )
      & ( ~ ( member6330420149250801815od_b_c @ A @ A4 )
       => ( ( inf_in4978071631833541052od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ A @ B4 ) )
          = ( inf_in4978071631833541052od_b_c @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_1061_Int__insert__left,axiom,
    ! [A: produc2669932201312201782od_c_a,C2: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( ( member5502040268208861279od_c_a @ A @ C2 )
       => ( ( inf_in3815988127369930372od_c_a @ ( insert8105206408023001606od_c_a @ A @ B4 ) @ C2 )
          = ( insert8105206408023001606od_c_a @ A @ ( inf_in3815988127369930372od_c_a @ B4 @ C2 ) ) ) )
      & ( ~ ( member5502040268208861279od_c_a @ A @ C2 )
       => ( ( inf_in3815988127369930372od_c_a @ ( insert8105206408023001606od_c_a @ A @ B4 ) @ C2 )
          = ( inf_in3815988127369930372od_c_a @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1062_Int__insert__left,axiom,
    ! [A: a,C2: set_a,B4: set_a] :
      ( ( ( member_a @ A @ C2 )
       => ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
          = ( insert_a @ A @ ( inf_inf_set_a @ B4 @ C2 ) ) ) )
      & ( ~ ( member_a @ A @ C2 )
       => ( ( inf_inf_set_a @ ( insert_a @ A @ B4 ) @ C2 )
          = ( inf_inf_set_a @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1063_Int__insert__left,axiom,
    ! [A: list_P903359562653991662od_b_c,C2: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ( member6330420149250801815od_b_c @ A @ C2 )
       => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ B4 ) @ C2 )
          = ( insert6227932334100060350od_b_c @ A @ ( inf_in4978071631833541052od_b_c @ B4 @ C2 ) ) ) )
      & ( ~ ( member6330420149250801815od_b_c @ A @ C2 )
       => ( ( inf_in4978071631833541052od_b_c @ ( insert6227932334100060350od_b_c @ A @ B4 ) @ C2 )
          = ( inf_in4978071631833541052od_b_c @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1064_Set_Oempty__def,axiom,
    ( bot_bo267491471446728962od_c_a
    = ( collec8405073525749625505od_c_a
      @ ^ [X4: produc2669932201312201782od_c_a] : $false ) ) ).

% Set.empty_def
thf(fact_1065_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X4: nat] : $false ) ) ).

% Set.empty_def
thf(fact_1066_insert__compr,axiom,
    ( insert_a
    = ( ^ [A6: a,B5: set_a] :
          ( collect_a
          @ ^ [X4: a] :
              ( ( X4 = A6 )
              | ( member_a @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_1067_insert__compr,axiom,
    ( insert6227932334100060350od_b_c
    = ( ^ [A6: list_P903359562653991662od_b_c,B5: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X4: list_P903359562653991662od_b_c] :
              ( ( X4 = A6 )
              | ( member6330420149250801815od_b_c @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_1068_insert__compr,axiom,
    ( insert8105206408023001606od_c_a
    = ( ^ [A6: produc2669932201312201782od_c_a,B5: set_Pr8357624050790843030od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [X4: produc2669932201312201782od_c_a] :
              ( ( X4 = A6 )
              | ( member5502040268208861279od_c_a @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_1069_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A6: nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( X4 = A6 )
              | ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_1070_insert__Collect,axiom,
    ! [A: produc2669932201312201782od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( insert8105206408023001606od_c_a @ A @ ( collec8405073525749625505od_c_a @ P ) )
      = ( collec8405073525749625505od_c_a
        @ ^ [U2: produc2669932201312201782od_c_a] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_1071_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_1072_Collect__conv__if,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,A: produc2669932201312201782od_c_a] :
      ( ( ( P @ A )
       => ( ( collec8405073525749625505od_c_a
            @ ^ [X4: produc2669932201312201782od_c_a] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert8105206408023001606od_c_a @ A @ bot_bo267491471446728962od_c_a ) ) )
      & ( ~ ( P @ A )
       => ( ( collec8405073525749625505od_c_a
            @ ^ [X4: produc2669932201312201782od_c_a] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bo267491471446728962od_c_a ) ) ) ).

% Collect_conv_if
thf(fact_1073_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_1074_Collect__conv__if2,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,A: produc2669932201312201782od_c_a] :
      ( ( ( P @ A )
       => ( ( collec8405073525749625505od_c_a
            @ ^ [X4: produc2669932201312201782od_c_a] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert8105206408023001606od_c_a @ A @ bot_bo267491471446728962od_c_a ) ) )
      & ( ~ ( P @ A )
       => ( ( collec8405073525749625505od_c_a
            @ ^ [X4: produc2669932201312201782od_c_a] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bo267491471446728962od_c_a ) ) ) ).

% Collect_conv_if2
thf(fact_1075_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_1076_mk__disjoint__insert,axiom,
    ! [A: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ A4 )
     => ? [B7: set_Pr8357624050790843030od_c_a] :
          ( ( A4
            = ( insert8105206408023001606od_c_a @ A @ B7 ) )
          & ~ ( member5502040268208861279od_c_a @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_1077_mk__disjoint__insert,axiom,
    ! [A: a,A4: set_a] :
      ( ( member_a @ A @ A4 )
     => ? [B7: set_a] :
          ( ( A4
            = ( insert_a @ A @ B7 ) )
          & ~ ( member_a @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_1078_mk__disjoint__insert,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ A4 )
     => ? [B7: set_li6436108459499378894od_b_c] :
          ( ( A4
            = ( insert6227932334100060350od_b_c @ A @ B7 ) )
          & ~ ( member6330420149250801815od_b_c @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_1079_singleton__iff,axiom,
    ! [B: produc2669932201312201782od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( insert8105206408023001606od_c_a @ A @ bot_bo267491471446728962od_c_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1080_singleton__iff,axiom,
    ! [B: a,A: a] :
      ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1081_singleton__iff,axiom,
    ! [B: list_P903359562653991662od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ B @ ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_1082_insert__eq__iff,axiom,
    ! [A: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ A @ A4 )
     => ( ~ ( member5502040268208861279od_c_a @ B @ B4 )
       => ( ( ( insert8105206408023001606od_c_a @ A @ A4 )
            = ( insert8105206408023001606od_c_a @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C6: set_Pr8357624050790843030od_c_a] :
                  ( ( A4
                    = ( insert8105206408023001606od_c_a @ B @ C6 ) )
                  & ~ ( member5502040268208861279od_c_a @ B @ C6 )
                  & ( B4
                    = ( insert8105206408023001606od_c_a @ A @ C6 ) )
                  & ~ ( member5502040268208861279od_c_a @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1083_insert__eq__iff,axiom,
    ! [A: a,A4: set_a,B: a,B4: set_a] :
      ( ~ ( member_a @ A @ A4 )
     => ( ~ ( member_a @ B @ B4 )
       => ( ( ( insert_a @ A @ A4 )
            = ( insert_a @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C6: set_a] :
                  ( ( A4
                    = ( insert_a @ B @ C6 ) )
                  & ~ ( member_a @ B @ C6 )
                  & ( B4
                    = ( insert_a @ A @ C6 ) )
                  & ~ ( member_a @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1084_insert__eq__iff,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ A @ A4 )
     => ( ~ ( member6330420149250801815od_b_c @ B @ B4 )
       => ( ( ( insert6227932334100060350od_b_c @ A @ A4 )
            = ( insert6227932334100060350od_b_c @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C6: set_li6436108459499378894od_b_c] :
                  ( ( A4
                    = ( insert6227932334100060350od_b_c @ B @ C6 ) )
                  & ~ ( member6330420149250801815od_b_c @ B @ C6 )
                  & ( B4
                    = ( insert6227932334100060350od_b_c @ A @ C6 ) )
                  & ~ ( member6330420149250801815od_b_c @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1085_insert__absorb,axiom,
    ! [A: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ A4 )
     => ( ( insert8105206408023001606od_c_a @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1086_insert__absorb,axiom,
    ! [A: a,A4: set_a] :
      ( ( member_a @ A @ A4 )
     => ( ( insert_a @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1087_insert__absorb,axiom,
    ! [A: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ A4 )
     => ( ( insert6227932334100060350od_b_c @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1088_insert__ident,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X @ A4 )
     => ( ~ ( member5502040268208861279od_c_a @ X @ B4 )
       => ( ( ( insert8105206408023001606od_c_a @ X @ A4 )
            = ( insert8105206408023001606od_c_a @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1089_insert__ident,axiom,
    ! [X: a,A4: set_a,B4: set_a] :
      ( ~ ( member_a @ X @ A4 )
     => ( ~ ( member_a @ X @ B4 )
       => ( ( ( insert_a @ X @ A4 )
            = ( insert_a @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1090_insert__ident,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X @ A4 )
     => ( ~ ( member6330420149250801815od_b_c @ X @ B4 )
       => ( ( ( insert6227932334100060350od_b_c @ X @ A4 )
            = ( insert6227932334100060350od_b_c @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1091_singletonD,axiom,
    ! [B: produc2669932201312201782od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( insert8105206408023001606od_c_a @ A @ bot_bo267491471446728962od_c_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1092_singletonD,axiom,
    ! [B: a,A: a] :
      ( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1093_singletonD,axiom,
    ! [B: list_P903359562653991662od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ B @ ( insert6227932334100060350od_b_c @ A @ bot_bo4166481423041325370od_b_c ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_1094_Set_Oset__insert,axiom,
    ! [X: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ A4 )
     => ~ ! [B7: set_Pr8357624050790843030od_c_a] :
            ( ( A4
              = ( insert8105206408023001606od_c_a @ X @ B7 ) )
           => ( member5502040268208861279od_c_a @ X @ B7 ) ) ) ).

% Set.set_insert
thf(fact_1095_Set_Oset__insert,axiom,
    ! [X: a,A4: set_a] :
      ( ( member_a @ X @ A4 )
     => ~ ! [B7: set_a] :
            ( ( A4
              = ( insert_a @ X @ B7 ) )
           => ( member_a @ X @ B7 ) ) ) ).

% Set.set_insert
thf(fact_1096_Set_Oset__insert,axiom,
    ! [X: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ A4 )
     => ~ ! [B7: set_li6436108459499378894od_b_c] :
            ( ( A4
              = ( insert6227932334100060350od_b_c @ X @ B7 ) )
           => ( member6330420149250801815od_b_c @ X @ B7 ) ) ) ).

% Set.set_insert
thf(fact_1097_ex__in__conv,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a] :
      ( ( ? [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ A4 ) )
      = ( A4 != bot_bo267491471446728962od_c_a ) ) ).

% ex_in_conv
thf(fact_1098_ex__in__conv,axiom,
    ! [A4: set_a] :
      ( ( ? [X4: a] : ( member_a @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_1099_ex__in__conv,axiom,
    ! [A4: set_li6436108459499378894od_b_c] :
      ( ( ? [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ A4 ) )
      = ( A4 != bot_bo4166481423041325370od_b_c ) ) ).

% ex_in_conv
thf(fact_1100_insertI2,axiom,
    ! [A: produc2669932201312201782od_c_a,B4: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ B4 )
     => ( member5502040268208861279od_c_a @ A @ ( insert8105206408023001606od_c_a @ B @ B4 ) ) ) ).

% insertI2
thf(fact_1101_insertI2,axiom,
    ! [A: a,B4: set_a,B: a] :
      ( ( member_a @ A @ B4 )
     => ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).

% insertI2
thf(fact_1102_insertI2,axiom,
    ! [A: list_P903359562653991662od_b_c,B4: set_li6436108459499378894od_b_c,B: list_P903359562653991662od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ B4 )
     => ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ B @ B4 ) ) ) ).

% insertI2
thf(fact_1103_insertI1,axiom,
    ! [A: produc2669932201312201782od_c_a,B4: set_Pr8357624050790843030od_c_a] : ( member5502040268208861279od_c_a @ A @ ( insert8105206408023001606od_c_a @ A @ B4 ) ) ).

% insertI1
thf(fact_1104_insertI1,axiom,
    ! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).

% insertI1
thf(fact_1105_insertI1,axiom,
    ! [A: list_P903359562653991662od_b_c,B4: set_li6436108459499378894od_b_c] : ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ A @ B4 ) ) ).

% insertI1
thf(fact_1106_equals0I,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a] :
      ( ! [Y3: produc2669932201312201782od_c_a] :
          ~ ( member5502040268208861279od_c_a @ Y3 @ A4 )
     => ( A4 = bot_bo267491471446728962od_c_a ) ) ).

% equals0I
thf(fact_1107_equals0I,axiom,
    ! [A4: set_a] :
      ( ! [Y3: a] :
          ~ ( member_a @ Y3 @ A4 )
     => ( A4 = bot_bot_set_a ) ) ).

% equals0I
thf(fact_1108_equals0I,axiom,
    ! [A4: set_li6436108459499378894od_b_c] :
      ( ! [Y3: list_P903359562653991662od_b_c] :
          ~ ( member6330420149250801815od_b_c @ Y3 @ A4 )
     => ( A4 = bot_bo4166481423041325370od_b_c ) ) ).

% equals0I
thf(fact_1109_equals0D,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( A4 = bot_bo267491471446728962od_c_a )
     => ~ ( member5502040268208861279od_c_a @ A @ A4 ) ) ).

% equals0D
thf(fact_1110_equals0D,axiom,
    ! [A4: set_a,A: a] :
      ( ( A4 = bot_bot_set_a )
     => ~ ( member_a @ A @ A4 ) ) ).

% equals0D
thf(fact_1111_equals0D,axiom,
    ! [A4: set_li6436108459499378894od_b_c,A: list_P903359562653991662od_b_c] :
      ( ( A4 = bot_bo4166481423041325370od_b_c )
     => ~ ( member6330420149250801815od_b_c @ A @ A4 ) ) ).

% equals0D
thf(fact_1112_insertE,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a,A4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ ( insert8105206408023001606od_c_a @ B @ A4 ) )
     => ( ( A != B )
       => ( member5502040268208861279od_c_a @ A @ A4 ) ) ) ).

% insertE
thf(fact_1113_insertE,axiom,
    ! [A: a,B: a,A4: set_a] :
      ( ( member_a @ A @ ( insert_a @ B @ A4 ) )
     => ( ( A != B )
       => ( member_a @ A @ A4 ) ) ) ).

% insertE
thf(fact_1114_insertE,axiom,
    ! [A: list_P903359562653991662od_b_c,B: list_P903359562653991662od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( member6330420149250801815od_b_c @ A @ ( insert6227932334100060350od_b_c @ B @ A4 ) )
     => ( ( A != B )
       => ( member6330420149250801815od_b_c @ A @ A4 ) ) ) ).

% insertE
thf(fact_1115_emptyE,axiom,
    ! [A: produc2669932201312201782od_c_a] :
      ~ ( member5502040268208861279od_c_a @ A @ bot_bo267491471446728962od_c_a ) ).

% emptyE
thf(fact_1116_emptyE,axiom,
    ! [A: a] :
      ~ ( member_a @ A @ bot_bot_set_a ) ).

% emptyE
thf(fact_1117_emptyE,axiom,
    ! [A: list_P903359562653991662od_b_c] :
      ~ ( member6330420149250801815od_b_c @ A @ bot_bo4166481423041325370od_b_c ) ).

% emptyE
thf(fact_1118_image__constant,axiom,
    ! [X: a,A4: set_a,C3: list_P903359562653991662od_b_c] :
      ( ( member_a @ X @ A4 )
     => ( ( image_8656872703560917237od_b_c
          @ ^ [X4: a] : C3
          @ A4 )
        = ( insert6227932334100060350od_b_c @ C3 @ bot_bo4166481423041325370od_b_c ) ) ) ).

% image_constant
thf(fact_1119_image__constant,axiom,
    ! [X: a,A4: set_a,C3: set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ X @ A4 )
     => ( ( image_2946797484331234845od_c_a
          @ ^ [X4: a] : C3
          @ A4 )
        = ( insert7672950595763708006od_c_a @ C3 @ bot_bo3605389620251476834od_c_a ) ) ) ).

% image_constant
thf(fact_1120_image__constant__conv,axiom,
    ! [A4: set_a,C3: list_P903359562653991662od_b_c] :
      ( ( ( A4 = bot_bot_set_a )
       => ( ( image_8656872703560917237od_b_c
            @ ^ [X4: a] : C3
            @ A4 )
          = bot_bo4166481423041325370od_b_c ) )
      & ( ( A4 != bot_bot_set_a )
       => ( ( image_8656872703560917237od_b_c
            @ ^ [X4: a] : C3
            @ A4 )
          = ( insert6227932334100060350od_b_c @ C3 @ bot_bo4166481423041325370od_b_c ) ) ) ) ).

% image_constant_conv
thf(fact_1121_image__constant__conv,axiom,
    ! [A4: set_a,C3: set_Pr8357624050790843030od_c_a] :
      ( ( ( A4 = bot_bot_set_a )
       => ( ( image_2946797484331234845od_c_a
            @ ^ [X4: a] : C3
            @ A4 )
          = bot_bo3605389620251476834od_c_a ) )
      & ( ( A4 != bot_bot_set_a )
       => ( ( image_2946797484331234845od_c_a
            @ ^ [X4: a] : C3
            @ A4 )
          = ( insert7672950595763708006od_c_a @ C3 @ bot_bo3605389620251476834od_c_a ) ) ) ) ).

% image_constant_conv
thf(fact_1122_Int__emptyI,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X3 @ A4 )
         => ~ ( member5502040268208861279od_c_a @ X3 @ B4 ) )
     => ( ( inf_in3815988127369930372od_c_a @ A4 @ B4 )
        = bot_bo267491471446728962od_c_a ) ) ).

% Int_emptyI
thf(fact_1123_Int__emptyI,axiom,
    ! [A4: set_a,B4: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A4 )
         => ~ ( member_a @ X3 @ B4 ) )
     => ( ( inf_inf_set_a @ A4 @ B4 )
        = bot_bot_set_a ) ) ).

% Int_emptyI
thf(fact_1124_Int__emptyI,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A4 )
         => ~ ( member6330420149250801815od_b_c @ X3 @ B4 ) )
     => ( ( inf_in4978071631833541052od_b_c @ A4 @ B4 )
        = bot_bo4166481423041325370od_b_c ) ) ).

% Int_emptyI
thf(fact_1125_disjoint__iff,axiom,
    ! [A4: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
      ( ( ( inf_in3815988127369930372od_c_a @ A4 @ B4 )
        = bot_bo267491471446728962od_c_a )
      = ( ! [X4: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X4 @ A4 )
           => ~ ( member5502040268208861279od_c_a @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1126_disjoint__iff,axiom,
    ! [A4: set_a,B4: set_a] :
      ( ( ( inf_inf_set_a @ A4 @ B4 )
        = bot_bot_set_a )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A4 )
           => ~ ( member_a @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1127_disjoint__iff,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ A4 @ B4 )
        = bot_bo4166481423041325370od_b_c )
      = ( ! [X4: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X4 @ A4 )
           => ~ ( member6330420149250801815od_b_c @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1128_Int__empty__left,axiom,
    ! [B4: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ bot_bo4166481423041325370od_b_c @ B4 )
      = bot_bo4166481423041325370od_b_c ) ).

% Int_empty_left
thf(fact_1129_Int__empty__right,axiom,
    ! [A4: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ A4 @ bot_bo4166481423041325370od_b_c )
      = bot_bo4166481423041325370od_b_c ) ).

% Int_empty_right
thf(fact_1130_disjoint__iff__not__equal,axiom,
    ! [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
      ( ( ( inf_in4978071631833541052od_b_c @ A4 @ B4 )
        = bot_bo4166481423041325370od_b_c )
      = ( ! [X4: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ X4 @ A4 )
           => ! [Y5: list_P903359562653991662od_b_c] :
                ( ( member6330420149250801815od_b_c @ Y5 @ B4 )
               => ( X4 != Y5 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1131_preserves__divergence__insertE_I1_J,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,Beta: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( preser261684820601417592_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ Beta @ X7 ) )
     => ( preser261684820601417592_b_c_d @ M12 @ M23 @ X7 ) ) ).

% preserves_divergence_insertE(1)
thf(fact_1132_language__empty__io,axiom,
    ! [M2: fsm_d_b_c] :
      ( ( ( ( inputs_d_b_c @ M2 )
          = bot_bot_set_b )
        | ( ( outputs_d_b_c @ M2 )
          = bot_bot_set_c ) )
     => ( ( lS_d_b_c @ M2 @ ( initial_d_b_c @ M2 ) )
        = ( insert6227932334100060350od_b_c @ nil_Product_prod_b_c @ bot_bo4166481423041325370od_b_c ) ) ) ).

% language_empty_io
thf(fact_1133_language__empty__io,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( ( ( inputs_a_b_c @ M2 )
          = bot_bot_set_b )
        | ( ( outputs_a_b_c @ M2 )
          = bot_bot_set_c ) )
     => ( ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) )
        = ( insert6227932334100060350od_b_c @ nil_Product_prod_b_c @ bot_bo4166481423041325370od_b_c ) ) ) ).

% language_empty_io
thf(fact_1134_empty__set,axiom,
    ( bot_bo2863795366942399540od_b_c
    = ( set_Product_prod_b_c2 @ nil_Product_prod_b_c ) ) ).

% empty_set
thf(fact_1135_empty__set,axiom,
    ( bot_bo267491471446728962od_c_a
    = ( set_Pr5664641634098845387od_c_a @ nil_Pr1342775757158464060od_c_a ) ) ).

% empty_set
thf(fact_1136_preserves__divergence__converge__replace__iff,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_d_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M12 )
     => ( ( observable_d_b_c @ M23 )
       => ( ( minimal_d_b_c @ M12 )
         => ( ( minimal_d_b_c @ M23 )
           => ( ( converge_d_b_c @ M12 @ U @ V2 )
             => ( ( converge_d_b_c @ M23 @ U @ V2 )
               => ( ( preser4725570932485813173_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ U @ X7 ) )
                  = ( preser4725570932485813173_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_replace_iff
thf(fact_1137_preserves__divergence__converge__replace__iff,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M12 )
     => ( ( observable_a_b_c @ M23 )
       => ( ( minimal_d_b_c @ M12 )
         => ( ( minimal_a_b_c @ M23 )
           => ( ( converge_d_b_c @ M12 @ U @ V2 )
             => ( ( converge_a_b_c @ M23 @ U @ V2 )
               => ( ( preser4725570932485813170_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ U @ X7 ) )
                  = ( preser4725570932485813170_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_replace_iff
thf(fact_1138_preserves__divergence__converge__replace__iff,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M12 )
     => ( ( observable_a_b_c @ M23 )
       => ( ( minimal_a_b_c @ M12 )
         => ( ( minimal_a_b_c @ M23 )
           => ( ( converge_a_b_c @ M12 @ U @ V2 )
             => ( ( converge_a_b_c @ M23 @ U @ V2 )
               => ( ( preser261684820601417589_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ U @ X7 ) )
                  = ( preser261684820601417589_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_replace_iff
thf(fact_1139_preserves__divergence__converge__replace__iff,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M12 )
     => ( ( observable_d_b_c @ M23 )
       => ( ( minimal_a_b_c @ M12 )
         => ( ( minimal_d_b_c @ M23 )
           => ( ( converge_a_b_c @ M12 @ U @ V2 )
             => ( ( converge_d_b_c @ M23 @ U @ V2 )
               => ( ( preser261684820601417592_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ U @ X7 ) )
                  = ( preser261684820601417592_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_replace_iff
thf(fact_1140_preserves__divergence__converge__replace,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_d_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M12 )
     => ( ( observable_d_b_c @ M23 )
       => ( ( minimal_d_b_c @ M12 )
         => ( ( minimal_d_b_c @ M23 )
           => ( ( converge_d_b_c @ M12 @ U @ V2 )
             => ( ( converge_d_b_c @ M23 @ U @ V2 )
               => ( ( preser4725570932485813173_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ U @ X7 ) )
                 => ( preser4725570932485813173_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_replace
thf(fact_1141_preserves__divergence__converge__replace,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M12 )
     => ( ( observable_a_b_c @ M23 )
       => ( ( minimal_d_b_c @ M12 )
         => ( ( minimal_a_b_c @ M23 )
           => ( ( converge_d_b_c @ M12 @ U @ V2 )
             => ( ( converge_a_b_c @ M23 @ U @ V2 )
               => ( ( preser4725570932485813170_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ U @ X7 ) )
                 => ( preser4725570932485813170_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_replace
thf(fact_1142_preserves__divergence__converge__replace,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M12 )
     => ( ( observable_a_b_c @ M23 )
       => ( ( minimal_a_b_c @ M12 )
         => ( ( minimal_a_b_c @ M23 )
           => ( ( converge_a_b_c @ M12 @ U @ V2 )
             => ( ( converge_a_b_c @ M23 @ U @ V2 )
               => ( ( preser261684820601417589_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ U @ X7 ) )
                 => ( preser261684820601417589_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_replace
thf(fact_1143_preserves__divergence__converge__replace,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M12 )
     => ( ( observable_d_b_c @ M23 )
       => ( ( minimal_a_b_c @ M12 )
         => ( ( minimal_d_b_c @ M23 )
           => ( ( converge_a_b_c @ M12 @ U @ V2 )
             => ( ( converge_d_b_c @ M23 @ U @ V2 )
               => ( ( preser261684820601417592_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ U @ X7 ) )
                 => ( preser261684820601417592_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_replace
thf(fact_1144_preserves__divergence__converge__insert,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_d_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M12 )
     => ( ( observable_d_b_c @ M23 )
       => ( ( minimal_d_b_c @ M12 )
         => ( ( minimal_d_b_c @ M23 )
           => ( ( converge_d_b_c @ M12 @ U @ V2 )
             => ( ( converge_d_b_c @ M23 @ U @ V2 )
               => ( ( preser4725570932485813173_b_c_d @ M12 @ M23 @ X7 )
                 => ( ( member6330420149250801815od_b_c @ U @ X7 )
                   => ( preser4725570932485813173_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_insert
thf(fact_1145_preserves__divergence__converge__insert,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M12 )
     => ( ( observable_a_b_c @ M23 )
       => ( ( minimal_d_b_c @ M12 )
         => ( ( minimal_a_b_c @ M23 )
           => ( ( converge_d_b_c @ M12 @ U @ V2 )
             => ( ( converge_a_b_c @ M23 @ U @ V2 )
               => ( ( preser4725570932485813170_b_c_a @ M12 @ M23 @ X7 )
                 => ( ( member6330420149250801815od_b_c @ U @ X7 )
                   => ( preser4725570932485813170_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_insert
thf(fact_1146_preserves__divergence__converge__insert,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_a_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M12 )
     => ( ( observable_a_b_c @ M23 )
       => ( ( minimal_a_b_c @ M12 )
         => ( ( minimal_a_b_c @ M23 )
           => ( ( converge_a_b_c @ M12 @ U @ V2 )
             => ( ( converge_a_b_c @ M23 @ U @ V2 )
               => ( ( preser261684820601417589_b_c_a @ M12 @ M23 @ X7 )
                 => ( ( member6330420149250801815od_b_c @ U @ X7 )
                   => ( preser261684820601417589_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_insert
thf(fact_1147_preserves__divergence__converge__insert,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,U: list_P903359562653991662od_b_c,V2: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M12 )
     => ( ( observable_d_b_c @ M23 )
       => ( ( minimal_a_b_c @ M12 )
         => ( ( minimal_d_b_c @ M23 )
           => ( ( converge_a_b_c @ M12 @ U @ V2 )
             => ( ( converge_d_b_c @ M23 @ U @ V2 )
               => ( ( preser261684820601417592_b_c_d @ M12 @ M23 @ X7 )
                 => ( ( member6330420149250801815od_b_c @ U @ X7 )
                   => ( preser261684820601417592_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ V2 @ X7 ) ) ) ) ) ) ) ) ) ) ).

% preserves_divergence_converge_insert
thf(fact_1148_pred__equals__eq2,axiom,
    ! [R: set_Pr3237851069733663483_b_c_e,S3: set_Pr3237851069733663483_b_c_e] :
      ( ( ( ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) @ R ) )
        = ( ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1149_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_b_c,S3: set_Product_prod_b_c] :
      ( ( ( ^ [X4: b,Y5: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ R ) )
        = ( ^ [X4: b,Y5: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1150_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_c_a,S3: set_Product_prod_c_a] :
      ( ( ( ^ [X4: c,Y5: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y5 ) @ R ) )
        = ( ^ [X4: c,Y5: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y5 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1151_pred__equals__eq2,axiom,
    ! [R: set_Pr272002107461315949od_c_a,S3: set_Pr272002107461315949od_c_a] :
      ( ( ( ^ [X4: b,Y5: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y5 ) @ R ) )
        = ( ^ [X4: b,Y5: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y5 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1152_pred__equals__eq2,axiom,
    ! [R: set_Pr8357624050790843030od_c_a,S3: set_Pr8357624050790843030od_c_a] :
      ( ( ( ^ [X4: a,Y5: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y5 ) @ R ) )
        = ( ^ [X4: a,Y5: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y5 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_1153_inf__Int__eq2,axiom,
    ! [R: set_Pr3237851069733663483_b_c_e,S3: set_Pr3237851069733663483_b_c_e] :
      ( ( inf_in245022678971466472_c_e_o
        @ ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) @ R )
        @ ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) @ ( inf_in1201650487001137129_b_c_e @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_1154_inf__Int__eq2,axiom,
    ! [R: set_Product_prod_b_c,S3: set_Product_prod_b_c] :
      ( ( inf_inf_b_c_o
        @ ^ [X4: b,Y5: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ R )
        @ ^ [X4: b,Y5: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: b,Y5: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ ( inf_in8411426679332095414od_b_c @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_1155_inf__Int__eq2,axiom,
    ! [R: set_Product_prod_c_a,S3: set_Product_prod_c_a] :
      ( ( inf_inf_c_a_o
        @ ^ [X4: c,Y5: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y5 ) @ R )
        @ ^ [X4: c,Y5: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: c,Y5: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y5 ) @ ( inf_in7633709598634416755od_c_a @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_1156_inf__Int__eq2,axiom,
    ! [R: set_Pr272002107461315949od_c_a,S3: set_Pr272002107461315949od_c_a] :
      ( ( inf_in1711295913471491574_c_a_o
        @ ^ [X4: b,Y5: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y5 ) @ R )
        @ ^ [X4: b,Y5: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: b,Y5: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y5 ) @ ( inf_in1309429386321702747od_c_a @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_1157_inf__Int__eq2,axiom,
    ! [R: set_Pr8357624050790843030od_c_a,S3: set_Pr8357624050790843030od_c_a] :
      ( ( inf_in8721996393322500493_c_a_o
        @ ^ [X4: a,Y5: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y5 ) @ R )
        @ ^ [X4: a,Y5: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: a,Y5: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y5 ) @ ( inf_in3815988127369930372od_c_a @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_1158_preserves__divergence__insertI,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_d_b_c,X7: set_li6436108459499378894od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( preser4725570932485813173_b_c_d @ M12 @ M23 @ X7 )
     => ( ! [Alpha3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ Alpha3 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) @ X7 ) )
           => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) )
             => ( ~ ( converge_d_b_c @ M12 @ Alpha3 @ Beta )
               => ~ ( converge_d_b_c @ M23 @ Alpha3 @ Beta ) ) ) )
       => ( preser4725570932485813173_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ Beta @ X7 ) ) ) ) ).

% preserves_divergence_insertI
thf(fact_1159_preserves__divergence__insertI,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_a_b_c,X7: set_li6436108459499378894od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( preser4725570932485813170_b_c_a @ M12 @ M23 @ X7 )
     => ( ! [Alpha3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ Alpha3 @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) @ X7 ) )
           => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) )
             => ( ~ ( converge_d_b_c @ M12 @ Alpha3 @ Beta )
               => ~ ( converge_a_b_c @ M23 @ Alpha3 @ Beta ) ) ) )
       => ( preser4725570932485813170_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ Beta @ X7 ) ) ) ) ).

% preserves_divergence_insertI
thf(fact_1160_preserves__divergence__insertI,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_a_b_c,X7: set_li6436108459499378894od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( preser261684820601417589_b_c_a @ M12 @ M23 @ X7 )
     => ( ! [Alpha3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ Alpha3 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) @ X7 ) )
           => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) )
             => ( ~ ( converge_a_b_c @ M12 @ Alpha3 @ Beta )
               => ~ ( converge_a_b_c @ M23 @ Alpha3 @ Beta ) ) ) )
       => ( preser261684820601417589_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ Beta @ X7 ) ) ) ) ).

% preserves_divergence_insertI
thf(fact_1161_preserves__divergence__insertI,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,X7: set_li6436108459499378894od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( preser261684820601417592_b_c_d @ M12 @ M23 @ X7 )
     => ( ! [Alpha3: list_P903359562653991662od_b_c] :
            ( ( member6330420149250801815od_b_c @ Alpha3 @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) @ X7 ) )
           => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) )
             => ( ~ ( converge_a_b_c @ M12 @ Alpha3 @ Beta )
               => ~ ( converge_d_b_c @ M23 @ Alpha3 @ Beta ) ) ) )
       => ( preser261684820601417592_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ Beta @ X7 ) ) ) ) ).

% preserves_divergence_insertI
thf(fact_1162_preserves__divergence__insertE_I2_J,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_d_b_c,Beta: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c,Alpha: list_P903359562653991662od_b_c] :
      ( ( preser4725570932485813173_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ Beta @ X7 ) )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) @ X7 ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) )
         => ( ~ ( converge_d_b_c @ M12 @ Alpha @ Beta )
           => ~ ( converge_d_b_c @ M23 @ Alpha @ Beta ) ) ) ) ) ).

% preserves_divergence_insertE(2)
thf(fact_1163_preserves__divergence__insertE_I2_J,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_a_b_c,Beta: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c,Alpha: list_P903359562653991662od_b_c] :
      ( ( preser4725570932485813170_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ Beta @ X7 ) )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) @ X7 ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) )
         => ( ~ ( converge_d_b_c @ M12 @ Alpha @ Beta )
           => ~ ( converge_a_b_c @ M23 @ Alpha @ Beta ) ) ) ) ) ).

% preserves_divergence_insertE(2)
thf(fact_1164_preserves__divergence__insertE_I2_J,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_a_b_c,Beta: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c,Alpha: list_P903359562653991662od_b_c] :
      ( ( preser261684820601417589_b_c_a @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ Beta @ X7 ) )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) @ X7 ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) )
         => ( ~ ( converge_a_b_c @ M12 @ Alpha @ Beta )
           => ~ ( converge_a_b_c @ M23 @ Alpha @ Beta ) ) ) ) ) ).

% preserves_divergence_insertE(2)
thf(fact_1165_preserves__divergence__insertE_I2_J,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,Beta: list_P903359562653991662od_b_c,X7: set_li6436108459499378894od_b_c,Alpha: list_P903359562653991662od_b_c] :
      ( ( preser261684820601417592_b_c_d @ M12 @ M23 @ ( insert6227932334100060350od_b_c @ Beta @ X7 ) )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) @ X7 ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) )
         => ( ~ ( converge_a_b_c @ M12 @ Alpha @ Beta )
           => ~ ( converge_d_b_c @ M23 @ Alpha @ Beta ) ) ) ) ) ).

% preserves_divergence_insertE(2)
thf(fact_1166_linear__order__from__list__position_H_Ocases,axiom,
    ! [X: list_P903359562653991662od_b_c] :
      ( ( X != nil_Product_prod_b_c )
     => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
            ( X
           != ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_1167_find__index_Ocases,axiom,
    ! [X: produc7816210539016439438od_b_c] :
      ( ! [F3: product_prod_b_c > $o] :
          ( X
         != ( produc7392021306980666816od_b_c @ F3 @ nil_Product_prod_b_c ) )
     => ~ ! [F3: product_prod_b_c > $o,X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
            ( X
           != ( produc7392021306980666816od_b_c @ F3 @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) ) ) ) ).

% find_index.cases
thf(fact_1168_list__set__sym,axiom,
    ! [X: list_P903359562653991662od_b_c,Y: list_P903359562653991662od_b_c] :
      ( ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ X @ Y ) )
      = ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Y @ X ) ) ) ).

% list_set_sym
thf(fact_1169_list__set__sym,axiom,
    ! [X: list_P6327159017948738492od_c_a,Y: list_P6327159017948738492od_c_a] :
      ( ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ X @ Y ) )
      = ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Y @ X ) ) ) ).

% list_set_sym
thf(fact_1170_list__prefix__elem,axiom,
    ! [X: a,Xs2: list_a,Ys: list_a] :
      ( ( member_a @ X @ ( set_a2 @ ( append_a @ Xs2 @ Ys ) ) )
     => ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
       => ( member_a @ X @ ( set_a2 @ Xs2 ) ) ) ) ).

% list_prefix_elem
thf(fact_1171_list__prefix__elem,axiom,
    ! [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c] :
      ( ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs2 @ Ys ) ) )
     => ( ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Ys ) )
       => ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs2 ) ) ) ) ).

% list_prefix_elem
thf(fact_1172_list__prefix__elem,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs2 @ Ys ) ) )
     => ( ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Ys ) )
       => ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs2 ) ) ) ) ).

% list_prefix_elem
thf(fact_1173_list__prefix__elem,axiom,
    ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs2 @ Ys ) ) )
     => ( ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) ) ) ) ).

% list_prefix_elem
thf(fact_1174_list__concat__non__elem,axiom,
    ! [X: a,Xs2: list_a,Ys: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
       => ~ ( member_a @ X @ ( set_a2 @ ( append_a @ Xs2 @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_1175_list__concat__non__elem,axiom,
    ! [X: list_P903359562653991662od_b_c,Xs2: list_l8907847357763382004od_b_c,Ys: list_l8907847357763382004od_b_c] :
      ( ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Xs2 ) )
     => ( ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ Ys ) )
       => ~ ( member6330420149250801815od_b_c @ X @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs2 @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_1176_list__concat__non__elem,axiom,
    ! [X: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs2 ) )
     => ( ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Ys ) )
       => ~ ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs2 @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_1177_list__concat__non__elem,axiom,
    ! [X: produc2669932201312201782od_c_a,Xs2: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) )
     => ( ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ Ys ) )
       => ~ ( member5502040268208861279od_c_a @ X @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs2 @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_1178_subrelI,axiom,
    ! [R2: set_Pr3237851069733663483_b_c_e,S4: set_Pr3237851069733663483_b_c_e] :
      ( ! [X3: prefix425356687782099470od_b_c,Y3: e] :
          ( ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) @ R2 )
         => ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le1795193733426387611_b_c_e @ R2 @ S4 ) ) ).

% subrelI
thf(fact_1179_subrelI,axiom,
    ! [R2: set_Product_prod_b_c,S4: set_Product_prod_b_c] :
      ( ! [X3: b,Y3: c] :
          ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ R2 )
         => ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le253122037897321832od_b_c @ R2 @ S4 ) ) ).

% subrelI
thf(fact_1180_subrelI,axiom,
    ! [R2: set_Product_prod_c_a,S4: set_Product_prod_c_a] :
      ( ! [X3: c,Y3: a] :
          ( ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) @ R2 )
         => ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le8698776994054418981od_c_a @ R2 @ S4 ) ) ).

% subrelI
thf(fact_1181_subrelI,axiom,
    ! [R2: set_Pr272002107461315949od_c_a,S4: set_Pr272002107461315949od_c_a] :
      ( ! [X3: b,Y3: product_prod_c_a] :
          ( ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) @ R2 )
         => ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le4080174072498570509od_c_a @ R2 @ S4 ) ) ).

% subrelI
thf(fact_1182_subrelI,axiom,
    ! [R2: set_Pr8357624050790843030od_c_a,S4: set_Pr8357624050790843030od_c_a] :
      ( ! [X3: a,Y3: produc2133359284458116493od_c_a] :
          ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) @ R2 )
         => ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y3 ) @ S4 ) )
     => ( ord_le3226338165915859510od_c_a @ R2 @ S4 ) ) ).

% subrelI
thf(fact_1183_pred__subset__eq,axiom,
    ! [R: set_Pr8357624050790843030od_c_a,S3: set_Pr8357624050790843030od_c_a] :
      ( ( ord_le753561201022589991_c_a_o
        @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ R )
        @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ S3 ) )
      = ( ord_le3226338165915859510od_c_a @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_1184_pred__subset__eq,axiom,
    ! [R: set_a,S3: set_a] :
      ( ( ord_less_eq_a_o
        @ ^ [X4: a] : ( member_a @ X4 @ R )
        @ ^ [X4: a] : ( member_a @ X4 @ S3 ) )
      = ( ord_less_eq_set_a @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_1185_pred__subset__eq,axiom,
    ! [R: set_li6436108459499378894od_b_c,S3: set_li6436108459499378894od_b_c] :
      ( ( ord_le1261880702084804079_b_c_o
        @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ R )
        @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ S3 ) )
      = ( ord_le282488521294790766od_b_c @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_1186_inf__Int__eq,axiom,
    ! [R: set_Pr8357624050790843030od_c_a,S3: set_Pr8357624050790843030od_c_a] :
      ( ( inf_in5491342246039287129_c_a_o
        @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ R )
        @ ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ S3 ) )
      = ( ^ [X4: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X4 @ ( inf_in3815988127369930372od_c_a @ R @ S3 ) ) ) ) ).

% inf_Int_eq
thf(fact_1187_inf__Int__eq,axiom,
    ! [R: set_a,S3: set_a] :
      ( ( inf_inf_a_o
        @ ^ [X4: a] : ( member_a @ X4 @ R )
        @ ^ [X4: a] : ( member_a @ X4 @ S3 ) )
      = ( ^ [X4: a] : ( member_a @ X4 @ ( inf_inf_set_a @ R @ S3 ) ) ) ) ).

% inf_Int_eq
thf(fact_1188_inf__Int__eq,axiom,
    ! [R: set_li6436108459499378894od_b_c,S3: set_li6436108459499378894od_b_c] :
      ( ( inf_in3324677931821484577_b_c_o
        @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ R )
        @ ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ S3 ) )
      = ( ^ [X4: list_P903359562653991662od_b_c] : ( member6330420149250801815od_b_c @ X4 @ ( inf_in4978071631833541052od_b_c @ R @ S3 ) ) ) ) ).

% inf_Int_eq
thf(fact_1189_list__prefix__subset,axiom,
    ! [Ts: list_P903359562653991662od_b_c,Xs2: list_P903359562653991662od_b_c] :
      ( ? [Ys6: list_P903359562653991662od_b_c] :
          ( Ts
          = ( append2547753245680614915od_b_c @ Xs2 @ Ys6 ) )
     => ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs2 ) @ ( set_Product_prod_b_c2 @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_1190_list__prefix__subset,axiom,
    ! [Ts: list_P6327159017948738492od_c_a,Xs2: list_P6327159017948738492od_c_a] :
      ( ? [Ys6: list_P6327159017948738492od_c_a] :
          ( Ts
          = ( append3884164603879732689od_c_a @ Xs2 @ Ys6 ) )
     => ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs2 ) @ ( set_Pr5664641634098845387od_c_a @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_1191_list__prefix__subset,axiom,
    ! [Ts: list_l8907847357763382004od_b_c,Xs2: list_l8907847357763382004od_b_c] :
      ( ? [Ys6: list_l8907847357763382004od_b_c] :
          ( Ts
          = ( append6992831555715836169od_b_c @ Xs2 @ Ys6 ) )
     => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs2 ) @ ( set_li4480668622519654659od_b_c @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_1192_list__append__subset3,axiom,
    ! [Xs1: list_P903359562653991662od_b_c,Ys1: list_P903359562653991662od_b_c,Xs22: list_P903359562653991662od_b_c,Ys22: list_P903359562653991662od_b_c,Xs32: list_P903359562653991662od_b_c,Ys32: 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 @ Xs22 ) @ ( set_Product_prod_b_c2 @ Ys22 ) )
       => ( ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ Xs32 ) @ ( set_Product_prod_b_c2 @ Ys32 ) )
         => ( ord_le253122037897321832od_b_c @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Xs1 @ ( append2547753245680614915od_b_c @ Xs22 @ Xs32 ) ) ) @ ( set_Product_prod_b_c2 @ ( append2547753245680614915od_b_c @ Ys1 @ ( append2547753245680614915od_b_c @ Ys22 @ Ys32 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_1193_list__append__subset3,axiom,
    ! [Xs1: list_P6327159017948738492od_c_a,Ys1: list_P6327159017948738492od_c_a,Xs22: list_P6327159017948738492od_c_a,Ys22: list_P6327159017948738492od_c_a,Xs32: list_P6327159017948738492od_c_a,Ys32: list_P6327159017948738492od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs1 ) @ ( set_Pr5664641634098845387od_c_a @ Ys1 ) )
     => ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs22 ) @ ( set_Pr5664641634098845387od_c_a @ Ys22 ) )
       => ( ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ Xs32 ) @ ( set_Pr5664641634098845387od_c_a @ Ys32 ) )
         => ( ord_le3226338165915859510od_c_a @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Xs1 @ ( append3884164603879732689od_c_a @ Xs22 @ Xs32 ) ) ) @ ( set_Pr5664641634098845387od_c_a @ ( append3884164603879732689od_c_a @ Ys1 @ ( append3884164603879732689od_c_a @ Ys22 @ Ys32 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_1194_list__append__subset3,axiom,
    ! [Xs1: list_l8907847357763382004od_b_c,Ys1: list_l8907847357763382004od_b_c,Xs22: list_l8907847357763382004od_b_c,Ys22: list_l8907847357763382004od_b_c,Xs32: list_l8907847357763382004od_b_c,Ys32: 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 @ Xs22 ) @ ( set_li4480668622519654659od_b_c @ Ys22 ) )
       => ( ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ Xs32 ) @ ( set_li4480668622519654659od_b_c @ Ys32 ) )
         => ( ord_le282488521294790766od_b_c @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Xs1 @ ( append6992831555715836169od_b_c @ Xs22 @ Xs32 ) ) ) @ ( set_li4480668622519654659od_b_c @ ( append6992831555715836169od_b_c @ Ys1 @ ( append6992831555715836169od_b_c @ Ys22 @ Ys32 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_1195_find__remove_H_Ocases,axiom,
    ! [X: produc5559707038353031495od_b_c] :
      ( ! [P3: product_prod_b_c > $o,Uu2: list_P903359562653991662od_b_c] :
          ( X
         != ( produc1005531610559690873od_b_c @ P3 @ ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Uu2 ) ) )
     => ~ ! [P3: product_prod_b_c > $o,X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Prev: list_P903359562653991662od_b_c] :
            ( X
           != ( produc1005531610559690873od_b_c @ P3 @ ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_1196_is__prefix_Ocases,axiom,
    ! [X: produc5878855887281848999od_b_c] :
      ( ! [Uu2: list_P903359562653991662od_b_c] :
          ( X
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Uu2 ) )
     => ( ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c] :
            ( X
           != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ nil_Product_prod_b_c ) )
       => ~ ! [X3: product_prod_b_c,Xs3: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
              ( X
             != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs3 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys2 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_1197_h__out_Osimps,axiom,
    ! [M2: fsm_a_b_c,Q2: a,X: b] :
      ( ( h_out_a_b_c @ M2 @ ( product_Pair_a_b @ Q2 @ X ) )
      = ( collect_c
        @ ^ [Y5: c] :
          ? [Q5: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q2 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y5 @ Q5 ) ) ) @ ( transitions_a_b_c @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1198_h__out_Osimps,axiom,
    ! [M2: fsm_b_c_nat,Q2: b,X: c] :
      ( ( h_out_b_c_nat @ M2 @ ( product_Pair_b_c @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Y5: nat] :
          ? [Q5: b] : ( member7205605469418551960_nat_b @ ( produc2952021607952343931_nat_b @ Q2 @ ( produc1482701325287381453_nat_b @ X @ ( product_Pair_nat_b @ Y5 @ Q5 ) ) ) @ ( transitions_b_c_nat @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1199_h__out_Osimps,axiom,
    ! [M2: fsm_c_a_nat,Q2: c,X: a] :
      ( ( h_out_c_a_nat @ M2 @ ( product_Pair_c_a @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Y5: nat] :
          ? [Q5: c] : ( member3840034767840562712_nat_c @ ( produc5068050717085582971_nat_c @ Q2 @ ( produc5715557445571560268_nat_c @ X @ ( product_Pair_nat_c @ Y5 @ Q5 ) ) ) @ ( transitions_c_a_nat @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1200_h__out_Osimps,axiom,
    ! [M2: fsm_c_a_b,Q2: c,X: a] :
      ( ( h_out_c_a_b @ M2 @ ( product_Pair_c_a @ Q2 @ X ) )
      = ( collect_b
        @ ^ [Y5: b] :
          ? [Q5: c] : ( member265788643176044257od_b_c @ ( produc8300696160155326504od_b_c @ Q2 @ ( produc6867761804911831425od_b_c @ X @ ( product_Pair_b_c @ Y5 @ Q5 ) ) ) @ ( transitions_c_a_b @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1201_h__out_Osimps,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,Q2: product_prod_c_a,X: a] :
      ( ( h_out_8774345398997575435_a_a_b @ M2 @ ( produc7661602042108997680_c_a_a @ Q2 @ X ) )
      = ( collect_b
        @ ^ [Y5: b] :
          ? [Q5: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q2 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y5 @ Q5 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1202_h__out_Osimps,axiom,
    ! [M2: fsm_b_8881565123868762728_a_nat,Q2: b,X: product_prod_c_a] :
      ( ( h_out_7043617306241251473_a_nat @ M2 @ ( produc3440865914000881661od_c_a @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Y5: nat] :
          ? [Q5: b] : ( member6037585776607591613_nat_b @ ( produc5408764996987726368_nat_b @ Q2 @ ( produc7397790810943070066_nat_b @ X @ ( product_Pair_nat_b @ Y5 @ Q5 ) ) ) @ ( transi4389418769516128417_a_nat @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1203_h__out_Osimps,axiom,
    ! [M2: fsm_Pr1631622435462234230_e_nat,Q2: prefix425356687782099470od_b_c,X: e] :
      ( ( h_out_7860023044450888735_e_nat @ M2 @ ( produc2511600812177743883_b_c_e @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Y5: nat] :
          ? [Q5: prefix425356687782099470od_b_c] : ( member4348767779791260600od_b_c @ ( produc8644425055493693723od_b_c @ Q2 @ ( produc3386770699685700318od_b_c @ X @ ( produc5018994884486087269od_b_c @ Y5 @ Q5 ) ) ) @ ( transi6891536161183548463_e_nat @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1204_h__out_Osimps,axiom,
    ! [M2: fsm_a_1723543254342851473_a_nat,Q2: a,X: produc2133359284458116493od_c_a] :
      ( ( h_out_5121717952982053178_a_nat @ M2 @ ( produc5762801557034676134od_c_a @ Q2 @ X ) )
      = ( collect_nat
        @ ^ [Y5: nat] :
          ? [Q5: a] : ( member9210739242824242467_nat_a @ ( produc7625539614740449286_nat_a @ Q2 @ ( produc3827177206403408857_nat_a @ X @ ( product_Pair_nat_a @ Y5 @ Q5 ) ) ) @ ( transi8412874649895244106_a_nat @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1205_h__out_Osimps,axiom,
    ! [M2: fsm_a_836089231483141141_c_a_c,Q2: a,X: produc2133359284458116493od_c_a] :
      ( ( h_out_1068671350592293910_c_a_c @ M2 @ ( produc5762801557034676134od_c_a @ Q2 @ X ) )
      = ( collect_c
        @ ^ [Y5: c] :
          ? [Q5: a] : ( member8587115632268026449od_c_a @ ( produc8441414538239260312od_c_a @ Q2 @ ( produc1679040934828577519od_c_a @ X @ ( product_Pair_c_a @ Y5 @ Q5 ) ) ) @ ( transi4754753175853207558_c_a_c @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1206_h__out_Osimps,axiom,
    ! [M2: fsm_b_5636252348735175577od_c_a,Q2: b,X: c] :
      ( ( h_out_165584265889796634od_c_a @ M2 @ ( product_Pair_b_c @ Q2 @ X ) )
      = ( collec8405073525749625505od_c_a
        @ ^ [Y5: produc2669932201312201782od_c_a] :
          ? [Q5: b] : ( member5468748651639462952_c_a_b @ ( produc7794378412417055343_c_a_b @ Q2 @ ( produc1210583608432325383_c_a_b @ X @ ( produc174534808117382112_c_a_b @ Y5 @ Q5 ) ) ) @ ( transi4984929337990653450od_c_a @ M2 ) ) ) ) ).

% h_out.simps
thf(fact_1207_h__out_Oelims,axiom,
    ! [X: fsm_a_b_c,Xa3: product_prod_a_b,Y: set_c] :
      ( ( ( h_out_a_b_c @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: a,X3: b] :
            ( ( Xa3
              = ( product_Pair_a_b @ Q3 @ X3 ) )
           => ( Y
             != ( collect_c
                @ ^ [Y5: c] :
                  ? [Q5: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q3 @ ( produc3440865914000881661od_c_a @ X3 @ ( product_Pair_c_a @ Y5 @ Q5 ) ) ) @ ( transitions_a_b_c @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1208_h__out_Oelims,axiom,
    ! [X: fsm_b_c_nat,Xa3: product_prod_b_c,Y: set_nat] :
      ( ( ( h_out_b_c_nat @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: b,X3: c] :
            ( ( Xa3
              = ( product_Pair_b_c @ Q3 @ X3 ) )
           => ( Y
             != ( collect_nat
                @ ^ [Y5: nat] :
                  ? [Q5: b] : ( member7205605469418551960_nat_b @ ( produc2952021607952343931_nat_b @ Q3 @ ( produc1482701325287381453_nat_b @ X3 @ ( product_Pair_nat_b @ Y5 @ Q5 ) ) ) @ ( transitions_b_c_nat @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1209_h__out_Oelims,axiom,
    ! [X: fsm_c_a_nat,Xa3: product_prod_c_a,Y: set_nat] :
      ( ( ( h_out_c_a_nat @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: c,X3: a] :
            ( ( Xa3
              = ( product_Pair_c_a @ Q3 @ X3 ) )
           => ( Y
             != ( collect_nat
                @ ^ [Y5: nat] :
                  ? [Q5: c] : ( member3840034767840562712_nat_c @ ( produc5068050717085582971_nat_c @ Q3 @ ( produc5715557445571560268_nat_c @ X3 @ ( product_Pair_nat_c @ Y5 @ Q5 ) ) ) @ ( transitions_c_a_nat @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1210_h__out_Oelims,axiom,
    ! [X: fsm_c_a_b,Xa3: product_prod_c_a,Y: set_b] :
      ( ( ( h_out_c_a_b @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: c,X3: a] :
            ( ( Xa3
              = ( product_Pair_c_a @ Q3 @ X3 ) )
           => ( Y
             != ( collect_b
                @ ^ [Y5: b] :
                  ? [Q5: c] : ( member265788643176044257od_b_c @ ( produc8300696160155326504od_b_c @ Q3 @ ( produc6867761804911831425od_b_c @ X3 @ ( product_Pair_b_c @ Y5 @ Q5 ) ) ) @ ( transitions_c_a_b @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1211_h__out_Oelims,axiom,
    ! [X: fsm_Pr1886043760914421514_a_a_b,Xa3: produc5771957243474376000_c_a_a,Y: set_b] :
      ( ( ( h_out_8774345398997575435_a_a_b @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: product_prod_c_a,X3: a] :
            ( ( Xa3
              = ( produc7661602042108997680_c_a_a @ Q3 @ X3 ) )
           => ( Y
             != ( collect_b
                @ ^ [Y5: b] :
                  ? [Q5: product_prod_c_a] : ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q3 @ ( produc5762801557034676134od_c_a @ X3 @ ( produc3440865914000881661od_c_a @ Y5 @ Q5 ) ) ) @ ( transi2941759273464634107_a_a_b @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1212_h__out_Oelims,axiom,
    ! [X: fsm_b_8881565123868762728_a_nat,Xa3: produc2133359284458116493od_c_a,Y: set_nat] :
      ( ( ( h_out_7043617306241251473_a_nat @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: b,X3: product_prod_c_a] :
            ( ( Xa3
              = ( produc3440865914000881661od_c_a @ Q3 @ X3 ) )
           => ( Y
             != ( collect_nat
                @ ^ [Y5: nat] :
                  ? [Q5: b] : ( member6037585776607591613_nat_b @ ( produc5408764996987726368_nat_b @ Q3 @ ( produc7397790810943070066_nat_b @ X3 @ ( product_Pair_nat_b @ Y5 @ Q5 ) ) ) @ ( transi4389418769516128417_a_nat @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1213_h__out_Oelims,axiom,
    ! [X: fsm_Pr1631622435462234230_e_nat,Xa3: produc4534891796109904283_b_c_e,Y: set_nat] :
      ( ( ( h_out_7860023044450888735_e_nat @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: prefix425356687782099470od_b_c,X3: e] :
            ( ( Xa3
              = ( produc2511600812177743883_b_c_e @ Q3 @ X3 ) )
           => ( Y
             != ( collect_nat
                @ ^ [Y5: nat] :
                  ? [Q5: prefix425356687782099470od_b_c] : ( member4348767779791260600od_b_c @ ( produc8644425055493693723od_b_c @ Q3 @ ( produc3386770699685700318od_b_c @ X3 @ ( produc5018994884486087269od_b_c @ Y5 @ Q5 ) ) ) @ ( transi6891536161183548463_e_nat @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1214_h__out_Oelims,axiom,
    ! [X: fsm_a_1723543254342851473_a_nat,Xa3: produc2669932201312201782od_c_a,Y: set_nat] :
      ( ( ( h_out_5121717952982053178_a_nat @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: a,X3: produc2133359284458116493od_c_a] :
            ( ( Xa3
              = ( produc5762801557034676134od_c_a @ Q3 @ X3 ) )
           => ( Y
             != ( collect_nat
                @ ^ [Y5: nat] :
                  ? [Q5: a] : ( member9210739242824242467_nat_a @ ( produc7625539614740449286_nat_a @ Q3 @ ( produc3827177206403408857_nat_a @ X3 @ ( product_Pair_nat_a @ Y5 @ Q5 ) ) ) @ ( transi8412874649895244106_a_nat @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1215_h__out_Oelims,axiom,
    ! [X: fsm_a_836089231483141141_c_a_c,Xa3: produc2669932201312201782od_c_a,Y: set_c] :
      ( ( ( h_out_1068671350592293910_c_a_c @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: a,X3: produc2133359284458116493od_c_a] :
            ( ( Xa3
              = ( produc5762801557034676134od_c_a @ Q3 @ X3 ) )
           => ( Y
             != ( collect_c
                @ ^ [Y5: c] :
                  ? [Q5: a] : ( member8587115632268026449od_c_a @ ( produc8441414538239260312od_c_a @ Q3 @ ( produc1679040934828577519od_c_a @ X3 @ ( product_Pair_c_a @ Y5 @ Q5 ) ) ) @ ( transi4754753175853207558_c_a_c @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1216_h__out_Oelims,axiom,
    ! [X: fsm_b_5636252348735175577od_c_a,Xa3: product_prod_b_c,Y: set_Pr8357624050790843030od_c_a] :
      ( ( ( h_out_165584265889796634od_c_a @ X @ Xa3 )
        = Y )
     => ~ ! [Q3: b,X3: c] :
            ( ( Xa3
              = ( product_Pair_b_c @ Q3 @ X3 ) )
           => ( Y
             != ( collec8405073525749625505od_c_a
                @ ^ [Y5: produc2669932201312201782od_c_a] :
                  ? [Q5: b] : ( member5468748651639462952_c_a_b @ ( produc7794378412417055343_c_a_b @ Q3 @ ( produc1210583608432325383_c_a_b @ X3 @ ( produc174534808117382112_c_a_b @ Y5 @ Q5 ) ) ) @ ( transi4984929337990653450od_c_a @ X ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1217_set__Cons__sing__Nil,axiom,
    ! [A4: set_Product_prod_b_c] :
      ( ( set_Co7706362885935805726od_b_c @ A4 @ ( insert6227932334100060350od_b_c @ nil_Product_prod_b_c @ bot_bo4166481423041325370od_b_c ) )
      = ( image_5872333176446027019od_b_c
        @ ^ [X4: product_prod_b_c] : ( cons_P4529483553340347422od_b_c @ X4 @ nil_Product_prod_b_c )
        @ A4 ) ) ).

% set_Cons_sing_Nil
thf(fact_1218_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ X @ bot_bo4166481423041325370od_b_c )
      = bot_bo4166481423041325370od_b_c ) ).

% boolean_algebra.conj_zero_right
thf(fact_1219_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_li6436108459499378894od_b_c] :
      ( ( inf_in4978071631833541052od_b_c @ bot_bo4166481423041325370od_b_c @ X )
      = bot_bo4166481423041325370od_b_c ) ).

% boolean_algebra.conj_zero_left
thf(fact_1220_remove__proper__prefixes__def,axiom,
    ( remove5339202614008355685od_b_c
    = ( ^ [Xs: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [X4: list_P903359562653991662od_b_c] :
              ( ( member6330420149250801815od_b_c @ X4 @ Xs )
              & ~ ? [X6: list_P903359562653991662od_b_c] :
                    ( ( X6 != nil_Product_prod_b_c )
                    & ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ X4 @ X6 ) @ Xs ) ) ) ) ) ) ).

% remove_proper_prefixes_def
thf(fact_1221_add__transition__with__components__simps_I3_J,axiom,
    ! [M2: fsm_d_b_c,T: produc1399170498009216118od_c_d] :
      ( ( outputs_d_b_c @ ( add_tr6566484180594966275_d_b_c @ M2 @ T ) )
      = ( insert_c @ ( product_fst_c_d @ ( produc5221446337070271918od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) ) @ ( outputs_d_b_c @ M2 ) ) ) ).

% add_transition_with_components_simps(3)
thf(fact_1222_add__transition__with__components__simps_I3_J,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T: produc3205353004677322104od_c_a] :
      ( ( output2371653660386375239_a_a_b @ ( add_tr5093536917197733991_a_a_b @ M2 @ T ) )
      = ( insert_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( output2371653660386375239_a_a_b @ M2 ) ) ) ).

% add_transition_with_components_simps(3)
thf(fact_1223_add__transition__with__components__simps_I3_J,axiom,
    ! [M2: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( outputs_a_b_c @ ( add_tr8224572408740125824_a_b_c @ M2 @ T ) )
      = ( insert_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M2 ) ) ) ).

% add_transition_with_components_simps(3)
thf(fact_1224_add__transition__with__components__simps_I1_J,axiom,
    ! [M2: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( initial_a_b_c @ ( add_tr8224572408740125824_a_b_c @ M2 @ T ) )
      = ( initial_a_b_c @ M2 ) ) ).

% add_transition_with_components_simps(1)
thf(fact_1225_add__transition__with__components__simps_I1_J,axiom,
    ! [M2: fsm_d_b_c,T: produc1399170498009216118od_c_d] :
      ( ( initial_d_b_c @ ( add_tr6566484180594966275_d_b_c @ M2 @ T ) )
      = ( initial_d_b_c @ M2 ) ) ).

% add_transition_with_components_simps(1)
thf(fact_1226_add__transition__with__components__simps_I4_J,axiom,
    ! [M2: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( transitions_a_b_c @ ( add_tr8224572408740125824_a_b_c @ M2 @ T ) )
      = ( insert8105206408023001606od_c_a @ T @ ( transitions_a_b_c @ M2 ) ) ) ).

% add_transition_with_components_simps(4)
thf(fact_1227_add__transition__with__components__simps_I2_J,axiom,
    ! [M2: fsm_d_b_c,T: produc1399170498009216118od_c_d] :
      ( ( inputs_d_b_c @ ( add_tr6566484180594966275_d_b_c @ M2 @ T ) )
      = ( insert_b @ ( produc301949087656271980od_c_d @ ( produc5370652324230910612od_c_d @ T ) ) @ ( inputs_d_b_c @ M2 ) ) ) ).

% add_transition_with_components_simps(2)
thf(fact_1228_add__transition__with__components__simps_I2_J,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T: produc3205353004677322104od_c_a] :
      ( ( inputs8636541066849307272_a_a_b @ ( add_tr5093536917197733991_a_a_b @ M2 @ T ) )
      = ( insert_a @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( inputs8636541066849307272_a_a_b @ M2 ) ) ) ).

% add_transition_with_components_simps(2)
thf(fact_1229_add__transition__with__components__simps_I2_J,axiom,
    ! [M2: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( inputs_a_b_c @ ( add_tr8224572408740125824_a_b_c @ M2 @ T ) )
      = ( insert_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M2 ) ) ) ).

% add_transition_with_components_simps(2)
thf(fact_1230_bot__empty__eq2,axiom,
    ( bot_bo7972094606718898790_c_e_o
    = ( ^ [X4: prefix425356687782099470od_b_c,Y5: e] : ( member323891770073680196_b_c_e @ ( produc2511600812177743883_b_c_e @ X4 @ Y5 ) @ bot_bo5533743774279703399_b_c_e ) ) ) ).

% bot_empty_eq2
thf(fact_1231_bot__empty__eq2,axiom,
    ( bot_bot_b_c_o
    = ( ^ [X4: b,Y5: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y5 ) @ bot_bo2863795366942399540od_b_c ) ) ) ).

% bot_empty_eq2
thf(fact_1232_bot__empty__eq2,axiom,
    ( bot_bot_c_a_o
    = ( ^ [X4: c,Y5: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ X4 @ Y5 ) @ bot_bo2086078286244720881od_c_a ) ) ) ).

% bot_empty_eq2
thf(fact_1233_bot__empty__eq2,axiom,
    ( bot_bo970769885656619636_c_a_o
    = ( ^ [X4: b,Y5: product_prod_c_a] : ( member1160383752307813046od_c_a @ ( produc3440865914000881661od_c_a @ X4 @ Y5 ) @ bot_bo8402128546026624985od_c_a ) ) ) ).

% bot_empty_eq2
thf(fact_1234_bot__empty__eq2,axiom,
    ( bot_bo2582192026207158283_c_a_o
    = ( ^ [X4: a,Y5: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X4 @ Y5 ) @ bot_bo267491471446728962od_c_a ) ) ) ).

% bot_empty_eq2
thf(fact_1235_bot__set__def,axiom,
    ( bot_bo267491471446728962od_c_a
    = ( collec8405073525749625505od_c_a @ bot_bo4987066436690517083_c_a_o ) ) ).

% bot_set_def
thf(fact_1236_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_1237_boolean__algebra__cancel_Oinf2,axiom,
    ! [B4: set_li6436108459499378894od_b_c,K3: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c] :
      ( ( B4
        = ( inf_in4978071631833541052od_b_c @ K3 @ B ) )
     => ( ( inf_in4978071631833541052od_b_c @ A @ B4 )
        = ( inf_in4978071631833541052od_b_c @ K3 @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_1238_boolean__algebra__cancel_Oinf1,axiom,
    ! [A4: set_li6436108459499378894od_b_c,K3: set_li6436108459499378894od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( A4
        = ( inf_in4978071631833541052od_b_c @ K3 @ A ) )
     => ( ( inf_in4978071631833541052od_b_c @ A4 @ B )
        = ( inf_in4978071631833541052od_b_c @ K3 @ ( inf_in4978071631833541052od_b_c @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_1239_set__Cons__def,axiom,
    ( set_Co7906722538203907820od_c_a
    = ( ^ [A5: set_Pr8357624050790843030od_c_a,XS: set_li1159382662694783132od_c_a] :
          ( collec6273869032445462695od_c_a
          @ ^ [Z4: list_P6327159017948738492od_c_a] :
            ? [X4: produc2669932201312201782od_c_a,Xs: list_P6327159017948738492od_c_a] :
              ( ( Z4
                = ( cons_P8659537020926305772od_c_a @ X4 @ Xs ) )
              & ( member5502040268208861279od_c_a @ X4 @ A5 )
              & ( member7410604586820865893od_c_a @ Xs @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_1240_set__Cons__def,axiom,
    ( set_Cons_a
    = ( ^ [A5: set_a,XS: set_list_a] :
          ( collect_list_a
          @ ^ [Z4: list_a] :
            ? [X4: a,Xs: list_a] :
              ( ( Z4
                = ( cons_a @ X4 @ Xs ) )
              & ( member_a @ X4 @ A5 )
              & ( member_list_a @ Xs @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_1241_set__Cons__def,axiom,
    ( set_Co4997685901046503076od_b_c
    = ( ^ [A5: set_li6436108459499378894od_b_c,XS: set_li336664667694595924od_b_c] :
          ( collec686158730324120671od_b_c
          @ ^ [Z4: list_l8907847357763382004od_b_c] :
            ? [X4: list_P903359562653991662od_b_c,Xs: list_l8907847357763382004od_b_c] :
              ( ( Z4
                = ( cons_l342771267123639716od_b_c @ X4 @ Xs ) )
              & ( member6330420149250801815od_b_c @ X4 @ A5 )
              & ( member7934871825613078557od_b_c @ Xs @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_1242_set__Cons__def,axiom,
    ( set_Co7706362885935805726od_b_c
    = ( ^ [A5: set_Product_prod_b_c,XS: set_li6436108459499378894od_b_c] :
          ( collec2280997390073109977od_b_c
          @ ^ [Z4: list_P903359562653991662od_b_c] :
            ? [X4: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
              ( ( Z4
                = ( cons_P4529483553340347422od_b_c @ X4 @ Xs ) )
              & ( member7862447941013992593od_b_c @ X4 @ A5 )
              & ( member6330420149250801815od_b_c @ Xs @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_1243_initialised__convergence__preserving__transition__cover__is__complete,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_a_b_c,T5: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M12 )
     => ( ( observable_a_b_c @ M23 )
       => ( ( minimal_a_b_c @ M12 )
         => ( ( minimal_a_b_c @ M23 )
           => ( ( ( inputs_a_b_c @ M23 )
                = ( inputs_a_b_c @ M12 ) )
             => ( ( ( outputs_a_b_c @ M23 )
                  = ( outputs_a_b_c @ M12 ) )
               => ( ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) @ T5 )
                    = ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M23 @ ( initial_a_b_c @ M23 ) ) @ T5 ) )
                 => ( ( ord_le282488521294790766od_b_c @ A4 @ T5 )
                   => ( ( transi6365780447273092801_a_b_c @ M12 @ A4 )
                     => ( ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ A4 )
                       => ( ( preser1543909642583041678_b_c_a @ M12 @ M23 @ A4 )
                         => ( ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) )
                            = ( lS_a_b_c @ M23 @ ( initial_a_b_c @ M23 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% initialised_convergence_preserving_transition_cover_is_complete
thf(fact_1244_initialised__convergence__preserving__transition__cover__is__complete,axiom,
    ! [M12: fsm_a_b_c,M23: fsm_d_b_c,T5: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( observable_a_b_c @ M12 )
     => ( ( observable_d_b_c @ M23 )
       => ( ( minimal_a_b_c @ M12 )
         => ( ( minimal_d_b_c @ M23 )
           => ( ( ( inputs_d_b_c @ M23 )
                = ( inputs_a_b_c @ M12 ) )
             => ( ( ( outputs_d_b_c @ M23 )
                  = ( outputs_a_b_c @ M12 ) )
               => ( ( ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) ) @ T5 )
                    = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M23 @ ( initial_d_b_c @ M23 ) ) @ T5 ) )
                 => ( ( ord_le282488521294790766od_b_c @ A4 @ T5 )
                   => ( ( transi6365780447273092801_a_b_c @ M12 @ A4 )
                     => ( ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ A4 )
                       => ( ( preser1543909642583041681_b_c_d @ M12 @ M23 @ A4 )
                         => ( ( lS_a_b_c @ M12 @ ( initial_a_b_c @ M12 ) )
                            = ( lS_d_b_c @ M23 @ ( initial_d_b_c @ M23 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% initialised_convergence_preserving_transition_cover_is_complete
thf(fact_1245_initialised__convergence__preserving__transition__cover__is__complete,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_a_b_c,T5: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M12 )
     => ( ( observable_a_b_c @ M23 )
       => ( ( minimal_d_b_c @ M12 )
         => ( ( minimal_a_b_c @ M23 )
           => ( ( ( inputs_a_b_c @ M23 )
                = ( inputs_d_b_c @ M12 ) )
             => ( ( ( outputs_a_b_c @ M23 )
                  = ( outputs_d_b_c @ M12 ) )
               => ( ( ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) @ T5 )
                    = ( inf_in4978071631833541052od_b_c @ ( lS_a_b_c @ M23 @ ( initial_a_b_c @ M23 ) ) @ T5 ) )
                 => ( ( ord_le282488521294790766od_b_c @ A4 @ T5 )
                   => ( ( transi4707692219127933252_d_b_c @ M12 @ A4 )
                     => ( ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ A4 )
                       => ( ( preser6007795754467437259_b_c_a @ M12 @ M23 @ A4 )
                         => ( ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) )
                            = ( lS_a_b_c @ M23 @ ( initial_a_b_c @ M23 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% initialised_convergence_preserving_transition_cover_is_complete
thf(fact_1246_initialised__convergence__preserving__transition__cover__is__complete,axiom,
    ! [M12: fsm_d_b_c,M23: fsm_d_b_c,T5: set_li6436108459499378894od_b_c,A4: set_li6436108459499378894od_b_c] :
      ( ( observable_d_b_c @ M12 )
     => ( ( observable_d_b_c @ M23 )
       => ( ( minimal_d_b_c @ M12 )
         => ( ( minimal_d_b_c @ M23 )
           => ( ( ( inputs_d_b_c @ M23 )
                = ( inputs_d_b_c @ M12 ) )
             => ( ( ( outputs_d_b_c @ M23 )
                  = ( outputs_d_b_c @ M12 ) )
               => ( ( ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) ) @ T5 )
                    = ( inf_in4978071631833541052od_b_c @ ( lS_d_b_c @ M23 @ ( initial_d_b_c @ M23 ) ) @ T5 ) )
                 => ( ( ord_le282488521294790766od_b_c @ A4 @ T5 )
                   => ( ( transi4707692219127933252_d_b_c @ M12 @ A4 )
                     => ( ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ A4 )
                       => ( ( preser6007795754467437262_b_c_d @ M12 @ M23 @ A4 )
                         => ( ( lS_d_b_c @ M12 @ ( initial_d_b_c @ M12 ) )
                            = ( lS_d_b_c @ M23 @ ( initial_d_b_c @ M23 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% initialised_convergence_preserving_transition_cover_is_complete
thf(fact_1247_unverified__transitions,axiom,
    ( unveri8662357380954777460itions
    = ( sort_u5845992193009891830itions @ m1 @ v
      @ ( filter3395910357047317171od_c_a
        @ ^ [T2: produc2669932201312201782od_c_a] :
            ( ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ ( reacha1620305530751930115_a_b_c @ m1 ) )
            & ~ ( member5502040268208861279od_c_a @ T2 @ sc_cov38355202683674721itions ) )
        @ ( transi4013027331527956996_a_b_c @ m1 ) ) ) ) ).

% unverified_transitions
thf(fact_1248_sc__covered__transitions,axiom,
    ( sc_cov38355202683674721itions
    = ( comple6722362287630405891od_c_a
      @ ( image_2946797484331234845od_c_a
        @ ^ [Q4: a] : ( state_7626543070582903172_a_b_c @ m1 @ v @ ( v @ Q4 ) )
        @ ( reacha1620305530751930115_a_b_c @ m1 ) ) ) ) ).

% sc_covered_transitions
thf(fact_1249_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat

% Helper facts (5)
thf(help_If_2_1_If_001t__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_T,axiom,
    ! [X: prefix425356687782099470od_b_c,Y: prefix425356687782099470od_b_c] :
      ( ( if_Pre4314254153497284564od_b_c @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_T,axiom,
    ! [X: prefix425356687782099470od_b_c,Y: prefix425356687782099470od_b_c] :
      ( ( if_Pre4314254153497284564od_b_c @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__e_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__e_J_T,axiom,
    ! [X: produc4534891796109904283_b_c_e,Y: produc4534891796109904283_b_c_e] :
      ( ( if_Pro509616640563734881_b_c_e @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Prefix____Tree__Oprefix____tree_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J_Mtf__e_J_T,axiom,
    ! [X: produc4534891796109904283_b_c_e,Y: produc4534891796109904283_b_c_e] :
      ( ( if_Pro509616640563734881_b_c_e @ $true @ X @ Y )
      = X ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [T8: prefix425356687782099470od_b_c,G5: e] :
      ( ( ( produc2511600812177743883_b_c_e @ T8 @ G5 )
        = ( establ5936990325632390625rgence @ m1 @ v @ ( produc1447154562020694007_b_c_e @ tGxy ) @ ( produc6779600879576379961_b_c_e @ tGxy ) @ cg_insert @ cg_lookup @ m @ t4 ) )
     => thesis ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------