%------------------------------------------------------------------------------
% 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/0045_State_Separator/prob_02674_138616__19895510_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2129 ( 588 unt; 852 typ;   0 def)
%            Number of atoms       : 4135 (2153 equ;   0 cnn)
%            Maximal formula atoms :   14 (   3 avg)
%            Number of connectives : 16265 ( 443   ~;  77   |; 703   &;13436   @)
%                                         (   0 <=>;1606  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   7 avg)
%            Number of types       :  221 ( 220 usr)
%            Number of type conns  : 1515 (1515   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  635 ( 632 usr;  37 con; 0-5 aty)
%            Number of variables   : 4440 ( 492   ^;3487   !; 461   ?;4440   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:28:34.095
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__b_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__b_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__b_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__b_J_J_J_J,type,
    member5726013431686633943_a_a_b: produc6221990862682875182_a_a_b > set_Pr7305169452869355022_a_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__b_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
    member7629133530163241470_a_a_a: produc8677025873505188437_a_a_a > set_Pr7404925299755511477_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_J_J,type,
    member4643618012521571351_a_a_a: produc6800189296593107456_a_a_a > set_Pr2800860636864108854_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J,type,
    member4183500612887236192_a_a_a: produc5928450642596067255_a_a_a > set_Pr4956056921713532439_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Sum____Type__Osum_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_J_J_J,type,
    member2160747886658840973_a_a_a: produc2172198381475533558_a_a_a > set_Pr4521937885902694828_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_J_J,type,
    member3530913115812046943_a_a_a: produc1185401179908872758_a_a_a > set_Pr6602633843840758422_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_J_J,type,
    member5834302193316035974_a_a_a: produc7863193890929681519_a_a_a > set_Pr7080229523859472165_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_J,type,
    member6311545829279235415_a_a_a: produc8681278163233174848_a_a_a > set_Pr8976229215889504886_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__b_J_J,type,
    member2904354503647999742_a_a_b: produc3952246846989946709_a_a_b > set_Pr5180098504696546229_a_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J_J,type,
    member8362999488266169055_a_a_a: produc884577481120224310_a_a_a > set_Pr4618682941230328598_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
    member6836137003679210882_a_a_a: produc2766379329844667609_a_a_a > set_Pr5813531400943588409_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__b_J,type,
    member642552543386773071_a_a_b: produc4770514982935793080_a_a_b > set_Pr6246767948713639278_a_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
    member1536750027164364322od_a_a: produc7928013997122480633od_a_a > set_Pr625025929086095065od_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_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__b_J_J_J,type,
    member8152910379392791327od_c_b: produc5320802312496131830od_c_b > set_Pr2571968302941365590od_c_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J_J,type,
    member54949653122719408_a_a_a: produc7810058533641270553_a_a_a > set_Pr7557400373007319375_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__c_J_J_J,type,
    member5460219220941260647od_b_c: produc8442256252373339344od_b_c > set_Pr3755630826337151622od_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
    member6990616694304513154_a_a_a: produc1895207096998757355_a_a_a > set_Pr904891338250078753_a_a_a > $o ).

thf(sy_c_member_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
    member5679158588528110291_a_a_a: sum_su5154756782089200444_a_a_a > set_Su8591296761904255986_a_a_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_v_M,type,
    m: fsm_a_b_c ).

thf(sy_v_cs,type,
    cs: list_P7155397214832950472_a_a_b ).

thf(sy_v_i____,type,
    i: nat ).

thf(sy_v_q1,type,
    q1: a ).

thf(sy_v_q1_H____,type,
    q12: a ).

thf(sy_v_q2,type,
    q2: a ).

thf(sy_v_q2_H____,type,
    q22: a ).

thf(sy_v_t,type,
    t: produc884577481120224310_a_a_a ).

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

thf(sy_v_x____,type,
    x: b ).

% Relevant facts (1274)
thf(fact_0_that_I2_J,axiom,
    ( ( member5679158588528110291_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ t ) ) ) @ ( insert5061175218292437356_a_a_a @ ( sum_In6345099547827038749od_a_a @ q1 ) @ ( insert5061175218292437356_a_a_a @ ( sum_In6345099547827038749od_a_a @ q2 ) @ bot_bo2676859181106700934_a_a_a ) ) )
   => thesis ) ).

% that(2)
thf(fact_1_that_I1_J,axiom,
    ( ( member5679158588528110291_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ t ) ) ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ ( take_P6639847873373218378_a_a_b @ i @ cs ) ) ) )
   => thesis ) ).

% that(1)
thf(fact_2__092_060open_062t__target_At_A_092_060in_062_Aset_A_Imap_Afst_A_Itake_Ai_Acs_J_J_A_092_060union_062_A_123Inr_Aq1_M_AInr_Aq2_125_092_060close_062,axiom,
    member5679158588528110291_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ t ) ) ) @ ( sup_su7096409449263244574_a_a_a @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ ( take_P6639847873373218378_a_a_b @ i @ cs ) ) ) @ ( insert5061175218292437356_a_a_a @ ( sum_In6345099547827038749od_a_a @ q1 ) @ ( insert5061175218292437356_a_a_a @ ( sum_In6345099547827038749od_a_a @ q2 ) @ bot_bo2676859181106700934_a_a_a ) ) ) ).

% \<open>t_target t \<in> set (map fst (take i cs)) \<union> {Inr q1, Inr q2}\<close>
thf(fact_3__092_060open_062t__input_At_A_061_Ax_092_060close_062,axiom,
    ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ t ) )
    = x ) ).

% \<open>t_input t = x\<close>
thf(fact_4_map__eq__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,G: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = ( map_Pr1074589380674644461_a_a_a @ G @ Xs ) )
      = ( ! [X: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ X @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_5_map__eq__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,G: produc1895207096998757355_a_a_a > c] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = ( map_Pr2287363492687751055_a_a_c @ G @ Xs ) )
      = ( ! [X: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ X @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_6_map__eq__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a,G: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ( ( map_Pr3321722608670633418_a_a_a @ F @ Xs )
        = ( map_Pr3321722608670633418_a_a_a @ G @ Xs ) )
      = ( ! [X: produc7810058533641270553_a_a_a] :
            ( ( member54949653122719408_a_a_a @ X @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_7_map__eq__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,G: produc7810058533641270553_a_a_a > b] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = ( map_Pr3622043693041699552_a_a_b @ G @ Xs ) )
      = ( ! [X: produc7810058533641270553_a_a_a] :
            ( ( member54949653122719408_a_a_a @ X @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_8_map__eq__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,G: produc4770514982935793080_a_a_b > b] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = ( map_Pr8925698157833054977_a_b_b @ G @ Xs ) )
      = ( ! [X: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_9_map__eq__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b,G: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ( ( map_Pr7354669405448068282_a_a_a @ F @ Xs )
        = ( map_Pr7354669405448068282_a_a_a @ G @ Xs ) )
      = ( ! [X: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_10_singletonI,axiom,
    ! [A: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ A @ ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) ) ).

% singletonI
thf(fact_11_singletonI,axiom,
    ! [A: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ A @ ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) ) ).

% singletonI
thf(fact_12_singletonI,axiom,
    ! [A: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ A @ ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) ) ).

% singletonI
thf(fact_13_pair__list__eqI,axiom,
    ! [Xs: list_P1813520604369389947_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a @ Xs )
        = ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a @ Ys ) )
     => ( ( ( map_Pr1074589380674644461_a_a_a @ produc1172910294343954743_a_a_a @ Xs )
          = ( map_Pr1074589380674644461_a_a_a @ produc1172910294343954743_a_a_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_14_pair__list__eqI,axiom,
    ! [Xs: list_P7200181061086074537_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ produc5842599967860340391_a_a_a @ Xs )
        = ( map_Pr3622043693041699552_a_a_b @ produc5842599967860340391_a_a_a @ Ys ) )
     => ( ( ( map_Pr3321722608670633418_a_a_a @ produc6236129573204179685_a_a_a @ Xs )
          = ( map_Pr3321722608670633418_a_a_a @ produc6236129573204179685_a_a_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_15_pair__list__eqI,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,Ys: list_P8197596098214363964_a_a_a] :
      ( ( ( map_Pr1409128909316453038_a_a_a @ produc7477144754405272026_a_a_a @ Xs )
        = ( map_Pr1409128909316453038_a_a_a @ produc7477144754405272026_a_a_a @ Ys ) )
     => ( ( ( map_Pr3477903750503579787_a_a_a @ produc1629229363059081756_a_a_a @ Xs )
          = ( map_Pr3477903750503579787_a_a_a @ produc1629229363059081756_a_a_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_16_pair__list__eqI,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Xs )
        = ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Ys ) )
     => ( ( ( map_Pr8925698157833054977_a_b_b @ produc7675857798057594300_a_a_b @ Xs )
          = ( map_Pr8925698157833054977_a_b_b @ produc7675857798057594300_a_a_b @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_17__092_060open_062t__source_At_A_061_Afst_A_Ics_A_B_Ai_J_092_060close_062,axiom,
    ( ( produc7477144754405272026_a_a_a @ t )
    = ( produc4064411471574014846_a_a_b @ ( nth_Pr167341066936870769_a_a_b @ cs @ i ) ) ) ).

% \<open>t_source t = fst (cs ! i)\<close>
thf(fact_18_sum_Oinject_I2_J,axiom,
    ! [X2: a,Y2: a] :
      ( ( ( sum_In6345099547827038749od_a_a @ X2 )
        = ( sum_In6345099547827038749od_a_a @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% sum.inject(2)
thf(fact_19_old_Osum_Oinject_I2_J,axiom,
    ! [B: a,B2: a] :
      ( ( ( sum_In6345099547827038749od_a_a @ B )
        = ( sum_In6345099547827038749od_a_a @ B2 ) )
      = ( B = B2 ) ) ).

% old.sum.inject(2)
thf(fact_20_insertCI,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B3: set_Su8591296761904255986_a_a_a,B: sum_su5154756782089200444_a_a_a] :
      ( ( ~ ( member5679158588528110291_a_a_a @ A @ B3 )
       => ( A = B ) )
     => ( member5679158588528110291_a_a_a @ A @ ( insert5061175218292437356_a_a_a @ B @ B3 ) ) ) ).

% insertCI
thf(fact_21_insertCI,axiom,
    ! [A: produc884577481120224310_a_a_a,B3: set_Pr4618682941230328598_a_a_a,B: produc884577481120224310_a_a_a] :
      ( ( ~ ( member8362999488266169055_a_a_a @ A @ B3 )
       => ( A = B ) )
     => ( member8362999488266169055_a_a_a @ A @ ( insert2249508923356641798_a_a_a @ B @ B3 ) ) ) ).

% insertCI
thf(fact_22_insertCI,axiom,
    ! [A: produc4770514982935793080_a_a_b,B3: set_Pr6246767948713639278_a_a_b,B: produc4770514982935793080_a_a_b] :
      ( ( ~ ( member642552543386773071_a_a_b @ A @ B3 )
       => ( A = B ) )
     => ( member642552543386773071_a_a_b @ A @ ( insert8042272945010224104_a_a_b @ B @ B3 ) ) ) ).

% insertCI
thf(fact_23_insert__iff,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ A @ ( insert5061175218292437356_a_a_a @ B @ A2 ) )
      = ( ( A = B )
        | ( member5679158588528110291_a_a_a @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_24_insert__iff,axiom,
    ! [A: produc884577481120224310_a_a_a,B: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ A @ ( insert2249508923356641798_a_a_a @ B @ A2 ) )
      = ( ( A = B )
        | ( member8362999488266169055_a_a_a @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_25_insert__iff,axiom,
    ! [A: produc4770514982935793080_a_a_b,B: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( member642552543386773071_a_a_b @ A @ ( insert8042272945010224104_a_a_b @ B @ A2 ) )
      = ( ( A = B )
        | ( member642552543386773071_a_a_b @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_26_insert__absorb2,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( insert5061175218292437356_a_a_a @ X3 @ ( insert5061175218292437356_a_a_a @ X3 @ A2 ) )
      = ( insert5061175218292437356_a_a_a @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_27_insert__absorb2,axiom,
    ! [X3: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( insert8042272945010224104_a_a_b @ X3 @ ( insert8042272945010224104_a_a_b @ X3 @ A2 ) )
      = ( insert8042272945010224104_a_a_b @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_28_insert__absorb2,axiom,
    ! [X3: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( insert2249508923356641798_a_a_a @ X3 @ ( insert2249508923356641798_a_a_a @ X3 @ A2 ) )
      = ( insert2249508923356641798_a_a_a @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_29_empty__iff,axiom,
    ! [C: sum_su5154756782089200444_a_a_a] :
      ~ ( member5679158588528110291_a_a_a @ C @ bot_bo2676859181106700934_a_a_a ) ).

% empty_iff
thf(fact_30_empty__iff,axiom,
    ! [C: produc4770514982935793080_a_a_b] :
      ~ ( member642552543386773071_a_a_b @ C @ bot_bo5052542052834138114_a_a_b ) ).

% empty_iff
thf(fact_31_empty__iff,axiom,
    ! [C: produc884577481120224310_a_a_a] :
      ~ ( member8362999488266169055_a_a_a @ C @ bot_bo2898616998271641474_a_a_a ) ).

% empty_iff
thf(fact_32_all__not__in__conv,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ( ( ! [X: sum_su5154756782089200444_a_a_a] :
            ~ ( member5679158588528110291_a_a_a @ X @ A2 ) )
      = ( A2 = bot_bo2676859181106700934_a_a_a ) ) ).

% all_not_in_conv
thf(fact_33_all__not__in__conv,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] :
      ( ( ! [X: produc4770514982935793080_a_a_b] :
            ~ ( member642552543386773071_a_a_b @ X @ A2 ) )
      = ( A2 = bot_bo5052542052834138114_a_a_b ) ) ).

% all_not_in_conv
thf(fact_34_all__not__in__conv,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] :
      ( ( ! [X: produc884577481120224310_a_a_a] :
            ~ ( member8362999488266169055_a_a_a @ X @ A2 ) )
      = ( A2 = bot_bo2898616998271641474_a_a_a ) ) ).

% all_not_in_conv
thf(fact_35_empty__Collect__eq,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( bot_bo2676859181106700934_a_a_a
        = ( collec299858375640228881_a_a_a @ P ) )
      = ( ! [X: sum_su5154756782089200444_a_a_a] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_36_empty__Collect__eq,axiom,
    ! [P: produc4770514982935793080_a_a_b > $o] :
      ( ( bot_bo5052542052834138114_a_a_b
        = ( collec4985708871760228749_a_a_b @ P ) )
      = ( ! [X: produc4770514982935793080_a_a_b] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_37_empty__Collect__eq,axiom,
    ! [P: produc884577481120224310_a_a_a > $o] :
      ( ( bot_bo2898616998271641474_a_a_a
        = ( collec3562233214436895265_a_a_a @ P ) )
      = ( ! [X: produc884577481120224310_a_a_a] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_38_Collect__empty__eq,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( ( collec299858375640228881_a_a_a @ P )
        = bot_bo2676859181106700934_a_a_a )
      = ( ! [X: sum_su5154756782089200444_a_a_a] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_39_Collect__empty__eq,axiom,
    ! [P: produc4770514982935793080_a_a_b > $o] :
      ( ( ( collec4985708871760228749_a_a_b @ P )
        = bot_bo5052542052834138114_a_a_b )
      = ( ! [X: produc4770514982935793080_a_a_b] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_40_Collect__empty__eq,axiom,
    ! [P: produc884577481120224310_a_a_a > $o] :
      ( ( ( collec3562233214436895265_a_a_a @ P )
        = bot_bo2898616998271641474_a_a_a )
      = ( ! [X: produc884577481120224310_a_a_a] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_41_UnCI,axiom,
    ! [C: produc884577481120224310_a_a_a,B3: set_Pr4618682941230328598_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( ~ ( member8362999488266169055_a_a_a @ C @ B3 )
       => ( member8362999488266169055_a_a_a @ C @ A2 ) )
     => ( member8362999488266169055_a_a_a @ C @ ( sup_su9011528761648865002_a_a_a @ A2 @ B3 ) ) ) ).

% UnCI
thf(fact_42_UnCI,axiom,
    ! [C: produc4770514982935793080_a_a_b,B3: set_Pr6246767948713639278_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( ~ ( member642552543386773071_a_a_b @ C @ B3 )
       => ( member642552543386773071_a_a_b @ C @ A2 ) )
     => ( member642552543386773071_a_a_b @ C @ ( sup_su2660408029157294746_a_a_b @ A2 @ B3 ) ) ) ).

% UnCI
thf(fact_43_UnCI,axiom,
    ! [C: sum_su5154756782089200444_a_a_a,B3: set_Su8591296761904255986_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( ~ ( member5679158588528110291_a_a_a @ C @ B3 )
       => ( member5679158588528110291_a_a_a @ C @ A2 ) )
     => ( member5679158588528110291_a_a_a @ C @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) ) ) ).

% UnCI
thf(fact_44_Un__iff,axiom,
    ! [C: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ C @ ( sup_su9011528761648865002_a_a_a @ A2 @ B3 ) )
      = ( ( member8362999488266169055_a_a_a @ C @ A2 )
        | ( member8362999488266169055_a_a_a @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_45_Un__iff,axiom,
    ! [C: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ( member642552543386773071_a_a_b @ C @ ( sup_su2660408029157294746_a_a_b @ A2 @ B3 ) )
      = ( ( member642552543386773071_a_a_b @ C @ A2 )
        | ( member642552543386773071_a_a_b @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_46_Un__iff,axiom,
    ! [C: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ C @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) )
      = ( ( member5679158588528110291_a_a_a @ C @ A2 )
        | ( member5679158588528110291_a_a_a @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_47__092_060open_062t__input_At_A_061_Asnd_A_Ics_A_B_Ai_J_092_060close_062,axiom,
    ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ t ) )
    = ( produc7675857798057594300_a_a_b @ ( nth_Pr167341066936870769_a_a_b @ cs @ i ) ) ) ).

% \<open>t_input t = snd (cs ! i)\<close>
thf(fact_48_Un__empty,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( ( sup_su7096409449263244574_a_a_a @ A2 @ B3 )
        = bot_bo2676859181106700934_a_a_a )
      = ( ( A2 = bot_bo2676859181106700934_a_a_a )
        & ( B3 = bot_bo2676859181106700934_a_a_a ) ) ) ).

% Un_empty
thf(fact_49_Un__empty,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ( ( sup_su2660408029157294746_a_a_b @ A2 @ B3 )
        = bot_bo5052542052834138114_a_a_b )
      = ( ( A2 = bot_bo5052542052834138114_a_a_b )
        & ( B3 = bot_bo5052542052834138114_a_a_b ) ) ) ).

% Un_empty
thf(fact_50_Un__empty,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ( ( sup_su9011528761648865002_a_a_a @ A2 @ B3 )
        = bot_bo2898616998271641474_a_a_a )
      = ( ( A2 = bot_bo2898616998271641474_a_a_a )
        & ( B3 = bot_bo2898616998271641474_a_a_a ) ) ) ).

% Un_empty
thf(fact_51_Un__insert__right,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,A: produc4770514982935793080_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2660408029157294746_a_a_b @ A2 @ ( insert8042272945010224104_a_a_b @ A @ B3 ) )
      = ( insert8042272945010224104_a_a_b @ A @ ( sup_su2660408029157294746_a_a_b @ A2 @ B3 ) ) ) ).

% Un_insert_right
thf(fact_52_Un__insert__right,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,A: produc884577481120224310_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su9011528761648865002_a_a_a @ A2 @ ( insert2249508923356641798_a_a_a @ A @ B3 ) )
      = ( insert2249508923356641798_a_a_a @ A @ ( sup_su9011528761648865002_a_a_a @ A2 @ B3 ) ) ) ).

% Un_insert_right
thf(fact_53_Un__insert__right,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,A: sum_su5154756782089200444_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ A2 @ ( insert5061175218292437356_a_a_a @ A @ B3 ) )
      = ( insert5061175218292437356_a_a_a @ A @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) ) ) ).

% Un_insert_right
thf(fact_54_Un__insert__left,axiom,
    ! [A: produc4770514982935793080_a_a_b,B3: set_Pr6246767948713639278_a_a_b,C2: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2660408029157294746_a_a_b @ ( insert8042272945010224104_a_a_b @ A @ B3 ) @ C2 )
      = ( insert8042272945010224104_a_a_b @ A @ ( sup_su2660408029157294746_a_a_b @ B3 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_55_Un__insert__left,axiom,
    ! [A: produc884577481120224310_a_a_a,B3: set_Pr4618682941230328598_a_a_a,C2: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su9011528761648865002_a_a_a @ ( insert2249508923356641798_a_a_a @ A @ B3 ) @ C2 )
      = ( insert2249508923356641798_a_a_a @ A @ ( sup_su9011528761648865002_a_a_a @ B3 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_56_Un__insert__left,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B3: set_Su8591296761904255986_a_a_a,C2: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ ( insert5061175218292437356_a_a_a @ A @ B3 ) @ C2 )
      = ( insert5061175218292437356_a_a_a @ A @ ( sup_su7096409449263244574_a_a_a @ B3 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_57_UnE,axiom,
    ! [C: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ C @ ( sup_su9011528761648865002_a_a_a @ A2 @ B3 ) )
     => ( ~ ( member8362999488266169055_a_a_a @ C @ A2 )
       => ( member8362999488266169055_a_a_a @ C @ B3 ) ) ) ).

% UnE
thf(fact_58_UnE,axiom,
    ! [C: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ( member642552543386773071_a_a_b @ C @ ( sup_su2660408029157294746_a_a_b @ A2 @ B3 ) )
     => ( ~ ( member642552543386773071_a_a_b @ C @ A2 )
       => ( member642552543386773071_a_a_b @ C @ B3 ) ) ) ).

% UnE
thf(fact_59_UnE,axiom,
    ! [C: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ C @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) )
     => ( ~ ( member5679158588528110291_a_a_a @ C @ A2 )
       => ( member5679158588528110291_a_a_a @ C @ B3 ) ) ) ).

% UnE
thf(fact_60_UnI1,axiom,
    ! [C: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ C @ A2 )
     => ( member8362999488266169055_a_a_a @ C @ ( sup_su9011528761648865002_a_a_a @ A2 @ B3 ) ) ) ).

% UnI1
thf(fact_61_UnI1,axiom,
    ! [C: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ( member642552543386773071_a_a_b @ C @ A2 )
     => ( member642552543386773071_a_a_b @ C @ ( sup_su2660408029157294746_a_a_b @ A2 @ B3 ) ) ) ).

% UnI1
thf(fact_62_UnI1,axiom,
    ! [C: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ C @ A2 )
     => ( member5679158588528110291_a_a_a @ C @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) ) ) ).

% UnI1
thf(fact_63_UnI2,axiom,
    ! [C: produc884577481120224310_a_a_a,B3: set_Pr4618682941230328598_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ C @ B3 )
     => ( member8362999488266169055_a_a_a @ C @ ( sup_su9011528761648865002_a_a_a @ A2 @ B3 ) ) ) ).

% UnI2
thf(fact_64_UnI2,axiom,
    ! [C: produc4770514982935793080_a_a_b,B3: set_Pr6246767948713639278_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( member642552543386773071_a_a_b @ C @ B3 )
     => ( member642552543386773071_a_a_b @ C @ ( sup_su2660408029157294746_a_a_b @ A2 @ B3 ) ) ) ).

% UnI2
thf(fact_65_UnI2,axiom,
    ! [C: sum_su5154756782089200444_a_a_a,B3: set_Su8591296761904255986_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ C @ B3 )
     => ( member5679158588528110291_a_a_a @ C @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) ) ) ).

% UnI2
thf(fact_66_bex__Un,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( ? [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) )
            & ( P @ X ) ) )
      = ( ? [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ A2 )
            & ( P @ X ) )
        | ? [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ B3 )
            & ( P @ X ) ) ) ) ).

% bex_Un
thf(fact_67_ball__Un,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( ! [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) )
           => ( P @ X ) ) )
      = ( ! [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ A2 )
           => ( P @ X ) )
        & ! [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ B3 )
           => ( P @ X ) ) ) ) ).

% ball_Un
thf(fact_68_Un__assoc,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,C2: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) @ C2 )
      = ( sup_su7096409449263244574_a_a_a @ A2 @ ( sup_su7096409449263244574_a_a_a @ B3 @ C2 ) ) ) ).

% Un_assoc
thf(fact_69_Un__absorb,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ A2 @ A2 )
      = A2 ) ).

% Un_absorb
thf(fact_70_Un__commute,axiom,
    ( sup_su7096409449263244574_a_a_a
    = ( ^ [A3: set_Su8591296761904255986_a_a_a,B4: set_Su8591296761904255986_a_a_a] : ( sup_su7096409449263244574_a_a_a @ B4 @ A3 ) ) ) ).

% Un_commute
thf(fact_71_Un__left__absorb,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ A2 @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) )
      = ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) ) ).

% Un_left_absorb
thf(fact_72_Un__left__commute,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,C2: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ A2 @ ( sup_su7096409449263244574_a_a_a @ B3 @ C2 ) )
      = ( sup_su7096409449263244574_a_a_a @ B3 @ ( sup_su7096409449263244574_a_a_a @ A2 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_73_Un__empty__right,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ A2 @ bot_bo2676859181106700934_a_a_a )
      = A2 ) ).

% Un_empty_right
thf(fact_74_Un__empty__right,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2660408029157294746_a_a_b @ A2 @ bot_bo5052542052834138114_a_a_b )
      = A2 ) ).

% Un_empty_right
thf(fact_75_Un__empty__right,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su9011528761648865002_a_a_a @ A2 @ bot_bo2898616998271641474_a_a_a )
      = A2 ) ).

% Un_empty_right
thf(fact_76_Un__empty__left,axiom,
    ! [B3: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ bot_bo2676859181106700934_a_a_a @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_77_Un__empty__left,axiom,
    ! [B3: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2660408029157294746_a_a_b @ bot_bo5052542052834138114_a_a_b @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_78_Un__empty__left,axiom,
    ! [B3: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su9011528761648865002_a_a_a @ bot_bo2898616998271641474_a_a_a @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_79_ex__in__conv,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ( ( ? [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ A2 ) )
      = ( A2 != bot_bo2676859181106700934_a_a_a ) ) ).

% ex_in_conv
thf(fact_80_ex__in__conv,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] :
      ( ( ? [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ A2 ) )
      = ( A2 != bot_bo5052542052834138114_a_a_b ) ) ).

% ex_in_conv
thf(fact_81_ex__in__conv,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] :
      ( ( ? [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ A2 ) )
      = ( A2 != bot_bo2898616998271641474_a_a_a ) ) ).

% ex_in_conv
thf(fact_82_equals0I,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ( ! [Y: sum_su5154756782089200444_a_a_a] :
          ~ ( member5679158588528110291_a_a_a @ Y @ A2 )
     => ( A2 = bot_bo2676859181106700934_a_a_a ) ) ).

% equals0I
thf(fact_83_equals0I,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] :
      ( ! [Y: produc4770514982935793080_a_a_b] :
          ~ ( member642552543386773071_a_a_b @ Y @ A2 )
     => ( A2 = bot_bo5052542052834138114_a_a_b ) ) ).

% equals0I
thf(fact_84_equals0I,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] :
      ( ! [Y: produc884577481120224310_a_a_a] :
          ~ ( member8362999488266169055_a_a_a @ Y @ A2 )
     => ( A2 = bot_bo2898616998271641474_a_a_a ) ) ).

% equals0I
thf(fact_85_equals0D,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,A: sum_su5154756782089200444_a_a_a] :
      ( ( A2 = bot_bo2676859181106700934_a_a_a )
     => ~ ( member5679158588528110291_a_a_a @ A @ A2 ) ) ).

% equals0D
thf(fact_86_equals0D,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,A: produc4770514982935793080_a_a_b] :
      ( ( A2 = bot_bo5052542052834138114_a_a_b )
     => ~ ( member642552543386773071_a_a_b @ A @ A2 ) ) ).

% equals0D
thf(fact_87_equals0D,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,A: produc884577481120224310_a_a_a] :
      ( ( A2 = bot_bo2898616998271641474_a_a_a )
     => ~ ( member8362999488266169055_a_a_a @ A @ A2 ) ) ).

% equals0D
thf(fact_88_emptyE,axiom,
    ! [A: sum_su5154756782089200444_a_a_a] :
      ~ ( member5679158588528110291_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) ).

% emptyE
thf(fact_89_emptyE,axiom,
    ! [A: produc4770514982935793080_a_a_b] :
      ~ ( member642552543386773071_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) ).

% emptyE
thf(fact_90_emptyE,axiom,
    ! [A: produc884577481120224310_a_a_a] :
      ~ ( member8362999488266169055_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) ).

% emptyE
thf(fact_91_mk__disjoint__insert,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ A @ A2 )
     => ? [B5: set_Su8591296761904255986_a_a_a] :
          ( ( A2
            = ( insert5061175218292437356_a_a_a @ A @ B5 ) )
          & ~ ( member5679158588528110291_a_a_a @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_92_mk__disjoint__insert,axiom,
    ! [A: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ A @ A2 )
     => ? [B5: set_Pr4618682941230328598_a_a_a] :
          ( ( A2
            = ( insert2249508923356641798_a_a_a @ A @ B5 ) )
          & ~ ( member8362999488266169055_a_a_a @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_93_mk__disjoint__insert,axiom,
    ! [A: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( member642552543386773071_a_a_b @ A @ A2 )
     => ? [B5: set_Pr6246767948713639278_a_a_b] :
          ( ( A2
            = ( insert8042272945010224104_a_a_b @ A @ B5 ) )
          & ~ ( member642552543386773071_a_a_b @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_94_insert__commute,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( insert5061175218292437356_a_a_a @ X3 @ ( insert5061175218292437356_a_a_a @ Y3 @ A2 ) )
      = ( insert5061175218292437356_a_a_a @ Y3 @ ( insert5061175218292437356_a_a_a @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_95_insert__commute,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Y3: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( insert8042272945010224104_a_a_b @ X3 @ ( insert8042272945010224104_a_a_b @ Y3 @ A2 ) )
      = ( insert8042272945010224104_a_a_b @ Y3 @ ( insert8042272945010224104_a_a_b @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_96_insert__commute,axiom,
    ! [X3: produc884577481120224310_a_a_a,Y3: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( insert2249508923356641798_a_a_a @ X3 @ ( insert2249508923356641798_a_a_a @ Y3 @ A2 ) )
      = ( insert2249508923356641798_a_a_a @ Y3 @ ( insert2249508923356641798_a_a_a @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_97_insert__eq__iff,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a,B: sum_su5154756782089200444_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ~ ( member5679158588528110291_a_a_a @ A @ A2 )
     => ( ~ ( member5679158588528110291_a_a_a @ B @ B3 )
       => ( ( ( insert5061175218292437356_a_a_a @ A @ A2 )
            = ( insert5061175218292437356_a_a_a @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A2 = B3 ) )
            & ( ( A != B )
             => ? [C3: set_Su8591296761904255986_a_a_a] :
                  ( ( A2
                    = ( insert5061175218292437356_a_a_a @ B @ C3 ) )
                  & ~ ( member5679158588528110291_a_a_a @ B @ C3 )
                  & ( B3
                    = ( insert5061175218292437356_a_a_a @ A @ C3 ) )
                  & ~ ( member5679158588528110291_a_a_a @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_98_insert__eq__iff,axiom,
    ! [A: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a,B: produc884577481120224310_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ~ ( member8362999488266169055_a_a_a @ A @ A2 )
     => ( ~ ( member8362999488266169055_a_a_a @ B @ B3 )
       => ( ( ( insert2249508923356641798_a_a_a @ A @ A2 )
            = ( insert2249508923356641798_a_a_a @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A2 = B3 ) )
            & ( ( A != B )
             => ? [C3: set_Pr4618682941230328598_a_a_a] :
                  ( ( A2
                    = ( insert2249508923356641798_a_a_a @ B @ C3 ) )
                  & ~ ( member8362999488266169055_a_a_a @ B @ C3 )
                  & ( B3
                    = ( insert2249508923356641798_a_a_a @ A @ C3 ) )
                  & ~ ( member8362999488266169055_a_a_a @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_99_insert__eq__iff,axiom,
    ! [A: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b,B: produc4770514982935793080_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ~ ( member642552543386773071_a_a_b @ A @ A2 )
     => ( ~ ( member642552543386773071_a_a_b @ B @ B3 )
       => ( ( ( insert8042272945010224104_a_a_b @ A @ A2 )
            = ( insert8042272945010224104_a_a_b @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A2 = B3 ) )
            & ( ( A != B )
             => ? [C3: set_Pr6246767948713639278_a_a_b] :
                  ( ( A2
                    = ( insert8042272945010224104_a_a_b @ B @ C3 ) )
                  & ~ ( member642552543386773071_a_a_b @ B @ C3 )
                  & ( B3
                    = ( insert8042272945010224104_a_a_b @ A @ C3 ) )
                  & ~ ( member642552543386773071_a_a_b @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_100_insert__absorb,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ A @ A2 )
     => ( ( insert5061175218292437356_a_a_a @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_101_insert__absorb,axiom,
    ! [A: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ A @ A2 )
     => ( ( insert2249508923356641798_a_a_a @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_102_insert__absorb,axiom,
    ! [A: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( member642552543386773071_a_a_b @ A @ A2 )
     => ( ( insert8042272945010224104_a_a_b @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_103_insert__ident,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ~ ( member5679158588528110291_a_a_a @ X3 @ A2 )
     => ( ~ ( member5679158588528110291_a_a_a @ X3 @ B3 )
       => ( ( ( insert5061175218292437356_a_a_a @ X3 @ A2 )
            = ( insert5061175218292437356_a_a_a @ X3 @ B3 ) )
          = ( A2 = B3 ) ) ) ) ).

% insert_ident
thf(fact_104_insert__ident,axiom,
    ! [X3: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ~ ( member8362999488266169055_a_a_a @ X3 @ A2 )
     => ( ~ ( member8362999488266169055_a_a_a @ X3 @ B3 )
       => ( ( ( insert2249508923356641798_a_a_a @ X3 @ A2 )
            = ( insert2249508923356641798_a_a_a @ X3 @ B3 ) )
          = ( A2 = B3 ) ) ) ) ).

% insert_ident
thf(fact_105_insert__ident,axiom,
    ! [X3: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ~ ( member642552543386773071_a_a_b @ X3 @ A2 )
     => ( ~ ( member642552543386773071_a_a_b @ X3 @ B3 )
       => ( ( ( insert8042272945010224104_a_a_b @ X3 @ A2 )
            = ( insert8042272945010224104_a_a_b @ X3 @ B3 ) )
          = ( A2 = B3 ) ) ) ) ).

% insert_ident
thf(fact_106_Set_Oset__insert,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ A2 )
     => ~ ! [B5: set_Su8591296761904255986_a_a_a] :
            ( ( A2
              = ( insert5061175218292437356_a_a_a @ X3 @ B5 ) )
           => ( member5679158588528110291_a_a_a @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_107_Set_Oset__insert,axiom,
    ! [X3: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ A2 )
     => ~ ! [B5: set_Pr4618682941230328598_a_a_a] :
            ( ( A2
              = ( insert2249508923356641798_a_a_a @ X3 @ B5 ) )
           => ( member8362999488266169055_a_a_a @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_108_Set_Oset__insert,axiom,
    ! [X3: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ A2 )
     => ~ ! [B5: set_Pr6246767948713639278_a_a_b] :
            ( ( A2
              = ( insert8042272945010224104_a_a_b @ X3 @ B5 ) )
           => ( member642552543386773071_a_a_b @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_109_mem__Collect__eq,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( member5679158588528110291_a_a_a @ A @ ( collec299858375640228881_a_a_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
    ! [A: produc884577481120224310_a_a_a,P: produc884577481120224310_a_a_a > $o] :
      ( ( member8362999488266169055_a_a_a @ A @ ( collec3562233214436895265_a_a_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
    ! [A: produc4770514982935793080_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ( member642552543386773071_a_a_b @ A @ ( collec4985708871760228749_a_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_112_Collect__mem__eq,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ( ( collec299858375640228881_a_a_a
        @ ^ [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] :
      ( ( collec3562233214436895265_a_a_a
        @ ^ [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] :
      ( ( collec4985708871760228749_a_a_b
        @ ^ [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_115_insertI2,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B3: set_Su8591296761904255986_a_a_a,B: sum_su5154756782089200444_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ A @ B3 )
     => ( member5679158588528110291_a_a_a @ A @ ( insert5061175218292437356_a_a_a @ B @ B3 ) ) ) ).

% insertI2
thf(fact_116_insertI2,axiom,
    ! [A: produc884577481120224310_a_a_a,B3: set_Pr4618682941230328598_a_a_a,B: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ A @ B3 )
     => ( member8362999488266169055_a_a_a @ A @ ( insert2249508923356641798_a_a_a @ B @ B3 ) ) ) ).

% insertI2
thf(fact_117_insertI2,axiom,
    ! [A: produc4770514982935793080_a_a_b,B3: set_Pr6246767948713639278_a_a_b,B: produc4770514982935793080_a_a_b] :
      ( ( member642552543386773071_a_a_b @ A @ B3 )
     => ( member642552543386773071_a_a_b @ A @ ( insert8042272945010224104_a_a_b @ B @ B3 ) ) ) ).

% insertI2
thf(fact_118_insertI1,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B3: set_Su8591296761904255986_a_a_a] : ( member5679158588528110291_a_a_a @ A @ ( insert5061175218292437356_a_a_a @ A @ B3 ) ) ).

% insertI1
thf(fact_119_insertI1,axiom,
    ! [A: produc884577481120224310_a_a_a,B3: set_Pr4618682941230328598_a_a_a] : ( member8362999488266169055_a_a_a @ A @ ( insert2249508923356641798_a_a_a @ A @ B3 ) ) ).

% insertI1
thf(fact_120_insertI1,axiom,
    ! [A: produc4770514982935793080_a_a_b,B3: set_Pr6246767948713639278_a_a_b] : ( member642552543386773071_a_a_b @ A @ ( insert8042272945010224104_a_a_b @ A @ B3 ) ) ).

% insertI1
thf(fact_121_insertE,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ A @ ( insert5061175218292437356_a_a_a @ B @ A2 ) )
     => ( ( A != B )
       => ( member5679158588528110291_a_a_a @ A @ A2 ) ) ) ).

% insertE
thf(fact_122_insertE,axiom,
    ! [A: produc884577481120224310_a_a_a,B: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ A @ ( insert2249508923356641798_a_a_a @ B @ A2 ) )
     => ( ( A != B )
       => ( member8362999488266169055_a_a_a @ A @ A2 ) ) ) ).

% insertE
thf(fact_123_insertE,axiom,
    ! [A: produc4770514982935793080_a_a_b,B: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( member642552543386773071_a_a_b @ A @ ( insert8042272945010224104_a_a_b @ B @ A2 ) )
     => ( ( A != B )
       => ( member642552543386773071_a_a_b @ A @ A2 ) ) ) ).

% insertE
thf(fact_124_singleton__Un__iff,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a )
        = ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) )
      = ( ( ( A2 = bot_bo2676859181106700934_a_a_a )
          & ( B3
            = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) ) )
        | ( ( A2
            = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) )
          & ( B3 = bot_bo2676859181106700934_a_a_a ) )
        | ( ( A2
            = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) )
          & ( B3
            = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_125_singleton__Un__iff,axiom,
    ! [X3: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ( ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b )
        = ( sup_su2660408029157294746_a_a_b @ A2 @ B3 ) )
      = ( ( ( A2 = bot_bo5052542052834138114_a_a_b )
          & ( B3
            = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) ) )
        | ( ( A2
            = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) )
          & ( B3 = bot_bo5052542052834138114_a_a_b ) )
        | ( ( A2
            = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) )
          & ( B3
            = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_126_singleton__Un__iff,axiom,
    ! [X3: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ( ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a )
        = ( sup_su9011528761648865002_a_a_a @ A2 @ B3 ) )
      = ( ( ( A2 = bot_bo2898616998271641474_a_a_a )
          & ( B3
            = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) ) )
        | ( ( A2
            = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) )
          & ( B3 = bot_bo2898616998271641474_a_a_a ) )
        | ( ( A2
            = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) )
          & ( B3
            = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_127_Un__singleton__iff,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,X3: sum_su5154756782089200444_a_a_a] :
      ( ( ( sup_su7096409449263244574_a_a_a @ A2 @ B3 )
        = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) )
      = ( ( ( A2 = bot_bo2676859181106700934_a_a_a )
          & ( B3
            = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) ) )
        | ( ( A2
            = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) )
          & ( B3 = bot_bo2676859181106700934_a_a_a ) )
        | ( ( A2
            = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) )
          & ( B3
            = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_128_Un__singleton__iff,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b,X3: produc4770514982935793080_a_a_b] :
      ( ( ( sup_su2660408029157294746_a_a_b @ A2 @ B3 )
        = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) )
      = ( ( ( A2 = bot_bo5052542052834138114_a_a_b )
          & ( B3
            = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) ) )
        | ( ( A2
            = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) )
          & ( B3 = bot_bo5052542052834138114_a_a_b ) )
        | ( ( A2
            = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) )
          & ( B3
            = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_129_Un__singleton__iff,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a,X3: produc884577481120224310_a_a_a] :
      ( ( ( sup_su9011528761648865002_a_a_a @ A2 @ B3 )
        = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) )
      = ( ( ( A2 = bot_bo2898616998271641474_a_a_a )
          & ( B3
            = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) ) )
        | ( ( A2
            = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) )
          & ( B3 = bot_bo2898616998271641474_a_a_a ) )
        | ( ( A2
            = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) )
          & ( B3
            = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_130_insert__is__Un,axiom,
    ( insert5061175218292437356_a_a_a
    = ( ^ [A4: sum_su5154756782089200444_a_a_a] : ( sup_su7096409449263244574_a_a_a @ ( insert5061175218292437356_a_a_a @ A4 @ bot_bo2676859181106700934_a_a_a ) ) ) ) ).

% insert_is_Un
thf(fact_131_insert__is__Un,axiom,
    ( insert8042272945010224104_a_a_b
    = ( ^ [A4: produc4770514982935793080_a_a_b] : ( sup_su2660408029157294746_a_a_b @ ( insert8042272945010224104_a_a_b @ A4 @ bot_bo5052542052834138114_a_a_b ) ) ) ) ).

% insert_is_Un
thf(fact_132_insert__is__Un,axiom,
    ( insert2249508923356641798_a_a_a
    = ( ^ [A4: produc884577481120224310_a_a_a] : ( sup_su9011528761648865002_a_a_a @ ( insert2249508923356641798_a_a_a @ A4 @ bot_bo2898616998271641474_a_a_a ) ) ) ) ).

% insert_is_Un
thf(fact_133_Inr__inject,axiom,
    ! [X3: a,Y3: a] :
      ( ( ( sum_In6345099547827038749od_a_a @ X3 )
        = ( sum_In6345099547827038749od_a_a @ Y3 ) )
     => ( X3 = Y3 ) ) ).

% Inr_inject
thf(fact_134_take__equalityI,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ! [I: nat] :
          ( ( take_P6639847873373218378_a_a_b @ I @ Xs )
          = ( take_P6639847873373218378_a_a_b @ I @ Ys ) )
     => ( Xs = Ys ) ) ).

% take_equalityI
thf(fact_135_singleton__inject,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B: sum_su5154756782089200444_a_a_a] :
      ( ( ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a )
        = ( insert5061175218292437356_a_a_a @ B @ bot_bo2676859181106700934_a_a_a ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_136_singleton__inject,axiom,
    ! [A: produc4770514982935793080_a_a_b,B: produc4770514982935793080_a_a_b] :
      ( ( ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b )
        = ( insert8042272945010224104_a_a_b @ B @ bot_bo5052542052834138114_a_a_b ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_137_singleton__inject,axiom,
    ! [A: produc884577481120224310_a_a_a,B: produc884577481120224310_a_a_a] :
      ( ( ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a )
        = ( insert2249508923356641798_a_a_a @ B @ bot_bo2898616998271641474_a_a_a ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_138_insert__not__empty,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( insert5061175218292437356_a_a_a @ A @ A2 )
     != bot_bo2676859181106700934_a_a_a ) ).

% insert_not_empty
thf(fact_139_insert__not__empty,axiom,
    ! [A: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( insert8042272945010224104_a_a_b @ A @ A2 )
     != bot_bo5052542052834138114_a_a_b ) ).

% insert_not_empty
thf(fact_140_insert__not__empty,axiom,
    ! [A: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( insert2249508923356641798_a_a_a @ A @ A2 )
     != bot_bo2898616998271641474_a_a_a ) ).

% insert_not_empty
thf(fact_141_doubleton__eq__iff,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B: sum_su5154756782089200444_a_a_a,C: sum_su5154756782089200444_a_a_a,D: sum_su5154756782089200444_a_a_a] :
      ( ( ( insert5061175218292437356_a_a_a @ A @ ( insert5061175218292437356_a_a_a @ B @ bot_bo2676859181106700934_a_a_a ) )
        = ( insert5061175218292437356_a_a_a @ C @ ( insert5061175218292437356_a_a_a @ D @ bot_bo2676859181106700934_a_a_a ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_142_doubleton__eq__iff,axiom,
    ! [A: produc4770514982935793080_a_a_b,B: produc4770514982935793080_a_a_b,C: produc4770514982935793080_a_a_b,D: produc4770514982935793080_a_a_b] :
      ( ( ( insert8042272945010224104_a_a_b @ A @ ( insert8042272945010224104_a_a_b @ B @ bot_bo5052542052834138114_a_a_b ) )
        = ( insert8042272945010224104_a_a_b @ C @ ( insert8042272945010224104_a_a_b @ D @ bot_bo5052542052834138114_a_a_b ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_143_doubleton__eq__iff,axiom,
    ! [A: produc884577481120224310_a_a_a,B: produc884577481120224310_a_a_a,C: produc884577481120224310_a_a_a,D: produc884577481120224310_a_a_a] :
      ( ( ( insert2249508923356641798_a_a_a @ A @ ( insert2249508923356641798_a_a_a @ B @ bot_bo2898616998271641474_a_a_a ) )
        = ( insert2249508923356641798_a_a_a @ C @ ( insert2249508923356641798_a_a_a @ D @ bot_bo2898616998271641474_a_a_a ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_144_singleton__iff,axiom,
    ! [B: sum_su5154756782089200444_a_a_a,A: sum_su5154756782089200444_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ B @ ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_145_singleton__iff,axiom,
    ! [B: produc4770514982935793080_a_a_b,A: produc4770514982935793080_a_a_b] :
      ( ( member642552543386773071_a_a_b @ B @ ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_146_singleton__iff,axiom,
    ! [B: produc884577481120224310_a_a_a,A: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ B @ ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_147_singletonD,axiom,
    ! [B: sum_su5154756782089200444_a_a_a,A: sum_su5154756782089200444_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ B @ ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_148_singletonD,axiom,
    ! [B: produc4770514982935793080_a_a_b,A: produc4770514982935793080_a_a_b] :
      ( ( member642552543386773071_a_a_b @ B @ ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_149_singletonD,axiom,
    ! [B: produc884577481120224310_a_a_a,A: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ B @ ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_150_ex__map__conv,axiom,
    ! [Ys: list_b,F: produc4770514982935793080_a_a_b > b] :
      ( ( ? [Xs2: list_P7155397214832950472_a_a_b] :
            ( Ys
            = ( map_Pr8925698157833054977_a_b_b @ F @ Xs2 ) ) )
      = ( ! [X: b] :
            ( ( member_b @ X @ ( set_b2 @ Ys ) )
           => ? [Y4: produc4770514982935793080_a_a_b] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_151_ex__map__conv,axiom,
    ! [Ys: list_c,F: produc1895207096998757355_a_a_a > c] :
      ( ( ? [Xs2: list_P1813520604369389947_a_a_a] :
            ( Ys
            = ( map_Pr2287363492687751055_a_a_c @ F @ Xs2 ) ) )
      = ( ! [X: c] :
            ( ( member_c @ X @ ( set_c2 @ Ys ) )
           => ? [Y4: produc1895207096998757355_a_a_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_152_ex__map__conv,axiom,
    ! [Ys: list_P1813520604369389947_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ( ? [Xs2: list_P7200181061086074537_a_a_a] :
            ( Ys
            = ( map_Pr3321722608670633418_a_a_a @ F @ Xs2 ) ) )
      = ( ! [X: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ X @ ( set_Pr1702711106814914326_a_a_a @ Ys ) )
           => ? [Y4: produc7810058533641270553_a_a_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_153_ex__map__conv,axiom,
    ! [Ys: list_b,F: produc7810058533641270553_a_a_a > b] :
      ( ( ? [Xs2: list_P7200181061086074537_a_a_a] :
            ( Ys
            = ( map_Pr3622043693041699552_a_a_b @ F @ Xs2 ) ) )
      = ( ! [X: b] :
            ( ( member_b @ X @ ( set_b2 @ Ys ) )
           => ? [Y4: produc7810058533641270553_a_a_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_154_ex__map__conv,axiom,
    ! [Ys: list_S8072159660131056972_a_a_a,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ( ? [Xs2: list_P7155397214832950472_a_a_b] :
            ( Ys
            = ( map_Pr7354669405448068282_a_a_a @ F @ Xs2 ) ) )
      = ( ! [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ ( set_Su3113867532901839207_a_a_a @ Ys ) )
           => ? [Y4: produc4770514982935793080_a_a_b] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_155_ex__map__conv,axiom,
    ! [Ys: list_S8072159660131056972_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( ? [Xs2: list_P1813520604369389947_a_a_a] :
            ( Ys
            = ( map_Pr1074589380674644461_a_a_a @ F @ Xs2 ) ) )
      = ( ! [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ ( set_Su3113867532901839207_a_a_a @ Ys ) )
           => ? [Y4: produc1895207096998757355_a_a_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_156_map__cong,axiom,
    ! [Xs: list_P1813520604369389947_a_a_a,Ys: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,G: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( Xs = Ys )
     => ( ! [X4: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ X4 @ ( set_Pr1702711106814914326_a_a_a @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
          = ( map_Pr1074589380674644461_a_a_a @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_157_map__cong,axiom,
    ! [Xs: list_P1813520604369389947_a_a_a,Ys: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > c,G: produc1895207096998757355_a_a_a > c] :
      ( ( Xs = Ys )
     => ( ! [X4: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ X4 @ ( set_Pr1702711106814914326_a_a_a @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
          = ( map_Pr2287363492687751055_a_a_c @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_158_map__cong,axiom,
    ! [Xs: list_P7200181061086074537_a_a_a,Ys: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,G: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ( Xs = Ys )
     => ( ! [X4: produc7810058533641270553_a_a_a] :
            ( ( member54949653122719408_a_a_a @ X4 @ ( set_Pr6411044387506519364_a_a_a @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_Pr3321722608670633418_a_a_a @ F @ Xs )
          = ( map_Pr3321722608670633418_a_a_a @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_159_map__cong,axiom,
    ! [Xs: list_P7200181061086074537_a_a_a,Ys: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > b,G: produc7810058533641270553_a_a_a > b] :
      ( ( Xs = Ys )
     => ( ! [X4: produc7810058533641270553_a_a_a] :
            ( ( member54949653122719408_a_a_a @ X4 @ ( set_Pr6411044387506519364_a_a_a @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
          = ( map_Pr3622043693041699552_a_a_b @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_160_map__cong,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,G: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ( Xs = Ys )
     => ( ! [X4: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_Pr7354669405448068282_a_a_a @ F @ Xs )
          = ( map_Pr7354669405448068282_a_a_a @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_161_map__cong,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > b,G: produc4770514982935793080_a_a_b > b] :
      ( ( Xs = Ys )
     => ( ! [X4: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
          = ( map_Pr8925698157833054977_a_b_b @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_162_map__idI,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,F: produc884577481120224310_a_a_a > produc884577481120224310_a_a_a] :
      ( ! [X4: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X4 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_Pr8743395878510960708_a_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_163_map__idI,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,F: sum_su5154756782089200444_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ! [X4: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X4 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_Su7760560175274971966_a_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_164_map__idI,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > produc4770514982935793080_a_a_b] :
      ( ! [X4: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_Pr2859183448225536310_a_a_b @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_165_map__ext,axiom,
    ! [Xs: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,G: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ! [X4: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ X4 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = ( map_Pr1074589380674644461_a_a_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_166_map__ext,axiom,
    ! [Xs: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > c,G: produc1895207096998757355_a_a_a > c] :
      ( ! [X4: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ X4 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = ( map_Pr2287363492687751055_a_a_c @ G @ Xs ) ) ) ).

% map_ext
thf(fact_167_map__ext,axiom,
    ! [Xs: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,G: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ! [X4: produc7810058533641270553_a_a_a] :
          ( ( member54949653122719408_a_a_a @ X4 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_Pr3321722608670633418_a_a_a @ F @ Xs )
        = ( map_Pr3321722608670633418_a_a_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_168_map__ext,axiom,
    ! [Xs: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > b,G: produc7810058533641270553_a_a_a > b] :
      ( ! [X4: produc7810058533641270553_a_a_a] :
          ( ( member54949653122719408_a_a_a @ X4 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = ( map_Pr3622043693041699552_a_a_b @ G @ Xs ) ) ) ).

% map_ext
thf(fact_169_map__ext,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,G: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ! [X4: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_Pr7354669405448068282_a_a_a @ F @ Xs )
        = ( map_Pr7354669405448068282_a_a_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_170_map__ext,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > b,G: produc4770514982935793080_a_a_b > b] :
      ( ! [X4: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = ( map_Pr8925698157833054977_a_b_b @ G @ Xs ) ) ) ).

% map_ext
thf(fact_171_list_Omap__ident__strong,axiom,
    ! [T: list_P8197596098214363964_a_a_a,F: produc884577481120224310_a_a_a > produc884577481120224310_a_a_a] :
      ( ! [Z: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ Z @ ( set_Pr8067384617587748683_a_a_a @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_Pr8743395878510960708_a_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_172_list_Omap__ident__strong,axiom,
    ! [T: list_S8072159660131056972_a_a_a,F: sum_su5154756782089200444_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ! [Z: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ Z @ ( set_Su3113867532901839207_a_a_a @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_Su7760560175274971966_a_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_173_list_Omap__ident__strong,axiom,
    ! [T: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > produc4770514982935793080_a_a_b] :
      ( ! [Z: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ Z @ ( set_Pr4578018992751950051_a_a_b @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_Pr2859183448225536310_a_a_b @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_174_list_Oinj__map__strong,axiom,
    ! [X3: list_P1813520604369389947_a_a_a,Xa: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Fa: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ! [Z: produc1895207096998757355_a_a_a,Za: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ Z @ ( set_Pr1702711106814914326_a_a_a @ X3 ) )
         => ( ( member6990616694304513154_a_a_a @ Za @ ( set_Pr1702711106814914326_a_a_a @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_Pr1074589380674644461_a_a_a @ F @ X3 )
          = ( map_Pr1074589380674644461_a_a_a @ Fa @ Xa ) )
       => ( X3 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_175_list_Oinj__map__strong,axiom,
    ! [X3: list_P1813520604369389947_a_a_a,Xa: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > c,Fa: produc1895207096998757355_a_a_a > c] :
      ( ! [Z: produc1895207096998757355_a_a_a,Za: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ Z @ ( set_Pr1702711106814914326_a_a_a @ X3 ) )
         => ( ( member6990616694304513154_a_a_a @ Za @ ( set_Pr1702711106814914326_a_a_a @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_Pr2287363492687751055_a_a_c @ F @ X3 )
          = ( map_Pr2287363492687751055_a_a_c @ Fa @ Xa ) )
       => ( X3 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_176_list_Oinj__map__strong,axiom,
    ! [X3: list_P7200181061086074537_a_a_a,Xa: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Fa: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ! [Z: produc7810058533641270553_a_a_a,Za: produc7810058533641270553_a_a_a] :
          ( ( member54949653122719408_a_a_a @ Z @ ( set_Pr6411044387506519364_a_a_a @ X3 ) )
         => ( ( member54949653122719408_a_a_a @ Za @ ( set_Pr6411044387506519364_a_a_a @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_Pr3321722608670633418_a_a_a @ F @ X3 )
          = ( map_Pr3321722608670633418_a_a_a @ Fa @ Xa ) )
       => ( X3 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_177_list_Oinj__map__strong,axiom,
    ! [X3: list_P7200181061086074537_a_a_a,Xa: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > b,Fa: produc7810058533641270553_a_a_a > b] :
      ( ! [Z: produc7810058533641270553_a_a_a,Za: produc7810058533641270553_a_a_a] :
          ( ( member54949653122719408_a_a_a @ Z @ ( set_Pr6411044387506519364_a_a_a @ X3 ) )
         => ( ( member54949653122719408_a_a_a @ Za @ ( set_Pr6411044387506519364_a_a_a @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_Pr3622043693041699552_a_a_b @ F @ X3 )
          = ( map_Pr3622043693041699552_a_a_b @ Fa @ Xa ) )
       => ( X3 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_178_list_Oinj__map__strong,axiom,
    ! [X3: list_P7155397214832950472_a_a_b,Xa: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Fa: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ! [Z: produc4770514982935793080_a_a_b,Za: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ Z @ ( set_Pr4578018992751950051_a_a_b @ X3 ) )
         => ( ( member642552543386773071_a_a_b @ Za @ ( set_Pr4578018992751950051_a_a_b @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_Pr7354669405448068282_a_a_a @ F @ X3 )
          = ( map_Pr7354669405448068282_a_a_a @ Fa @ Xa ) )
       => ( X3 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_179_list_Oinj__map__strong,axiom,
    ! [X3: list_P7155397214832950472_a_a_b,Xa: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > b,Fa: produc4770514982935793080_a_a_b > b] :
      ( ! [Z: produc4770514982935793080_a_a_b,Za: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ Z @ ( set_Pr4578018992751950051_a_a_b @ X3 ) )
         => ( ( member642552543386773071_a_a_b @ Za @ ( set_Pr4578018992751950051_a_a_b @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_Pr8925698157833054977_a_b_b @ F @ X3 )
          = ( map_Pr8925698157833054977_a_b_b @ Fa @ Xa ) )
       => ( X3 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_180_list_Omap__cong0,axiom,
    ! [X3: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,G: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ! [Z: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ Z @ ( set_Pr1702711106814914326_a_a_a @ X3 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_Pr1074589380674644461_a_a_a @ F @ X3 )
        = ( map_Pr1074589380674644461_a_a_a @ G @ X3 ) ) ) ).

% list.map_cong0
thf(fact_181_list_Omap__cong0,axiom,
    ! [X3: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > c,G: produc1895207096998757355_a_a_a > c] :
      ( ! [Z: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ Z @ ( set_Pr1702711106814914326_a_a_a @ X3 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_Pr2287363492687751055_a_a_c @ F @ X3 )
        = ( map_Pr2287363492687751055_a_a_c @ G @ X3 ) ) ) ).

% list.map_cong0
thf(fact_182_list_Omap__cong0,axiom,
    ! [X3: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,G: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ! [Z: produc7810058533641270553_a_a_a] :
          ( ( member54949653122719408_a_a_a @ Z @ ( set_Pr6411044387506519364_a_a_a @ X3 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_Pr3321722608670633418_a_a_a @ F @ X3 )
        = ( map_Pr3321722608670633418_a_a_a @ G @ X3 ) ) ) ).

% list.map_cong0
thf(fact_183_list_Omap__cong0,axiom,
    ! [X3: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > b,G: produc7810058533641270553_a_a_a > b] :
      ( ! [Z: produc7810058533641270553_a_a_a] :
          ( ( member54949653122719408_a_a_a @ Z @ ( set_Pr6411044387506519364_a_a_a @ X3 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_Pr3622043693041699552_a_a_b @ F @ X3 )
        = ( map_Pr3622043693041699552_a_a_b @ G @ X3 ) ) ) ).

% list.map_cong0
thf(fact_184_list_Omap__cong0,axiom,
    ! [X3: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,G: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ! [Z: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ Z @ ( set_Pr4578018992751950051_a_a_b @ X3 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_Pr7354669405448068282_a_a_a @ F @ X3 )
        = ( map_Pr7354669405448068282_a_a_a @ G @ X3 ) ) ) ).

% list.map_cong0
thf(fact_185_list_Omap__cong0,axiom,
    ! [X3: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > b,G: produc4770514982935793080_a_a_b > b] :
      ( ! [Z: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ Z @ ( set_Pr4578018992751950051_a_a_b @ X3 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_Pr8925698157833054977_a_b_b @ F @ X3 )
        = ( map_Pr8925698157833054977_a_b_b @ G @ X3 ) ) ) ).

% list.map_cong0
thf(fact_186_list_Omap__cong,axiom,
    ! [X3: list_P1813520604369389947_a_a_a,Ya: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,G: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( X3 = Ya )
     => ( ! [Z: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ Z @ ( set_Pr1702711106814914326_a_a_a @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_Pr1074589380674644461_a_a_a @ F @ X3 )
          = ( map_Pr1074589380674644461_a_a_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_187_list_Omap__cong,axiom,
    ! [X3: list_P1813520604369389947_a_a_a,Ya: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > c,G: produc1895207096998757355_a_a_a > c] :
      ( ( X3 = Ya )
     => ( ! [Z: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ Z @ ( set_Pr1702711106814914326_a_a_a @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_Pr2287363492687751055_a_a_c @ F @ X3 )
          = ( map_Pr2287363492687751055_a_a_c @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_188_list_Omap__cong,axiom,
    ! [X3: list_P7200181061086074537_a_a_a,Ya: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,G: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ( X3 = Ya )
     => ( ! [Z: produc7810058533641270553_a_a_a] :
            ( ( member54949653122719408_a_a_a @ Z @ ( set_Pr6411044387506519364_a_a_a @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_Pr3321722608670633418_a_a_a @ F @ X3 )
          = ( map_Pr3321722608670633418_a_a_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_189_list_Omap__cong,axiom,
    ! [X3: list_P7200181061086074537_a_a_a,Ya: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > b,G: produc7810058533641270553_a_a_a > b] :
      ( ( X3 = Ya )
     => ( ! [Z: produc7810058533641270553_a_a_a] :
            ( ( member54949653122719408_a_a_a @ Z @ ( set_Pr6411044387506519364_a_a_a @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_Pr3622043693041699552_a_a_b @ F @ X3 )
          = ( map_Pr3622043693041699552_a_a_b @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_190_list_Omap__cong,axiom,
    ! [X3: list_P7155397214832950472_a_a_b,Ya: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,G: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ( X3 = Ya )
     => ( ! [Z: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ Z @ ( set_Pr4578018992751950051_a_a_b @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_Pr7354669405448068282_a_a_a @ F @ X3 )
          = ( map_Pr7354669405448068282_a_a_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_191_list_Omap__cong,axiom,
    ! [X3: list_P7155397214832950472_a_a_b,Ya: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > b,G: produc4770514982935793080_a_a_b > b] :
      ( ( X3 = Ya )
     => ( ! [Z: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ Z @ ( set_Pr4578018992751950051_a_a_b @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_Pr8925698157833054977_a_b_b @ F @ X3 )
          = ( map_Pr8925698157833054977_a_b_b @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_192_in__set__takeD,axiom,
    ! [X3: produc884577481120224310_a_a_a,N: nat,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ ( take_P6848719478162124068_a_a_a @ N @ Xs ) ) )
     => ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) ) ) ).

% in_set_takeD
thf(fact_193_in__set__takeD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,N: nat,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( take_S1842584226280693198_a_a_a @ N @ Xs ) ) )
     => ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ).

% in_set_takeD
thf(fact_194_in__set__takeD,axiom,
    ! [X3: produc4770514982935793080_a_a_b,N: nat,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ ( take_P6639847873373218378_a_a_b @ N @ Xs ) ) )
     => ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ).

% in_set_takeD
thf(fact_195_take__map,axiom,
    ! [N: nat,F: produc4770514982935793080_a_a_b > produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( take_P6639847873373218378_a_a_b @ N @ ( map_Pr2859183448225536310_a_a_b @ F @ Xs ) )
      = ( map_Pr2859183448225536310_a_a_b @ F @ ( take_P6639847873373218378_a_a_b @ N @ Xs ) ) ) ).

% take_map
thf(fact_196_take__map,axiom,
    ! [N: nat,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b] :
      ( ( take_S1842584226280693198_a_a_a @ N @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) )
      = ( map_Pr7354669405448068282_a_a_a @ F @ ( take_P6639847873373218378_a_a_b @ N @ Xs ) ) ) ).

% take_map
thf(fact_197_take__map,axiom,
    ! [N: nat,F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( take_b @ N @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) )
      = ( map_Pr8925698157833054977_a_b_b @ F @ ( take_P6639847873373218378_a_a_b @ N @ Xs ) ) ) ).

% take_map
thf(fact_198_take__map,axiom,
    ! [N: nat,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a] :
      ( ( take_S1842584226280693198_a_a_a @ N @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) )
      = ( map_Pr1074589380674644461_a_a_a @ F @ ( take_P3764539987436182653_a_a_a @ N @ Xs ) ) ) ).

% take_map
thf(fact_199_take__map,axiom,
    ! [N: nat,F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a] :
      ( ( take_c @ N @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) )
      = ( map_Pr2287363492687751055_a_a_c @ F @ ( take_P3764539987436182653_a_a_a @ N @ Xs ) ) ) ).

% take_map
thf(fact_200_take__map,axiom,
    ! [N: nat,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a] :
      ( ( take_P3764539987436182653_a_a_a @ N @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) )
      = ( map_Pr3321722608670633418_a_a_a @ F @ ( take_P2578698388176104363_a_a_a @ N @ Xs ) ) ) ).

% take_map
thf(fact_201_take__map,axiom,
    ! [N: nat,F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a] :
      ( ( take_b @ N @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) )
      = ( map_Pr3622043693041699552_a_a_b @ F @ ( take_P2578698388176104363_a_a_a @ N @ Xs ) ) ) ).

% take_map
thf(fact_202_sup__bot_Oright__neutral,axiom,
    ! [A: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ A @ bot_bo2676859181106700934_a_a_a )
      = A ) ).

% sup_bot.right_neutral
thf(fact_203_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2660408029157294746_a_a_b @ A @ bot_bo5052542052834138114_a_a_b )
      = A ) ).

% sup_bot.right_neutral
thf(fact_204_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su9011528761648865002_a_a_a @ A @ bot_bo2898616998271641474_a_a_a )
      = A ) ).

% sup_bot.right_neutral
thf(fact_205_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( bot_bo2676859181106700934_a_a_a
        = ( sup_su7096409449263244574_a_a_a @ A @ B ) )
      = ( ( A = bot_bo2676859181106700934_a_a_a )
        & ( B = bot_bo2676859181106700934_a_a_a ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_206_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr6246767948713639278_a_a_b,B: set_Pr6246767948713639278_a_a_b] :
      ( ( bot_bo5052542052834138114_a_a_b
        = ( sup_su2660408029157294746_a_a_b @ A @ B ) )
      = ( ( A = bot_bo5052542052834138114_a_a_b )
        & ( B = bot_bo5052542052834138114_a_a_b ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_207_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr4618682941230328598_a_a_a,B: set_Pr4618682941230328598_a_a_a] :
      ( ( bot_bo2898616998271641474_a_a_a
        = ( sup_su9011528761648865002_a_a_a @ A @ B ) )
      = ( ( A = bot_bo2898616998271641474_a_a_a )
        & ( B = bot_bo2898616998271641474_a_a_a ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_208_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ bot_bo2676859181106700934_a_a_a @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_209_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2660408029157294746_a_a_b @ bot_bo5052542052834138114_a_a_b @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_210_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su9011528761648865002_a_a_a @ bot_bo2898616998271641474_a_a_a @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_211_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( ( sup_su7096409449263244574_a_a_a @ A @ B )
        = bot_bo2676859181106700934_a_a_a )
      = ( ( A = bot_bo2676859181106700934_a_a_a )
        & ( B = bot_bo2676859181106700934_a_a_a ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_212_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr6246767948713639278_a_a_b,B: set_Pr6246767948713639278_a_a_b] :
      ( ( ( sup_su2660408029157294746_a_a_b @ A @ B )
        = bot_bo5052542052834138114_a_a_b )
      = ( ( A = bot_bo5052542052834138114_a_a_b )
        & ( B = bot_bo5052542052834138114_a_a_b ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_213_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr4618682941230328598_a_a_a,B: set_Pr4618682941230328598_a_a_a] :
      ( ( ( sup_su9011528761648865002_a_a_a @ A @ B )
        = bot_bo2898616998271641474_a_a_a )
      = ( ( A = bot_bo2898616998271641474_a_a_a )
        & ( B = bot_bo2898616998271641474_a_a_a ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_214_sup__eq__bot__iff,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a] :
      ( ( ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 )
        = bot_bo2676859181106700934_a_a_a )
      = ( ( X3 = bot_bo2676859181106700934_a_a_a )
        & ( Y3 = bot_bo2676859181106700934_a_a_a ) ) ) ).

% sup_eq_bot_iff
thf(fact_215_sup__eq__bot__iff,axiom,
    ! [X3: set_Pr6246767948713639278_a_a_b,Y3: set_Pr6246767948713639278_a_a_b] :
      ( ( ( sup_su2660408029157294746_a_a_b @ X3 @ Y3 )
        = bot_bo5052542052834138114_a_a_b )
      = ( ( X3 = bot_bo5052542052834138114_a_a_b )
        & ( Y3 = bot_bo5052542052834138114_a_a_b ) ) ) ).

% sup_eq_bot_iff
thf(fact_216_sup__eq__bot__iff,axiom,
    ! [X3: set_Pr4618682941230328598_a_a_a,Y3: set_Pr4618682941230328598_a_a_a] :
      ( ( ( sup_su9011528761648865002_a_a_a @ X3 @ Y3 )
        = bot_bo2898616998271641474_a_a_a )
      = ( ( X3 = bot_bo2898616998271641474_a_a_a )
        & ( Y3 = bot_bo2898616998271641474_a_a_a ) ) ) ).

% sup_eq_bot_iff
thf(fact_217_bot__eq__sup__iff,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a] :
      ( ( bot_bo2676859181106700934_a_a_a
        = ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) )
      = ( ( X3 = bot_bo2676859181106700934_a_a_a )
        & ( Y3 = bot_bo2676859181106700934_a_a_a ) ) ) ).

% bot_eq_sup_iff
thf(fact_218_bot__eq__sup__iff,axiom,
    ! [X3: set_Pr6246767948713639278_a_a_b,Y3: set_Pr6246767948713639278_a_a_b] :
      ( ( bot_bo5052542052834138114_a_a_b
        = ( sup_su2660408029157294746_a_a_b @ X3 @ Y3 ) )
      = ( ( X3 = bot_bo5052542052834138114_a_a_b )
        & ( Y3 = bot_bo5052542052834138114_a_a_b ) ) ) ).

% bot_eq_sup_iff
thf(fact_219_bot__eq__sup__iff,axiom,
    ! [X3: set_Pr4618682941230328598_a_a_a,Y3: set_Pr4618682941230328598_a_a_a] :
      ( ( bot_bo2898616998271641474_a_a_a
        = ( sup_su9011528761648865002_a_a_a @ X3 @ Y3 ) )
      = ( ( X3 = bot_bo2898616998271641474_a_a_a )
        & ( Y3 = bot_bo2898616998271641474_a_a_a ) ) ) ).

% bot_eq_sup_iff
thf(fact_220_sup__bot__right,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a )
      = X3 ) ).

% sup_bot_right
thf(fact_221_sup__bot__right,axiom,
    ! [X3: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2660408029157294746_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b )
      = X3 ) ).

% sup_bot_right
thf(fact_222_sup__bot__right,axiom,
    ! [X3: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su9011528761648865002_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a )
      = X3 ) ).

% sup_bot_right
thf(fact_223_sup__bot__left,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ bot_bo2676859181106700934_a_a_a @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_224_sup__bot__left,axiom,
    ! [X3: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2660408029157294746_a_a_b @ bot_bo5052542052834138114_a_a_b @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_225_sup__bot__left,axiom,
    ! [X3: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su9011528761648865002_a_a_a @ bot_bo2898616998271641474_a_a_a @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_226_set__union,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( set_Pr4578018992751950051_a_a_b @ ( union_5970663100036501310_a_a_b @ Xs @ Ys ) )
      = ( sup_su2660408029157294746_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ Xs ) @ ( set_Pr4578018992751950051_a_a_b @ Ys ) ) ) ).

% set_union
thf(fact_227_set__union,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( set_Su3113867532901839207_a_a_a @ ( union_2037822112265804482_a_a_a @ Xs @ Ys ) )
      = ( sup_su7096409449263244574_a_a_a @ ( set_Su3113867532901839207_a_a_a @ Xs ) @ ( set_Su3113867532901839207_a_a_a @ Ys ) ) ) ).

% set_union
thf(fact_228_sup_Oidem,axiom,
    ! [A: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ A @ A )
      = A ) ).

% sup.idem
thf(fact_229_sup__idem,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ X3 @ X3 )
      = X3 ) ).

% sup_idem
thf(fact_230_sup_Oleft__idem,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ A @ ( sup_su7096409449263244574_a_a_a @ A @ B ) )
      = ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ).

% sup.left_idem
thf(fact_231_sup_Oright__idem,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ ( sup_su7096409449263244574_a_a_a @ A @ B ) @ B )
      = ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ).

% sup.right_idem
thf(fact_232_sup__left__idem,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) )
      = ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) ) ).

% sup_left_idem
thf(fact_233_bot__set__def,axiom,
    ( bot_bo2676859181106700934_a_a_a
    = ( collec299858375640228881_a_a_a @ bot_bo2398072012024994175_a_a_o ) ) ).

% bot_set_def
thf(fact_234_bot__set__def,axiom,
    ( bot_bo5052542052834138114_a_a_b
    = ( collec4985708871760228749_a_a_b @ bot_bo992258580158221059_a_b_o ) ) ).

% bot_set_def
thf(fact_235_bot__set__def,axiom,
    ( bot_bo2898616998271641474_a_a_a
    = ( collec3562233214436895265_a_a_a @ bot_bo6766449244381574107_a_a_o ) ) ).

% bot_set_def
thf(fact_236_sup__left__commute,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a,Z2: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ Y3 @ Z2 ) )
      = ( sup_su7096409449263244574_a_a_a @ Y3 @ ( sup_su7096409449263244574_a_a_a @ X3 @ Z2 ) ) ) ).

% sup_left_commute
thf(fact_237_sup_Oleft__commute,axiom,
    ! [B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a,C: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ B @ ( sup_su7096409449263244574_a_a_a @ A @ C ) )
      = ( sup_su7096409449263244574_a_a_a @ A @ ( sup_su7096409449263244574_a_a_a @ B @ C ) ) ) ).

% sup.left_commute
thf(fact_238_sup__commute,axiom,
    ( sup_su7096409449263244574_a_a_a
    = ( ^ [X: set_Su8591296761904255986_a_a_a,Y4: set_Su8591296761904255986_a_a_a] : ( sup_su7096409449263244574_a_a_a @ Y4 @ X ) ) ) ).

% sup_commute
thf(fact_239_sup_Ocommute,axiom,
    ( sup_su7096409449263244574_a_a_a
    = ( ^ [A4: set_Su8591296761904255986_a_a_a,B6: set_Su8591296761904255986_a_a_a] : ( sup_su7096409449263244574_a_a_a @ B6 @ A4 ) ) ) ).

% sup.commute
thf(fact_240_sup__assoc,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a,Z2: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) @ Z2 )
      = ( sup_su7096409449263244574_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ Y3 @ Z2 ) ) ) ).

% sup_assoc
thf(fact_241_sup_Oassoc,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a,C: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ ( sup_su7096409449263244574_a_a_a @ A @ B ) @ C )
      = ( sup_su7096409449263244574_a_a_a @ A @ ( sup_su7096409449263244574_a_a_a @ B @ C ) ) ) ).

% sup.assoc
thf(fact_242_inf__sup__aci_I5_J,axiom,
    ( sup_su7096409449263244574_a_a_a
    = ( ^ [X: set_Su8591296761904255986_a_a_a,Y4: set_Su8591296761904255986_a_a_a] : ( sup_su7096409449263244574_a_a_a @ Y4 @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_243_inf__sup__aci_I6_J,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a,Z2: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) @ Z2 )
      = ( sup_su7096409449263244574_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ Y3 @ Z2 ) ) ) ).

% inf_sup_aci(6)
thf(fact_244_inf__sup__aci_I7_J,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a,Z2: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ Y3 @ Z2 ) )
      = ( sup_su7096409449263244574_a_a_a @ Y3 @ ( sup_su7096409449263244574_a_a_a @ X3 @ Z2 ) ) ) ).

% inf_sup_aci(7)
thf(fact_245_inf__sup__aci_I8_J,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) )
      = ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) ) ).

% inf_sup_aci(8)
thf(fact_246__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062i_O_A_092_060lbrakk_062i_A_060_Alength_Acs_059_At__source_At_A_061_Afst_A_Ics_A_B_Ai_J_059_At__input_At_A_061_Asnd_A_Ics_A_B_Ai_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [I: nat] :
        ( ( ord_less_nat @ I @ ( size_s1546398270689948508_a_a_b @ cs ) )
       => ( ( ( produc7477144754405272026_a_a_a @ t )
            = ( produc4064411471574014846_a_a_b @ ( nth_Pr167341066936870769_a_a_b @ cs @ I ) ) )
         => ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ t ) )
           != ( produc7675857798057594300_a_a_b @ ( nth_Pr167341066936870769_a_a_b @ cs @ I ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>i. \<lbrakk>i < length cs; t_source t = fst (cs ! i); t_input t = snd (cs ! i)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_247_distinguishing__transitions__left__observable__against__distinguishing__transitions__left,axiom,
    ! [T1: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a,T2: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ T1 @ ( state_251499256903769654_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member8362999488266169055_a_a_a @ T2 @ ( state_251499256903769654_a_b_c @ M @ Q1 @ Q2 ) )
       => ( ( ( ( produc7477144754405272026_a_a_a @ T1 )
              = ( produc7477144754405272026_a_a_a @ T2 ) )
            & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) )
              = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) )
            & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) ) )
              = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) ) ) )
         => ( T1 = T2 ) ) ) ) ).

% distinguishing_transitions_left_observable_against_distinguishing_transitions_left
thf(fact_248_distinguishing__transitions__right__observable__against__distinguishing__transitions__right,axiom,
    ! [T1: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a,T2: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ T1 @ ( state_581478702265048387_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member8362999488266169055_a_a_a @ T2 @ ( state_581478702265048387_a_b_c @ M @ Q1 @ Q2 ) )
       => ( ( ( ( produc7477144754405272026_a_a_a @ T1 )
              = ( produc7477144754405272026_a_a_a @ T2 ) )
            & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) )
              = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) )
            & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) ) )
              = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) ) ) )
         => ( T1 = T2 ) ) ) ) ).

% distinguishing_transitions_right_observable_against_distinguishing_transitions_right
thf(fact_249_assms_I1_J,axiom,
    ! [I2: nat,T: produc884577481120224310_a_a_a] :
      ( ( ord_less_nat @ I2 @ ( size_s1546398270689948508_a_a_b @ cs ) )
     => ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ m @ q1 @ q2 ) ) )
       => ( ( ( produc7477144754405272026_a_a_a @ T )
            = ( produc4064411471574014846_a_a_b @ ( nth_Pr167341066936870769_a_a_b @ cs @ I2 ) ) )
         => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) )
              = ( produc7675857798057594300_a_a_b @ ( nth_Pr167341066936870769_a_a_b @ cs @ I2 ) ) )
           => ( member5679158588528110291_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) @ ( sup_su7096409449263244574_a_a_a @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ ( take_P6639847873373218378_a_a_b @ I2 @ cs ) ) ) @ ( insert5061175218292437356_a_a_a @ ( sum_In6345099547827038749od_a_a @ q1 ) @ ( insert5061175218292437356_a_a_a @ ( sum_In6345099547827038749od_a_a @ q2 ) @ bot_bo2676859181106700934_a_a_a ) ) ) ) ) ) ) ) ).

% assms(1)
thf(fact_250_is__singletonI,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a] : ( is_sin2843750684005750600_a_a_a @ ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) ) ).

% is_singletonI
thf(fact_251_is__singletonI,axiom,
    ! [X3: produc4770514982935793080_a_a_b] : ( is_sin8179762472146605508_a_a_b @ ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) ) ).

% is_singletonI
thf(fact_252_is__singletonI,axiom,
    ! [X3: produc884577481120224310_a_a_a] : ( is_sin128316332050476330_a_a_a @ ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) ) ).

% is_singletonI
thf(fact_253_map__set,axiom,
    ! [X3: produc1895207096998757355_a_a_a,Xs: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > c] :
      ( ( member6990616694304513154_a_a_a @ X3 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
     => ( member_c @ ( F @ X3 ) @ ( set_c2 @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_254_map__set,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ( member_b @ ( F @ X3 ) @ ( set_b2 @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_255_map__set,axiom,
    ! [X3: produc7810058533641270553_a_a_a,Xs: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > b] :
      ( ( member54949653122719408_a_a_a @ X3 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
     => ( member_b @ ( F @ X3 ) @ ( set_b2 @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_256_map__set,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,F: sum_su5154756782089200444_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
     => ( member5679158588528110291_a_a_a @ ( F @ X3 ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Su7760560175274971966_a_a_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_257_map__set,axiom,
    ! [X3: produc1895207096998757355_a_a_a,Xs: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( member6990616694304513154_a_a_a @ X3 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
     => ( member5679158588528110291_a_a_a @ ( F @ X3 ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_258_map__set,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,F: sum_su5154756782089200444_a_a_a > produc4770514982935793080_a_a_b] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
     => ( member642552543386773071_a_a_b @ ( F @ X3 ) @ ( set_Pr4578018992751950051_a_a_b @ ( map_Su3117778036294953914_a_a_b @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_259_map__set,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ( member5679158588528110291_a_a_a @ ( F @ X3 ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_260_map__set,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > produc4770514982935793080_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ( member642552543386773071_a_a_b @ ( F @ X3 ) @ ( set_Pr4578018992751950051_a_a_b @ ( map_Pr2859183448225536310_a_a_b @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_261_map__set,axiom,
    ! [X3: produc7810058533641270553_a_a_a,Xs: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ( member54949653122719408_a_a_a @ X3 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
     => ( member6990616694304513154_a_a_a @ ( F @ X3 ) @ ( set_Pr1702711106814914326_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_262_map__set,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a,F: produc884577481120224310_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
     => ( member5679158588528110291_a_a_a @ ( F @ X3 ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr1409128909316453038_a_a_a @ F @ Xs ) ) ) ) ).

% map_set
thf(fact_263_set__map__elem,axiom,
    ! [Y3: c,F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a] :
      ( ( member_c @ Y3 @ ( set_c2 @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) ) )
     => ~ ! [X4: produc1895207096998757355_a_a_a] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member6990616694304513154_a_a_a @ X4 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_264_set__map__elem,axiom,
    ! [Y3: b,F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member_b @ Y3 @ ( set_b2 @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) ) )
     => ~ ! [X4: produc4770514982935793080_a_a_b] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_265_set__map__elem,axiom,
    ! [Y3: b,F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a] :
      ( ( member_b @ Y3 @ ( set_b2 @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) ) )
     => ~ ! [X4: produc7810058533641270553_a_a_a] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member54949653122719408_a_a_a @ X4 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_266_set__map__elem,axiom,
    ! [Y3: sum_su5154756782089200444_a_a_a,F: sum_su5154756782089200444_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ ( map_Su7760560175274971966_a_a_a @ F @ Xs ) ) )
     => ~ ! [X4: sum_su5154756782089200444_a_a_a] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member5679158588528110291_a_a_a @ X4 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_267_set__map__elem,axiom,
    ! [Y3: sum_su5154756782089200444_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) ) )
     => ~ ! [X4: produc1895207096998757355_a_a_a] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member6990616694304513154_a_a_a @ X4 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_268_set__map__elem,axiom,
    ! [Y3: sum_su5154756782089200444_a_a_a,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) ) )
     => ~ ! [X4: produc4770514982935793080_a_a_b] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_269_set__map__elem,axiom,
    ! [Y3: produc4770514982935793080_a_a_b,F: sum_su5154756782089200444_a_a_a > produc4770514982935793080_a_a_b,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ ( map_Su3117778036294953914_a_a_b @ F @ Xs ) ) )
     => ~ ! [X4: sum_su5154756782089200444_a_a_a] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member5679158588528110291_a_a_a @ X4 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_270_set__map__elem,axiom,
    ! [Y3: produc4770514982935793080_a_a_b,F: produc4770514982935793080_a_a_b > produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ ( map_Pr2859183448225536310_a_a_b @ F @ Xs ) ) )
     => ~ ! [X4: produc4770514982935793080_a_a_b] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_271_set__map__elem,axiom,
    ! [Y3: produc1895207096998757355_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a] :
      ( ( member6990616694304513154_a_a_a @ Y3 @ ( set_Pr1702711106814914326_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) ) )
     => ~ ! [X4: produc7810058533641270553_a_a_a] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member54949653122719408_a_a_a @ X4 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_272_set__map__elem,axiom,
    ! [Y3: produc884577481120224310_a_a_a,F: sum_su5154756782089200444_a_a_a > produc884577481120224310_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ Y3 @ ( set_Pr8067384617587748683_a_a_a @ ( map_Su2265628143642336180_a_a_a @ F @ Xs ) ) )
     => ~ ! [X4: sum_su5154756782089200444_a_a_a] :
            ( ( Y3
              = ( F @ X4 ) )
           => ~ ( member5679158588528110291_a_a_a @ X4 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_273_list__map__set__prop,axiom,
    ! [X3: b,F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,P: b > $o] :
      ( ( member_b @ X3 @ ( set_b2 @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) ) )
     => ( ! [Y: produc4770514982935793080_a_a_b] : ( P @ ( F @ Y ) )
       => ( P @ X3 ) ) ) ).

% list_map_set_prop
thf(fact_274_list__map__set__prop,axiom,
    ! [X3: c,F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,P: c > $o] :
      ( ( member_c @ X3 @ ( set_c2 @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) ) )
     => ( ! [Y: produc1895207096998757355_a_a_a] : ( P @ ( F @ Y ) )
       => ( P @ X3 ) ) ) ).

% list_map_set_prop
thf(fact_275_list__map__set__prop,axiom,
    ! [X3: produc1895207096998757355_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a,P: produc1895207096998757355_a_a_a > $o] :
      ( ( member6990616694304513154_a_a_a @ X3 @ ( set_Pr1702711106814914326_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) ) )
     => ( ! [Y: produc7810058533641270553_a_a_a] : ( P @ ( F @ Y ) )
       => ( P @ X3 ) ) ) ).

% list_map_set_prop
thf(fact_276_list__map__set__prop,axiom,
    ! [X3: b,F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,P: b > $o] :
      ( ( member_b @ X3 @ ( set_b2 @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) ) )
     => ( ! [Y: produc7810058533641270553_a_a_a] : ( P @ ( F @ Y ) )
       => ( P @ X3 ) ) ) ).

% list_map_set_prop
thf(fact_277_list__map__set__prop,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) ) )
     => ( ! [Y: produc4770514982935793080_a_a_b] : ( P @ ( F @ Y ) )
       => ( P @ X3 ) ) ) ).

% list_map_set_prop
thf(fact_278_list__map__set__prop,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) ) )
     => ( ! [Y: produc1895207096998757355_a_a_a] : ( P @ ( F @ Y ) )
       => ( P @ X3 ) ) ) ).

% list_map_set_prop
thf(fact_279__092_060open_062i_A_060_Alength_Acs_092_060close_062,axiom,
    ord_less_nat @ i @ ( size_s1546398270689948508_a_a_b @ cs ) ).

% \<open>i < length cs\<close>
thf(fact_280_assms_I2_J,axiom,
    member8362999488266169055_a_a_a @ t @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ m @ q1 @ q2 ) ) ).

% assms(2)
thf(fact_281_length__map,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a] :
      ( ( size_s7186666008910096352_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) )
      = ( size_s5427893697081163791_a_a_a @ Xs ) ) ).

% length_map
thf(fact_282_length__map,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a] :
      ( ( size_size_list_c @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) )
      = ( size_s5427893697081163791_a_a_a @ Xs ) ) ).

% length_map
thf(fact_283_length__map,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a] :
      ( ( size_s5427893697081163791_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) )
      = ( size_s3598275489302632765_a_a_a @ Xs ) ) ).

% length_map
thf(fact_284_length__map,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a] :
      ( ( size_size_list_b @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) )
      = ( size_s3598275489302632765_a_a_a @ Xs ) ) ).

% length_map
thf(fact_285_length__map,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b] :
      ( ( size_s7186666008910096352_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) )
      = ( size_s1546398270689948508_a_a_b @ Xs ) ) ).

% length_map
thf(fact_286_length__map,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( size_size_list_b @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) )
      = ( size_s1546398270689948508_a_a_b @ Xs ) ) ).

% length_map
thf(fact_287_length__map,axiom,
    ! [F: produc4770514982935793080_a_a_b > produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( size_s1546398270689948508_a_a_b @ ( map_Pr2859183448225536310_a_a_b @ F @ Xs ) )
      = ( size_s1546398270689948508_a_a_b @ Xs ) ) ).

% length_map
thf(fact_288_nth__take,axiom,
    ! [I2: nat,N: nat,Xs: list_P7155397214832950472_a_a_b] :
      ( ( ord_less_nat @ I2 @ N )
     => ( ( nth_Pr167341066936870769_a_a_b @ ( take_P6639847873373218378_a_a_b @ N @ Xs ) @ I2 )
        = ( nth_Pr167341066936870769_a_a_b @ Xs @ I2 ) ) ) ).

% nth_take
thf(fact_289_nth__map,axiom,
    ! [N: nat,Xs: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( ord_less_nat @ N @ ( size_s5427893697081163791_a_a_a @ Xs ) )
     => ( ( nth_Su7097239896299986421_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr6515405217854610852_a_a_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_290_nth__map,axiom,
    ! [N: nat,Xs: list_P1813520604369389947_a_a_a,F: produc1895207096998757355_a_a_a > c] :
      ( ( ord_less_nat @ N @ ( size_s5427893697081163791_a_a_a @ Xs ) )
     => ( ( nth_c @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr6515405217854610852_a_a_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_291_nth__map,axiom,
    ! [N: nat,Xs: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ( ord_less_nat @ N @ ( size_s3598275489302632765_a_a_a @ Xs ) )
     => ( ( nth_Pr6515405217854610852_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr6430744503775489490_a_a_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_292_nth__map,axiom,
    ! [N: nat,Xs: list_P7200181061086074537_a_a_a,F: produc7810058533641270553_a_a_a > b] :
      ( ( ord_less_nat @ N @ ( size_s3598275489302632765_a_a_a @ Xs ) )
     => ( ( nth_b @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr6430744503775489490_a_a_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_293_nth__map,axiom,
    ! [N: nat,Xs: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > produc4770514982935793080_a_a_b] :
      ( ( ord_less_nat @ N @ ( size_s1546398270689948508_a_a_b @ Xs ) )
     => ( ( nth_Pr167341066936870769_a_a_b @ ( map_Pr2859183448225536310_a_a_b @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr167341066936870769_a_a_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_294_nth__map,axiom,
    ! [N: nat,Xs: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ( ord_less_nat @ N @ ( size_s1546398270689948508_a_a_b @ Xs ) )
     => ( ( nth_Su7097239896299986421_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr167341066936870769_a_a_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_295_nth__map,axiom,
    ! [N: nat,Xs: list_P7155397214832950472_a_a_b,F: produc4770514982935793080_a_a_b > b] :
      ( ( ord_less_nat @ N @ ( size_s1546398270689948508_a_a_b @ Xs ) )
     => ( ( nth_b @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr167341066936870769_a_a_b @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_296_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_297_less__imp__neq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_298_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_299_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_300_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_301_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X4 )
             => ( P @ Y5 ) )
         => ( P @ X4 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_302_antisym__conv3,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_303_linorder__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_304_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_305_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_306_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X5: nat] : ( P2 @ X5 ) )
    = ( ^ [P3: nat > $o] :
        ? [N2: nat] :
          ( ( P3 @ N2 )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ~ ( P3 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_307_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B7: nat] :
          ( ( ord_less_nat @ A5 @ B7 )
         => ( P @ A5 @ B7 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B7: nat] :
              ( ( P @ B7 @ A5 )
             => ( P @ A5 @ B7 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_308_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_309_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_310_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_311_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_312_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_313_linorder__neqE,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_314_order__less__asym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_315_linorder__neq__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
      = ( ( ord_less_nat @ X3 @ Y3 )
        | ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_316_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_317_order__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_318_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y: nat] :
              ( ( ord_less_nat @ X4 @ Y )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_319_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y: nat] :
              ( ( ord_less_nat @ X4 @ Y )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_320_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_321_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y: nat] :
              ( ( ord_less_nat @ X4 @ Y )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_322_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y: nat] :
              ( ( ord_less_nat @ X4 @ Y )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_323_order__less__not__sym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_324_order__less__imp__triv,axiom,
    ! [X3: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_325_linorder__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_326_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_327_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_328_order__less__imp__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_329_neq__if__length__neq,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
       != ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_330_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_P7155397214832950472_a_a_b] :
      ( ( size_s1546398270689948508_a_a_b @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_331_length__induct,axiom,
    ! [P: list_P7155397214832950472_a_a_b > $o,Xs: list_P7155397214832950472_a_a_b] :
      ( ! [Xs3: list_P7155397214832950472_a_a_b] :
          ( ! [Ys2: list_P7155397214832950472_a_a_b] :
              ( ( ord_less_nat @ ( size_s1546398270689948508_a_a_b @ Ys2 ) @ ( size_s1546398270689948508_a_a_b @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_332_nth__equalityI,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1546398270689948508_a_a_b @ Xs ) )
           => ( ( nth_Pr167341066936870769_a_a_b @ Xs @ I )
              = ( nth_Pr167341066936870769_a_a_b @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_333_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > produc4770514982935793080_a_a_b > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: produc4770514982935793080_a_a_b] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs2: list_P7155397214832950472_a_a_b] :
            ( ( ( size_s1546398270689948508_a_a_b @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_Pr167341066936870769_a_a_b @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_334_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_P7155397214832950472_a_a_b,Z3: list_P7155397214832950472_a_a_b] : ( Y6 = Z3 ) )
    = ( ^ [Xs2: list_P7155397214832950472_a_a_b,Ys3: list_P7155397214832950472_a_a_b] :
          ( ( ( size_s1546398270689948508_a_a_b @ Xs2 )
            = ( size_s1546398270689948508_a_a_b @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s1546398270689948508_a_a_b @ Xs2 ) )
             => ( ( nth_Pr167341066936870769_a_a_b @ Xs2 @ I3 )
                = ( nth_Pr167341066936870769_a_a_b @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_335_bot_Onot__eq__extremum,axiom,
    ! [A: set_Su8591296761904255986_a_a_a] :
      ( ( A != bot_bo2676859181106700934_a_a_a )
      = ( ord_le1107997805182519878_a_a_a @ bot_bo2676859181106700934_a_a_a @ A ) ) ).

% bot.not_eq_extremum
thf(fact_336_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr6246767948713639278_a_a_b] :
      ( ( A != bot_bo5052542052834138114_a_a_b )
      = ( ord_le8627993822118182850_a_a_b @ bot_bo5052542052834138114_a_a_b @ A ) ) ).

% bot.not_eq_extremum
thf(fact_337_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr4618682941230328598_a_a_a] :
      ( ( A != bot_bo2898616998271641474_a_a_a )
      = ( ord_le6196359254896381890_a_a_a @ bot_bo2898616998271641474_a_a_a @ A ) ) ).

% bot.not_eq_extremum
thf(fact_338_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_339_bot_Oextremum__strict,axiom,
    ! [A: set_Su8591296761904255986_a_a_a] :
      ~ ( ord_le1107997805182519878_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) ).

% bot.extremum_strict
thf(fact_340_bot_Oextremum__strict,axiom,
    ! [A: set_Pr6246767948713639278_a_a_b] :
      ~ ( ord_le8627993822118182850_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) ).

% bot.extremum_strict
thf(fact_341_bot_Oextremum__strict,axiom,
    ! [A: set_Pr4618682941230328598_a_a_a] :
      ~ ( ord_le6196359254896381890_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) ).

% bot.extremum_strict
thf(fact_342_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_343_list__property__from__index__property,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,P: produc884577481120224310_a_a_a > $o,X3: produc884577481120224310_a_a_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s4783218027424981928_a_a_a @ Xs ) )
         => ( P @ ( nth_Pr7648848803501811517_a_a_a @ Xs @ I ) ) )
     => ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
       => ( P @ X3 ) ) ) ).

% list_property_from_index_property
thf(fact_344_list__property__from__index__property,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o,X3: sum_su5154756782089200444_a_a_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7186666008910096352_a_a_a @ Xs ) )
         => ( P @ ( nth_Su7097239896299986421_a_a_a @ Xs @ I ) ) )
     => ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
       => ( P @ X3 ) ) ) ).

% list_property_from_index_property
thf(fact_345_list__property__from__index__property,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o,X3: produc4770514982935793080_a_a_b] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s1546398270689948508_a_a_b @ Xs ) )
         => ( P @ ( nth_Pr167341066936870769_a_a_b @ Xs @ I ) ) )
     => ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
       => ( P @ X3 ) ) ) ).

% list_property_from_index_property
thf(fact_346_nth__mem,axiom,
    ! [N: nat,Xs: list_P8197596098214363964_a_a_a] :
      ( ( ord_less_nat @ N @ ( size_s4783218027424981928_a_a_a @ Xs ) )
     => ( member8362999488266169055_a_a_a @ ( nth_Pr7648848803501811517_a_a_a @ Xs @ N ) @ ( set_Pr8067384617587748683_a_a_a @ Xs ) ) ) ).

% nth_mem
thf(fact_347_nth__mem,axiom,
    ! [N: nat,Xs: list_S8072159660131056972_a_a_a] :
      ( ( ord_less_nat @ N @ ( size_s7186666008910096352_a_a_a @ Xs ) )
     => ( member5679158588528110291_a_a_a @ ( nth_Su7097239896299986421_a_a_a @ Xs @ N ) @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ).

% nth_mem
thf(fact_348_nth__mem,axiom,
    ! [N: nat,Xs: list_P7155397214832950472_a_a_b] :
      ( ( ord_less_nat @ N @ ( size_s1546398270689948508_a_a_b @ Xs ) )
     => ( member642552543386773071_a_a_b @ ( nth_Pr167341066936870769_a_a_b @ Xs @ N ) @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ).

% nth_mem
thf(fact_349_list__ball__nth,axiom,
    ! [N: nat,Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( ord_less_nat @ N @ ( size_s7186666008910096352_a_a_a @ Xs ) )
     => ( ! [X4: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X4 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_Su7097239896299986421_a_a_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_350_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ( ord_less_nat @ N @ ( size_s1546398270689948508_a_a_b @ Xs ) )
     => ( ! [X4: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_Pr167341066936870769_a_a_b @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_351_in__set__conv__nth,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s4783218027424981928_a_a_a @ Xs ) )
            & ( ( nth_Pr7648848803501811517_a_a_a @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_352_in__set__conv__nth,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7186666008910096352_a_a_a @ Xs ) )
            & ( ( nth_Su7097239896299986421_a_a_a @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_353_in__set__conv__nth,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
            & ( ( nth_Pr167341066936870769_a_a_b @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_354_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,P: produc884577481120224310_a_a_a > $o,X3: produc884577481120224310_a_a_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s4783218027424981928_a_a_a @ Xs ) )
         => ( P @ ( nth_Pr7648848803501811517_a_a_a @ Xs @ I ) ) )
     => ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_355_all__nth__imp__all__set,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o,X3: sum_su5154756782089200444_a_a_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7186666008910096352_a_a_a @ Xs ) )
         => ( P @ ( nth_Su7097239896299986421_a_a_a @ Xs @ I ) ) )
     => ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_356_all__nth__imp__all__set,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o,X3: produc4770514982935793080_a_a_b] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s1546398270689948508_a_a_b @ Xs ) )
         => ( P @ ( nth_Pr167341066936870769_a_a_b @ Xs @ I ) ) )
     => ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_357_all__set__conv__all__nth,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( ! [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7186666008910096352_a_a_a @ Xs ) )
           => ( P @ ( nth_Su7097239896299986421_a_a_a @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_358_all__set__conv__all__nth,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ( ! [X: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
           => ( P @ ( nth_Pr167341066936870769_a_a_b @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_359_map__equality__iff,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,G: produc1895207096998757355_a_a_a > c,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = ( map_Pr2287363492687751055_a_a_c @ G @ Ys ) )
      = ( ( ( size_s5427893697081163791_a_a_a @ Xs )
          = ( size_s5427893697081163791_a_a_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5427893697081163791_a_a_a @ Ys ) )
           => ( ( F @ ( nth_Pr6515405217854610852_a_a_a @ Xs @ I3 ) )
              = ( G @ ( nth_Pr6515405217854610852_a_a_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_360_map__equality__iff,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,G: produc4770514982935793080_a_a_b > c,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = ( map_Pr8925698157833054978_a_b_c @ G @ Ys ) )
      = ( ( ( size_s5427893697081163791_a_a_a @ Xs )
          = ( size_s1546398270689948508_a_a_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1546398270689948508_a_a_b @ Ys ) )
           => ( ( F @ ( nth_Pr6515405217854610852_a_a_a @ Xs @ I3 ) )
              = ( G @ ( nth_Pr167341066936870769_a_a_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_361_map__equality__iff,axiom,
    ! [F: produc4770514982935793080_a_a_b > c,Xs: list_P7155397214832950472_a_a_b,G: produc1895207096998757355_a_a_a > c,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr8925698157833054978_a_b_c @ F @ Xs )
        = ( map_Pr2287363492687751055_a_a_c @ G @ Ys ) )
      = ( ( ( size_s1546398270689948508_a_a_b @ Xs )
          = ( size_s5427893697081163791_a_a_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5427893697081163791_a_a_a @ Ys ) )
           => ( ( F @ ( nth_Pr167341066936870769_a_a_b @ Xs @ I3 ) )
              = ( G @ ( nth_Pr6515405217854610852_a_a_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_362_map__equality__iff,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,G: produc4770514982935793080_a_a_b > b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = ( map_Pr8925698157833054977_a_b_b @ G @ Ys ) )
      = ( ( ( size_s1546398270689948508_a_a_b @ Xs )
          = ( size_s1546398270689948508_a_a_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1546398270689948508_a_a_b @ Ys ) )
           => ( ( F @ ( nth_Pr167341066936870769_a_a_b @ Xs @ I3 ) )
              = ( G @ ( nth_Pr167341066936870769_a_a_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_363_map__equality__iff,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,G: produc4770514982935793080_a_a_b > b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = ( map_Pr8925698157833054977_a_b_b @ G @ Ys ) )
      = ( ( ( size_s3598275489302632765_a_a_a @ Xs )
          = ( size_s1546398270689948508_a_a_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1546398270689948508_a_a_b @ Ys ) )
           => ( ( F @ ( nth_Pr6430744503775489490_a_a_a @ Xs @ I3 ) )
              = ( G @ ( nth_Pr167341066936870769_a_a_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_364_map__equality__iff,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,G: produc7810058533641270553_a_a_a > b,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = ( map_Pr3622043693041699552_a_a_b @ G @ Ys ) )
      = ( ( ( size_s1546398270689948508_a_a_b @ Xs )
          = ( size_s3598275489302632765_a_a_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3598275489302632765_a_a_a @ Ys ) )
           => ( ( F @ ( nth_Pr167341066936870769_a_a_b @ Xs @ I3 ) )
              = ( G @ ( nth_Pr6430744503775489490_a_a_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_365_map__equality__iff,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,G: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = ( map_Pr1074589380674644461_a_a_a @ G @ Ys ) )
      = ( ( ( size_s5427893697081163791_a_a_a @ Xs )
          = ( size_s5427893697081163791_a_a_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5427893697081163791_a_a_a @ Ys ) )
           => ( ( F @ ( nth_Pr6515405217854610852_a_a_a @ Xs @ I3 ) )
              = ( G @ ( nth_Pr6515405217854610852_a_a_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_366_map__equality__iff,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,G: produc7810058533641270553_a_a_a > b,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = ( map_Pr3622043693041699552_a_a_b @ G @ Ys ) )
      = ( ( ( size_s3598275489302632765_a_a_a @ Xs )
          = ( size_s3598275489302632765_a_a_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3598275489302632765_a_a_a @ Ys ) )
           => ( ( F @ ( nth_Pr6430744503775489490_a_a_a @ Xs @ I3 ) )
              = ( G @ ( nth_Pr6430744503775489490_a_a_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_367_map__equality__iff,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,G: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = ( map_Pr7354669405448068282_a_a_a @ G @ Ys ) )
      = ( ( ( size_s5427893697081163791_a_a_a @ Xs )
          = ( size_s1546398270689948508_a_a_b @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1546398270689948508_a_a_b @ Ys ) )
           => ( ( F @ ( nth_Pr6515405217854610852_a_a_a @ Xs @ I3 ) )
              = ( G @ ( nth_Pr167341066936870769_a_a_b @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_368_map__equality__iff,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b,G: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr7354669405448068282_a_a_a @ F @ Xs )
        = ( map_Pr1074589380674644461_a_a_a @ G @ Ys ) )
      = ( ( ( size_s1546398270689948508_a_a_b @ Xs )
          = ( size_s5427893697081163791_a_a_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5427893697081163791_a_a_a @ Ys ) )
           => ( ( F @ ( nth_Pr167341066936870769_a_a_b @ Xs @ I3 ) )
              = ( G @ ( nth_Pr6515405217854610852_a_a_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_369_map__eq__imp__length__eq,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,G: produc1895207096998757355_a_a_a > c,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = ( map_Pr2287363492687751055_a_a_c @ G @ Ys ) )
     => ( ( size_s5427893697081163791_a_a_a @ Xs )
        = ( size_s5427893697081163791_a_a_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_370_map__eq__imp__length__eq,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,G: produc4770514982935793080_a_a_b > c,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = ( map_Pr8925698157833054978_a_b_c @ G @ Ys ) )
     => ( ( size_s5427893697081163791_a_a_a @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_371_map__eq__imp__length__eq,axiom,
    ! [F: produc4770514982935793080_a_a_b > c,Xs: list_P7155397214832950472_a_a_b,G: produc1895207096998757355_a_a_a > c,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr8925698157833054978_a_b_c @ F @ Xs )
        = ( map_Pr2287363492687751055_a_a_c @ G @ Ys ) )
     => ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s5427893697081163791_a_a_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_372_map__eq__imp__length__eq,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,G: produc4770514982935793080_a_a_b > b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = ( map_Pr8925698157833054977_a_b_b @ G @ Ys ) )
     => ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_373_map__eq__imp__length__eq,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,G: produc4770514982935793080_a_a_b > b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = ( map_Pr8925698157833054977_a_b_b @ G @ Ys ) )
     => ( ( size_s3598275489302632765_a_a_a @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_374_map__eq__imp__length__eq,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,G: produc7810058533641270553_a_a_a > b,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = ( map_Pr3622043693041699552_a_a_b @ G @ Ys ) )
     => ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s3598275489302632765_a_a_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_375_map__eq__imp__length__eq,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,G: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = ( map_Pr1074589380674644461_a_a_a @ G @ Ys ) )
     => ( ( size_s5427893697081163791_a_a_a @ Xs )
        = ( size_s5427893697081163791_a_a_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_376_map__eq__imp__length__eq,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,G: produc7810058533641270553_a_a_a > b,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = ( map_Pr3622043693041699552_a_a_b @ G @ Ys ) )
     => ( ( size_s3598275489302632765_a_a_a @ Xs )
        = ( size_s3598275489302632765_a_a_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_377_map__eq__imp__length__eq,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,G: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = ( map_Pr7354669405448068282_a_a_a @ G @ Ys ) )
     => ( ( size_s5427893697081163791_a_a_a @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_378_map__eq__imp__length__eq,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b,G: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr7354669405448068282_a_a_a @ F @ Xs )
        = ( map_Pr1074589380674644461_a_a_a @ G @ Ys ) )
     => ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s5427893697081163791_a_a_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_379_less__supI1,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le1107997805182519878_a_a_a @ X3 @ A )
     => ( ord_le1107997805182519878_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ).

% less_supI1
thf(fact_380_less__supI1,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ X3 @ A )
     => ( ord_less_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_381_less__supI2,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le1107997805182519878_a_a_a @ X3 @ B )
     => ( ord_le1107997805182519878_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ).

% less_supI2
thf(fact_382_less__supI2,axiom,
    ! [X3: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ X3 @ B )
     => ( ord_less_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_383_sup_Oabsorb3,axiom,
    ! [B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le1107997805182519878_a_a_a @ B @ A )
     => ( ( sup_su7096409449263244574_a_a_a @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_384_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_385_sup_Oabsorb4,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le1107997805182519878_a_a_a @ A @ B )
     => ( ( sup_su7096409449263244574_a_a_a @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_386_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_387_sup_Ostrict__boundedE,axiom,
    ! [B: set_Su8591296761904255986_a_a_a,C: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le1107997805182519878_a_a_a @ ( sup_su7096409449263244574_a_a_a @ B @ C ) @ A )
     => ~ ( ( ord_le1107997805182519878_a_a_a @ B @ A )
         => ~ ( ord_le1107997805182519878_a_a_a @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_388_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_389_sup_Ostrict__order__iff,axiom,
    ( ord_le1107997805182519878_a_a_a
    = ( ^ [B6: set_Su8591296761904255986_a_a_a,A4: set_Su8591296761904255986_a_a_a] :
          ( ( A4
            = ( sup_su7096409449263244574_a_a_a @ A4 @ B6 ) )
          & ( A4 != B6 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_390_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B6: nat,A4: nat] :
          ( ( A4
            = ( sup_sup_nat @ A4 @ B6 ) )
          & ( A4 != B6 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_391_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le1107997805182519878_a_a_a @ C @ A )
     => ( ord_le1107997805182519878_a_a_a @ C @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_392_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_393_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le1107997805182519878_a_a_a @ C @ B )
     => ( ord_le1107997805182519878_a_a_a @ C @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_394_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_395_is__singletonI_H,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ( ( A2 != bot_bo2676859181106700934_a_a_a )
     => ( ! [X4: sum_su5154756782089200444_a_a_a,Y: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X4 @ A2 )
           => ( ( member5679158588528110291_a_a_a @ Y @ A2 )
             => ( X4 = Y ) ) )
       => ( is_sin2843750684005750600_a_a_a @ A2 ) ) ) ).

% is_singletonI'
thf(fact_396_is__singletonI_H,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] :
      ( ( A2 != bot_bo5052542052834138114_a_a_b )
     => ( ! [X4: produc4770514982935793080_a_a_b,Y: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X4 @ A2 )
           => ( ( member642552543386773071_a_a_b @ Y @ A2 )
             => ( X4 = Y ) ) )
       => ( is_sin8179762472146605508_a_a_b @ A2 ) ) ) ).

% is_singletonI'
thf(fact_397_is__singletonI_H,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] :
      ( ( A2 != bot_bo2898616998271641474_a_a_a )
     => ( ! [X4: produc884577481120224310_a_a_a,Y: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X4 @ A2 )
           => ( ( member8362999488266169055_a_a_a @ Y @ A2 )
             => ( X4 = Y ) ) )
       => ( is_sin128316332050476330_a_a_a @ A2 ) ) ) ).

% is_singletonI'
thf(fact_398_distinguishing__transitions__left__observable__against__distinguishing__transitions__right,axiom,
    ! [T1: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a,T2: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ T1 @ ( state_251499256903769654_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member8362999488266169055_a_a_a @ T2 @ ( state_581478702265048387_a_b_c @ M @ Q1 @ Q2 ) )
       => ~ ( ( ( produc7477144754405272026_a_a_a @ T1 )
              = ( produc7477144754405272026_a_a_a @ T2 ) )
            & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) )
              = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) )
            & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) ) )
              = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) ) ) ) ) ) ).

% distinguishing_transitions_left_observable_against_distinguishing_transitions_right
thf(fact_399_is__singleton__def,axiom,
    ( is_sin2843750684005750600_a_a_a
    = ( ^ [A3: set_Su8591296761904255986_a_a_a] :
        ? [X: sum_su5154756782089200444_a_a_a] :
          ( A3
          = ( insert5061175218292437356_a_a_a @ X @ bot_bo2676859181106700934_a_a_a ) ) ) ) ).

% is_singleton_def
thf(fact_400_is__singleton__def,axiom,
    ( is_sin8179762472146605508_a_a_b
    = ( ^ [A3: set_Pr6246767948713639278_a_a_b] :
        ? [X: produc4770514982935793080_a_a_b] :
          ( A3
          = ( insert8042272945010224104_a_a_b @ X @ bot_bo5052542052834138114_a_a_b ) ) ) ) ).

% is_singleton_def
thf(fact_401_is__singleton__def,axiom,
    ( is_sin128316332050476330_a_a_a
    = ( ^ [A3: set_Pr4618682941230328598_a_a_a] :
        ? [X: produc884577481120224310_a_a_a] :
          ( A3
          = ( insert2249508923356641798_a_a_a @ X @ bot_bo2898616998271641474_a_a_a ) ) ) ) ).

% is_singleton_def
thf(fact_402_is__singletonE,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ( ( is_sin2843750684005750600_a_a_a @ A2 )
     => ~ ! [X4: sum_su5154756782089200444_a_a_a] :
            ( A2
           != ( insert5061175218292437356_a_a_a @ X4 @ bot_bo2676859181106700934_a_a_a ) ) ) ).

% is_singletonE
thf(fact_403_is__singletonE,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] :
      ( ( is_sin8179762472146605508_a_a_b @ A2 )
     => ~ ! [X4: produc4770514982935793080_a_a_b] :
            ( A2
           != ( insert8042272945010224104_a_a_b @ X4 @ bot_bo5052542052834138114_a_a_b ) ) ) ).

% is_singletonE
thf(fact_404_is__singletonE,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] :
      ( ( is_sin128316332050476330_a_a_a @ A2 )
     => ~ ! [X4: produc884577481120224310_a_a_a] :
            ( A2
           != ( insert2249508923356641798_a_a_a @ X4 @ bot_bo2898616998271641474_a_a_a ) ) ) ).

% is_singletonE
thf(fact_405_list__map__source__elem,axiom,
    ! [X3: c,F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a] :
      ( ( member_c @ X3 @ ( set_c2 @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) ) )
     => ? [X4: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ X4 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_406_list__map__source__elem,axiom,
    ! [X3: b,F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member_b @ X3 @ ( set_b2 @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) ) )
     => ? [X4: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_407_list__map__source__elem,axiom,
    ! [X3: b,F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a] :
      ( ( member_b @ X3 @ ( set_b2 @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) ) )
     => ? [X4: produc7810058533641270553_a_a_a] :
          ( ( member54949653122719408_a_a_a @ X4 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_408_list__map__source__elem,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,F: sum_su5154756782089200444_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( map_Su7760560175274971966_a_a_a @ F @ Xs ) ) )
     => ? [X4: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X4 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_409_list__map__source__elem,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) ) )
     => ? [X4: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ X4 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_410_list__map__source__elem,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) ) )
     => ? [X4: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_411_list__map__source__elem,axiom,
    ! [X3: produc4770514982935793080_a_a_b,F: sum_su5154756782089200444_a_a_a > produc4770514982935793080_a_a_b,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ ( map_Su3117778036294953914_a_a_b @ F @ Xs ) ) )
     => ? [X4: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X4 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_412_list__map__source__elem,axiom,
    ! [X3: produc4770514982935793080_a_a_b,F: produc4770514982935793080_a_a_b > produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ ( map_Pr2859183448225536310_a_a_b @ F @ Xs ) ) )
     => ? [X4: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_413_list__map__source__elem,axiom,
    ! [X3: produc1895207096998757355_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a] :
      ( ( member6990616694304513154_a_a_a @ X3 @ ( set_Pr1702711106814914326_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) ) )
     => ? [X4: produc7810058533641270553_a_a_a] :
          ( ( member54949653122719408_a_a_a @ X4 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_414_list__map__source__elem,axiom,
    ! [X3: produc884577481120224310_a_a_a,F: sum_su5154756782089200444_a_a_a > produc884577481120224310_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ ( map_Su2265628143642336180_a_a_a @ F @ Xs ) ) )
     => ? [X4: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X4 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
          & ( X3
            = ( F @ X4 ) ) ) ) ).

% list_map_source_elem
thf(fact_415_less__prod__def_H,axiom,
    ( ord_le1203424502768444845at_nat
    = ( ^ [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X ) @ ( product_fst_nat_nat @ Y4 ) )
          | ( ( ( product_fst_nat_nat @ X )
              = ( product_fst_nat_nat @ Y4 ) )
            & ( ord_less_nat @ ( product_snd_nat_nat @ X ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ) ).

% less_prod_def'
thf(fact_416_retains__outputs__for__states__and__inputs__def,axiom,
    ( retain3432474810896159496_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b,S: fsm_Pr6287166851770849218_a_a_b] :
        ! [X: produc6800189296593107456_a_a_a] :
          ( ( member4643618012521571351_a_a_a @ X @ ( transi7010299678732630507_a_a_b @ S ) )
         => ! [Y4: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ M3 ) )
             => ( ( ( ( produc2316789063618113990_a_a_a @ X )
                    = ( produc2316789063618113990_a_a_a @ Y4 ) )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) )
                    = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) )
               => ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ S ) ) ) ) ) ) ) ).

% retains_outputs_for_states_and_inputs_def
thf(fact_417_retains__outputs__for__states__and__inputs__def,axiom,
    ( retain8413675290114623755_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c,S: fsm_Su3173892059090327343_a_b_c] :
        ! [X: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X @ ( transi749694810082594024_a_b_c @ S ) )
         => ! [Y4: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ M3 ) )
             => ( ( ( ( produc7477144754405272026_a_a_a @ X )
                    = ( produc7477144754405272026_a_a_a @ Y4 ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) )
               => ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ S ) ) ) ) ) ) ) ).

% retains_outputs_for_states_and_inputs_def
thf(fact_418_Collect__empty__eq__bot,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( ( collec299858375640228881_a_a_a @ P )
        = bot_bo2676859181106700934_a_a_a )
      = ( P = bot_bo2398072012024994175_a_a_o ) ) ).

% Collect_empty_eq_bot
thf(fact_419_Collect__empty__eq__bot,axiom,
    ! [P: produc4770514982935793080_a_a_b > $o] :
      ( ( ( collec4985708871760228749_a_a_b @ P )
        = bot_bo5052542052834138114_a_a_b )
      = ( P = bot_bo992258580158221059_a_b_o ) ) ).

% Collect_empty_eq_bot
thf(fact_420_Collect__empty__eq__bot,axiom,
    ! [P: produc884577481120224310_a_a_a > $o] :
      ( ( ( collec3562233214436895265_a_a_a @ P )
        = bot_bo2898616998271641474_a_a_a )
      = ( P = bot_bo6766449244381574107_a_a_o ) ) ).

% Collect_empty_eq_bot
thf(fact_421_bot__empty__eq,axiom,
    ( bot_bo2398072012024994175_a_a_o
    = ( ^ [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ bot_bo2676859181106700934_a_a_a ) ) ) ).

% bot_empty_eq
thf(fact_422_bot__empty__eq,axiom,
    ( bot_bo992258580158221059_a_b_o
    = ( ^ [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ bot_bo5052542052834138114_a_a_b ) ) ) ).

% bot_empty_eq
thf(fact_423_bot__empty__eq,axiom,
    ( bot_bo6766449244381574107_a_a_o
    = ( ^ [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ bot_bo2898616998271641474_a_a_a ) ) ) ).

% bot_empty_eq
thf(fact_424_deterministic_Osimps,axiom,
    ( determ8774513197627008055_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b] :
        ! [X: produc6800189296593107456_a_a_a] :
          ( ( member4643618012521571351_a_a_a @ X @ ( transi7010299678732630507_a_a_b @ M3 ) )
         => ! [Y4: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ M3 ) )
             => ( ( ( ( produc2316789063618113990_a_a_a @ X )
                    = ( produc2316789063618113990_a_a_a @ Y4 ) )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) )
                    = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) )
               => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) )
                  & ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) ) )
                    = ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_425_deterministic_Osimps,axiom,
    ( determ6453466625651965340_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c] :
        ! [X: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X @ ( transi749694810082594024_a_b_c @ M3 ) )
         => ! [Y4: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ M3 ) )
             => ( ( ( ( produc7477144754405272026_a_a_a @ X )
                    = ( produc7477144754405272026_a_a_a @ Y4 ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) )
               => ( ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) ) )
                    = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) )
                  & ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) ) )
                    = ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_426_deterministic_Oelims_I1_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b,Y3: $o] :
      ( ( ( determ8774513197627008055_a_a_b @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ X @ ( transi7010299678732630507_a_a_b @ X3 ) )
             => ! [Y4: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ X3 ) )
                 => ( ( ( ( produc2316789063618113990_a_a_a @ X )
                        = ( produc2316789063618113990_a_a_a @ Y4 ) )
                      & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) )
                        = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) )
                   => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) ) )
                        = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) )
                      & ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) ) )
                        = ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_427_deterministic_Oelims_I1_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Y3: $o] :
      ( ( ( determ6453466625651965340_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ X @ ( transi749694810082594024_a_b_c @ X3 ) )
             => ! [Y4: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ X3 ) )
                 => ( ( ( ( produc7477144754405272026_a_a_a @ X )
                        = ( produc7477144754405272026_a_a_a @ Y4 ) )
                      & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) )
                        = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) )
                   => ( ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) ) )
                        = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) )
                      & ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) ) )
                        = ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_428_deterministic_Oelims_I2_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b] :
      ( ( determ8774513197627008055_a_a_b @ X3 )
     => ! [X7: produc6800189296593107456_a_a_a] :
          ( ( member4643618012521571351_a_a_a @ X7 @ ( transi7010299678732630507_a_a_b @ X3 ) )
         => ! [Xa2: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ Xa2 @ ( transi7010299678732630507_a_a_b @ X3 ) )
             => ( ( ( ( produc2316789063618113990_a_a_a @ X7 )
                    = ( produc2316789063618113990_a_a_a @ Xa2 ) )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) )
                    = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) ) ) )
               => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) ) ) )
                  & ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
                    = ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_429_deterministic_Oelims_I2_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c] :
      ( ( determ6453466625651965340_a_b_c @ X3 )
     => ! [X7: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X7 @ ( transi749694810082594024_a_b_c @ X3 ) )
         => ! [Xa2: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ Xa2 @ ( transi749694810082594024_a_b_c @ X3 ) )
             => ( ( ( ( produc7477144754405272026_a_a_a @ X7 )
                    = ( produc7477144754405272026_a_a_a @ Xa2 ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) ) ) )
               => ( ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
                    = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) ) ) )
                  & ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
                    = ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_430_deterministic_Oelims_I3_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b] :
      ( ~ ( determ8774513197627008055_a_a_b @ X3 )
     => ~ ! [X4: produc6800189296593107456_a_a_a] :
            ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ X3 ) )
           => ! [Xa3: produc6800189296593107456_a_a_a] :
                ( ( member4643618012521571351_a_a_a @ Xa3 @ ( transi7010299678732630507_a_a_b @ X3 ) )
               => ( ( ( ( produc2316789063618113990_a_a_a @ X4 )
                      = ( produc2316789063618113990_a_a_a @ Xa3 ) )
                    & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
                      = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa3 ) ) ) )
                 => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                      = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa3 ) ) ) )
                    & ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                      = ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa3 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_431_deterministic_Oelims_I3_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c] :
      ( ~ ( determ6453466625651965340_a_b_c @ X3 )
     => ~ ! [X4: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ X3 ) )
           => ! [Xa3: produc884577481120224310_a_a_a] :
                ( ( member8362999488266169055_a_a_a @ Xa3 @ ( transi749694810082594024_a_b_c @ X3 ) )
               => ( ( ( ( produc7477144754405272026_a_a_a @ X4 )
                      = ( produc7477144754405272026_a_a_a @ Xa3 ) )
                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
                      = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa3 ) ) ) )
                 => ( ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                      = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa3 ) ) ) )
                    & ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                      = ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa3 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_432_canonical__separator__targets__observable_I2_J,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( Q1 != Q2 )
           => ( ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                = ( sum_In6345099547827038749od_a_a @ Q1 ) )
             => ( member8362999488266169055_a_a_a @ T @ ( state_251499256903769654_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ) ) ).

% canonical_separator_targets_observable(2)
thf(fact_433_fsm__transition__source,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_Su3173892059090327343_a_b_c] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
     => ( member5679158588528110291_a_a_a @ ( produc7477144754405272026_a_a_a @ T ) @ ( states3035385874452377268_a_b_c @ M ) ) ) ).

% fsm_transition_source
thf(fact_434_fsm__transition__target,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_Su3173892059090327343_a_b_c] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
     => ( member5679158588528110291_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) @ ( states3035385874452377268_a_b_c @ M ) ) ) ).

% fsm_transition_target
thf(fact_435_fsm__transition__target,axiom,
    ! [T: produc6800189296593107456_a_a_a,M: fsm_Pr6287166851770849218_a_a_b] :
      ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ M ) )
     => ( member6990616694304513154_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) @ ( states8463590192175095071_a_a_b @ M ) ) ) ).

% fsm_transition_target
thf(fact_436_not__psubset__empty,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ~ ( ord_le1107997805182519878_a_a_a @ A2 @ bot_bo2676859181106700934_a_a_a ) ).

% not_psubset_empty
thf(fact_437_not__psubset__empty,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] :
      ~ ( ord_le8627993822118182850_a_a_b @ A2 @ bot_bo5052542052834138114_a_a_b ) ).

% not_psubset_empty
thf(fact_438_not__psubset__empty,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] :
      ~ ( ord_le6196359254896381890_a_a_a @ A2 @ bot_bo2898616998271641474_a_a_a ) ).

% not_psubset_empty
thf(fact_439_canonical__separator__distinguishing__transitions__right__containment,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( state_581478702265048387_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ) ).

% canonical_separator_distinguishing_transitions_right_containment
thf(fact_440_canonical__separator__distinguishing__transitions__left__containment,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( state_251499256903769654_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ) ).

% canonical_separator_distinguishing_transitions_left_containment
thf(fact_441_canonical__separator__targets__observable_I3_J,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( Q1 != Q2 )
           => ( ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                = ( sum_In6345099547827038749od_a_a @ Q2 ) )
             => ( member8362999488266169055_a_a_a @ T @ ( state_581478702265048387_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ) ) ).

% canonical_separator_targets_observable(3)
thf(fact_442_canonical__separator__transition__targets_I3_J,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
              = ( sum_In6345099547827038749od_a_a @ Q2 ) )
           => ( ( Q1 != Q2 )
             => ( member8362999488266169055_a_a_a @ T @ ( state_2881338517818858960_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ) ) ).

% canonical_separator_transition_targets(3)
thf(fact_443_canonical__separator__transition__targets_I2_J,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
              = ( sum_In6345099547827038749od_a_a @ Q1 ) )
           => ( ( Q1 != Q2 )
             => ( member8362999488266169055_a_a_a @ T @ ( state_2046487007905918109_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ) ) ).

% canonical_separator_transition_targets(2)
thf(fact_444_add__transition__with__components__simps_I5_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,T: produc884577481120224310_a_a_a] :
      ( ( states3035385874452377268_a_b_c @ ( add_tr191749633354624124_a_b_c @ M @ T ) )
      = ( insert5061175218292437356_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) @ ( insert5061175218292437356_a_a_a @ ( produc7477144754405272026_a_a_a @ T ) @ ( states3035385874452377268_a_b_c @ M ) ) ) ) ).

% add_transition_with_components_simps(5)
thf(fact_445_add__transition__with__components__simps_I5_J,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,T: produc6800189296593107456_a_a_a] :
      ( ( states8463590192175095071_a_a_b @ ( add_tr7313952166877789527_a_a_b @ M @ T ) )
      = ( insert5166965059073188379_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) @ ( insert5166965059073188379_a_a_a @ ( produc2316789063618113990_a_a_a @ T ) @ ( states8463590192175095071_a_a_b @ M ) ) ) ) ).

% add_transition_with_components_simps(5)
thf(fact_446_shifted__transitions__observable__against__distinguishing__transitions__right,axiom,
    ! [T1: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a,T2: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ T1 @ ( state_1345282896291928690_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member8362999488266169055_a_a_a @ T2 @ ( state_581478702265048387_a_b_c @ M @ Q1 @ Q2 ) )
       => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
         => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
           => ~ ( ( ( produc7477144754405272026_a_a_a @ T1 )
                  = ( produc7477144754405272026_a_a_a @ T2 ) )
                & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) )
                  = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) )
                & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) ) )
                  = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) ) ) ) ) ) ) ) ).

% shifted_transitions_observable_against_distinguishing_transitions_right
thf(fact_447_shifted__transitions__observable__against__distinguishing__transitions__left,axiom,
    ! [T1: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a,T2: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ T1 @ ( state_1345282896291928690_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member8362999488266169055_a_a_a @ T2 @ ( state_251499256903769654_a_b_c @ M @ Q1 @ Q2 ) )
       => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
         => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
           => ~ ( ( ( produc7477144754405272026_a_a_a @ T1 )
                  = ( produc7477144754405272026_a_a_a @ T2 ) )
                & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) )
                  = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) )
                & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) ) )
                  = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) ) ) ) ) ) ) ) ).

% shifted_transitions_observable_against_distinguishing_transitions_left
thf(fact_448_canonical__separator__transition__targets_I4_J,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( sum_is4240621810064316983_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) )
            | ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
              = ( sum_In6345099547827038749od_a_a @ Q1 ) )
            | ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
              = ( sum_In6345099547827038749od_a_a @ Q2 ) ) ) ) ) ) ).

% canonical_separator_transition_targets(4)
thf(fact_449_canonical__separator__transition__ex,axiom,
    ! [T: produc2172198381475533558_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a,S1: produc1895207096998757355_a_a_a,S2: produc1895207096998757355_a_a_a] :
      ( ( member2160747886658840973_a_a_a @ T @ ( transi1397692036513875993_a_a_b @ ( state_7689045803801065988_a_a_b @ M @ Q1 @ Q2 ) ) )
     => ( ( member6990616694304513154_a_a_a @ Q1 @ ( states8463590192175095071_a_a_b @ M ) )
       => ( ( member6990616694304513154_a_a_a @ Q2 @ ( states8463590192175095071_a_a_b @ M ) )
         => ( ( ( produc4882268871966557188_a_a_a @ T )
              = ( sum_In122109773348503092_a_a_a @ ( produc8806572864378474095_a_a_a @ S1 @ S2 ) ) )
           => ( ? [X4: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ M ) )
                  & ( ( produc2316789063618113990_a_a_a @ X4 )
                    = S1 )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
                    = ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                    = ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) ) ) )
              | ? [X4: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ M ) )
                  & ( ( produc2316789063618113990_a_a_a @ X4 )
                    = S2 )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
                    = ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                    = ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) ) ) ) ) ) ) ) ) ).

% canonical_separator_transition_ex
thf(fact_450_canonical__separator__transition__ex,axiom,
    ! [T: produc1185401179908872758_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,S1: sum_su5154756782089200444_a_a_a,S2: sum_su5154756782089200444_a_a_a] :
      ( ( member3530913115812046943_a_a_a @ T @ ( transi5838560229626784988_a_b_c @ ( state_4431404365493471503_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3035385874452377268_a_b_c @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3035385874452377268_a_b_c @ M ) )
         => ( ( ( produc3221131896138471826_a_a_a @ T )
              = ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ S1 @ S2 ) ) )
           => ( ? [X4: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ M ) )
                  & ( ( produc7477144754405272026_a_a_a @ X4 )
                    = S1 )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
                    = ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) )
                  & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                    = ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) ) ) )
              | ? [X4: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ M ) )
                  & ( ( produc7477144754405272026_a_a_a @ X4 )
                    = S2 )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
                    = ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) )
                  & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                    = ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) ) ) ) ) ) ) ) ) ).

% canonical_separator_transition_ex
thf(fact_451_canonical__separator__transition__ex,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a,S1: a,S2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( ( produc7477144754405272026_a_a_a @ T )
              = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ S1 @ S2 ) ) )
           => ( ? [X4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M ) )
                  & ( ( produc4225943634317630866od_c_a @ X4 )
                    = S1 )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) ) )
              | ? [X4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M ) )
                  & ( ( produc4225943634317630866od_c_a @ X4 )
                    = S2 )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) ) ) ) ) ) ) ) ).

% canonical_separator_transition_ex
thf(fact_452_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su7096409449263244574_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_453_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2660408029157294746_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_454_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su9011528761648865002_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_455_assms_I3_J,axiom,
    ? [Q12: a,Q22: a,X4: b] :
      ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q12 @ Q22 ) ) @ X4 ) @ ( set_Pr4578018992751950051_a_a_b @ cs ) )
      & ( ( produc7477144754405272026_a_a_a @ t )
        = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q12 @ Q22 ) ) )
      & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ t ) )
        = X4 ) ) ).

% assms(3)
thf(fact_456_old_Osum_Oinject_I1_J,axiom,
    ! [A: product_prod_a_a,A6: product_prod_a_a] :
      ( ( ( sum_In6412843608951811077_a_a_a @ A )
        = ( sum_In6412843608951811077_a_a_a @ A6 ) )
      = ( A = A6 ) ) ).

% old.sum.inject(1)
thf(fact_457_sum_Oinject_I1_J,axiom,
    ! [X1: product_prod_a_a,Y1: product_prod_a_a] :
      ( ( ( sum_In6412843608951811077_a_a_a @ X1 )
        = ( sum_In6412843608951811077_a_a_a @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% sum.inject(1)
thf(fact_458__092_060open_062t__source_At_A_061_AInl_A_Iq1_H_M_Aq2_H_J_092_060close_062,axiom,
    ( ( produc7477144754405272026_a_a_a @ t )
    = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ q12 @ q22 ) ) ) ).

% \<open>t_source t = Inl (q1', q2')\<close>
thf(fact_459__092_060open_062_IInl_A_Iq1_H_M_Aq2_H_J_M_Ax_J_A_092_060in_062_Aset_Acs_092_060close_062,axiom,
    member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ q12 @ q22 ) ) @ x ) @ ( set_Pr4578018992751950051_a_a_b @ cs ) ).

% \<open>(Inl (q1', q2'), x) \<in> set cs\<close>
thf(fact_460__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q1_H_Aq2_H_Ax_O_A_092_060lbrakk_062_IInl_A_Iq1_H_M_Aq2_H_J_M_Ax_J_A_092_060in_062_Aset_Acs_059_At__source_At_A_061_AInl_A_Iq1_H_M_Aq2_H_J_059_At__input_At_A_061_Ax_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Q12: a,Q22: a,X4: b] :
        ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q12 @ Q22 ) ) @ X4 ) @ ( set_Pr4578018992751950051_a_a_b @ cs ) )
       => ( ( ( produc7477144754405272026_a_a_a @ t )
            = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q12 @ Q22 ) ) )
         => ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ t ) )
           != X4 ) ) ) ).

% \<open>\<And>thesis. (\<And>q1' q2' x. \<lbrakk>(Inl (q1', q2'), x) \<in> set cs; t_source t = Inl (q1', q2'); t_input t = x\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_461_add__transition__with__components__simps_I4_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,T: produc884577481120224310_a_a_a] :
      ( ( transi749694810082594024_a_b_c @ ( add_tr191749633354624124_a_b_c @ M @ T ) )
      = ( insert2249508923356641798_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) ) ) ).

% add_transition_with_components_simps(4)
thf(fact_462_psubsetD,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,C: sum_su5154756782089200444_a_a_a] :
      ( ( ord_le1107997805182519878_a_a_a @ A2 @ B3 )
     => ( ( member5679158588528110291_a_a_a @ C @ A2 )
       => ( member5679158588528110291_a_a_a @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_463_psubsetD,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a,C: produc884577481120224310_a_a_a] :
      ( ( ord_le6196359254896381890_a_a_a @ A2 @ B3 )
     => ( ( member8362999488266169055_a_a_a @ C @ A2 )
       => ( member8362999488266169055_a_a_a @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_464_psubsetD,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b,C: produc4770514982935793080_a_a_b] :
      ( ( ord_le8627993822118182850_a_a_b @ A2 @ B3 )
     => ( ( member642552543386773071_a_a_b @ C @ A2 )
       => ( member642552543386773071_a_a_b @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_465_Inl__inject,axiom,
    ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
      ( ( ( sum_In6412843608951811077_a_a_a @ X3 )
        = ( sum_In6412843608951811077_a_a_a @ Y3 ) )
     => ( X3 = Y3 ) ) ).

% Inl_inject
thf(fact_466_isl__def,axiom,
    ( sum_is4240621810064316983_a_a_a
    = ( ^ [Sum: sum_su5154756782089200444_a_a_a] :
        ? [X12: product_prod_a_a] :
          ( Sum
          = ( sum_In6412843608951811077_a_a_a @ X12 ) ) ) ) ).

% isl_def
thf(fact_467_sum_OdiscI_I1_J,axiom,
    ! [Sum2: sum_su5154756782089200444_a_a_a,X1: product_prod_a_a] :
      ( ( Sum2
        = ( sum_In6412843608951811077_a_a_a @ X1 ) )
     => ( sum_is4240621810064316983_a_a_a @ Sum2 ) ) ).

% sum.discI(1)
thf(fact_468_sum_Odisc_I1_J,axiom,
    ! [X1: product_prod_a_a] : ( sum_is4240621810064316983_a_a_a @ ( sum_In6412843608951811077_a_a_a @ X1 ) ) ).

% sum.disc(1)
thf(fact_469_deterministic__alt__def,axiom,
    ( determ6453466625651965340_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c] :
        ! [Q13: sum_su5154756782089200444_a_a_a,X: b,Y7: c,Y8: c,Q14: sum_su5154756782089200444_a_a_a,Q15: sum_su5154756782089200444_a_a_a] :
          ( ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q13 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y7 @ Q14 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) )
            & ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q13 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y8 @ Q15 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) ) )
         => ( ( Y7 = Y8 )
            & ( Q14 = Q15 ) ) ) ) ) ).

% deterministic_alt_def
thf(fact_470_split__sum__all,axiom,
    ( ( ^ [P2: sum_su5154756782089200444_a_a_a > $o] :
        ! [X5: sum_su5154756782089200444_a_a_a] : ( P2 @ X5 ) )
    = ( ^ [P3: sum_su5154756782089200444_a_a_a > $o] :
          ( ! [X: product_prod_a_a] : ( P3 @ ( sum_In6412843608951811077_a_a_a @ X ) )
          & ! [X: a] : ( P3 @ ( sum_In6345099547827038749od_a_a @ X ) ) ) ) ) ).

% split_sum_all
thf(fact_471_split__sum__ex,axiom,
    ( ( ^ [P2: sum_su5154756782089200444_a_a_a > $o] :
        ? [X5: sum_su5154756782089200444_a_a_a] : ( P2 @ X5 ) )
    = ( ^ [P3: sum_su5154756782089200444_a_a_a > $o] :
          ( ? [X: product_prod_a_a] : ( P3 @ ( sum_In6412843608951811077_a_a_a @ X ) )
          | ? [X: a] : ( P3 @ ( sum_In6345099547827038749od_a_a @ X ) ) ) ) ) ).

% split_sum_ex
thf(fact_472_Inr__not__Inl,axiom,
    ! [B: a,A: product_prod_a_a] :
      ( ( sum_In6345099547827038749od_a_a @ B )
     != ( sum_In6412843608951811077_a_a_a @ A ) ) ).

% Inr_not_Inl
thf(fact_473_sumE,axiom,
    ! [S3: sum_su5154756782089200444_a_a_a] :
      ( ! [X4: product_prod_a_a] :
          ( S3
         != ( sum_In6412843608951811077_a_a_a @ X4 ) )
     => ~ ! [Y: a] :
            ( S3
           != ( sum_In6345099547827038749od_a_a @ Y ) ) ) ).

% sumE
thf(fact_474_old_Osum_Oexhaust,axiom,
    ! [Y3: sum_su5154756782089200444_a_a_a] :
      ( ! [A5: product_prod_a_a] :
          ( Y3
         != ( sum_In6412843608951811077_a_a_a @ A5 ) )
     => ~ ! [B7: a] :
            ( Y3
           != ( sum_In6345099547827038749od_a_a @ B7 ) ) ) ).

% old.sum.exhaust
thf(fact_475_old_Osum_Odistinct_I1_J,axiom,
    ! [A: product_prod_a_a,B2: a] :
      ( ( sum_In6412843608951811077_a_a_a @ A )
     != ( sum_In6345099547827038749od_a_a @ B2 ) ) ).

% old.sum.distinct(1)
thf(fact_476_old_Osum_Odistinct_I2_J,axiom,
    ! [B2: a,A: product_prod_a_a] :
      ( ( sum_In6345099547827038749od_a_a @ B2 )
     != ( sum_In6412843608951811077_a_a_a @ A ) ) ).

% old.sum.distinct(2)
thf(fact_477_sum_Odistinct_I1_J,axiom,
    ! [X1: product_prod_a_a,X2: a] :
      ( ( sum_In6412843608951811077_a_a_a @ X1 )
     != ( sum_In6345099547827038749od_a_a @ X2 ) ) ).

% sum.distinct(1)
thf(fact_478_bot__prod__def,axiom,
    ( bot_bo2512674912376662449_a_a_a
    = ( produc975303698946120061_a_a_a @ bot_bo2676859181106700934_a_a_a @ bot_bo2676859181106700934_a_a_a ) ) ).

% bot_prod_def
thf(fact_479_bot__prod__def,axiom,
    ( bot_bo6932056865985061165_a_a_b
    = ( produc1782693847520575481_a_a_b @ bot_bo2676859181106700934_a_a_a @ bot_bo5052542052834138114_a_a_b ) ) ).

% bot_prod_def
thf(fact_480_bot__prod__def,axiom,
    ( bot_bo2462133688595082157_a_a_a
    = ( produc2779947338657330891_a_a_a @ bot_bo2676859181106700934_a_a_a @ bot_bo2898616998271641474_a_a_a ) ) ).

% bot_prod_def
thf(fact_481_bot__prod__def,axiom,
    ( bot_bo1017558259406586413_a_a_a
    = ( produc5347324035342034681_a_a_a @ bot_bo5052542052834138114_a_a_b @ bot_bo2676859181106700934_a_a_a ) ) ).

% bot_prod_def
thf(fact_482_bot__prod__def,axiom,
    ( bot_bo7991703675616730537_a_a_b
    = ( produc2477030430842270069_a_a_b @ bot_bo5052542052834138114_a_a_b @ bot_bo5052542052834138114_a_a_b ) ) ).

% bot_prod_def
thf(fact_483_bot__prod__def,axiom,
    ( bot_bo1385750253692666929_a_a_a
    = ( produc691750705766406735_a_a_a @ bot_bo5052542052834138114_a_a_b @ bot_bo2898616998271641474_a_a_a ) ) ).

% bot_prod_def
thf(fact_484_bot__prod__def,axiom,
    ( bot_bo2269467187568170187_a_a_a
    = ( produc3002878504517184177_a_a_a @ bot_bo2898616998271641474_a_a_a @ bot_bo2676859181106700934_a_a_a ) ) ).

% bot_prod_def
thf(fact_485_bot__prod__def,axiom,
    ( bot_bo7638358668229304903_a_a_b
    = ( produc7104200029644651821_a_a_b @ bot_bo2898616998271641474_a_a_a @ bot_bo5052542052834138114_a_a_b ) ) ).

% bot_prod_def
thf(fact_486_bot__prod__def,axiom,
    ( bot_bo7033759339267084819_a_a_a
    = ( produc3361136090342852759_a_a_a @ bot_bo2898616998271641474_a_a_a @ bot_bo2898616998271641474_a_a_a ) ) ).

% bot_prod_def
thf(fact_487_sum_OdiscI_I2_J,axiom,
    ! [Sum2: sum_su5154756782089200444_a_a_a,X2: a] :
      ( ( Sum2
        = ( sum_In6345099547827038749od_a_a @ X2 ) )
     => ~ ( sum_is4240621810064316983_a_a_a @ Sum2 ) ) ).

% sum.discI(2)
thf(fact_488_sum_Odisc_I2_J,axiom,
    ! [X2: a] :
      ~ ( sum_is4240621810064316983_a_a_a @ ( sum_In6345099547827038749od_a_a @ X2 ) ) ).

% sum.disc(2)
thf(fact_489_canonical__separator__transitions__alt__def,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) )
          = ( sup_su9011528761648865002_a_a_a @ ( sup_su9011528761648865002_a_a_a @ ( state_1345282896291928690_a_b_c @ M @ Q1 @ Q2 ) @ ( state_2046487007905918109_a_b_c @ M @ Q1 @ Q2 ) ) @ ( state_2881338517818858960_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ).

% canonical_separator_transitions_alt_def
thf(fact_490_boolean__algebra__cancel_Osup1,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,K: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( A2
        = ( sup_su7096409449263244574_a_a_a @ K @ A ) )
     => ( ( sup_su7096409449263244574_a_a_a @ A2 @ B )
        = ( sup_su7096409449263244574_a_a_a @ K @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_491_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_Su8591296761904255986_a_a_a,K: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( B3
        = ( sup_su7096409449263244574_a_a_a @ K @ B ) )
     => ( ( sup_su7096409449263244574_a_a_a @ A @ B3 )
        = ( sup_su7096409449263244574_a_a_a @ K @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_492_canonical__separator__targets__observable_I1_J,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( Q1 != Q2 )
           => ( ( sum_is4240621810064316983_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) )
             => ( member8362999488266169055_a_a_a @ T @ ( state_1345282896291928690_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ) ) ).

% canonical_separator_targets_observable(1)
thf(fact_493_canonical__separator__t__source__isl,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( sum_is4240621810064316983_a_a_a @ ( produc7477144754405272026_a_a_a @ T ) ) ) ) ) ).

% canonical_separator_t_source_isl
thf(fact_494_distinguishing__transitions__right__sources__targets,axiom,
    ! [T: produc2172198381475533558_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a] :
      ( ( member2160747886658840973_a_a_a @ T @ ( state_234223814407059111_a_a_b @ M @ Q1 @ Q2 ) )
     => ( ( member6990616694304513154_a_a_a @ Q1 @ ( states8463590192175095071_a_a_b @ M ) )
       => ~ ! [Q12: produc1895207096998757355_a_a_a,Q22: produc1895207096998757355_a_a_a] :
              ( ( ( produc4882268871966557188_a_a_a @ T )
                = ( sum_In122109773348503092_a_a_a @ ( produc8806572864378474095_a_a_a @ Q12 @ Q22 ) ) )
             => ( ( member6990616694304513154_a_a_a @ Q12 @ ( states8463590192175095071_a_a_b @ M ) )
               => ( ( member6990616694304513154_a_a_a @ Q22 @ ( states8463590192175095071_a_a_b @ M ) )
                 => ! [T3: produc6800189296593107456_a_a_a] :
                      ( ( member4643618012521571351_a_a_a @ T3 @ ( transi7010299678732630507_a_a_b @ M ) )
                     => ( ( ( produc2316789063618113990_a_a_a @ T3 )
                          = Q22 )
                       => ( ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T3 ) )
                            = ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) )
                         => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T3 ) ) )
                              = ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) ) )
                           => ( ~ ? [X7: produc6800189296593107456_a_a_a] :
                                    ( ( member4643618012521571351_a_a_a @ X7 @ ( transi7010299678732630507_a_a_b @ M ) )
                                    & ( ( produc2316789063618113990_a_a_a @ X7 )
                                      = Q12 )
                                    & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) )
                                      = ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) )
                                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
                                      = ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) ) ) )
                             => ( ( produc151198060758000217_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) )
                               != ( sum_In4154768905285699222_a_a_a @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% distinguishing_transitions_right_sources_targets
thf(fact_495_distinguishing__transitions__right__sources__targets,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( state_2881338517818858960_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ~ ! [Q12: a,Q22: a] :
              ( ( ( produc7477144754405272026_a_a_a @ T )
                = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q12 @ Q22 ) ) )
             => ( ( member_a @ Q12 @ ( states_a_b_c @ M ) )
               => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
                 => ! [T3: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ T3 @ ( transitions_a_b_c @ M ) )
                     => ( ( ( produc4225943634317630866od_c_a @ T3 )
                          = Q22 )
                       => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) )
                            = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                         => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) )
                              = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) )
                           => ( ~ ? [X7: produc2669932201312201782od_c_a] :
                                    ( ( member5502040268208861279od_c_a @ X7 @ ( transitions_a_b_c @ M ) )
                                    & ( ( produc4225943634317630866od_c_a @ X7 )
                                      = Q12 )
                                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X7 ) )
                                      = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X7 ) ) )
                                      = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) ) )
                             => ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                               != ( sum_In6345099547827038749od_a_a @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% distinguishing_transitions_right_sources_targets
thf(fact_496_distinguishing__transitions__right__sources__targets,axiom,
    ! [T: produc1185401179908872758_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a] :
      ( ( member3530913115812046943_a_a_a @ T @ ( state_5647776831386003244_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3035385874452377268_a_b_c @ M ) )
       => ~ ! [Q12: sum_su5154756782089200444_a_a_a,Q22: sum_su5154756782089200444_a_a_a] :
              ( ( ( produc3221131896138471826_a_a_a @ T )
                = ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ Q12 @ Q22 ) ) )
             => ( ( member5679158588528110291_a_a_a @ Q12 @ ( states3035385874452377268_a_b_c @ M ) )
               => ( ( member5679158588528110291_a_a_a @ Q22 @ ( states3035385874452377268_a_b_c @ M ) )
                 => ! [T3: produc884577481120224310_a_a_a] :
                      ( ( member8362999488266169055_a_a_a @ T3 @ ( transi749694810082594024_a_b_c @ M ) )
                     => ( ( ( produc7477144754405272026_a_a_a @ T3 )
                          = Q22 )
                       => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T3 ) )
                            = ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) )
                         => ( ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T3 ) ) )
                              = ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) ) )
                           => ( ~ ? [X7: produc884577481120224310_a_a_a] :
                                    ( ( member8362999488266169055_a_a_a @ X7 @ ( transi749694810082594024_a_b_c @ M ) )
                                    & ( ( produc7477144754405272026_a_a_a @ X7 )
                                      = Q12 )
                                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) )
                                      = ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) )
                                    & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
                                      = ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) ) ) )
                             => ( ( produc4853709320028323467_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) )
                               != ( sum_In5332043013833219623_a_a_a @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% distinguishing_transitions_right_sources_targets
thf(fact_497_distinguishing__transitions__left__sources__targets,axiom,
    ! [T: produc2172198381475533558_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a] :
      ( ( member2160747886658840973_a_a_a @ T @ ( state_8712764720510717172_a_a_b @ M @ Q1 @ Q2 ) )
     => ( ( member6990616694304513154_a_a_a @ Q2 @ ( states8463590192175095071_a_a_b @ M ) )
       => ~ ! [Q12: produc1895207096998757355_a_a_a,Q22: produc1895207096998757355_a_a_a] :
              ( ( ( produc4882268871966557188_a_a_a @ T )
                = ( sum_In122109773348503092_a_a_a @ ( produc8806572864378474095_a_a_a @ Q12 @ Q22 ) ) )
             => ( ( member6990616694304513154_a_a_a @ Q12 @ ( states8463590192175095071_a_a_b @ M ) )
               => ( ( member6990616694304513154_a_a_a @ Q22 @ ( states8463590192175095071_a_a_b @ M ) )
                 => ! [T3: produc6800189296593107456_a_a_a] :
                      ( ( member4643618012521571351_a_a_a @ T3 @ ( transi7010299678732630507_a_a_b @ M ) )
                     => ( ( ( produc2316789063618113990_a_a_a @ T3 )
                          = Q12 )
                       => ( ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T3 ) )
                            = ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) )
                         => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T3 ) ) )
                              = ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) ) )
                           => ( ~ ? [X7: produc6800189296593107456_a_a_a] :
                                    ( ( member4643618012521571351_a_a_a @ X7 @ ( transi7010299678732630507_a_a_b @ M ) )
                                    & ( ( produc2316789063618113990_a_a_a @ X7 )
                                      = Q22 )
                                    & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) )
                                      = ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) )
                                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
                                      = ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) ) ) )
                             => ( ( produc151198060758000217_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T ) ) )
                               != ( sum_In4154768905285699222_a_a_a @ Q1 ) ) ) ) ) ) ) ) ) ) ) ) ).

% distinguishing_transitions_left_sources_targets
thf(fact_498_distinguishing__transitions__left__sources__targets,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( state_2046487007905918109_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ~ ! [Q12: a,Q22: a] :
              ( ( ( produc7477144754405272026_a_a_a @ T )
                = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q12 @ Q22 ) ) )
             => ( ( member_a @ Q12 @ ( states_a_b_c @ M ) )
               => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
                 => ! [T3: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ T3 @ ( transitions_a_b_c @ M ) )
                     => ( ( ( produc4225943634317630866od_c_a @ T3 )
                          = Q12 )
                       => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) )
                            = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                         => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) )
                              = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) )
                           => ( ~ ? [X7: produc2669932201312201782od_c_a] :
                                    ( ( member5502040268208861279od_c_a @ X7 @ ( transitions_a_b_c @ M ) )
                                    & ( ( produc4225943634317630866od_c_a @ X7 )
                                      = Q22 )
                                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X7 ) )
                                      = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X7 ) ) )
                                      = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) ) )
                             => ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                               != ( sum_In6345099547827038749od_a_a @ Q1 ) ) ) ) ) ) ) ) ) ) ) ) ).

% distinguishing_transitions_left_sources_targets
thf(fact_499_distinguishing__transitions__left__sources__targets,axiom,
    ! [T: produc1185401179908872758_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a] :
      ( ( member3530913115812046943_a_a_a @ T @ ( state_1106402805630587423_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3035385874452377268_a_b_c @ M ) )
       => ~ ! [Q12: sum_su5154756782089200444_a_a_a,Q22: sum_su5154756782089200444_a_a_a] :
              ( ( ( produc3221131896138471826_a_a_a @ T )
                = ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ Q12 @ Q22 ) ) )
             => ( ( member5679158588528110291_a_a_a @ Q12 @ ( states3035385874452377268_a_b_c @ M ) )
               => ( ( member5679158588528110291_a_a_a @ Q22 @ ( states3035385874452377268_a_b_c @ M ) )
                 => ! [T3: produc884577481120224310_a_a_a] :
                      ( ( member8362999488266169055_a_a_a @ T3 @ ( transi749694810082594024_a_b_c @ M ) )
                     => ( ( ( produc7477144754405272026_a_a_a @ T3 )
                          = Q12 )
                       => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T3 ) )
                            = ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) )
                         => ( ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T3 ) ) )
                              = ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) ) )
                           => ( ~ ? [X7: produc884577481120224310_a_a_a] :
                                    ( ( member8362999488266169055_a_a_a @ X7 @ ( transi749694810082594024_a_b_c @ M ) )
                                    & ( ( produc7477144754405272026_a_a_a @ X7 )
                                      = Q22 )
                                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) )
                                      = ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) )
                                    & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
                                      = ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) ) ) )
                             => ( ( produc4853709320028323467_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) )
                               != ( sum_In5332043013833219623_a_a_a @ Q1 ) ) ) ) ) ) ) ) ) ) ) ) ).

% distinguishing_transitions_left_sources_targets
thf(fact_500_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_a] :
      ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_501_prod_Ocollapse,axiom,
    ! [Prod: produc1895207096998757355_a_a_a] :
      ( ( produc8806894744968565733_a_a_a @ ( produc6784836004715151097_a_a_a @ Prod ) @ ( produc1172910294343954743_a_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_502_prod_Ocollapse,axiom,
    ! [Prod: produc7810058533641270553_a_a_a] :
      ( ( produc1730628064597884819_a_a_a @ ( produc5842599967860340391_a_a_a @ Prod ) @ ( produc6236129573204179685_a_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_503_prod_Ocollapse,axiom,
    ! [Prod: produc884577481120224310_a_a_a] :
      ( ( produc6863513614254923246_a_a_a @ ( produc7477144754405272026_a_a_a @ Prod ) @ ( produc1629229363059081756_a_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_504_prod_Ocollapse,axiom,
    ! [Prod: produc4770514982935793080_a_a_b] :
      ( ( produc6086470211827429482_a_a_b @ ( produc4064411471574014846_a_a_b @ Prod ) @ ( produc7675857798057594300_a_a_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_505_Inl__Inr__False,axiom,
    ! [X3: product_prod_a_a,Y3: a] :
      ( ( sum_In6412843608951811077_a_a_a @ X3 )
     != ( sum_In6345099547827038749od_a_a @ Y3 ) ) ).

% Inl_Inr_False
thf(fact_506_Inr__Inl__False,axiom,
    ! [X3: a,Y3: product_prod_a_a] :
      ( ( sum_In6345099547827038749od_a_a @ X3 )
     != ( sum_In6412843608951811077_a_a_a @ Y3 ) ) ).

% Inr_Inl_False
thf(fact_507_canonical__separator__transition_I1_J,axiom,
    ! [T: produc1185401179908872758_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,S1: sum_su5154756782089200444_a_a_a,S2: sum_su5154756782089200444_a_a_a,S12: sum_su5154756782089200444_a_a_a,S22: sum_su5154756782089200444_a_a_a] :
      ( ( member3530913115812046943_a_a_a @ T @ ( transi5838560229626784988_a_b_c @ ( state_4431404365493471503_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3035385874452377268_a_b_c @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3035385874452377268_a_b_c @ M ) )
         => ( ( ( produc3221131896138471826_a_a_a @ T )
              = ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ S1 @ S2 ) ) )
           => ( ( observ6810049560328217077_a_b_c @ M )
             => ( ( Q1 != Q2 )
               => ( ( ( produc4853709320028323467_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) )
                    = ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ S12 @ S22 ) ) )
                 => ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ S1 @ ( produc1730628064597884819_a_a_a @ ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) @ ( produc8806894744968565733_a_a_a @ ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) ) @ S12 ) ) ) @ ( transi749694810082594024_a_b_c @ M ) )
                    & ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ S2 @ ( produc1730628064597884819_a_a_a @ ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) @ ( produc8806894744968565733_a_a_a @ ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T ) ) ) @ S22 ) ) ) @ ( transi749694810082594024_a_b_c @ M ) ) ) ) ) ) ) ) ) ) ).

% canonical_separator_transition(1)
thf(fact_508_canonical__separator__transition_I1_J,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a,S1: a,S2: a,S12: a,S22: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( ( produc7477144754405272026_a_a_a @ T )
              = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ S1 @ S2 ) ) )
           => ( ( observable_a_b_c @ M )
             => ( ( Q1 != Q2 )
               => ( ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                    = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ S12 @ S22 ) ) )
                 => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ S1 @ ( produc3440865914000881661od_c_a @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) @ ( product_Pair_c_a @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) @ S12 ) ) ) @ ( transitions_a_b_c @ M ) )
                    & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ S2 @ ( produc3440865914000881661od_c_a @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) @ ( product_Pair_c_a @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) @ S22 ) ) ) @ ( transitions_a_b_c @ M ) ) ) ) ) ) ) ) ) ) ).

% canonical_separator_transition(1)
thf(fact_509_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: a > a > $o,X3: a,Y3: a,A: product_prod_a_a] :
      ( ( P @ X3 @ Y3 )
     => ( ( A
          = ( product_Pair_a_a @ X3 @ Y3 ) )
       => ( P @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_510_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: c > sum_su5154756782089200444_a_a_a > $o,X3: c,Y3: sum_su5154756782089200444_a_a_a,A: produc1895207096998757355_a_a_a] :
      ( ( P @ X3 @ Y3 )
     => ( ( A
          = ( produc8806894744968565733_a_a_a @ X3 @ Y3 ) )
       => ( P @ ( produc6784836004715151097_a_a_a @ A ) @ ( produc1172910294343954743_a_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_511_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: b > produc1895207096998757355_a_a_a > $o,X3: b,Y3: produc1895207096998757355_a_a_a,A: produc7810058533641270553_a_a_a] :
      ( ( P @ X3 @ Y3 )
     => ( ( A
          = ( produc1730628064597884819_a_a_a @ X3 @ Y3 ) )
       => ( P @ ( produc5842599967860340391_a_a_a @ A ) @ ( produc6236129573204179685_a_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_512_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > produc7810058533641270553_a_a_a > $o,X3: sum_su5154756782089200444_a_a_a,Y3: produc7810058533641270553_a_a_a,A: produc884577481120224310_a_a_a] :
      ( ( P @ X3 @ Y3 )
     => ( ( A
          = ( produc6863513614254923246_a_a_a @ X3 @ Y3 ) )
       => ( P @ ( produc7477144754405272026_a_a_a @ A ) @ ( produc1629229363059081756_a_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_513_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > b > $o,X3: sum_su5154756782089200444_a_a_a,Y3: b,A: produc4770514982935793080_a_a_b] :
      ( ( P @ X3 @ Y3 )
     => ( ( A
          = ( produc6086470211827429482_a_a_b @ X3 @ Y3 ) )
       => ( P @ ( produc4064411471574014846_a_a_b @ A ) @ ( produc7675857798057594300_a_a_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_514_surjective__pairing,axiom,
    ! [T: product_prod_a_a] :
      ( T
      = ( product_Pair_a_a @ ( product_fst_a_a @ T ) @ ( product_snd_a_a @ T ) ) ) ).

% surjective_pairing
thf(fact_515_surjective__pairing,axiom,
    ! [T: produc1895207096998757355_a_a_a] :
      ( T
      = ( produc8806894744968565733_a_a_a @ ( produc6784836004715151097_a_a_a @ T ) @ ( produc1172910294343954743_a_a_a @ T ) ) ) ).

% surjective_pairing
thf(fact_516_surjective__pairing,axiom,
    ! [T: produc7810058533641270553_a_a_a] :
      ( T
      = ( produc1730628064597884819_a_a_a @ ( produc5842599967860340391_a_a_a @ T ) @ ( produc6236129573204179685_a_a_a @ T ) ) ) ).

% surjective_pairing
thf(fact_517_surjective__pairing,axiom,
    ! [T: produc884577481120224310_a_a_a] :
      ( T
      = ( produc6863513614254923246_a_a_a @ ( produc7477144754405272026_a_a_a @ T ) @ ( produc1629229363059081756_a_a_a @ T ) ) ) ).

% surjective_pairing
thf(fact_518_surjective__pairing,axiom,
    ! [T: produc4770514982935793080_a_a_b] :
      ( T
      = ( produc6086470211827429482_a_a_b @ ( produc4064411471574014846_a_a_b @ T ) @ ( produc7675857798057594300_a_a_b @ T ) ) ) ).

% surjective_pairing
thf(fact_519_old_Oprod_Oinject,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B: b,A6: sum_su5154756782089200444_a_a_a,B2: b] :
      ( ( ( produc6086470211827429482_a_a_b @ A @ B )
        = ( produc6086470211827429482_a_a_b @ A6 @ B2 ) )
      = ( ( A = A6 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_520_old_Oprod_Oinject,axiom,
    ! [A: a,B: a,A6: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A6 @ B2 ) )
      = ( ( A = A6 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_521_prod_Oinject,axiom,
    ! [X1: sum_su5154756782089200444_a_a_a,X2: b,Y1: sum_su5154756782089200444_a_a_a,Y2: b] :
      ( ( ( produc6086470211827429482_a_a_b @ X1 @ X2 )
        = ( produc6086470211827429482_a_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_522_prod_Oinject,axiom,
    ! [X1: a,X2: a,Y1: a,Y2: a] :
      ( ( ( product_Pair_a_a @ X1 @ X2 )
        = ( product_Pair_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_523_canonical__separator__observable,axiom,
    ! [M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( observ6810049560328217077_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ).

% canonical_separator_observable
thf(fact_524_observable__alt__def,axiom,
    ( observ6810049560328217077_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c] :
        ! [Q13: sum_su5154756782089200444_a_a_a,X: b,Y4: c,Q14: sum_su5154756782089200444_a_a_a,Q15: sum_su5154756782089200444_a_a_a] :
          ( ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q13 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q14 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) )
            & ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q13 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q15 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) ) )
         => ( Q14 = Q15 ) ) ) ) ).

% observable_alt_def
thf(fact_525_Pair__inject,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B: b,A6: sum_su5154756782089200444_a_a_a,B2: b] :
      ( ( ( produc6086470211827429482_a_a_b @ A @ B )
        = ( produc6086470211827429482_a_a_b @ A6 @ B2 ) )
     => ~ ( ( A = A6 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_526_Pair__inject,axiom,
    ! [A: a,B: a,A6: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A6 @ B2 ) )
     => ~ ( ( A = A6 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_527_prod__cases,axiom,
    ! [P: produc4770514982935793080_a_a_b > $o,P4: produc4770514982935793080_a_a_b] :
      ( ! [A5: sum_su5154756782089200444_a_a_a,B7: b] : ( P @ ( produc6086470211827429482_a_a_b @ A5 @ B7 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_528_prod__cases,axiom,
    ! [P: product_prod_a_a > $o,P4: product_prod_a_a] :
      ( ! [A5: a,B7: a] : ( P @ ( product_Pair_a_a @ A5 @ B7 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_529_surj__pair,axiom,
    ! [P4: produc4770514982935793080_a_a_b] :
    ? [X4: sum_su5154756782089200444_a_a_a,Y: b] :
      ( P4
      = ( produc6086470211827429482_a_a_b @ X4 @ Y ) ) ).

% surj_pair
thf(fact_530_surj__pair,axiom,
    ! [P4: product_prod_a_a] :
    ? [X4: a,Y: a] :
      ( P4
      = ( product_Pair_a_a @ X4 @ Y ) ) ).

% surj_pair
thf(fact_531_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc4770514982935793080_a_a_b] :
      ~ ! [A5: sum_su5154756782089200444_a_a_a,B7: b] :
          ( Y3
         != ( produc6086470211827429482_a_a_b @ A5 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_532_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_a_a] :
      ~ ! [A5: a,B7: a] :
          ( Y3
         != ( product_Pair_a_a @ A5 @ B7 ) ) ).

% old.prod.exhaust
thf(fact_533_not__arg__cong__Inr,axiom,
    ! [X3: a,Y3: a] :
      ( ( X3 != Y3 )
     => ( ( sum_In6345099547827038749od_a_a @ X3 )
       != ( sum_In6345099547827038749od_a_a @ Y3 ) ) ) ).

% not_arg_cong_Inr
thf(fact_534_observable__transition__unique,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,T: produc6800189296593107456_a_a_a] :
      ( ( observ7609480127267726238_a_a_b @ M )
     => ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ M ) )
       => ? [X4: produc6800189296593107456_a_a_a] :
            ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ M ) )
            & ( ( produc2316789063618113990_a_a_a @ X4 )
              = ( produc2316789063618113990_a_a_a @ T ) )
            & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
              = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) )
            & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
              = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) )
            & ! [Y5: produc6800189296593107456_a_a_a] :
                ( ( ( member4643618012521571351_a_a_a @ Y5 @ ( transi7010299678732630507_a_a_b @ M ) )
                  & ( ( produc2316789063618113990_a_a_a @ Y5 )
                    = ( produc2316789063618113990_a_a_a @ T ) )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y5 ) )
                    = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Y5 ) ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_535_observable__transition__unique,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,T: produc884577481120224310_a_a_a] :
      ( ( observ6810049560328217077_a_b_c @ M )
     => ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
       => ? [X4: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ M ) )
            & ( ( produc7477144754405272026_a_a_a @ X4 )
              = ( produc7477144754405272026_a_a_a @ T ) )
            & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
              = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
            & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
              = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) )
            & ! [Y5: produc884577481120224310_a_a_a] :
                ( ( ( member8362999488266169055_a_a_a @ Y5 @ ( transi749694810082594024_a_b_c @ M ) )
                  & ( ( produc7477144754405272026_a_a_a @ Y5 )
                    = ( produc7477144754405272026_a_a_a @ T ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y5 ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                  & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Y5 ) ) )
                    = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_536_observable_Oelims_I3_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b] :
      ( ~ ( observ7609480127267726238_a_a_b @ X3 )
     => ~ ! [X4: produc6800189296593107456_a_a_a] :
            ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ X3 ) )
           => ! [Xa3: produc6800189296593107456_a_a_a] :
                ( ( member4643618012521571351_a_a_a @ Xa3 @ ( transi7010299678732630507_a_a_b @ X3 ) )
               => ( ( ( ( produc2316789063618113990_a_a_a @ X4 )
                      = ( produc2316789063618113990_a_a_a @ Xa3 ) )
                    & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
                      = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa3 ) ) )
                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                      = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa3 ) ) ) ) )
                 => ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                    = ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa3 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_537_observable_Oelims_I3_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c] :
      ( ~ ( observ6810049560328217077_a_b_c @ X3 )
     => ~ ! [X4: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ X3 ) )
           => ! [Xa3: produc884577481120224310_a_a_a] :
                ( ( member8362999488266169055_a_a_a @ Xa3 @ ( transi749694810082594024_a_b_c @ X3 ) )
               => ( ( ( ( produc7477144754405272026_a_a_a @ X4 )
                      = ( produc7477144754405272026_a_a_a @ Xa3 ) )
                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
                      = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa3 ) ) )
                    & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                      = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa3 ) ) ) ) )
                 => ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                    = ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa3 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_538_observable_Oelims_I2_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b] :
      ( ( observ7609480127267726238_a_a_b @ X3 )
     => ! [X7: produc6800189296593107456_a_a_a] :
          ( ( member4643618012521571351_a_a_a @ X7 @ ( transi7010299678732630507_a_a_b @ X3 ) )
         => ! [Xa2: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ Xa2 @ ( transi7010299678732630507_a_a_b @ X3 ) )
             => ( ( ( ( produc2316789063618113990_a_a_a @ X7 )
                    = ( produc2316789063618113990_a_a_a @ Xa2 ) )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) )
                    = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) ) ) ) )
               => ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
                  = ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_539_observable_Oelims_I2_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c] :
      ( ( observ6810049560328217077_a_b_c @ X3 )
     => ! [X7: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X7 @ ( transi749694810082594024_a_b_c @ X3 ) )
         => ! [Xa2: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ Xa2 @ ( transi749694810082594024_a_b_c @ X3 ) )
             => ( ( ( ( produc7477144754405272026_a_a_a @ X7 )
                    = ( produc7477144754405272026_a_a_a @ Xa2 ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) ) )
                  & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
                    = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) ) ) ) )
               => ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
                  = ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_540_observable_Oelims_I1_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b,Y3: $o] :
      ( ( ( observ7609480127267726238_a_a_b @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ X @ ( transi7010299678732630507_a_a_b @ X3 ) )
             => ! [Y4: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ X3 ) )
                 => ( ( ( ( produc2316789063618113990_a_a_a @ X )
                        = ( produc2316789063618113990_a_a_a @ Y4 ) )
                      & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) )
                        = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) )
                      & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) ) )
                        = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) ) )
                   => ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) ) )
                      = ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_541_observable_Oelims_I1_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Y3: $o] :
      ( ( ( observ6810049560328217077_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ X @ ( transi749694810082594024_a_b_c @ X3 ) )
             => ! [Y4: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ X3 ) )
                 => ( ( ( ( produc7477144754405272026_a_a_a @ X )
                        = ( produc7477144754405272026_a_a_a @ Y4 ) )
                      & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) )
                        = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) )
                      & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) ) )
                        = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) ) )
                   => ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) ) )
                      = ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_542_observable_Osimps,axiom,
    ( observ7609480127267726238_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b] :
        ! [X: produc6800189296593107456_a_a_a] :
          ( ( member4643618012521571351_a_a_a @ X @ ( transi7010299678732630507_a_a_b @ M3 ) )
         => ! [Y4: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ M3 ) )
             => ( ( ( ( produc2316789063618113990_a_a_a @ X )
                    = ( produc2316789063618113990_a_a_a @ Y4 ) )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) )
                    = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) ) )
               => ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) ) )
                  = ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_543_observable_Osimps,axiom,
    ( observ6810049560328217077_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c] :
        ! [X: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X @ ( transi749694810082594024_a_b_c @ M3 ) )
         => ! [Y4: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ M3 ) )
             => ( ( ( ( produc7477144754405272026_a_a_a @ X )
                    = ( produc7477144754405272026_a_a_a @ Y4 ) )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) )
                  & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) ) )
                    = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) ) )
               => ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) ) )
                  = ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_544_shifted__transitions__observable__against__shifted__transitions,axiom,
    ! [T1: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a,T2: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ T1 @ ( state_1345282896291928690_a_b_c @ M @ Q1 @ Q2 ) )
     => ( ( member8362999488266169055_a_a_a @ T2 @ ( state_1345282896291928690_a_b_c @ M @ Q1 @ Q2 ) )
       => ( ( observable_a_b_c @ M )
         => ( ( ( ( produc7477144754405272026_a_a_a @ T1 )
                = ( produc7477144754405272026_a_a_a @ T2 ) )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) )
                = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) )
              & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T1 ) ) )
                = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T2 ) ) ) ) )
           => ( T1 = T2 ) ) ) ) ) ).

% shifted_transitions_observable_against_shifted_transitions
thf(fact_545_snd__conv,axiom,
    ! [X1: a,X2: a] :
      ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_546_snd__conv,axiom,
    ! [X1: c,X2: sum_su5154756782089200444_a_a_a] :
      ( ( produc1172910294343954743_a_a_a @ ( produc8806894744968565733_a_a_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_547_snd__conv,axiom,
    ! [X1: b,X2: produc1895207096998757355_a_a_a] :
      ( ( produc6236129573204179685_a_a_a @ ( produc1730628064597884819_a_a_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_548_snd__conv,axiom,
    ! [X1: sum_su5154756782089200444_a_a_a,X2: produc7810058533641270553_a_a_a] :
      ( ( produc1629229363059081756_a_a_a @ ( produc6863513614254923246_a_a_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_549_snd__conv,axiom,
    ! [X1: sum_su5154756782089200444_a_a_a,X2: b] :
      ( ( produc7675857798057594300_a_a_b @ ( produc6086470211827429482_a_a_b @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_550_snd__eqD,axiom,
    ! [X3: a,Y3: a,A: a] :
      ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_551_snd__eqD,axiom,
    ! [X3: c,Y3: sum_su5154756782089200444_a_a_a,A: sum_su5154756782089200444_a_a_a] :
      ( ( ( produc1172910294343954743_a_a_a @ ( produc8806894744968565733_a_a_a @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_552_snd__eqD,axiom,
    ! [X3: b,Y3: produc1895207096998757355_a_a_a,A: produc1895207096998757355_a_a_a] :
      ( ( ( produc6236129573204179685_a_a_a @ ( produc1730628064597884819_a_a_a @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_553_snd__eqD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: produc7810058533641270553_a_a_a,A: produc7810058533641270553_a_a_a] :
      ( ( ( produc1629229363059081756_a_a_a @ ( produc6863513614254923246_a_a_a @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_554_snd__eqD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: b,A: b] :
      ( ( ( produc7675857798057594300_a_a_b @ ( produc6086470211827429482_a_a_b @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_555_fst__conv,axiom,
    ! [X1: a,X2: a] :
      ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_556_fst__conv,axiom,
    ! [X1: sum_su5154756782089200444_a_a_a,X2: b] :
      ( ( produc4064411471574014846_a_a_b @ ( produc6086470211827429482_a_a_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_557_fst__conv,axiom,
    ! [X1: b,X2: produc1895207096998757355_a_a_a] :
      ( ( produc5842599967860340391_a_a_a @ ( produc1730628064597884819_a_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_558_fst__conv,axiom,
    ! [X1: sum_su5154756782089200444_a_a_a,X2: produc7810058533641270553_a_a_a] :
      ( ( produc7477144754405272026_a_a_a @ ( produc6863513614254923246_a_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_559_fst__conv,axiom,
    ! [X1: c,X2: sum_su5154756782089200444_a_a_a] :
      ( ( produc6784836004715151097_a_a_a @ ( produc8806894744968565733_a_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_560_fst__eqD,axiom,
    ! [X3: a,Y3: a,A: a] :
      ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_561_fst__eqD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: b,A: sum_su5154756782089200444_a_a_a] :
      ( ( ( produc4064411471574014846_a_a_b @ ( produc6086470211827429482_a_a_b @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_562_fst__eqD,axiom,
    ! [X3: b,Y3: produc1895207096998757355_a_a_a,A: b] :
      ( ( ( produc5842599967860340391_a_a_a @ ( produc1730628064597884819_a_a_a @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_563_fst__eqD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: produc7810058533641270553_a_a_a,A: sum_su5154756782089200444_a_a_a] :
      ( ( ( produc7477144754405272026_a_a_a @ ( produc6863513614254923246_a_a_a @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_564_fst__eqD,axiom,
    ! [X3: c,Y3: sum_su5154756782089200444_a_a_a,A: c] :
      ( ( ( produc6784836004715151097_a_a_a @ ( produc8806894744968565733_a_a_a @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_565_prod_Oexpand,axiom,
    ! [Prod: produc1895207096998757355_a_a_a,Prod2: produc1895207096998757355_a_a_a] :
      ( ( ( ( produc6784836004715151097_a_a_a @ Prod )
          = ( produc6784836004715151097_a_a_a @ Prod2 ) )
        & ( ( produc1172910294343954743_a_a_a @ Prod )
          = ( produc1172910294343954743_a_a_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_566_prod_Oexpand,axiom,
    ! [Prod: produc7810058533641270553_a_a_a,Prod2: produc7810058533641270553_a_a_a] :
      ( ( ( ( produc5842599967860340391_a_a_a @ Prod )
          = ( produc5842599967860340391_a_a_a @ Prod2 ) )
        & ( ( produc6236129573204179685_a_a_a @ Prod )
          = ( produc6236129573204179685_a_a_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_567_prod_Oexpand,axiom,
    ! [Prod: produc884577481120224310_a_a_a,Prod2: produc884577481120224310_a_a_a] :
      ( ( ( ( produc7477144754405272026_a_a_a @ Prod )
          = ( produc7477144754405272026_a_a_a @ Prod2 ) )
        & ( ( produc1629229363059081756_a_a_a @ Prod )
          = ( produc1629229363059081756_a_a_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_568_prod_Oexpand,axiom,
    ! [Prod: produc4770514982935793080_a_a_b,Prod2: produc4770514982935793080_a_a_b] :
      ( ( ( ( produc4064411471574014846_a_a_b @ Prod )
          = ( produc4064411471574014846_a_a_b @ Prod2 ) )
        & ( ( produc7675857798057594300_a_a_b @ Prod )
          = ( produc7675857798057594300_a_a_b @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_569_prod__eqI,axiom,
    ! [P4: produc1895207096998757355_a_a_a,Q: produc1895207096998757355_a_a_a] :
      ( ( ( produc6784836004715151097_a_a_a @ P4 )
        = ( produc6784836004715151097_a_a_a @ Q ) )
     => ( ( ( produc1172910294343954743_a_a_a @ P4 )
          = ( produc1172910294343954743_a_a_a @ Q ) )
       => ( P4 = Q ) ) ) ).

% prod_eqI
thf(fact_570_prod__eqI,axiom,
    ! [P4: produc7810058533641270553_a_a_a,Q: produc7810058533641270553_a_a_a] :
      ( ( ( produc5842599967860340391_a_a_a @ P4 )
        = ( produc5842599967860340391_a_a_a @ Q ) )
     => ( ( ( produc6236129573204179685_a_a_a @ P4 )
          = ( produc6236129573204179685_a_a_a @ Q ) )
       => ( P4 = Q ) ) ) ).

% prod_eqI
thf(fact_571_prod__eqI,axiom,
    ! [P4: produc884577481120224310_a_a_a,Q: produc884577481120224310_a_a_a] :
      ( ( ( produc7477144754405272026_a_a_a @ P4 )
        = ( produc7477144754405272026_a_a_a @ Q ) )
     => ( ( ( produc1629229363059081756_a_a_a @ P4 )
          = ( produc1629229363059081756_a_a_a @ Q ) )
       => ( P4 = Q ) ) ) ).

% prod_eqI
thf(fact_572_prod__eqI,axiom,
    ! [P4: produc4770514982935793080_a_a_b,Q: produc4770514982935793080_a_a_b] :
      ( ( ( produc4064411471574014846_a_a_b @ P4 )
        = ( produc4064411471574014846_a_a_b @ Q ) )
     => ( ( ( produc7675857798057594300_a_a_b @ P4 )
          = ( produc7675857798057594300_a_a_b @ Q ) )
       => ( P4 = Q ) ) ) ).

% prod_eqI
thf(fact_573_prod__eq__iff,axiom,
    ( ( ^ [Y6: produc1895207096998757355_a_a_a,Z3: produc1895207096998757355_a_a_a] : ( Y6 = Z3 ) )
    = ( ^ [S4: produc1895207096998757355_a_a_a,T4: produc1895207096998757355_a_a_a] :
          ( ( ( produc6784836004715151097_a_a_a @ S4 )
            = ( produc6784836004715151097_a_a_a @ T4 ) )
          & ( ( produc1172910294343954743_a_a_a @ S4 )
            = ( produc1172910294343954743_a_a_a @ T4 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_574_prod__eq__iff,axiom,
    ( ( ^ [Y6: produc7810058533641270553_a_a_a,Z3: produc7810058533641270553_a_a_a] : ( Y6 = Z3 ) )
    = ( ^ [S4: produc7810058533641270553_a_a_a,T4: produc7810058533641270553_a_a_a] :
          ( ( ( produc5842599967860340391_a_a_a @ S4 )
            = ( produc5842599967860340391_a_a_a @ T4 ) )
          & ( ( produc6236129573204179685_a_a_a @ S4 )
            = ( produc6236129573204179685_a_a_a @ T4 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_575_prod__eq__iff,axiom,
    ( ( ^ [Y6: produc884577481120224310_a_a_a,Z3: produc884577481120224310_a_a_a] : ( Y6 = Z3 ) )
    = ( ^ [S4: produc884577481120224310_a_a_a,T4: produc884577481120224310_a_a_a] :
          ( ( ( produc7477144754405272026_a_a_a @ S4 )
            = ( produc7477144754405272026_a_a_a @ T4 ) )
          & ( ( produc1629229363059081756_a_a_a @ S4 )
            = ( produc1629229363059081756_a_a_a @ T4 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_576_prod__eq__iff,axiom,
    ( ( ^ [Y6: produc4770514982935793080_a_a_b,Z3: produc4770514982935793080_a_a_b] : ( Y6 = Z3 ) )
    = ( ^ [S4: produc4770514982935793080_a_a_b,T4: produc4770514982935793080_a_a_b] :
          ( ( ( produc4064411471574014846_a_a_b @ S4 )
            = ( produc4064411471574014846_a_a_b @ T4 ) )
          & ( ( produc7675857798057594300_a_a_b @ S4 )
            = ( produc7675857798057594300_a_a_b @ T4 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_577_obj__sumE,axiom,
    ! [S3: sum_su5154756782089200444_a_a_a] :
      ( ! [X4: product_prod_a_a] :
          ( S3
         != ( sum_In6412843608951811077_a_a_a @ X4 ) )
     => ~ ! [X4: a] :
            ( S3
           != ( sum_In6345099547827038749od_a_a @ X4 ) ) ) ).

% obj_sumE
thf(fact_578_canonical__separator__transition_I4_J,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a,S1: a,S2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( ( produc7477144754405272026_a_a_a @ T )
              = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ S1 @ S2 ) ) )
           => ( ( observable_a_b_c @ M )
             => ( ( Q1 != Q2 )
               => ( ? [S13: a,S23: a] :
                      ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                      = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ S13 @ S23 ) ) )
                  | ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                    = ( sum_In6345099547827038749od_a_a @ Q1 ) )
                  | ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) )
                    = ( sum_In6345099547827038749od_a_a @ Q2 ) ) ) ) ) ) ) ) ) ).

% canonical_separator_transition(4)
thf(fact_579_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_a] :
      ( Prod
      = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_580_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1895207096998757355_a_a_a] :
      ( Prod
      = ( produc8806894744968565733_a_a_a @ ( produc6784836004715151097_a_a_a @ Prod ) @ ( produc1172910294343954743_a_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_581_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7810058533641270553_a_a_a] :
      ( Prod
      = ( produc1730628064597884819_a_a_a @ ( produc5842599967860340391_a_a_a @ Prod ) @ ( produc6236129573204179685_a_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_582_prod_Oexhaust__sel,axiom,
    ! [Prod: produc884577481120224310_a_a_a] :
      ( Prod
      = ( produc6863513614254923246_a_a_a @ ( produc7477144754405272026_a_a_a @ Prod ) @ ( produc1629229363059081756_a_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_583_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4770514982935793080_a_a_b] :
      ( Prod
      = ( produc6086470211827429482_a_a_b @ ( produc4064411471574014846_a_a_b @ Prod ) @ ( produc7675857798057594300_a_a_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_584_exI__realizer,axiom,
    ! [P: a > a > $o,Y3: a,X3: a] :
      ( ( P @ Y3 @ X3 )
     => ( P @ ( product_snd_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_585_exI__realizer,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > c > $o,Y3: sum_su5154756782089200444_a_a_a,X3: c] :
      ( ( P @ Y3 @ X3 )
     => ( P @ ( produc1172910294343954743_a_a_a @ ( produc8806894744968565733_a_a_a @ X3 @ Y3 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc8806894744968565733_a_a_a @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_586_exI__realizer,axiom,
    ! [P: produc1895207096998757355_a_a_a > b > $o,Y3: produc1895207096998757355_a_a_a,X3: b] :
      ( ( P @ Y3 @ X3 )
     => ( P @ ( produc6236129573204179685_a_a_a @ ( produc1730628064597884819_a_a_a @ X3 @ Y3 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1730628064597884819_a_a_a @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_587_exI__realizer,axiom,
    ! [P: produc7810058533641270553_a_a_a > sum_su5154756782089200444_a_a_a > $o,Y3: produc7810058533641270553_a_a_a,X3: sum_su5154756782089200444_a_a_a] :
      ( ( P @ Y3 @ X3 )
     => ( P @ ( produc1629229363059081756_a_a_a @ ( produc6863513614254923246_a_a_a @ X3 @ Y3 ) ) @ ( produc7477144754405272026_a_a_a @ ( produc6863513614254923246_a_a_a @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_588_exI__realizer,axiom,
    ! [P: b > sum_su5154756782089200444_a_a_a > $o,Y3: b,X3: sum_su5154756782089200444_a_a_a] :
      ( ( P @ Y3 @ X3 )
     => ( P @ ( produc7675857798057594300_a_a_b @ ( produc6086470211827429482_a_a_b @ X3 @ Y3 ) ) @ ( produc4064411471574014846_a_a_b @ ( produc6086470211827429482_a_a_b @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_589_conjI__realizer,axiom,
    ! [P: a > $o,P4: a,Q3: a > $o,Q: a] :
      ( ( P @ P4 )
     => ( ( Q3 @ Q )
       => ( ( P @ ( product_fst_a_a @ ( product_Pair_a_a @ P4 @ Q ) ) )
          & ( Q3 @ ( product_snd_a_a @ ( product_Pair_a_a @ P4 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_590_conjI__realizer,axiom,
    ! [P: c > $o,P4: c,Q3: sum_su5154756782089200444_a_a_a > $o,Q: sum_su5154756782089200444_a_a_a] :
      ( ( P @ P4 )
     => ( ( Q3 @ Q )
       => ( ( P @ ( produc6784836004715151097_a_a_a @ ( produc8806894744968565733_a_a_a @ P4 @ Q ) ) )
          & ( Q3 @ ( produc1172910294343954743_a_a_a @ ( produc8806894744968565733_a_a_a @ P4 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_591_conjI__realizer,axiom,
    ! [P: b > $o,P4: b,Q3: produc1895207096998757355_a_a_a > $o,Q: produc1895207096998757355_a_a_a] :
      ( ( P @ P4 )
     => ( ( Q3 @ Q )
       => ( ( P @ ( produc5842599967860340391_a_a_a @ ( produc1730628064597884819_a_a_a @ P4 @ Q ) ) )
          & ( Q3 @ ( produc6236129573204179685_a_a_a @ ( produc1730628064597884819_a_a_a @ P4 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_592_conjI__realizer,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > $o,P4: sum_su5154756782089200444_a_a_a,Q3: produc7810058533641270553_a_a_a > $o,Q: produc7810058533641270553_a_a_a] :
      ( ( P @ P4 )
     => ( ( Q3 @ Q )
       => ( ( P @ ( produc7477144754405272026_a_a_a @ ( produc6863513614254923246_a_a_a @ P4 @ Q ) ) )
          & ( Q3 @ ( produc1629229363059081756_a_a_a @ ( produc6863513614254923246_a_a_a @ P4 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_593_conjI__realizer,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > $o,P4: sum_su5154756782089200444_a_a_a,Q3: b > $o,Q: b] :
      ( ( P @ P4 )
     => ( ( Q3 @ Q )
       => ( ( P @ ( produc4064411471574014846_a_a_b @ ( produc6086470211827429482_a_a_b @ P4 @ Q ) ) )
          & ( Q3 @ ( produc7675857798057594300_a_a_b @ ( produc6086470211827429482_a_a_b @ P4 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_594_state__separator__from__input__choices__simps_I2_J,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a,Cs: list_P7155397214832950472_a_a_b] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ! [Qq: sum_su5154756782089200444_a_a_a,X4: b] :
              ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ Qq @ X4 ) @ ( set_Pr4578018992751950051_a_a_b @ Cs ) )
             => ( ( member5679158588528110291_a_a_a @ Qq @ ( states3035385874452377268_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
                & ( member_b @ X4 @ ( inputs_a_b_c @ M ) ) ) )
         => ( ( member5679158588528110291_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q1 @ Q2 ) ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
           => ( ! [Qq: sum_su5154756782089200444_a_a_a] :
                  ( ( member5679158588528110291_a_a_a @ Qq @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
                 => ? [Q16: a,Q23: a] :
                      ( Qq
                      = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q16 @ Q23 ) ) ) )
             => ( ( states3035385874452377268_a_b_c @ ( state_3459771070710583608_a_b_c @ M @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ Q1 @ Q2 @ Cs ) )
                = ( sup_su7096409449263244574_a_a_a @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) @ ( insert5061175218292437356_a_a_a @ ( sum_In6345099547827038749od_a_a @ Q1 ) @ ( insert5061175218292437356_a_a_a @ ( sum_In6345099547827038749od_a_a @ Q2 ) @ bot_bo2676859181106700934_a_a_a ) ) ) ) ) ) ) ) ) ).

% state_separator_from_input_choices_simps(2)
thf(fact_595_shifted__transitions__targets,axiom,
    ! [T: produc884577481120224310_a_a_a,Ts: set_Pr1636130888832580182od_a_a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( fSM_sh2270962973017757008_b_c_a @ Ts ) )
     => ( sum_is4240621810064316983_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) ) ) ).

% shifted_transitions_targets
thf(fact_596_fsm__transition__input,axiom,
    ! [T: produc6800189296593107456_a_a_a,M: fsm_Pr6287166851770849218_a_a_b] :
      ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ M ) )
     => ( member5679158588528110291_a_a_a @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) @ ( inputs925126750808843768_a_a_b @ M ) ) ) ).

% fsm_transition_input
thf(fact_597_fsm__transition__input,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_Su3173892059090327343_a_b_c] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
     => ( member_b @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) @ ( inputs3694848887974842907_a_b_c @ M ) ) ) ).

% fsm_transition_input
thf(fact_598_add__transition__with__components__simps_I2_J,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,T: produc6800189296593107456_a_a_a] :
      ( ( inputs925126750808843768_a_a_b @ ( add_tr7313952166877789527_a_a_b @ M @ T ) )
      = ( insert5061175218292437356_a_a_a @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) @ ( inputs925126750808843768_a_a_b @ M ) ) ) ).

% add_transition_with_components_simps(2)
thf(fact_599_add__transition__with__components__simps_I2_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,T: produc884577481120224310_a_a_a] :
      ( ( inputs3694848887974842907_a_b_c @ ( add_tr191749633354624124_a_b_c @ M @ T ) )
      = ( insert_b @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) @ ( inputs3694848887974842907_a_b_c @ M ) ) ) ).

% add_transition_with_components_simps(2)
thf(fact_600_canonical__separator__simps_I3_J,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ( inputs3694848887974842907_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) )
          = ( inputs_a_b_c @ M ) ) ) ) ).

% canonical_separator_simps(3)
thf(fact_601_state__separator__from__input__choices__simps_I3_J,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a,Cs: list_P7155397214832950472_a_a_b] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ! [Qq: sum_su5154756782089200444_a_a_a,X4: b] :
              ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ Qq @ X4 ) @ ( set_Pr4578018992751950051_a_a_b @ Cs ) )
             => ( ( member5679158588528110291_a_a_a @ Qq @ ( states3035385874452377268_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
                & ( member_b @ X4 @ ( inputs_a_b_c @ M ) ) ) )
         => ( ( member5679158588528110291_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q1 @ Q2 ) ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
           => ( ! [Qq: sum_su5154756782089200444_a_a_a] :
                  ( ( member5679158588528110291_a_a_a @ Qq @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
                 => ? [Q16: a,Q23: a] :
                      ( Qq
                      = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q16 @ Q23 ) ) ) )
             => ( ( inputs3694848887974842907_a_b_c @ ( state_3459771070710583608_a_b_c @ M @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ Q1 @ Q2 @ Cs ) )
                = ( inputs_a_b_c @ M ) ) ) ) ) ) ) ).

% state_separator_from_input_choices_simps(3)
thf(fact_602_exE__realizer_H,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > c > $o,P4: produc1895207096998757355_a_a_a] :
      ( ( P @ ( produc1172910294343954743_a_a_a @ P4 ) @ ( produc6784836004715151097_a_a_a @ P4 ) )
     => ~ ! [X4: c,Y: sum_su5154756782089200444_a_a_a] :
            ~ ( P @ Y @ X4 ) ) ).

% exE_realizer'
thf(fact_603_exE__realizer_H,axiom,
    ! [P: produc1895207096998757355_a_a_a > b > $o,P4: produc7810058533641270553_a_a_a] :
      ( ( P @ ( produc6236129573204179685_a_a_a @ P4 ) @ ( produc5842599967860340391_a_a_a @ P4 ) )
     => ~ ! [X4: b,Y: produc1895207096998757355_a_a_a] :
            ~ ( P @ Y @ X4 ) ) ).

% exE_realizer'
thf(fact_604_exE__realizer_H,axiom,
    ! [P: produc7810058533641270553_a_a_a > sum_su5154756782089200444_a_a_a > $o,P4: produc884577481120224310_a_a_a] :
      ( ( P @ ( produc1629229363059081756_a_a_a @ P4 ) @ ( produc7477144754405272026_a_a_a @ P4 ) )
     => ~ ! [X4: sum_su5154756782089200444_a_a_a,Y: produc7810058533641270553_a_a_a] :
            ~ ( P @ Y @ X4 ) ) ).

% exE_realizer'
thf(fact_605_exE__realizer_H,axiom,
    ! [P: b > sum_su5154756782089200444_a_a_a > $o,P4: produc4770514982935793080_a_a_b] :
      ( ( P @ ( produc7675857798057594300_a_a_b @ P4 ) @ ( produc4064411471574014846_a_a_b @ P4 ) )
     => ~ ! [X4: sum_su5154756782089200444_a_a_a,Y: b] :
            ~ ( P @ Y @ X4 ) ) ).

% exE_realizer'
thf(fact_606_state__separator__from__input__choices__submachine,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a,Cs: list_P7155397214832950472_a_a_b] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ! [Qq: sum_su5154756782089200444_a_a_a,X4: b] :
              ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ Qq @ X4 ) @ ( set_Pr4578018992751950051_a_a_b @ Cs ) )
             => ( ( member5679158588528110291_a_a_a @ Qq @ ( states3035385874452377268_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
                & ( member_b @ X4 @ ( inputs_a_b_c @ M ) ) ) )
         => ( ( member5679158588528110291_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q1 @ Q2 ) ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
           => ( ! [Qq: sum_su5154756782089200444_a_a_a] :
                  ( ( member5679158588528110291_a_a_a @ Qq @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
                 => ? [Q16: a,Q23: a] :
                      ( Qq
                      = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q16 @ Q23 ) ) ) )
             => ( is_sub8484252162375176803_a_b_c @ ( state_3459771070710583608_a_b_c @ M @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ Q1 @ Q2 @ Cs ) @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ) ) ).

% state_separator_from_input_choices_submachine
thf(fact_607_state__separator__from__input__choices__simps_I1_J,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a,Cs: list_P7155397214832950472_a_a_b] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ! [Qq: sum_su5154756782089200444_a_a_a,X4: b] :
              ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ Qq @ X4 ) @ ( set_Pr4578018992751950051_a_a_b @ Cs ) )
             => ( ( member5679158588528110291_a_a_a @ Qq @ ( states3035385874452377268_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
                & ( member_b @ X4 @ ( inputs_a_b_c @ M ) ) ) )
         => ( ( member5679158588528110291_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q1 @ Q2 ) ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
           => ( ! [Qq: sum_su5154756782089200444_a_a_a] :
                  ( ( member5679158588528110291_a_a_a @ Qq @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
                 => ? [Q16: a,Q23: a] :
                      ( Qq
                      = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q16 @ Q23 ) ) ) )
             => ( ( initia839594848065427214_a_b_c @ ( state_3459771070710583608_a_b_c @ M @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ Q1 @ Q2 @ Cs ) )
                = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q1 @ Q2 ) ) ) ) ) ) ) ) ).

% state_separator_from_input_choices_simps(1)
thf(fact_608_completely__specified__state_Oelims_I3_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b,Xa: produc1895207096998757355_a_a_a] :
      ( ~ ( comple2649600785675951450_a_a_b @ X3 @ Xa )
     => ~ ! [X4: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X4 @ ( inputs925126750808843768_a_a_b @ X3 ) )
           => ? [Xa2: produc6800189296593107456_a_a_a] :
                ( ( member4643618012521571351_a_a_a @ Xa2 @ ( transi7010299678732630507_a_a_b @ X3 ) )
                & ( ( produc2316789063618113990_a_a_a @ Xa2 )
                  = Xa )
                & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) )
                  = X4 ) ) ) ) ).

% completely_specified_state.elims(3)
thf(fact_609_completely__specified__state_Oelims_I3_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Xa: sum_su5154756782089200444_a_a_a] :
      ( ~ ( comple4819012629217656633_a_b_c @ X3 @ Xa )
     => ~ ! [X4: b] :
            ( ( member_b @ X4 @ ( inputs3694848887974842907_a_b_c @ X3 ) )
           => ? [Xa2: produc884577481120224310_a_a_a] :
                ( ( member8362999488266169055_a_a_a @ Xa2 @ ( transi749694810082594024_a_b_c @ X3 ) )
                & ( ( produc7477144754405272026_a_a_a @ Xa2 )
                  = Xa )
                & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) )
                  = X4 ) ) ) ) ).

% completely_specified_state.elims(3)
thf(fact_610_distinct__take,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,I2: nat] :
      ( ( distin1372936652459515135_a_a_b @ Xs )
     => ( distin1372936652459515135_a_a_b @ ( take_P6639847873373218378_a_a_b @ I2 @ Xs ) ) ) ).

% distinct_take
thf(fact_611_distinct__conv__nth,axiom,
    ( distin1372936652459515135_a_a_b
    = ( ^ [Xs2: list_P7155397214832950472_a_a_b] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s1546398270689948508_a_a_b @ Xs2 ) )
         => ! [J: nat] :
              ( ( ord_less_nat @ J @ ( size_s1546398270689948508_a_a_b @ Xs2 ) )
             => ( ( I3 != J )
               => ( ( nth_Pr167341066936870769_a_a_b @ Xs2 @ I3 )
                 != ( nth_Pr167341066936870769_a_a_b @ Xs2 @ J ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_612_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,I2: nat,J2: nat] :
      ( ( distin1372936652459515135_a_a_b @ Xs )
     => ( ( ord_less_nat @ I2 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
       => ( ( ord_less_nat @ J2 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
         => ( ( ( nth_Pr167341066936870769_a_a_b @ Xs @ I2 )
              = ( nth_Pr167341066936870769_a_a_b @ Xs @ J2 ) )
            = ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_613_non__distinct__repetition__indices,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b] :
      ( ~ ( distin1372936652459515135_a_a_b @ Xs )
     => ? [I: nat,J3: nat] :
          ( ( ord_less_nat @ I @ J3 )
          & ( ord_less_nat @ J3 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
          & ( ( nth_Pr167341066936870769_a_a_b @ Xs @ I )
            = ( nth_Pr167341066936870769_a_a_b @ Xs @ J3 ) ) ) ) ).

% non_distinct_repetition_indices
thf(fact_614_non__distinct__repetition__indices__rev,axiom,
    ! [I2: nat,J2: nat,Xs: list_P7155397214832950472_a_a_b] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ J2 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
       => ( ( ( nth_Pr167341066936870769_a_a_b @ Xs @ I2 )
            = ( nth_Pr167341066936870769_a_a_b @ Xs @ J2 ) )
         => ~ ( distin1372936652459515135_a_a_b @ Xs ) ) ) ) ).

% non_distinct_repetition_indices_rev
thf(fact_615_distinct__not__in__prefix,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a] :
      ( ! [I: nat,X4: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X4 @ ( set_Pr8067384617587748683_a_a_a @ ( take_P6848719478162124068_a_a_a @ I @ Xs ) ) )
         => ( ( nth_Pr7648848803501811517_a_a_a @ Xs @ I )
           != X4 ) )
     => ( distin5648706335258269743_a_a_a @ Xs ) ) ).

% distinct_not_in_prefix
thf(fact_616_distinct__not__in__prefix,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a] :
      ( ! [I: nat,X4: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X4 @ ( set_Su3113867532901839207_a_a_a @ ( take_S1842584226280693198_a_a_a @ I @ Xs ) ) )
         => ( ( nth_Su7097239896299986421_a_a_a @ Xs @ I )
           != X4 ) )
     => ( distin5156058890512421251_a_a_a @ Xs ) ) ).

% distinct_not_in_prefix
thf(fact_617_distinct__not__in__prefix,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b] :
      ( ! [I: nat,X4: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X4 @ ( set_Pr4578018992751950051_a_a_b @ ( take_P6639847873373218378_a_a_b @ I @ Xs ) ) )
         => ( ( nth_Pr167341066936870769_a_a_b @ Xs @ I )
           != X4 ) )
     => ( distin1372936652459515135_a_a_b @ Xs ) ) ).

% distinct_not_in_prefix
thf(fact_618_distinct__Ex1,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,X3: produc884577481120224310_a_a_a] :
      ( ( distin5648706335258269743_a_a_a @ Xs )
     => ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s4783218027424981928_a_a_a @ Xs ) )
            & ( ( nth_Pr7648848803501811517_a_a_a @ Xs @ X4 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s4783218027424981928_a_a_a @ Xs ) )
                  & ( ( nth_Pr7648848803501811517_a_a_a @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_619_distinct__Ex1,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,X3: sum_su5154756782089200444_a_a_a] :
      ( ( distin5156058890512421251_a_a_a @ Xs )
     => ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s7186666008910096352_a_a_a @ Xs ) )
            & ( ( nth_Su7097239896299986421_a_a_a @ Xs @ X4 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s7186666008910096352_a_a_a @ Xs ) )
                  & ( ( nth_Su7097239896299986421_a_a_a @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_620_distinct__Ex1,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,X3: produc4770514982935793080_a_a_b] :
      ( ( distin1372936652459515135_a_a_b @ Xs )
     => ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
            & ( ( nth_Pr167341066936870769_a_a_b @ Xs @ X4 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
                  & ( ( nth_Pr167341066936870769_a_a_b @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_621_canonical__separator__initial,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) )
          = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q1 @ Q2 ) ) ) ) ) ).

% canonical_separator_initial
thf(fact_622_list__distinct__prefix,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s4783218027424981928_a_a_a @ Xs ) )
         => ~ ( member8362999488266169055_a_a_a @ ( nth_Pr7648848803501811517_a_a_a @ Xs @ I ) @ ( set_Pr8067384617587748683_a_a_a @ ( take_P6848719478162124068_a_a_a @ I @ Xs ) ) ) )
     => ( distin5648706335258269743_a_a_a @ Xs ) ) ).

% list_distinct_prefix
thf(fact_623_list__distinct__prefix,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7186666008910096352_a_a_a @ Xs ) )
         => ~ ( member5679158588528110291_a_a_a @ ( nth_Su7097239896299986421_a_a_a @ Xs @ I ) @ ( set_Su3113867532901839207_a_a_a @ ( take_S1842584226280693198_a_a_a @ I @ Xs ) ) ) )
     => ( distin5156058890512421251_a_a_a @ Xs ) ) ).

% list_distinct_prefix
thf(fact_624_list__distinct__prefix,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s1546398270689948508_a_a_b @ Xs ) )
         => ~ ( member642552543386773071_a_a_b @ ( nth_Pr167341066936870769_a_a_b @ Xs @ I ) @ ( set_Pr4578018992751950051_a_a_b @ ( take_P6639847873373218378_a_a_b @ I @ Xs ) ) ) )
     => ( distin1372936652459515135_a_a_b @ Xs ) ) ).

% list_distinct_prefix
thf(fact_625_completely__specified__alt__def,axiom,
    ( comple3605256123784756372_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c] :
        ! [X: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X @ ( states3035385874452377268_a_b_c @ M3 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs3694848887974842907_a_b_c @ M3 ) )
             => ? [Q4: sum_su5154756782089200444_a_a_a,Z4: c] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X @ ( produc1730628064597884819_a_a_a @ Y4 @ ( produc8806894744968565733_a_a_a @ Z4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) ) ) ) ) ) ).

% completely_specified_alt_def
thf(fact_626_completely__specified_Osimps,axiom,
    ( comple3249000580571322175_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b] :
        ! [X: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ X @ ( states8463590192175095071_a_a_b @ M3 ) )
         => ! [Y4: sum_su5154756782089200444_a_a_a] :
              ( ( member5679158588528110291_a_a_a @ Y4 @ ( inputs925126750808843768_a_a_b @ M3 ) )
             => ? [Z4: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ Z4 @ ( transi7010299678732630507_a_a_b @ M3 ) )
                  & ( ( produc2316789063618113990_a_a_a @ Z4 )
                    = X )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Z4 ) )
                    = Y4 ) ) ) ) ) ) ).

% completely_specified.simps
thf(fact_627_completely__specified_Osimps,axiom,
    ( comple3605256123784756372_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c] :
        ! [X: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X @ ( states3035385874452377268_a_b_c @ M3 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs3694848887974842907_a_b_c @ M3 ) )
             => ? [Z4: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ Z4 @ ( transi749694810082594024_a_b_c @ M3 ) )
                  & ( ( produc7477144754405272026_a_a_a @ Z4 )
                    = X )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Z4 ) )
                    = Y4 ) ) ) ) ) ) ).

% completely_specified.simps
thf(fact_628_completely__specified_Oelims_I1_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b,Y3: $o] :
      ( ( ( comple3249000580571322175_a_a_b @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X: produc1895207096998757355_a_a_a] :
              ( ( member6990616694304513154_a_a_a @ X @ ( states8463590192175095071_a_a_b @ X3 ) )
             => ! [Y4: sum_su5154756782089200444_a_a_a] :
                  ( ( member5679158588528110291_a_a_a @ Y4 @ ( inputs925126750808843768_a_a_b @ X3 ) )
                 => ? [Z4: produc6800189296593107456_a_a_a] :
                      ( ( member4643618012521571351_a_a_a @ Z4 @ ( transi7010299678732630507_a_a_b @ X3 ) )
                      & ( ( produc2316789063618113990_a_a_a @ Z4 )
                        = X )
                      & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Z4 ) )
                        = Y4 ) ) ) ) ) ) ) ).

% completely_specified.elims(1)
thf(fact_629_completely__specified_Oelims_I1_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Y3: $o] :
      ( ( ( comple3605256123784756372_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X: sum_su5154756782089200444_a_a_a] :
              ( ( member5679158588528110291_a_a_a @ X @ ( states3035385874452377268_a_b_c @ X3 ) )
             => ! [Y4: b] :
                  ( ( member_b @ Y4 @ ( inputs3694848887974842907_a_b_c @ X3 ) )
                 => ? [Z4: produc884577481120224310_a_a_a] :
                      ( ( member8362999488266169055_a_a_a @ Z4 @ ( transi749694810082594024_a_b_c @ X3 ) )
                      & ( ( produc7477144754405272026_a_a_a @ Z4 )
                        = X )
                      & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Z4 ) )
                        = Y4 ) ) ) ) ) ) ) ).

% completely_specified.elims(1)
thf(fact_630_completely__specified_Oelims_I2_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b] :
      ( ( comple3249000580571322175_a_a_b @ X3 )
     => ! [X7: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ X7 @ ( states8463590192175095071_a_a_b @ X3 ) )
         => ! [Xa2: sum_su5154756782089200444_a_a_a] :
              ( ( member5679158588528110291_a_a_a @ Xa2 @ ( inputs925126750808843768_a_a_b @ X3 ) )
             => ? [Xb: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ Xb @ ( transi7010299678732630507_a_a_b @ X3 ) )
                  & ( ( produc2316789063618113990_a_a_a @ Xb )
                    = X7 )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xb ) )
                    = Xa2 ) ) ) ) ) ).

% completely_specified.elims(2)
thf(fact_631_completely__specified_Oelims_I2_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c] :
      ( ( comple3605256123784756372_a_b_c @ X3 )
     => ! [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ ( states3035385874452377268_a_b_c @ X3 ) )
         => ! [Xa2: b] :
              ( ( member_b @ Xa2 @ ( inputs3694848887974842907_a_b_c @ X3 ) )
             => ? [Xb: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ Xb @ ( transi749694810082594024_a_b_c @ X3 ) )
                  & ( ( produc7477144754405272026_a_a_a @ Xb )
                    = X7 )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xb ) )
                    = Xa2 ) ) ) ) ) ).

% completely_specified.elims(2)
thf(fact_632_completely__specified_Oelims_I3_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b] :
      ( ~ ( comple3249000580571322175_a_a_b @ X3 )
     => ~ ! [X4: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ X4 @ ( states8463590192175095071_a_a_b @ X3 ) )
           => ! [Xa3: sum_su5154756782089200444_a_a_a] :
                ( ( member5679158588528110291_a_a_a @ Xa3 @ ( inputs925126750808843768_a_a_b @ X3 ) )
               => ? [Xb2: produc6800189296593107456_a_a_a] :
                    ( ( member4643618012521571351_a_a_a @ Xb2 @ ( transi7010299678732630507_a_a_b @ X3 ) )
                    & ( ( produc2316789063618113990_a_a_a @ Xb2 )
                      = X4 )
                    & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xb2 ) )
                      = Xa3 ) ) ) ) ) ).

% completely_specified.elims(3)
thf(fact_633_completely__specified_Oelims_I3_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c] :
      ( ~ ( comple3605256123784756372_a_b_c @ X3 )
     => ~ ! [X4: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X4 @ ( states3035385874452377268_a_b_c @ X3 ) )
           => ! [Xa3: b] :
                ( ( member_b @ Xa3 @ ( inputs3694848887974842907_a_b_c @ X3 ) )
               => ? [Xb2: produc884577481120224310_a_a_a] :
                    ( ( member8362999488266169055_a_a_a @ Xb2 @ ( transi749694810082594024_a_b_c @ X3 ) )
                    & ( ( produc7477144754405272026_a_a_a @ Xb2 )
                      = X4 )
                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xb2 ) )
                      = Xa3 ) ) ) ) ) ).

% completely_specified.elims(3)
thf(fact_634_completely__specified__state_Osimps,axiom,
    ( comple2649600785675951450_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b,Q5: produc1895207096998757355_a_a_a] :
        ! [X: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X @ ( inputs925126750808843768_a_a_b @ M3 ) )
         => ? [Y4: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ M3 ) )
              & ( ( produc2316789063618113990_a_a_a @ Y4 )
                = Q5 )
              & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) )
                = X ) ) ) ) ) ).

% completely_specified_state.simps
thf(fact_635_completely__specified__state_Osimps,axiom,
    ( comple4819012629217656633_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c,Q5: sum_su5154756782089200444_a_a_a] :
        ! [X: b] :
          ( ( member_b @ X @ ( inputs3694848887974842907_a_b_c @ M3 ) )
         => ? [Y4: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ M3 ) )
              & ( ( produc7477144754405272026_a_a_a @ Y4 )
                = Q5 )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) )
                = X ) ) ) ) ) ).

% completely_specified_state.simps
thf(fact_636_completely__specified__state_Oelims_I1_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b,Xa: produc1895207096998757355_a_a_a,Y3: $o] :
      ( ( ( comple2649600785675951450_a_a_b @ X3 @ Xa )
        = Y3 )
     => ( Y3
        = ( ! [X: sum_su5154756782089200444_a_a_a] :
              ( ( member5679158588528110291_a_a_a @ X @ ( inputs925126750808843768_a_a_b @ X3 ) )
             => ? [Y4: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ X3 ) )
                  & ( ( produc2316789063618113990_a_a_a @ Y4 )
                    = Xa )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) )
                    = X ) ) ) ) ) ) ).

% completely_specified_state.elims(1)
thf(fact_637_completely__specified__state_Oelims_I1_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Xa: sum_su5154756782089200444_a_a_a,Y3: $o] :
      ( ( ( comple4819012629217656633_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( Y3
        = ( ! [X: b] :
              ( ( member_b @ X @ ( inputs3694848887974842907_a_b_c @ X3 ) )
             => ? [Y4: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ X3 ) )
                  & ( ( produc7477144754405272026_a_a_a @ Y4 )
                    = Xa )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) )
                    = X ) ) ) ) ) ) ).

% completely_specified_state.elims(1)
thf(fact_638_completely__specified__state_Oelims_I2_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b,Xa: produc1895207096998757355_a_a_a] :
      ( ( comple2649600785675951450_a_a_b @ X3 @ Xa )
     => ! [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ ( inputs925126750808843768_a_a_b @ X3 ) )
         => ? [Xa3: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ Xa3 @ ( transi7010299678732630507_a_a_b @ X3 ) )
              & ( ( produc2316789063618113990_a_a_a @ Xa3 )
                = Xa )
              & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa3 ) )
                = X7 ) ) ) ) ).

% completely_specified_state.elims(2)
thf(fact_639_completely__specified__state_Oelims_I2_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Xa: sum_su5154756782089200444_a_a_a] :
      ( ( comple4819012629217656633_a_b_c @ X3 @ Xa )
     => ! [X7: b] :
          ( ( member_b @ X7 @ ( inputs3694848887974842907_a_b_c @ X3 ) )
         => ? [Xa3: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ Xa3 @ ( transi749694810082594024_a_b_c @ X3 ) )
              & ( ( produc7477144754405272026_a_a_a @ Xa3 )
                = Xa )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa3 ) )
                = X7 ) ) ) ) ).

% completely_specified_state.elims(2)
thf(fact_640_state__separator__from__input__choices__single__input,axiom,
    ! [Cs: list_P7155397214832950472_a_a_b,Q1: a,M: fsm_a_b_c,Q2: a] :
      ( ( distin5156058890512421251_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ! [Qq: sum_su5154756782089200444_a_a_a,X4: b] :
                ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ Qq @ X4 ) @ ( set_Pr4578018992751950051_a_a_b @ Cs ) )
               => ( ( member5679158588528110291_a_a_a @ Qq @ ( states3035385874452377268_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
                  & ( member_b @ X4 @ ( inputs_a_b_c @ M ) ) ) )
           => ( ( member5679158588528110291_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q1 @ Q2 ) ) @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
             => ( ! [Qq: sum_su5154756782089200444_a_a_a] :
                    ( ( member5679158588528110291_a_a_a @ Qq @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Cs ) ) )
                   => ? [Q16: a,Q23: a] :
                        ( Qq
                        = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q16 @ Q23 ) ) ) )
               => ( single6641483607941472940_a_b_c @ ( state_3459771070710583608_a_b_c @ M @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ Q1 @ Q2 @ Cs ) ) ) ) ) ) ) ) ).

% state_separator_from_input_choices_single_input
thf(fact_641_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,K: a,V1: a,V2: a] :
      ( ( distinct_a @ ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a @ Xs ) )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ K @ V1 ) @ ( set_Product_prod_a_a2 @ Xs ) )
       => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ K @ V2 ) @ ( set_Product_prod_a_a2 @ Xs ) )
         => ( V1 = V2 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_642_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,K: sum_su5154756782089200444_a_a_a,V1: b,V2: b] :
      ( ( distin5156058890512421251_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Xs ) )
     => ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ K @ V1 ) @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
       => ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ K @ V2 ) @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
         => ( V1 = V2 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_643_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P7200181061086074537_a_a_a,K: b,V1: produc1895207096998757355_a_a_a,V2: produc1895207096998757355_a_a_a] :
      ( ( distinct_b @ ( map_Pr3622043693041699552_a_a_b @ produc5842599967860340391_a_a_a @ Xs ) )
     => ( ( member54949653122719408_a_a_a @ ( produc1730628064597884819_a_a_a @ K @ V1 ) @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
       => ( ( member54949653122719408_a_a_a @ ( produc1730628064597884819_a_a_a @ K @ V2 ) @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
         => ( V1 = V2 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_644_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,K: sum_su5154756782089200444_a_a_a,V1: produc7810058533641270553_a_a_a,V2: produc7810058533641270553_a_a_a] :
      ( ( distin5156058890512421251_a_a_a @ ( map_Pr1409128909316453038_a_a_a @ produc7477144754405272026_a_a_a @ Xs ) )
     => ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ K @ V1 ) @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
       => ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ K @ V2 ) @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
         => ( V1 = V2 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_645_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P1813520604369389947_a_a_a,K: c,V1: sum_su5154756782089200444_a_a_a,V2: sum_su5154756782089200444_a_a_a] :
      ( ( distinct_c @ ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a @ Xs ) )
     => ( ( member6990616694304513154_a_a_a @ ( produc8806894744968565733_a_a_a @ K @ V1 ) @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
       => ( ( member6990616694304513154_a_a_a @ ( produc8806894744968565733_a_a_a @ K @ V2 ) @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
         => ( V1 = V2 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_646_completely__specified__io__targets,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Io: list_P7155397214832950472_a_a_b] :
      ( ( comple3249000580571322175_a_a_b @ M )
     => ! [X7: produc1895207096998757355_a_a_a] :
          ( ( member6990616694304513154_a_a_a @ X7 @ ( io_tar1879755107597495661_a_a_b @ M @ Io @ ( initia5013972905137889861_a_a_b @ M ) ) )
         => ! [Xa2: sum_su5154756782089200444_a_a_a] :
              ( ( member5679158588528110291_a_a_a @ Xa2 @ ( inputs925126750808843768_a_a_b @ M ) )
             => ? [Xb: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ Xb @ ( transi7010299678732630507_a_a_b @ M ) )
                  & ( ( produc2316789063618113990_a_a_a @ Xb )
                    = X7 )
                  & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xb ) )
                    = Xa2 ) ) ) ) ) ).

% completely_specified_io_targets
thf(fact_647_completely__specified__io__targets,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( comple3605256123784756372_a_b_c @ M )
     => ! [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ ( io_tar1727080856364058086_a_b_c @ M @ Io @ ( initia839594848065427214_a_b_c @ M ) ) )
         => ! [Xa2: b] :
              ( ( member_b @ Xa2 @ ( inputs3694848887974842907_a_b_c @ M ) )
             => ? [Xb: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ Xb @ ( transi749694810082594024_a_b_c @ M ) )
                  & ( ( produc7477144754405272026_a_a_a @ Xb )
                    = X7 )
                  & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xb ) )
                    = Xa2 ) ) ) ) ) ).

% completely_specified_io_targets
thf(fact_648_non__sym__dist__pairs_H__indices,axiom,
    ! [Xs: list_a,X3: a,Y3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
       => ? [I: nat,J3: nat] :
            ( ( ( nth_a @ Xs @ I )
              = X3 )
            & ( ( nth_a @ Xs @ J3 )
              = Y3 )
            & ( ord_less_nat @ I @ J3 )
            & ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
            & ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs'_indices
thf(fact_649_non__sym__dist__pairs_H__indices,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,X3: produc4770514982935793080_a_a_b,Y3: produc4770514982935793080_a_a_b] :
      ( ( distin1372936652459515135_a_a_b @ Xs )
     => ( ( member3114646429565181818_a_a_b @ ( produc5271149956529894921_a_a_b @ X3 @ Y3 ) @ ( set_Pr2819031558886761446_a_a_b @ ( non_sy5074223863352749421_a_a_b @ Xs ) ) )
       => ? [I: nat,J3: nat] :
            ( ( ( nth_Pr167341066936870769_a_a_b @ Xs @ I )
              = X3 )
            & ( ( nth_Pr167341066936870769_a_a_b @ Xs @ J3 )
              = Y3 )
            & ( ord_less_nat @ I @ J3 )
            & ( ord_less_nat @ I @ ( size_s1546398270689948508_a_a_b @ Xs ) )
            & ( ord_less_nat @ J3 @ ( size_s1546398270689948508_a_a_b @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs'_indices
thf(fact_650_map__is__Nil__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr7354669405448068282_a_a_a @ F @ Xs )
        = nil_Su3335689033985374134_a_a_a )
      = ( Xs = nil_Pr4580708590200510514_a_a_b ) ) ).

% map_is_Nil_conv
thf(fact_651_map__is__Nil__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = nil_b )
      = ( Xs = nil_Pr4580708590200510514_a_a_b ) ) ).

% map_is_Nil_conv
thf(fact_652_map__is__Nil__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = nil_Su3335689033985374134_a_a_a )
      = ( Xs = nil_Pr1705400704263474789_a_a_a ) ) ).

% map_is_Nil_conv
thf(fact_653_map__is__Nil__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = nil_c )
      = ( Xs = nil_Pr1705400704263474789_a_a_a ) ) ).

% map_is_Nil_conv
thf(fact_654_map__is__Nil__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a] :
      ( ( ( map_Pr3321722608670633418_a_a_a @ F @ Xs )
        = nil_Pr1705400704263474789_a_a_a )
      = ( Xs = nil_Pr1310111084261799315_a_a_a ) ) ).

% map_is_Nil_conv
thf(fact_655_map__is__Nil__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = nil_b )
      = ( Xs = nil_Pr1310111084261799315_a_a_a ) ) ).

% map_is_Nil_conv
thf(fact_656_Nil__is__map__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b] :
      ( ( nil_Su3335689033985374134_a_a_a
        = ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) )
      = ( Xs = nil_Pr4580708590200510514_a_a_b ) ) ).

% Nil_is_map_conv
thf(fact_657_Nil__is__map__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( nil_b
        = ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) )
      = ( Xs = nil_Pr4580708590200510514_a_a_b ) ) ).

% Nil_is_map_conv
thf(fact_658_Nil__is__map__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a] :
      ( ( nil_Su3335689033985374134_a_a_a
        = ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) )
      = ( Xs = nil_Pr1705400704263474789_a_a_a ) ) ).

% Nil_is_map_conv
thf(fact_659_Nil__is__map__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a] :
      ( ( nil_c
        = ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) )
      = ( Xs = nil_Pr1705400704263474789_a_a_a ) ) ).

% Nil_is_map_conv
thf(fact_660_Nil__is__map__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a] :
      ( ( nil_Pr1705400704263474789_a_a_a
        = ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) )
      = ( Xs = nil_Pr1310111084261799315_a_a_a ) ) ).

% Nil_is_map_conv
thf(fact_661_Nil__is__map__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a] :
      ( ( nil_b
        = ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) )
      = ( Xs = nil_Pr1310111084261799315_a_a_a ) ) ).

% Nil_is_map_conv
thf(fact_662_list_Omap__disc__iff,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,A: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr7354669405448068282_a_a_a @ F @ A )
        = nil_Su3335689033985374134_a_a_a )
      = ( A = nil_Pr4580708590200510514_a_a_b ) ) ).

% list.map_disc_iff
thf(fact_663_list_Omap__disc__iff,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,A: list_P7155397214832950472_a_a_b] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ A )
        = nil_b )
      = ( A = nil_Pr4580708590200510514_a_a_b ) ) ).

% list.map_disc_iff
thf(fact_664_list_Omap__disc__iff,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,A: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ A )
        = nil_Su3335689033985374134_a_a_a )
      = ( A = nil_Pr1705400704263474789_a_a_a ) ) ).

% list.map_disc_iff
thf(fact_665_list_Omap__disc__iff,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,A: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ A )
        = nil_c )
      = ( A = nil_Pr1705400704263474789_a_a_a ) ) ).

% list.map_disc_iff
thf(fact_666_list_Omap__disc__iff,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,A: list_P7200181061086074537_a_a_a] :
      ( ( ( map_Pr3321722608670633418_a_a_a @ F @ A )
        = nil_Pr1705400704263474789_a_a_a )
      = ( A = nil_Pr1310111084261799315_a_a_a ) ) ).

% list.map_disc_iff
thf(fact_667_list_Omap__disc__iff,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,A: list_P7200181061086074537_a_a_a] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ A )
        = nil_b )
      = ( A = nil_Pr1310111084261799315_a_a_a ) ) ).

% list.map_disc_iff
thf(fact_668_set__empty2,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a] :
      ( ( bot_bo2676859181106700934_a_a_a
        = ( set_Su3113867532901839207_a_a_a @ Xs ) )
      = ( Xs = nil_Su3335689033985374134_a_a_a ) ) ).

% set_empty2
thf(fact_669_set__empty2,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b] :
      ( ( bot_bo5052542052834138114_a_a_b
        = ( set_Pr4578018992751950051_a_a_b @ Xs ) )
      = ( Xs = nil_Pr4580708590200510514_a_a_b ) ) ).

% set_empty2
thf(fact_670_set__empty2,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a] :
      ( ( bot_bo2898616998271641474_a_a_a
        = ( set_Pr8067384617587748683_a_a_a @ Xs ) )
      = ( Xs = nil_Pr6029279252097497404_a_a_a ) ) ).

% set_empty2
thf(fact_671_set__empty,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a] :
      ( ( ( set_Su3113867532901839207_a_a_a @ Xs )
        = bot_bo2676859181106700934_a_a_a )
      = ( Xs = nil_Su3335689033985374134_a_a_a ) ) ).

% set_empty
thf(fact_672_set__empty,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b] :
      ( ( ( set_Pr4578018992751950051_a_a_b @ Xs )
        = bot_bo5052542052834138114_a_a_b )
      = ( Xs = nil_Pr4580708590200510514_a_a_b ) ) ).

% set_empty
thf(fact_673_set__empty,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a] :
      ( ( ( set_Pr8067384617587748683_a_a_a @ Xs )
        = bot_bo2898616998271641474_a_a_a )
      = ( Xs = nil_Pr6029279252097497404_a_a_a ) ) ).

% set_empty
thf(fact_674_list_Osimps_I8_J,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ( map_Pr7354669405448068282_a_a_a @ F @ nil_Pr4580708590200510514_a_a_b )
      = nil_Su3335689033985374134_a_a_a ) ).

% list.simps(8)
thf(fact_675_list_Osimps_I8_J,axiom,
    ! [F: produc4770514982935793080_a_a_b > b] :
      ( ( map_Pr8925698157833054977_a_b_b @ F @ nil_Pr4580708590200510514_a_a_b )
      = nil_b ) ).

% list.simps(8)
thf(fact_676_list_Osimps_I8_J,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( map_Pr1074589380674644461_a_a_a @ F @ nil_Pr1705400704263474789_a_a_a )
      = nil_Su3335689033985374134_a_a_a ) ).

% list.simps(8)
thf(fact_677_list_Osimps_I8_J,axiom,
    ! [F: produc1895207096998757355_a_a_a > c] :
      ( ( map_Pr2287363492687751055_a_a_c @ F @ nil_Pr1705400704263474789_a_a_a )
      = nil_c ) ).

% list.simps(8)
thf(fact_678_list_Osimps_I8_J,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ( map_Pr3321722608670633418_a_a_a @ F @ nil_Pr1310111084261799315_a_a_a )
      = nil_Pr1705400704263474789_a_a_a ) ).

% list.simps(8)
thf(fact_679_list_Osimps_I8_J,axiom,
    ! [F: produc7810058533641270553_a_a_a > b] :
      ( ( map_Pr3622043693041699552_a_a_b @ F @ nil_Pr1310111084261799315_a_a_a )
      = nil_b ) ).

% list.simps(8)
thf(fact_680_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_P6639847873373218378_a_a_b @ N @ nil_Pr4580708590200510514_a_a_b )
      = nil_Pr4580708590200510514_a_a_b ) ).

% take_Nil
thf(fact_681_empty__set,axiom,
    ( bot_bo2676859181106700934_a_a_a
    = ( set_Su3113867532901839207_a_a_a @ nil_Su3335689033985374134_a_a_a ) ) ).

% empty_set
thf(fact_682_empty__set,axiom,
    ( bot_bo5052542052834138114_a_a_b
    = ( set_Pr4578018992751950051_a_a_b @ nil_Pr4580708590200510514_a_a_b ) ) ).

% empty_set
thf(fact_683_empty__set,axiom,
    ( bot_bo2898616998271641474_a_a_a
    = ( set_Pr8067384617587748683_a_a_a @ nil_Pr6029279252097497404_a_a_a ) ) ).

% empty_set
thf(fact_684_cycle__incoming__transition__ex,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,T: produc884577481120224310_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
     => ( ( P4 != nil_Pr6029279252097497404_a_a_a )
       => ( ( ( target5023745622805013891_a_b_c @ Q @ P4 )
            = Q )
         => ( ( member8362999488266169055_a_a_a @ T @ ( set_Pr8067384617587748683_a_a_a @ P4 ) )
           => ? [X4: produc884577481120224310_a_a_a] :
                ( ( member8362999488266169055_a_a_a @ X4 @ ( set_Pr8067384617587748683_a_a_a @ P4 ) )
                & ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                  = ( produc7477144754405272026_a_a_a @ T ) ) ) ) ) ) ) ).

% cycle_incoming_transition_ex
thf(fact_685_cycle__incoming__transition__ex,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a,P4: list_P3520633289758869904_a_a_a,T: produc6800189296593107456_a_a_a] :
      ( ( path_P4574188132515142236_a_a_b @ M @ Q @ P4 )
     => ( ( P4 != nil_Pr3590249381822989690_a_a_a )
       => ( ( ( target4721681511209552016_a_a_b @ Q @ P4 )
            = Q )
         => ( ( member4643618012521571351_a_a_a @ T @ ( set_Pr3535892682048719019_a_a_a @ P4 ) )
           => ? [X4: produc6800189296593107456_a_a_a] :
                ( ( member4643618012521571351_a_a_a @ X4 @ ( set_Pr3535892682048719019_a_a_a @ P4 ) )
                & ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                  = ( produc2316789063618113990_a_a_a @ T ) ) ) ) ) ) ) ).

% cycle_incoming_transition_ex
thf(fact_686_cyclic__path__transition__states__property_I1_J,axiom,
    ! [P4: list_P8197596098214363964_a_a_a,P: sum_su5154756782089200444_a_a_a > $o,M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a] :
      ( ? [X7: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X7 @ ( set_Pr8067384617587748683_a_a_a @ P4 ) )
          & ( P @ ( produc7477144754405272026_a_a_a @ X7 ) ) )
     => ( ! [X4: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X4 @ ( set_Pr8067384617587748683_a_a_a @ P4 ) )
           => ( ( P @ ( produc7477144754405272026_a_a_a @ X4 ) )
             => ( P @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) ) ) ) )
       => ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
         => ( ( ( target5023745622805013891_a_b_c @ Q @ P4 )
              = Q )
           => ! [X7: produc884577481120224310_a_a_a] :
                ( ( member8362999488266169055_a_a_a @ X7 @ ( set_Pr8067384617587748683_a_a_a @ P4 ) )
               => ( P @ ( produc7477144754405272026_a_a_a @ X7 ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(1)
thf(fact_687_cyclic__path__transition__states__property_I1_J,axiom,
    ! [P4: list_P3520633289758869904_a_a_a,P: produc1895207096998757355_a_a_a > $o,M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a] :
      ( ? [X7: produc6800189296593107456_a_a_a] :
          ( ( member4643618012521571351_a_a_a @ X7 @ ( set_Pr3535892682048719019_a_a_a @ P4 ) )
          & ( P @ ( produc2316789063618113990_a_a_a @ X7 ) ) )
     => ( ! [X4: produc6800189296593107456_a_a_a] :
            ( ( member4643618012521571351_a_a_a @ X4 @ ( set_Pr3535892682048719019_a_a_a @ P4 ) )
           => ( ( P @ ( produc2316789063618113990_a_a_a @ X4 ) )
             => ( P @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) ) ) ) )
       => ( ( path_P4574188132515142236_a_a_b @ M @ Q @ P4 )
         => ( ( ( target4721681511209552016_a_a_b @ Q @ P4 )
              = Q )
           => ! [X7: produc6800189296593107456_a_a_a] :
                ( ( member4643618012521571351_a_a_a @ X7 @ ( set_Pr3535892682048719019_a_a_a @ P4 ) )
               => ( P @ ( produc2316789063618113990_a_a_a @ X7 ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(1)
thf(fact_688_cyclic__path__transition__states__property_I2_J,axiom,
    ! [P4: list_P8197596098214363964_a_a_a,P: sum_su5154756782089200444_a_a_a > $o,M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a] :
      ( ? [X7: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X7 @ ( set_Pr8067384617587748683_a_a_a @ P4 ) )
          & ( P @ ( produc7477144754405272026_a_a_a @ X7 ) ) )
     => ( ! [X4: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X4 @ ( set_Pr8067384617587748683_a_a_a @ P4 ) )
           => ( ( P @ ( produc7477144754405272026_a_a_a @ X4 ) )
             => ( P @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) ) ) ) )
       => ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
         => ( ( ( target5023745622805013891_a_b_c @ Q @ P4 )
              = Q )
           => ! [X7: produc884577481120224310_a_a_a] :
                ( ( member8362999488266169055_a_a_a @ X7 @ ( set_Pr8067384617587748683_a_a_a @ P4 ) )
               => ( P @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(2)
thf(fact_689_cyclic__path__transition__states__property_I2_J,axiom,
    ! [P4: list_P3520633289758869904_a_a_a,P: produc1895207096998757355_a_a_a > $o,M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a] :
      ( ? [X7: produc6800189296593107456_a_a_a] :
          ( ( member4643618012521571351_a_a_a @ X7 @ ( set_Pr3535892682048719019_a_a_a @ P4 ) )
          & ( P @ ( produc2316789063618113990_a_a_a @ X7 ) ) )
     => ( ! [X4: produc6800189296593107456_a_a_a] :
            ( ( member4643618012521571351_a_a_a @ X4 @ ( set_Pr3535892682048719019_a_a_a @ P4 ) )
           => ( ( P @ ( produc2316789063618113990_a_a_a @ X4 ) )
             => ( P @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) ) ) ) )
       => ( ( path_P4574188132515142236_a_a_b @ M @ Q @ P4 )
         => ( ( ( target4721681511209552016_a_a_b @ Q @ P4 )
              = Q )
           => ! [X7: produc6800189296593107456_a_a_a] :
                ( ( member4643618012521571351_a_a_a @ X7 @ ( set_Pr3535892682048719019_a_a_a @ P4 ) )
               => ( P @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) ) ) ) ) ) ) ) ).

% cyclic_path_transition_states_property(2)
thf(fact_690_non__sym__dist__pairs_H__elems__non__refl,axiom,
    ! [Xs: list_a,X3: a,Y3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) ) ) ) ).

% non_sym_dist_pairs'_elems_non_refl
thf(fact_691_non__sym__dist__pairs_H__elems__distinct_I3_J,axiom,
    ! [Xs: list_a,X3: a,Y3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
       => ( X3 != Y3 ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(3)
thf(fact_692_non__sym__dist__pairs_H__elems__distinct_I2_J,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,X3: produc884577481120224310_a_a_a,Y3: produc884577481120224310_a_a_a] :
      ( ( distin5648706335258269743_a_a_a @ Xs )
     => ( ( member4115638175673106640_a_a_a @ ( produc6861895844393288855_a_a_a @ X3 @ Y3 ) @ ( set_Pr3788371281935070012_a_a_a @ ( non_sy8288883172567897665_a_a_a @ Xs ) ) )
       => ( member8362999488266169055_a_a_a @ Y3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(2)
thf(fact_693_non__sym__dist__pairs_H__elems__distinct_I2_J,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,X3: sum_su5154756782089200444_a_a_a,Y3: sum_su5154756782089200444_a_a_a] :
      ( ( distin5156058890512421251_a_a_a @ Xs )
     => ( ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ X3 @ Y3 ) @ ( set_Pr6783383005027252718_a_a_a @ ( non_sy6705267804935307249_a_a_a @ Xs ) ) )
       => ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(2)
thf(fact_694_non__sym__dist__pairs_H__elems__distinct_I2_J,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,X3: produc4770514982935793080_a_a_b,Y3: produc4770514982935793080_a_a_b] :
      ( ( distin1372936652459515135_a_a_b @ Xs )
     => ( ( member3114646429565181818_a_a_b @ ( produc5271149956529894921_a_a_b @ X3 @ Y3 ) @ ( set_Pr2819031558886761446_a_a_b @ ( non_sy5074223863352749421_a_a_b @ Xs ) ) )
       => ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(2)
thf(fact_695_non__sym__dist__pairs_H__elems__distinct_I2_J,axiom,
    ! [Xs: list_a,X3: a,Y3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
       => ( member_a @ Y3 @ ( set_a2 @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(2)
thf(fact_696_non__sym__dist__pairs_H__elems__distinct_I1_J,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,X3: produc884577481120224310_a_a_a,Y3: produc884577481120224310_a_a_a] :
      ( ( distin5648706335258269743_a_a_a @ Xs )
     => ( ( member4115638175673106640_a_a_a @ ( produc6861895844393288855_a_a_a @ X3 @ Y3 ) @ ( set_Pr3788371281935070012_a_a_a @ ( non_sy8288883172567897665_a_a_a @ Xs ) ) )
       => ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(1)
thf(fact_697_non__sym__dist__pairs_H__elems__distinct_I1_J,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,X3: sum_su5154756782089200444_a_a_a,Y3: sum_su5154756782089200444_a_a_a] :
      ( ( distin5156058890512421251_a_a_a @ Xs )
     => ( ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ X3 @ Y3 ) @ ( set_Pr6783383005027252718_a_a_a @ ( non_sy6705267804935307249_a_a_a @ Xs ) ) )
       => ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(1)
thf(fact_698_non__sym__dist__pairs_H__elems__distinct_I1_J,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,X3: produc4770514982935793080_a_a_b,Y3: produc4770514982935793080_a_a_b] :
      ( ( distin1372936652459515135_a_a_b @ Xs )
     => ( ( member3114646429565181818_a_a_b @ ( produc5271149956529894921_a_a_b @ X3 @ Y3 ) @ ( set_Pr2819031558886761446_a_a_b @ ( non_sy5074223863352749421_a_a_b @ Xs ) ) )
       => ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(1)
thf(fact_699_non__sym__dist__pairs_H__elems__distinct_I1_J,axiom,
    ! [Xs: list_a,X3: a,Y3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( non_sym_dist_pairs_a @ Xs ) ) )
       => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ) ).

% non_sym_dist_pairs'_elems_distinct(1)
thf(fact_700_single__input__alt__def,axiom,
    ( single6641483607941472940_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c] :
        ! [Q13: sum_su5154756782089200444_a_a_a,X: b,X8: b,Y4: c,Y7: c,Q14: sum_su5154756782089200444_a_a_a,Q15: sum_su5154756782089200444_a_a_a] :
          ( ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q13 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q14 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) )
            & ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q13 @ ( produc1730628064597884819_a_a_a @ X8 @ ( produc8806894744968565733_a_a_a @ Y7 @ Q15 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) ) )
         => ( X = X8 ) ) ) ) ).

% single_input_alt_def
thf(fact_701_single__input_Oelims_I3_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b] :
      ( ~ ( single8564458778801658663_a_a_b @ X3 )
     => ~ ! [X4: produc6800189296593107456_a_a_a] :
            ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ X3 ) )
           => ! [Xa3: produc6800189296593107456_a_a_a] :
                ( ( member4643618012521571351_a_a_a @ Xa3 @ ( transi7010299678732630507_a_a_b @ X3 ) )
               => ( ( ( produc2316789063618113990_a_a_a @ X4 )
                    = ( produc2316789063618113990_a_a_a @ Xa3 ) )
                 => ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
                    = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa3 ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_702_single__input_Oelims_I3_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c] :
      ( ~ ( single6641483607941472940_a_b_c @ X3 )
     => ~ ! [X4: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ X3 ) )
           => ! [Xa3: produc884577481120224310_a_a_a] :
                ( ( member8362999488266169055_a_a_a @ Xa3 @ ( transi749694810082594024_a_b_c @ X3 ) )
               => ( ( ( produc7477144754405272026_a_a_a @ X4 )
                    = ( produc7477144754405272026_a_a_a @ Xa3 ) )
                 => ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
                    = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa3 ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_703_single__input_Oelims_I2_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b] :
      ( ( single8564458778801658663_a_a_b @ X3 )
     => ! [X7: produc6800189296593107456_a_a_a] :
          ( ( member4643618012521571351_a_a_a @ X7 @ ( transi7010299678732630507_a_a_b @ X3 ) )
         => ! [Xa2: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ Xa2 @ ( transi7010299678732630507_a_a_b @ X3 ) )
             => ( ( ( produc2316789063618113990_a_a_a @ X7 )
                  = ( produc2316789063618113990_a_a_a @ Xa2 ) )
               => ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) )
                  = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_704_single__input_Oelims_I2_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c] :
      ( ( single6641483607941472940_a_b_c @ X3 )
     => ! [X7: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X7 @ ( transi749694810082594024_a_b_c @ X3 ) )
         => ! [Xa2: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ Xa2 @ ( transi749694810082594024_a_b_c @ X3 ) )
             => ( ( ( produc7477144754405272026_a_a_a @ X7 )
                  = ( produc7477144754405272026_a_a_a @ Xa2 ) )
               => ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) )
                  = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_705_single__input_Oelims_I1_J,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b,Y3: $o] :
      ( ( ( single8564458778801658663_a_a_b @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ X @ ( transi7010299678732630507_a_a_b @ X3 ) )
             => ! [Y4: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ X3 ) )
                 => ( ( ( produc2316789063618113990_a_a_a @ X )
                      = ( produc2316789063618113990_a_a_a @ Y4 ) )
                   => ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) )
                      = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_706_single__input_Oelims_I1_J,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Y3: $o] :
      ( ( ( single6641483607941472940_a_b_c @ X3 )
        = Y3 )
     => ( Y3
        = ( ! [X: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ X @ ( transi749694810082594024_a_b_c @ X3 ) )
             => ! [Y4: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ X3 ) )
                 => ( ( ( produc7477144754405272026_a_a_a @ X )
                      = ( produc7477144754405272026_a_a_a @ Y4 ) )
                   => ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) )
                      = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_707_single__input_Osimps,axiom,
    ( single8564458778801658663_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b] :
        ! [X: produc6800189296593107456_a_a_a] :
          ( ( member4643618012521571351_a_a_a @ X @ ( transi7010299678732630507_a_a_b @ M3 ) )
         => ! [Y4: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ Y4 @ ( transi7010299678732630507_a_a_b @ M3 ) )
             => ( ( ( produc2316789063618113990_a_a_a @ X )
                  = ( produc2316789063618113990_a_a_a @ Y4 ) )
               => ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X ) )
                  = ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Y4 ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_708_single__input_Osimps,axiom,
    ( single6641483607941472940_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c] :
        ! [X: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X @ ( transi749694810082594024_a_b_c @ M3 ) )
         => ! [Y4: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ Y4 @ ( transi749694810082594024_a_b_c @ M3 ) )
             => ( ( ( produc7477144754405272026_a_a_a @ X )
                  = ( produc7477144754405272026_a_a_a @ Y4 ) )
               => ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X ) )
                  = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Y4 ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_709_acyclic__initial,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c] :
      ( ( acycli6334973534855091236_a_b_c @ M )
     => ~ ? [X7: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X7 @ ( transi749694810082594024_a_b_c @ M ) )
            & ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
              = ( initia839594848065427214_a_b_c @ M ) )
            & ? [P5: list_P8197596098214363964_a_a_a] :
                ( ( path_S221595587009941175_a_b_c @ M @ ( initia839594848065427214_a_b_c @ M ) @ P5 )
                & ( ( target5023745622805013891_a_b_c @ ( initia839594848065427214_a_b_c @ M ) @ P5 )
                  = ( produc7477144754405272026_a_a_a @ X7 ) ) ) ) ) ).

% acyclic_initial
thf(fact_710_acyclic__initial,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b] :
      ( ( acycli3019483576907578287_a_a_b @ M )
     => ~ ? [X7: produc6800189296593107456_a_a_a] :
            ( ( member4643618012521571351_a_a_a @ X7 @ ( transi7010299678732630507_a_a_b @ M ) )
            & ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
              = ( initia5013972905137889861_a_a_b @ M ) )
            & ? [P5: list_P3520633289758869904_a_a_a] :
                ( ( path_P4574188132515142236_a_a_b @ M @ ( initia5013972905137889861_a_a_b @ M ) @ P5 )
                & ( ( target4721681511209552016_a_a_b @ ( initia5013972905137889861_a_a_b @ M ) @ P5 )
                  = ( produc2316789063618113990_a_a_a @ X7 ) ) ) ) ) ).

% acyclic_initial
thf(fact_711_completely__specified__path__extension,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a,P4: list_P3520633289758869904_a_a_a,X3: sum_su5154756782089200444_a_a_a] :
      ( ( comple3249000580571322175_a_a_b @ M )
     => ( ( member6990616694304513154_a_a_a @ Q @ ( states8463590192175095071_a_a_b @ M ) )
       => ( ( path_P4574188132515142236_a_a_b @ M @ Q @ P4 )
         => ( ( member5679158588528110291_a_a_a @ X3 @ ( inputs925126750808843768_a_a_b @ M ) )
           => ~ ! [T5: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ T5 @ ( transi7010299678732630507_a_a_b @ M ) )
                 => ( ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T5 ) )
                      = X3 )
                   => ( ( produc2316789063618113990_a_a_a @ T5 )
                     != ( target4721681511209552016_a_a_b @ Q @ P4 ) ) ) ) ) ) ) ) ).

% completely_specified_path_extension
thf(fact_712_completely__specified__path__extension,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,X3: b] :
      ( ( comple3605256123784756372_a_b_c @ M )
     => ( ( member5679158588528110291_a_a_a @ Q @ ( states3035385874452377268_a_b_c @ M ) )
       => ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
         => ( ( member_b @ X3 @ ( inputs3694848887974842907_a_b_c @ M ) )
           => ~ ! [T5: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ T5 @ ( transi749694810082594024_a_b_c @ M ) )
                 => ( ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T5 ) )
                      = X3 )
                   => ( ( produc7477144754405272026_a_a_a @ T5 )
                     != ( target5023745622805013891_a_b_c @ Q @ P4 ) ) ) ) ) ) ) ) ).

% completely_specified_path_extension
thf(fact_713_canonical__separator__path__initial_I4_J,axiom,
    ! [M: fsm_a_b_c,Q1: a,Q2: a,P4: list_P8197596098214363964_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ( observable_a_b_c @ M )
           => ( ( Q1 != Q2 )
             => ( ? [S13: a,S23: a] :
                    ( ( target5023745622805013891_a_b_c @ ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) @ P4 )
                    = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ S13 @ S23 ) ) )
                | ( ( target5023745622805013891_a_b_c @ ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) @ P4 )
                  = ( sum_In6345099547827038749od_a_a @ Q1 ) )
                | ( ( target5023745622805013891_a_b_c @ ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) @ P4 )
                  = ( sum_In6345099547827038749od_a_a @ Q2 ) ) ) ) ) ) ) ) ).

% canonical_separator_path_initial(4)
thf(fact_714_path_Ocases,axiom,
    ! [A1: fsm_Su3173892059090327343_a_b_c,A22: sum_su5154756782089200444_a_a_a,A32: list_P8197596098214363964_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ A1 @ A22 @ A32 )
     => ( ( ( A32 = nil_Pr6029279252097497404_a_a_a )
         => ~ ( member5679158588528110291_a_a_a @ A22 @ ( states3035385874452377268_a_b_c @ A1 ) ) )
       => ~ ! [T5: produc884577481120224310_a_a_a] :
              ( ( A22
                = ( produc7477144754405272026_a_a_a @ T5 ) )
             => ! [Ts2: list_P8197596098214363964_a_a_a] :
                  ( ( A32
                    = ( cons_P6318101215856909548_a_a_a @ T5 @ Ts2 ) )
                 => ( ( member8362999488266169055_a_a_a @ T5 @ ( transi749694810082594024_a_b_c @ A1 ) )
                   => ~ ( path_S221595587009941175_a_b_c @ A1 @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T5 ) ) ) @ Ts2 ) ) ) ) ) ) ).

% path.cases
thf(fact_715_path_Ocases,axiom,
    ! [A1: fsm_Pr6287166851770849218_a_a_b,A22: produc1895207096998757355_a_a_a,A32: list_P3520633289758869904_a_a_a] :
      ( ( path_P4574188132515142236_a_a_b @ A1 @ A22 @ A32 )
     => ( ( ( A32 = nil_Pr3590249381822989690_a_a_a )
         => ~ ( member6990616694304513154_a_a_a @ A22 @ ( states8463590192175095071_a_a_b @ A1 ) ) )
       => ~ ! [T5: produc6800189296593107456_a_a_a] :
              ( ( A22
                = ( produc2316789063618113990_a_a_a @ T5 ) )
             => ! [Ts2: list_P3520633289758869904_a_a_a] :
                  ( ( A32
                    = ( cons_P5563258090397451722_a_a_a @ T5 @ Ts2 ) )
                 => ( ( member4643618012521571351_a_a_a @ T5 @ ( transi7010299678732630507_a_a_b @ A1 ) )
                   => ~ ( path_P4574188132515142236_a_a_b @ A1 @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T5 ) ) ) @ Ts2 ) ) ) ) ) ) ).

% path.cases
thf(fact_716_path_Osimps,axiom,
    ( path_S221595587009941175_a_b_c
    = ( ^ [A12: fsm_Su3173892059090327343_a_b_c,A23: sum_su5154756782089200444_a_a_a,A33: list_P8197596098214363964_a_a_a] :
          ( ? [Q5: sum_su5154756782089200444_a_a_a,M3: fsm_Su3173892059090327343_a_b_c] :
              ( ( A12 = M3 )
              & ( A23 = Q5 )
              & ( A33 = nil_Pr6029279252097497404_a_a_a )
              & ( member5679158588528110291_a_a_a @ Q5 @ ( states3035385874452377268_a_b_c @ M3 ) ) )
          | ? [T4: produc884577481120224310_a_a_a,M3: fsm_Su3173892059090327343_a_b_c,Ts3: list_P8197596098214363964_a_a_a] :
              ( ( A12 = M3 )
              & ( A23
                = ( produc7477144754405272026_a_a_a @ T4 ) )
              & ( A33
                = ( cons_P6318101215856909548_a_a_a @ T4 @ Ts3 ) )
              & ( member8362999488266169055_a_a_a @ T4 @ ( transi749694810082594024_a_b_c @ M3 ) )
              & ( path_S221595587009941175_a_b_c @ M3 @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) @ Ts3 ) ) ) ) ) ).

% path.simps
thf(fact_717_path_Osimps,axiom,
    ( path_P4574188132515142236_a_a_b
    = ( ^ [A12: fsm_Pr6287166851770849218_a_a_b,A23: produc1895207096998757355_a_a_a,A33: list_P3520633289758869904_a_a_a] :
          ( ? [Q5: produc1895207096998757355_a_a_a,M3: fsm_Pr6287166851770849218_a_a_b] :
              ( ( A12 = M3 )
              & ( A23 = Q5 )
              & ( A33 = nil_Pr3590249381822989690_a_a_a )
              & ( member6990616694304513154_a_a_a @ Q5 @ ( states8463590192175095071_a_a_b @ M3 ) ) )
          | ? [T4: produc6800189296593107456_a_a_a,M3: fsm_Pr6287166851770849218_a_a_b,Ts3: list_P3520633289758869904_a_a_a] :
              ( ( A12 = M3 )
              & ( A23
                = ( produc2316789063618113990_a_a_a @ T4 ) )
              & ( A33
                = ( cons_P5563258090397451722_a_a_a @ T4 @ Ts3 ) )
              & ( member4643618012521571351_a_a_a @ T4 @ ( transi7010299678732630507_a_a_b @ M3 ) )
              & ( path_P4574188132515142236_a_a_b @ M3 @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) @ Ts3 ) ) ) ) ) ).

% path.simps
thf(fact_718_singleton__conv2,axiom,
    ! [A: sum_su5154756782089200444_a_a_a] :
      ( ( collec299858375640228881_a_a_a
        @ ( ^ [Y6: sum_su5154756782089200444_a_a_a,Z3: sum_su5154756782089200444_a_a_a] : ( Y6 = Z3 )
          @ A ) )
      = ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) ) ).

% singleton_conv2
thf(fact_719_singleton__conv2,axiom,
    ! [A: produc4770514982935793080_a_a_b] :
      ( ( collec4985708871760228749_a_a_b
        @ ( ^ [Y6: produc4770514982935793080_a_a_b,Z3: produc4770514982935793080_a_a_b] : ( Y6 = Z3 )
          @ A ) )
      = ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) ) ).

% singleton_conv2
thf(fact_720_singleton__conv2,axiom,
    ! [A: produc884577481120224310_a_a_a] :
      ( ( collec3562233214436895265_a_a_a
        @ ( ^ [Y6: produc884577481120224310_a_a_a,Z3: produc884577481120224310_a_a_a] : ( Y6 = Z3 )
          @ A ) )
      = ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) ) ).

% singleton_conv2
thf(fact_721_singleton__conv,axiom,
    ! [A: sum_su5154756782089200444_a_a_a] :
      ( ( collec299858375640228881_a_a_a
        @ ^ [X: sum_su5154756782089200444_a_a_a] : ( X = A ) )
      = ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) ) ).

% singleton_conv
thf(fact_722_singleton__conv,axiom,
    ! [A: produc4770514982935793080_a_a_b] :
      ( ( collec4985708871760228749_a_a_b
        @ ^ [X: produc4770514982935793080_a_a_b] : ( X = A ) )
      = ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) ) ).

% singleton_conv
thf(fact_723_singleton__conv,axiom,
    ! [A: produc884577481120224310_a_a_a] :
      ( ( collec3562233214436895265_a_a_a
        @ ^ [X: produc884577481120224310_a_a_a] : ( X = A ) )
      = ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) ) ).

% singleton_conv
thf(fact_724_list_Osimps_I15_J,axiom,
    ! [X21: produc884577481120224310_a_a_a,X22: list_P8197596098214363964_a_a_a] :
      ( ( set_Pr8067384617587748683_a_a_a @ ( cons_P6318101215856909548_a_a_a @ X21 @ X22 ) )
      = ( insert2249508923356641798_a_a_a @ X21 @ ( set_Pr8067384617587748683_a_a_a @ X22 ) ) ) ).

% list.simps(15)
thf(fact_725_list_Osimps_I15_J,axiom,
    ! [X21: sum_su5154756782089200444_a_a_a,X22: list_S8072159660131056972_a_a_a] :
      ( ( set_Su3113867532901839207_a_a_a @ ( cons_S2140548516868915206_a_a_a @ X21 @ X22 ) )
      = ( insert5061175218292437356_a_a_a @ X21 @ ( set_Su3113867532901839207_a_a_a @ X22 ) ) ) ).

% list.simps(15)
thf(fact_726_list_Osimps_I15_J,axiom,
    ! [X21: produc4770514982935793080_a_a_b,X22: list_P7155397214832950472_a_a_b] :
      ( ( set_Pr4578018992751950051_a_a_b @ ( cons_P2262690024297812098_a_a_b @ X21 @ X22 ) )
      = ( insert8042272945010224104_a_a_b @ X21 @ ( set_Pr4578018992751950051_a_a_b @ X22 ) ) ) ).

% list.simps(15)
thf(fact_727_sup__set__def,axiom,
    ( sup_su9011528761648865002_a_a_a
    = ( ^ [A3: set_Pr4618682941230328598_a_a_a,B4: set_Pr4618682941230328598_a_a_a] :
          ( collec3562233214436895265_a_a_a
          @ ( sup_su6155851345211584115_a_a_o
            @ ^ [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ A3 )
            @ ^ [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_728_sup__set__def,axiom,
    ( sup_su2660408029157294746_a_a_b
    = ( ^ [A3: set_Pr6246767948713639278_a_a_b,B4: set_Pr6246767948713639278_a_a_b] :
          ( collec4985708871760228749_a_a_b
          @ ( sup_su2859647878392671851_a_b_o
            @ ^ [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ A3 )
            @ ^ [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_729_sup__set__def,axiom,
    ( sup_su7096409449263244574_a_a_a
    = ( ^ [A3: set_Su8591296761904255986_a_a_a,B4: set_Su8591296761904255986_a_a_a] :
          ( collec299858375640228881_a_a_a
          @ ( sup_su2021648564548494567_a_a_o
            @ ^ [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ A3 )
            @ ^ [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_730_Set_Oempty__def,axiom,
    ( bot_bo2676859181106700934_a_a_a
    = ( collec299858375640228881_a_a_a
      @ ^ [X: sum_su5154756782089200444_a_a_a] : $false ) ) ).

% Set.empty_def
thf(fact_731_Set_Oempty__def,axiom,
    ( bot_bo5052542052834138114_a_a_b
    = ( collec4985708871760228749_a_a_b
      @ ^ [X: produc4770514982935793080_a_a_b] : $false ) ) ).

% Set.empty_def
thf(fact_732_Set_Oempty__def,axiom,
    ( bot_bo2898616998271641474_a_a_a
    = ( collec3562233214436895265_a_a_a
      @ ^ [X: produc884577481120224310_a_a_a] : $false ) ) ).

% Set.empty_def
thf(fact_733_Collect__disj__eq,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > $o,Q3: sum_su5154756782089200444_a_a_a > $o] :
      ( ( collec299858375640228881_a_a_a
        @ ^ [X: sum_su5154756782089200444_a_a_a] :
            ( ( P @ X )
            | ( Q3 @ X ) ) )
      = ( sup_su7096409449263244574_a_a_a @ ( collec299858375640228881_a_a_a @ P ) @ ( collec299858375640228881_a_a_a @ Q3 ) ) ) ).

% Collect_disj_eq
thf(fact_734_Un__def,axiom,
    ( sup_su9011528761648865002_a_a_a
    = ( ^ [A3: set_Pr4618682941230328598_a_a_a,B4: set_Pr4618682941230328598_a_a_a] :
          ( collec3562233214436895265_a_a_a
          @ ^ [X: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ X @ A3 )
              | ( member8362999488266169055_a_a_a @ X @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_735_Un__def,axiom,
    ( sup_su2660408029157294746_a_a_b
    = ( ^ [A3: set_Pr6246767948713639278_a_a_b,B4: set_Pr6246767948713639278_a_a_b] :
          ( collec4985708871760228749_a_a_b
          @ ^ [X: produc4770514982935793080_a_a_b] :
              ( ( member642552543386773071_a_a_b @ X @ A3 )
              | ( member642552543386773071_a_a_b @ X @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_736_Un__def,axiom,
    ( sup_su7096409449263244574_a_a_a
    = ( ^ [A3: set_Su8591296761904255986_a_a_a,B4: set_Su8591296761904255986_a_a_a] :
          ( collec299858375640228881_a_a_a
          @ ^ [X: sum_su5154756782089200444_a_a_a] :
              ( ( member5679158588528110291_a_a_a @ X @ A3 )
              | ( member5679158588528110291_a_a_a @ X @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_737_insert__compr,axiom,
    ( insert5061175218292437356_a_a_a
    = ( ^ [A4: sum_su5154756782089200444_a_a_a,B4: set_Su8591296761904255986_a_a_a] :
          ( collec299858375640228881_a_a_a
          @ ^ [X: sum_su5154756782089200444_a_a_a] :
              ( ( X = A4 )
              | ( member5679158588528110291_a_a_a @ X @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_738_insert__compr,axiom,
    ( insert2249508923356641798_a_a_a
    = ( ^ [A4: produc884577481120224310_a_a_a,B4: set_Pr4618682941230328598_a_a_a] :
          ( collec3562233214436895265_a_a_a
          @ ^ [X: produc884577481120224310_a_a_a] :
              ( ( X = A4 )
              | ( member8362999488266169055_a_a_a @ X @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_739_insert__compr,axiom,
    ( insert8042272945010224104_a_a_b
    = ( ^ [A4: produc4770514982935793080_a_a_b,B4: set_Pr6246767948713639278_a_a_b] :
          ( collec4985708871760228749_a_a_b
          @ ^ [X: produc4770514982935793080_a_a_b] :
              ( ( X = A4 )
              | ( member642552543386773071_a_a_b @ X @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_740_insert__Collect,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( insert5061175218292437356_a_a_a @ A @ ( collec299858375640228881_a_a_a @ P ) )
      = ( collec299858375640228881_a_a_a
        @ ^ [U: sum_su5154756782089200444_a_a_a] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_741_insert__Collect,axiom,
    ! [A: produc4770514982935793080_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ( insert8042272945010224104_a_a_b @ A @ ( collec4985708871760228749_a_a_b @ P ) )
      = ( collec4985708871760228749_a_a_b
        @ ^ [U: produc4770514982935793080_a_a_b] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_742_insert__Collect,axiom,
    ! [A: produc884577481120224310_a_a_a,P: produc884577481120224310_a_a_a > $o] :
      ( ( insert2249508923356641798_a_a_a @ A @ ( collec3562233214436895265_a_a_a @ P ) )
      = ( collec3562233214436895265_a_a_a
        @ ^ [U: produc884577481120224310_a_a_a] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_743_sup__Un__eq,axiom,
    ! [R: set_Pr4618682941230328598_a_a_a,S5: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su6155851345211584115_a_a_o
        @ ^ [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ R )
        @ ^ [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ S5 ) )
      = ( ^ [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ ( sup_su9011528761648865002_a_a_a @ R @ S5 ) ) ) ) ).

% sup_Un_eq
thf(fact_744_sup__Un__eq,axiom,
    ! [R: set_Pr6246767948713639278_a_a_b,S5: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su2859647878392671851_a_b_o
        @ ^ [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ R )
        @ ^ [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ S5 ) )
      = ( ^ [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ ( sup_su2660408029157294746_a_a_b @ R @ S5 ) ) ) ) ).

% sup_Un_eq
thf(fact_745_sup__Un__eq,axiom,
    ! [R: set_Su8591296761904255986_a_a_a,S5: set_Su8591296761904255986_a_a_a] :
      ( ( sup_su2021648564548494567_a_a_o
        @ ^ [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ R )
        @ ^ [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ S5 ) )
      = ( ^ [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ ( sup_su7096409449263244574_a_a_a @ R @ S5 ) ) ) ) ).

% sup_Un_eq
thf(fact_746_Collect__conv__if2,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > $o,A: sum_su5154756782089200444_a_a_a] :
      ( ( ( P @ A )
       => ( ( collec299858375640228881_a_a_a
            @ ^ [X: sum_su5154756782089200444_a_a_a] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) ) )
      & ( ~ ( P @ A )
       => ( ( collec299858375640228881_a_a_a
            @ ^ [X: sum_su5154756782089200444_a_a_a] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo2676859181106700934_a_a_a ) ) ) ).

% Collect_conv_if2
thf(fact_747_Collect__conv__if2,axiom,
    ! [P: produc4770514982935793080_a_a_b > $o,A: produc4770514982935793080_a_a_b] :
      ( ( ( P @ A )
       => ( ( collec4985708871760228749_a_a_b
            @ ^ [X: produc4770514982935793080_a_a_b] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) ) )
      & ( ~ ( P @ A )
       => ( ( collec4985708871760228749_a_a_b
            @ ^ [X: produc4770514982935793080_a_a_b] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo5052542052834138114_a_a_b ) ) ) ).

% Collect_conv_if2
thf(fact_748_Collect__conv__if2,axiom,
    ! [P: produc884577481120224310_a_a_a > $o,A: produc884577481120224310_a_a_a] :
      ( ( ( P @ A )
       => ( ( collec3562233214436895265_a_a_a
            @ ^ [X: produc884577481120224310_a_a_a] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3562233214436895265_a_a_a
            @ ^ [X: produc884577481120224310_a_a_a] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo2898616998271641474_a_a_a ) ) ) ).

% Collect_conv_if2
thf(fact_749_Collect__conv__if,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > $o,A: sum_su5154756782089200444_a_a_a] :
      ( ( ( P @ A )
       => ( ( collec299858375640228881_a_a_a
            @ ^ [X: sum_su5154756782089200444_a_a_a] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) ) )
      & ( ~ ( P @ A )
       => ( ( collec299858375640228881_a_a_a
            @ ^ [X: sum_su5154756782089200444_a_a_a] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo2676859181106700934_a_a_a ) ) ) ).

% Collect_conv_if
thf(fact_750_Collect__conv__if,axiom,
    ! [P: produc4770514982935793080_a_a_b > $o,A: produc4770514982935793080_a_a_b] :
      ( ( ( P @ A )
       => ( ( collec4985708871760228749_a_a_b
            @ ^ [X: produc4770514982935793080_a_a_b] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) ) )
      & ( ~ ( P @ A )
       => ( ( collec4985708871760228749_a_a_b
            @ ^ [X: produc4770514982935793080_a_a_b] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo5052542052834138114_a_a_b ) ) ) ).

% Collect_conv_if
thf(fact_751_Collect__conv__if,axiom,
    ! [P: produc884577481120224310_a_a_a > $o,A: produc884577481120224310_a_a_a] :
      ( ( ( P @ A )
       => ( ( collec3562233214436895265_a_a_a
            @ ^ [X: produc884577481120224310_a_a_a] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3562233214436895265_a_a_a
            @ ^ [X: produc884577481120224310_a_a_a] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo2898616998271641474_a_a_a ) ) ) ).

% Collect_conv_if
thf(fact_752_insert__def,axiom,
    ( insert8042272945010224104_a_a_b
    = ( ^ [A4: produc4770514982935793080_a_a_b] :
          ( sup_su2660408029157294746_a_a_b
          @ ( collec4985708871760228749_a_a_b
            @ ^ [X: produc4770514982935793080_a_a_b] : ( X = A4 ) ) ) ) ) ).

% insert_def
thf(fact_753_insert__def,axiom,
    ( insert2249508923356641798_a_a_a
    = ( ^ [A4: produc884577481120224310_a_a_a] :
          ( sup_su9011528761648865002_a_a_a
          @ ( collec3562233214436895265_a_a_a
            @ ^ [X: produc884577481120224310_a_a_a] : ( X = A4 ) ) ) ) ) ).

% insert_def
thf(fact_754_insert__def,axiom,
    ( insert5061175218292437356_a_a_a
    = ( ^ [A4: sum_su5154756782089200444_a_a_a] :
          ( sup_su7096409449263244574_a_a_a
          @ ( collec299858375640228881_a_a_a
            @ ^ [X: sum_su5154756782089200444_a_a_a] : ( X = A4 ) ) ) ) ) ).

% insert_def
thf(fact_755_less__set__def,axiom,
    ( ord_le1107997805182519878_a_a_a
    = ( ^ [A3: set_Su8591296761904255986_a_a_a,B4: set_Su8591296761904255986_a_a_a] :
          ( ord_le7758598310295970239_a_a_o
          @ ^ [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ A3 )
          @ ^ [X: sum_su5154756782089200444_a_a_a] : ( member5679158588528110291_a_a_a @ X @ B4 ) ) ) ) ).

% less_set_def
thf(fact_756_less__set__def,axiom,
    ( ord_le6196359254896381890_a_a_a
    = ( ^ [A3: set_Pr4618682941230328598_a_a_a,B4: set_Pr4618682941230328598_a_a_a] :
          ( ord_le8611082066241723291_a_a_o
          @ ^ [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ A3 )
          @ ^ [X: produc884577481120224310_a_a_a] : ( member8362999488266169055_a_a_a @ X @ B4 ) ) ) ) ).

% less_set_def
thf(fact_757_less__set__def,axiom,
    ( ord_le8627993822118182850_a_a_b
    = ( ^ [A3: set_Pr6246767948713639278_a_a_b,B4: set_Pr6246767948713639278_a_a_b] :
          ( ord_le8377600345673339715_a_b_o
          @ ^ [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ A3 )
          @ ^ [X: produc4770514982935793080_a_a_b] : ( member642552543386773071_a_a_b @ X @ B4 ) ) ) ) ).

% less_set_def
thf(fact_758_sup__Un__eq2,axiom,
    ! [R: set_Pr4618682941230328598_a_a_a,S5: set_Pr4618682941230328598_a_a_a] :
      ( ( sup_su7102191406203184709_a_a_o
        @ ^ [X: sum_su5154756782089200444_a_a_a,Y4: produc7810058533641270553_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X @ Y4 ) @ R )
        @ ^ [X: sum_su5154756782089200444_a_a_a,Y4: produc7810058533641270553_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X @ Y4 ) @ S5 ) )
      = ( ^ [X: sum_su5154756782089200444_a_a_a,Y4: produc7810058533641270553_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X @ Y4 ) @ ( sup_su9011528761648865002_a_a_a @ R @ S5 ) ) ) ) ).

% sup_Un_eq2
thf(fact_759_sup__Un__eq2,axiom,
    ! [R: set_Pr6246767948713639278_a_a_b,S5: set_Pr6246767948713639278_a_a_b] :
      ( ( sup_su5268434625375998917_a_b_o
        @ ^ [X: sum_su5154756782089200444_a_a_a,Y4: b] : ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X @ Y4 ) @ R )
        @ ^ [X: sum_su5154756782089200444_a_a_a,Y4: b] : ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X @ Y4 ) @ S5 ) )
      = ( ^ [X: sum_su5154756782089200444_a_a_a,Y4: b] : ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X @ Y4 ) @ ( sup_su2660408029157294746_a_a_b @ R @ S5 ) ) ) ) ).

% sup_Un_eq2
thf(fact_760_sup__Un__eq2,axiom,
    ! [R: set_Product_prod_a_a,S5: set_Product_prod_a_a] :
      ( ( sup_sup_a_a_o
        @ ^ [X: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ R )
        @ ^ [X: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ S5 ) )
      = ( ^ [X: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ ( sup_su3048258781599657691od_a_a @ R @ S5 ) ) ) ) ).

% sup_Un_eq2
thf(fact_761_pred__equals__eq2,axiom,
    ! [R: set_Pr4618682941230328598_a_a_a,S5: set_Pr4618682941230328598_a_a_a] :
      ( ( ( ^ [X: sum_su5154756782089200444_a_a_a,Y4: produc7810058533641270553_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X @ Y4 ) @ R ) )
        = ( ^ [X: sum_su5154756782089200444_a_a_a,Y4: produc7810058533641270553_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X @ Y4 ) @ S5 ) ) )
      = ( R = S5 ) ) ).

% pred_equals_eq2
thf(fact_762_pred__equals__eq2,axiom,
    ! [R: set_Pr6246767948713639278_a_a_b,S5: set_Pr6246767948713639278_a_a_b] :
      ( ( ( ^ [X: sum_su5154756782089200444_a_a_a,Y4: b] : ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X @ Y4 ) @ R ) )
        = ( ^ [X: sum_su5154756782089200444_a_a_a,Y4: b] : ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X @ Y4 ) @ S5 ) ) )
      = ( R = S5 ) ) ).

% pred_equals_eq2
thf(fact_763_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_a_a,S5: set_Product_prod_a_a] :
      ( ( ( ^ [X: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ R ) )
        = ( ^ [X: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ S5 ) ) )
      = ( R = S5 ) ) ).

% pred_equals_eq2
thf(fact_764_bot__empty__eq2,axiom,
    ( bot_bot_a_a_o
    = ( ^ [X: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ bot_bo3357376287454694259od_a_a ) ) ) ).

% bot_empty_eq2
thf(fact_765_bot__empty__eq2,axiom,
    ( bot_bo2455107741248149805_a_b_o
    = ( ^ [X: sum_su5154756782089200444_a_a_a,Y4: b] : ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X @ Y4 ) @ bot_bo5052542052834138114_a_a_b ) ) ) ).

% bot_empty_eq2
thf(fact_766_bot__empty__eq2,axiom,
    ( bot_bo4083105413308890845_a_a_o
    = ( ^ [X: sum_su5154756782089200444_a_a_a,Y4: produc7810058533641270553_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X @ Y4 ) @ bot_bo2898616998271641474_a_a_a ) ) ) ).

% bot_empty_eq2
thf(fact_767_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc884577481120224310_a_a_a,X22: list_P8197596098214363964_a_a_a,X21: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ Y3 @ ( set_Pr8067384617587748683_a_a_a @ X22 ) )
     => ( member8362999488266169055_a_a_a @ Y3 @ ( set_Pr8067384617587748683_a_a_a @ ( cons_P6318101215856909548_a_a_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_768_list_Oset__intros_I2_J,axiom,
    ! [Y3: sum_su5154756782089200444_a_a_a,X22: list_S8072159660131056972_a_a_a,X21: sum_su5154756782089200444_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ X22 ) )
     => ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ ( cons_S2140548516868915206_a_a_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_769_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc4770514982935793080_a_a_b,X22: list_P7155397214832950472_a_a_b,X21: produc4770514982935793080_a_a_b] :
      ( ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ X22 ) )
     => ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ ( cons_P2262690024297812098_a_a_b @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_770_list_Oset__intros_I1_J,axiom,
    ! [X21: produc884577481120224310_a_a_a,X22: list_P8197596098214363964_a_a_a] : ( member8362999488266169055_a_a_a @ X21 @ ( set_Pr8067384617587748683_a_a_a @ ( cons_P6318101215856909548_a_a_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_771_list_Oset__intros_I1_J,axiom,
    ! [X21: sum_su5154756782089200444_a_a_a,X22: list_S8072159660131056972_a_a_a] : ( member5679158588528110291_a_a_a @ X21 @ ( set_Su3113867532901839207_a_a_a @ ( cons_S2140548516868915206_a_a_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_772_list_Oset__intros_I1_J,axiom,
    ! [X21: produc4770514982935793080_a_a_b,X22: list_P7155397214832950472_a_a_b] : ( member642552543386773071_a_a_b @ X21 @ ( set_Pr4578018992751950051_a_a_b @ ( cons_P2262690024297812098_a_a_b @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_773_list_Oset__cases,axiom,
    ! [E: produc884577481120224310_a_a_a,A: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ E @ ( set_Pr8067384617587748683_a_a_a @ A ) )
     => ( ! [Z22: list_P8197596098214363964_a_a_a] :
            ( A
           != ( cons_P6318101215856909548_a_a_a @ E @ Z22 ) )
       => ~ ! [Z1: produc884577481120224310_a_a_a,Z22: list_P8197596098214363964_a_a_a] :
              ( ( A
                = ( cons_P6318101215856909548_a_a_a @ Z1 @ Z22 ) )
             => ~ ( member8362999488266169055_a_a_a @ E @ ( set_Pr8067384617587748683_a_a_a @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_774_list_Oset__cases,axiom,
    ! [E: sum_su5154756782089200444_a_a_a,A: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ E @ ( set_Su3113867532901839207_a_a_a @ A ) )
     => ( ! [Z22: list_S8072159660131056972_a_a_a] :
            ( A
           != ( cons_S2140548516868915206_a_a_a @ E @ Z22 ) )
       => ~ ! [Z1: sum_su5154756782089200444_a_a_a,Z22: list_S8072159660131056972_a_a_a] :
              ( ( A
                = ( cons_S2140548516868915206_a_a_a @ Z1 @ Z22 ) )
             => ~ ( member5679158588528110291_a_a_a @ E @ ( set_Su3113867532901839207_a_a_a @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_775_list_Oset__cases,axiom,
    ! [E: produc4770514982935793080_a_a_b,A: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ E @ ( set_Pr4578018992751950051_a_a_b @ A ) )
     => ( ! [Z22: list_P7155397214832950472_a_a_b] :
            ( A
           != ( cons_P2262690024297812098_a_a_b @ E @ Z22 ) )
       => ~ ! [Z1: produc4770514982935793080_a_a_b,Z22: list_P7155397214832950472_a_a_b] :
              ( ( A
                = ( cons_P2262690024297812098_a_a_b @ Z1 @ Z22 ) )
             => ~ ( member642552543386773071_a_a_b @ E @ ( set_Pr4578018992751950051_a_a_b @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_776_set__ConsD,axiom,
    ! [Y3: produc884577481120224310_a_a_a,X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ Y3 @ ( set_Pr8067384617587748683_a_a_a @ ( cons_P6318101215856909548_a_a_a @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member8362999488266169055_a_a_a @ Y3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_777_set__ConsD,axiom,
    ! [Y3: sum_su5154756782089200444_a_a_a,X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ ( cons_S2140548516868915206_a_a_a @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_778_set__ConsD,axiom,
    ! [Y3: produc4770514982935793080_a_a_b,X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ ( cons_P2262690024297812098_a_a_b @ X3 @ Xs ) ) )
     => ( ( Y3 = X3 )
        | ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_779_list_Osimps_I9_J,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,X21: produc4770514982935793080_a_a_b,X22: list_P7155397214832950472_a_a_b] :
      ( ( map_Pr7354669405448068282_a_a_a @ F @ ( cons_P2262690024297812098_a_a_b @ X21 @ X22 ) )
      = ( cons_S2140548516868915206_a_a_a @ ( F @ X21 ) @ ( map_Pr7354669405448068282_a_a_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_780_list_Osimps_I9_J,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,X21: produc4770514982935793080_a_a_b,X22: list_P7155397214832950472_a_a_b] :
      ( ( map_Pr8925698157833054977_a_b_b @ F @ ( cons_P2262690024297812098_a_a_b @ X21 @ X22 ) )
      = ( cons_b @ ( F @ X21 ) @ ( map_Pr8925698157833054977_a_b_b @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_781_list_Osimps_I9_J,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,X21: produc1895207096998757355_a_a_a,X22: list_P1813520604369389947_a_a_a] :
      ( ( map_Pr1074589380674644461_a_a_a @ F @ ( cons_P8610754175215552181_a_a_a @ X21 @ X22 ) )
      = ( cons_S2140548516868915206_a_a_a @ ( F @ X21 ) @ ( map_Pr1074589380674644461_a_a_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_782_list_Osimps_I9_J,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,X21: produc1895207096998757355_a_a_a,X22: list_P1813520604369389947_a_a_a] :
      ( ( map_Pr2287363492687751055_a_a_c @ F @ ( cons_P8610754175215552181_a_a_a @ X21 @ X22 ) )
      = ( cons_c @ ( F @ X21 ) @ ( map_Pr2287363492687751055_a_a_c @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_783_list_Osimps_I9_J,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,X21: produc7810058533641270553_a_a_a,X22: list_P7200181061086074537_a_a_a] :
      ( ( map_Pr3321722608670633418_a_a_a @ F @ ( cons_P1662640394034814435_a_a_a @ X21 @ X22 ) )
      = ( cons_P8610754175215552181_a_a_a @ ( F @ X21 ) @ ( map_Pr3321722608670633418_a_a_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_784_list_Osimps_I9_J,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,X21: produc7810058533641270553_a_a_a,X22: list_P7200181061086074537_a_a_a] :
      ( ( map_Pr3622043693041699552_a_a_b @ F @ ( cons_P1662640394034814435_a_a_a @ X21 @ X22 ) )
      = ( cons_b @ ( F @ X21 ) @ ( map_Pr3622043693041699552_a_a_b @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_785_Cons__eq__map__D,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( cons_S2140548516868915206_a_a_a @ X3 @ Xs )
        = ( map_Pr7354669405448068282_a_a_a @ F @ Ys ) )
     => ? [Z: produc4770514982935793080_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
          ( ( Ys
            = ( cons_P2262690024297812098_a_a_b @ Z @ Zs ) )
          & ( X3
            = ( F @ Z ) )
          & ( Xs
            = ( map_Pr7354669405448068282_a_a_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_786_Cons__eq__map__D,axiom,
    ! [X3: b,Xs: list_b,F: produc4770514982935793080_a_a_b > b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( cons_b @ X3 @ Xs )
        = ( map_Pr8925698157833054977_a_b_b @ F @ Ys ) )
     => ? [Z: produc4770514982935793080_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
          ( ( Ys
            = ( cons_P2262690024297812098_a_a_b @ Z @ Zs ) )
          & ( X3
            = ( F @ Z ) )
          & ( Xs
            = ( map_Pr8925698157833054977_a_b_b @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_787_Cons__eq__map__D,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( cons_S2140548516868915206_a_a_a @ X3 @ Xs )
        = ( map_Pr1074589380674644461_a_a_a @ F @ Ys ) )
     => ? [Z: produc1895207096998757355_a_a_a,Zs: list_P1813520604369389947_a_a_a] :
          ( ( Ys
            = ( cons_P8610754175215552181_a_a_a @ Z @ Zs ) )
          & ( X3
            = ( F @ Z ) )
          & ( Xs
            = ( map_Pr1074589380674644461_a_a_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_788_Cons__eq__map__D,axiom,
    ! [X3: c,Xs: list_c,F: produc1895207096998757355_a_a_a > c,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( cons_c @ X3 @ Xs )
        = ( map_Pr2287363492687751055_a_a_c @ F @ Ys ) )
     => ? [Z: produc1895207096998757355_a_a_a,Zs: list_P1813520604369389947_a_a_a] :
          ( ( Ys
            = ( cons_P8610754175215552181_a_a_a @ Z @ Zs ) )
          & ( X3
            = ( F @ Z ) )
          & ( Xs
            = ( map_Pr2287363492687751055_a_a_c @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_789_Cons__eq__map__D,axiom,
    ! [X3: produc1895207096998757355_a_a_a,Xs: list_P1813520604369389947_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( cons_P8610754175215552181_a_a_a @ X3 @ Xs )
        = ( map_Pr3321722608670633418_a_a_a @ F @ Ys ) )
     => ? [Z: produc7810058533641270553_a_a_a,Zs: list_P7200181061086074537_a_a_a] :
          ( ( Ys
            = ( cons_P1662640394034814435_a_a_a @ Z @ Zs ) )
          & ( X3
            = ( F @ Z ) )
          & ( Xs
            = ( map_Pr3321722608670633418_a_a_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_790_Cons__eq__map__D,axiom,
    ! [X3: b,Xs: list_b,F: produc7810058533641270553_a_a_a > b,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( cons_b @ X3 @ Xs )
        = ( map_Pr3622043693041699552_a_a_b @ F @ Ys ) )
     => ? [Z: produc7810058533641270553_a_a_a,Zs: list_P7200181061086074537_a_a_a] :
          ( ( Ys
            = ( cons_P1662640394034814435_a_a_a @ Z @ Zs ) )
          & ( X3
            = ( F @ Z ) )
          & ( Xs
            = ( map_Pr3622043693041699552_a_a_b @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_791_map__eq__Cons__D,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b,Y3: sum_su5154756782089200444_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( ( map_Pr7354669405448068282_a_a_a @ F @ Xs )
        = ( cons_S2140548516868915206_a_a_a @ Y3 @ Ys ) )
     => ? [Z: produc4770514982935793080_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
          ( ( Xs
            = ( cons_P2262690024297812098_a_a_b @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y3 )
          & ( ( map_Pr7354669405448068282_a_a_a @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_792_map__eq__Cons__D,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,Y3: b,Ys: list_b] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = ( cons_b @ Y3 @ Ys ) )
     => ? [Z: produc4770514982935793080_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
          ( ( Xs
            = ( cons_P2262690024297812098_a_a_b @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y3 )
          & ( ( map_Pr8925698157833054977_a_b_b @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_793_map__eq__Cons__D,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,Y3: sum_su5154756782089200444_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = ( cons_S2140548516868915206_a_a_a @ Y3 @ Ys ) )
     => ? [Z: produc1895207096998757355_a_a_a,Zs: list_P1813520604369389947_a_a_a] :
          ( ( Xs
            = ( cons_P8610754175215552181_a_a_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y3 )
          & ( ( map_Pr1074589380674644461_a_a_a @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_794_map__eq__Cons__D,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,Y3: c,Ys: list_c] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = ( cons_c @ Y3 @ Ys ) )
     => ? [Z: produc1895207096998757355_a_a_a,Zs: list_P1813520604369389947_a_a_a] :
          ( ( Xs
            = ( cons_P8610754175215552181_a_a_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y3 )
          & ( ( map_Pr2287363492687751055_a_a_c @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_795_map__eq__Cons__D,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a,Y3: produc1895207096998757355_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr3321722608670633418_a_a_a @ F @ Xs )
        = ( cons_P8610754175215552181_a_a_a @ Y3 @ Ys ) )
     => ? [Z: produc7810058533641270553_a_a_a,Zs: list_P7200181061086074537_a_a_a] :
          ( ( Xs
            = ( cons_P1662640394034814435_a_a_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y3 )
          & ( ( map_Pr3321722608670633418_a_a_a @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_796_map__eq__Cons__D,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,Y3: b,Ys: list_b] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = ( cons_b @ Y3 @ Ys ) )
     => ? [Z: produc7810058533641270553_a_a_a,Zs: list_P7200181061086074537_a_a_a] :
          ( ( Xs
            = ( cons_P1662640394034814435_a_a_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y3 )
          & ( ( map_Pr3622043693041699552_a_a_b @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_797_Cons__eq__map__conv,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( cons_S2140548516868915206_a_a_a @ X3 @ Xs )
        = ( map_Pr7354669405448068282_a_a_a @ F @ Ys ) )
      = ( ? [Z4: produc4770514982935793080_a_a_b,Zs2: list_P7155397214832950472_a_a_b] :
            ( ( Ys
              = ( cons_P2262690024297812098_a_a_b @ Z4 @ Zs2 ) )
            & ( X3
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr7354669405448068282_a_a_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_798_Cons__eq__map__conv,axiom,
    ! [X3: b,Xs: list_b,F: produc4770514982935793080_a_a_b > b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( cons_b @ X3 @ Xs )
        = ( map_Pr8925698157833054977_a_b_b @ F @ Ys ) )
      = ( ? [Z4: produc4770514982935793080_a_a_b,Zs2: list_P7155397214832950472_a_a_b] :
            ( ( Ys
              = ( cons_P2262690024297812098_a_a_b @ Z4 @ Zs2 ) )
            & ( X3
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr8925698157833054977_a_b_b @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_799_Cons__eq__map__conv,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( cons_S2140548516868915206_a_a_a @ X3 @ Xs )
        = ( map_Pr1074589380674644461_a_a_a @ F @ Ys ) )
      = ( ? [Z4: produc1895207096998757355_a_a_a,Zs2: list_P1813520604369389947_a_a_a] :
            ( ( Ys
              = ( cons_P8610754175215552181_a_a_a @ Z4 @ Zs2 ) )
            & ( X3
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr1074589380674644461_a_a_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_800_Cons__eq__map__conv,axiom,
    ! [X3: c,Xs: list_c,F: produc1895207096998757355_a_a_a > c,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( cons_c @ X3 @ Xs )
        = ( map_Pr2287363492687751055_a_a_c @ F @ Ys ) )
      = ( ? [Z4: produc1895207096998757355_a_a_a,Zs2: list_P1813520604369389947_a_a_a] :
            ( ( Ys
              = ( cons_P8610754175215552181_a_a_a @ Z4 @ Zs2 ) )
            & ( X3
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr2287363492687751055_a_a_c @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_801_Cons__eq__map__conv,axiom,
    ! [X3: produc1895207096998757355_a_a_a,Xs: list_P1813520604369389947_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( cons_P8610754175215552181_a_a_a @ X3 @ Xs )
        = ( map_Pr3321722608670633418_a_a_a @ F @ Ys ) )
      = ( ? [Z4: produc7810058533641270553_a_a_a,Zs2: list_P7200181061086074537_a_a_a] :
            ( ( Ys
              = ( cons_P1662640394034814435_a_a_a @ Z4 @ Zs2 ) )
            & ( X3
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr3321722608670633418_a_a_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_802_Cons__eq__map__conv,axiom,
    ! [X3: b,Xs: list_b,F: produc7810058533641270553_a_a_a > b,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( cons_b @ X3 @ Xs )
        = ( map_Pr3622043693041699552_a_a_b @ F @ Ys ) )
      = ( ? [Z4: produc7810058533641270553_a_a_a,Zs2: list_P7200181061086074537_a_a_a] :
            ( ( Ys
              = ( cons_P1662640394034814435_a_a_a @ Z4 @ Zs2 ) )
            & ( X3
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr3622043693041699552_a_a_b @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_803_map__eq__Cons__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b,Y3: sum_su5154756782089200444_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( ( map_Pr7354669405448068282_a_a_a @ F @ Xs )
        = ( cons_S2140548516868915206_a_a_a @ Y3 @ Ys ) )
      = ( ? [Z4: produc4770514982935793080_a_a_b,Zs2: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( cons_P2262690024297812098_a_a_b @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y3 )
            & ( ( map_Pr7354669405448068282_a_a_a @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_804_map__eq__Cons__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,Y3: b,Ys: list_b] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = ( cons_b @ Y3 @ Ys ) )
      = ( ? [Z4: produc4770514982935793080_a_a_b,Zs2: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( cons_P2262690024297812098_a_a_b @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y3 )
            & ( ( map_Pr8925698157833054977_a_b_b @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_805_map__eq__Cons__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,Y3: sum_su5154756782089200444_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = ( cons_S2140548516868915206_a_a_a @ Y3 @ Ys ) )
      = ( ? [Z4: produc1895207096998757355_a_a_a,Zs2: list_P1813520604369389947_a_a_a] :
            ( ( Xs
              = ( cons_P8610754175215552181_a_a_a @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y3 )
            & ( ( map_Pr1074589380674644461_a_a_a @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_806_map__eq__Cons__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,Y3: c,Ys: list_c] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = ( cons_c @ Y3 @ Ys ) )
      = ( ? [Z4: produc1895207096998757355_a_a_a,Zs2: list_P1813520604369389947_a_a_a] :
            ( ( Xs
              = ( cons_P8610754175215552181_a_a_a @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y3 )
            & ( ( map_Pr2287363492687751055_a_a_c @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_807_map__eq__Cons__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a,Y3: produc1895207096998757355_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr3321722608670633418_a_a_a @ F @ Xs )
        = ( cons_P8610754175215552181_a_a_a @ Y3 @ Ys ) )
      = ( ? [Z4: produc7810058533641270553_a_a_a,Zs2: list_P7200181061086074537_a_a_a] :
            ( ( Xs
              = ( cons_P1662640394034814435_a_a_a @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y3 )
            & ( ( map_Pr3321722608670633418_a_a_a @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_808_map__eq__Cons__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,Y3: b,Ys: list_b] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = ( cons_b @ Y3 @ Ys ) )
      = ( ? [Z4: produc7810058533641270553_a_a_a,Zs2: list_P7200181061086074537_a_a_a] :
            ( ( Xs
              = ( cons_P1662640394034814435_a_a_a @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y3 )
            & ( ( map_Pr3622043693041699552_a_a_b @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_809_visited__states_Osimps,axiom,
    ( visite8040902618889214550_a_b_c
    = ( ^ [Q5: sum_su5154756782089200444_a_a_a,P6: list_P8197596098214363964_a_a_a] :
          ( cons_S2140548516868915206_a_a_a @ Q5
          @ ( map_Pr1409128909316453038_a_a_a
            @ ^ [A4: produc884577481120224310_a_a_a] : ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ A4 ) ) )
            @ P6 ) ) ) ) ).

% visited_states.simps
thf(fact_810_visited__states_Osimps,axiom,
    ( visite6172159911691885821_a_a_b
    = ( ^ [Q5: produc1895207096998757355_a_a_a,P6: list_P3520633289758869904_a_a_a] :
          ( cons_P8610754175215552181_a_a_a @ Q5
          @ ( map_Pr3998683999746195505_a_a_a
            @ ^ [A4: produc6800189296593107456_a_a_a] : ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ A4 ) ) )
            @ P6 ) ) ) ) ).

% visited_states.simps
thf(fact_811_visited__states_Oelims,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xa: list_P8197596098214363964_a_a_a,Y3: list_S8072159660131056972_a_a_a] :
      ( ( ( visite8040902618889214550_a_b_c @ X3 @ Xa )
        = Y3 )
     => ( Y3
        = ( cons_S2140548516868915206_a_a_a @ X3
          @ ( map_Pr1409128909316453038_a_a_a
            @ ^ [A4: produc884577481120224310_a_a_a] : ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ A4 ) ) )
            @ Xa ) ) ) ) ).

% visited_states.elims
thf(fact_812_visited__states_Oelims,axiom,
    ! [X3: produc1895207096998757355_a_a_a,Xa: list_P3520633289758869904_a_a_a,Y3: list_P1813520604369389947_a_a_a] :
      ( ( ( visite6172159911691885821_a_a_b @ X3 @ Xa )
        = Y3 )
     => ( Y3
        = ( cons_P8610754175215552181_a_a_a @ X3
          @ ( map_Pr3998683999746195505_a_a_a
            @ ^ [A4: produc6800189296593107456_a_a_a] : ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ A4 ) ) )
            @ Xa ) ) ) ) ).

% visited_states.elims
thf(fact_813_map__pair__fst,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a
        @ ( map_a_7860052162900579309od_a_a
          @ ^ [X: a] : ( product_Pair_a_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_814_map__pair__fst,axiom,
    ! [F: sum_su5154756782089200444_a_a_a > b,Xs: list_S8072159660131056972_a_a_a] :
      ( ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b
        @ ( map_Su3117778036294953914_a_a_b
          @ ^ [X: sum_su5154756782089200444_a_a_a] : ( produc6086470211827429482_a_a_b @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_815_map__pair__fst,axiom,
    ! [F: b > produc1895207096998757355_a_a_a,Xs: list_b] :
      ( ( map_Pr3622043693041699552_a_a_b @ produc5842599967860340391_a_a_a
        @ ( map_b_659468956176798676_a_a_a
          @ ^ [X: b] : ( produc1730628064597884819_a_a_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_816_map__pair__fst,axiom,
    ! [F: sum_su5154756782089200444_a_a_a > produc7810058533641270553_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( map_Pr1409128909316453038_a_a_a @ produc7477144754405272026_a_a_a
        @ ( map_Su2265628143642336180_a_a_a
          @ ^ [X: sum_su5154756782089200444_a_a_a] : ( produc6863513614254923246_a_a_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_817_map__pair__fst,axiom,
    ! [F: c > sum_su5154756782089200444_a_a_a,Xs: list_c] :
      ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a
        @ ( map_c_707547689978722727_a_a_a
          @ ^ [X: c] : ( produc8806894744968565733_a_a_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_818_list__induct4,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,Zs3: list_P7155397214832950472_a_a_b,Ws: list_P7155397214832950472_a_a_b,P: list_P7155397214832950472_a_a_b > list_P7155397214832950472_a_a_b > list_P7155397214832950472_a_a_b > list_P7155397214832950472_a_a_b > $o] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( ( size_s1546398270689948508_a_a_b @ Ys )
          = ( size_s1546398270689948508_a_a_b @ Zs3 ) )
       => ( ( ( size_s1546398270689948508_a_a_b @ Zs3 )
            = ( size_s1546398270689948508_a_a_b @ Ws ) )
         => ( ( P @ nil_Pr4580708590200510514_a_a_b @ nil_Pr4580708590200510514_a_a_b @ nil_Pr4580708590200510514_a_a_b @ nil_Pr4580708590200510514_a_a_b )
           => ( ! [X4: produc4770514982935793080_a_a_b,Xs3: list_P7155397214832950472_a_a_b,Y: produc4770514982935793080_a_a_b,Ys4: list_P7155397214832950472_a_a_b,Z: produc4770514982935793080_a_a_b,Zs: list_P7155397214832950472_a_a_b,W: produc4770514982935793080_a_a_b,Ws2: list_P7155397214832950472_a_a_b] :
                  ( ( ( size_s1546398270689948508_a_a_b @ Xs3 )
                    = ( size_s1546398270689948508_a_a_b @ Ys4 ) )
                 => ( ( ( size_s1546398270689948508_a_a_b @ Ys4 )
                      = ( size_s1546398270689948508_a_a_b @ Zs ) )
                   => ( ( ( size_s1546398270689948508_a_a_b @ Zs )
                        = ( size_s1546398270689948508_a_a_b @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws2 )
                       => ( P @ ( cons_P2262690024297812098_a_a_b @ X4 @ Xs3 ) @ ( cons_P2262690024297812098_a_a_b @ Y @ Ys4 ) @ ( cons_P2262690024297812098_a_a_b @ Z @ Zs ) @ ( cons_P2262690024297812098_a_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_819_list__induct3,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,Zs3: list_P7155397214832950472_a_a_b,P: list_P7155397214832950472_a_a_b > list_P7155397214832950472_a_a_b > list_P7155397214832950472_a_a_b > $o] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( ( size_s1546398270689948508_a_a_b @ Ys )
          = ( size_s1546398270689948508_a_a_b @ Zs3 ) )
       => ( ( P @ nil_Pr4580708590200510514_a_a_b @ nil_Pr4580708590200510514_a_a_b @ nil_Pr4580708590200510514_a_a_b )
         => ( ! [X4: produc4770514982935793080_a_a_b,Xs3: list_P7155397214832950472_a_a_b,Y: produc4770514982935793080_a_a_b,Ys4: list_P7155397214832950472_a_a_b,Z: produc4770514982935793080_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
                ( ( ( size_s1546398270689948508_a_a_b @ Xs3 )
                  = ( size_s1546398270689948508_a_a_b @ Ys4 ) )
               => ( ( ( size_s1546398270689948508_a_a_b @ Ys4 )
                    = ( size_s1546398270689948508_a_a_b @ Zs ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs )
                   => ( P @ ( cons_P2262690024297812098_a_a_b @ X4 @ Xs3 ) @ ( cons_P2262690024297812098_a_a_b @ Y @ Ys4 ) @ ( cons_P2262690024297812098_a_a_b @ Z @ Zs ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_820_list__induct2,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,P: list_P7155397214832950472_a_a_b > list_P7155397214832950472_a_a_b > $o] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( P @ nil_Pr4580708590200510514_a_a_b @ nil_Pr4580708590200510514_a_a_b )
       => ( ! [X4: produc4770514982935793080_a_a_b,Xs3: list_P7155397214832950472_a_a_b,Y: produc4770514982935793080_a_a_b,Ys4: list_P7155397214832950472_a_a_b] :
              ( ( ( size_s1546398270689948508_a_a_b @ Xs3 )
                = ( size_s1546398270689948508_a_a_b @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons_P2262690024297812098_a_a_b @ X4 @ Xs3 ) @ ( cons_P2262690024297812098_a_a_b @ Y @ Ys4 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_821_distinct_Osimps_I2_J,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( distin5648706335258269743_a_a_a @ ( cons_P6318101215856909548_a_a_a @ X3 @ Xs ) )
      = ( ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
        & ( distin5648706335258269743_a_a_a @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_822_distinct_Osimps_I2_J,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( distin5156058890512421251_a_a_a @ ( cons_S2140548516868915206_a_a_a @ X3 @ Xs ) )
      = ( ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
        & ( distin5156058890512421251_a_a_a @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_823_distinct_Osimps_I2_J,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( distin1372936652459515135_a_a_b @ ( cons_P2262690024297812098_a_a_b @ X3 @ Xs ) )
      = ( ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
        & ( distin1372936652459515135_a_a_b @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_824_observable__path__unique,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a,P4: list_P3520633289758869904_a_a_a,P7: list_P3520633289758869904_a_a_a] :
      ( ( observ7609480127267726238_a_a_b @ M )
     => ( ( path_P4574188132515142236_a_a_b @ M @ Q @ P4 )
       => ( ( path_P4574188132515142236_a_a_b @ M @ Q @ P7 )
         => ( ( ( map_Pr6873991885683231230_a_a_b
                @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                @ P4 )
              = ( map_Pr6873991885683231230_a_a_b
                @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                @ P7 ) )
           => ( P4 = P7 ) ) ) ) ) ).

% observable_path_unique
thf(fact_825_observable__path__unique,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,P7: list_P8197596098214363964_a_a_a] :
      ( ( observ6810049560328217077_a_b_c @ M )
     => ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
       => ( ( path_S221595587009941175_a_b_c @ M @ Q @ P7 )
         => ( ( ( map_Pr6504563922737449846od_b_c
                @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                @ P4 )
              = ( map_Pr6504563922737449846od_b_c
                @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                @ P7 ) )
           => ( P4 = P7 ) ) ) ) ) ).

% observable_path_unique
thf(fact_826_set__map__subset,axiom,
    ! [X3: produc1895207096998757355_a_a_a,Xs: list_P1813520604369389947_a_a_a,T: c,F: produc1895207096998757355_a_a_a > c] :
      ( ( member6990616694304513154_a_a_a @ X3 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
     => ( ( member_c @ T @ ( set_c2 @ ( map_Pr2287363492687751055_a_a_c @ F @ ( cons_P8610754175215552181_a_a_a @ X3 @ nil_Pr1705400704263474789_a_a_a ) ) ) )
       => ( member_c @ T @ ( set_c2 @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_827_set__map__subset,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,T: b,F: produc4770514982935793080_a_a_b > b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ( ( member_b @ T @ ( set_b2 @ ( map_Pr8925698157833054977_a_b_b @ F @ ( cons_P2262690024297812098_a_a_b @ X3 @ nil_Pr4580708590200510514_a_a_b ) ) ) )
       => ( member_b @ T @ ( set_b2 @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_828_set__map__subset,axiom,
    ! [X3: produc7810058533641270553_a_a_a,Xs: list_P7200181061086074537_a_a_a,T: b,F: produc7810058533641270553_a_a_a > b] :
      ( ( member54949653122719408_a_a_a @ X3 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
     => ( ( member_b @ T @ ( set_b2 @ ( map_Pr3622043693041699552_a_a_b @ F @ ( cons_P1662640394034814435_a_a_a @ X3 @ nil_Pr1310111084261799315_a_a_a ) ) ) )
       => ( member_b @ T @ ( set_b2 @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_829_set__map__subset,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,T: sum_su5154756782089200444_a_a_a,F: sum_su5154756782089200444_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
     => ( ( member5679158588528110291_a_a_a @ T @ ( set_Su3113867532901839207_a_a_a @ ( map_Su7760560175274971966_a_a_a @ F @ ( cons_S2140548516868915206_a_a_a @ X3 @ nil_Su3335689033985374134_a_a_a ) ) ) )
       => ( member5679158588528110291_a_a_a @ T @ ( set_Su3113867532901839207_a_a_a @ ( map_Su7760560175274971966_a_a_a @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_830_set__map__subset,axiom,
    ! [X3: produc1895207096998757355_a_a_a,Xs: list_P1813520604369389947_a_a_a,T: sum_su5154756782089200444_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( member6990616694304513154_a_a_a @ X3 @ ( set_Pr1702711106814914326_a_a_a @ Xs ) )
     => ( ( member5679158588528110291_a_a_a @ T @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ ( cons_P8610754175215552181_a_a_a @ X3 @ nil_Pr1705400704263474789_a_a_a ) ) ) )
       => ( member5679158588528110291_a_a_a @ T @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_831_set__map__subset,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,T: produc4770514982935793080_a_a_b,F: sum_su5154756782089200444_a_a_a > produc4770514982935793080_a_a_b] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
     => ( ( member642552543386773071_a_a_b @ T @ ( set_Pr4578018992751950051_a_a_b @ ( map_Su3117778036294953914_a_a_b @ F @ ( cons_S2140548516868915206_a_a_a @ X3 @ nil_Su3335689033985374134_a_a_a ) ) ) )
       => ( member642552543386773071_a_a_b @ T @ ( set_Pr4578018992751950051_a_a_b @ ( map_Su3117778036294953914_a_a_b @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_832_set__map__subset,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,T: sum_su5154756782089200444_a_a_a,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ( ( member5679158588528110291_a_a_a @ T @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ ( cons_P2262690024297812098_a_a_b @ X3 @ nil_Pr4580708590200510514_a_a_b ) ) ) )
       => ( member5679158588528110291_a_a_a @ T @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_833_set__map__subset,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,T: produc4770514982935793080_a_a_b,F: produc4770514982935793080_a_a_b > produc4770514982935793080_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ( ( member642552543386773071_a_a_b @ T @ ( set_Pr4578018992751950051_a_a_b @ ( map_Pr2859183448225536310_a_a_b @ F @ ( cons_P2262690024297812098_a_a_b @ X3 @ nil_Pr4580708590200510514_a_a_b ) ) ) )
       => ( member642552543386773071_a_a_b @ T @ ( set_Pr4578018992751950051_a_a_b @ ( map_Pr2859183448225536310_a_a_b @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_834_set__map__subset,axiom,
    ! [X3: produc7810058533641270553_a_a_a,Xs: list_P7200181061086074537_a_a_a,T: produc1895207096998757355_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a] :
      ( ( member54949653122719408_a_a_a @ X3 @ ( set_Pr6411044387506519364_a_a_a @ Xs ) )
     => ( ( member6990616694304513154_a_a_a @ T @ ( set_Pr1702711106814914326_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ ( cons_P1662640394034814435_a_a_a @ X3 @ nil_Pr1310111084261799315_a_a_a ) ) ) )
       => ( member6990616694304513154_a_a_a @ T @ ( set_Pr1702711106814914326_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_835_set__map__subset,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a,T: sum_su5154756782089200444_a_a_a,F: produc884577481120224310_a_a_a > sum_su5154756782089200444_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
     => ( ( member5679158588528110291_a_a_a @ T @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr1409128909316453038_a_a_a @ F @ ( cons_P6318101215856909548_a_a_a @ X3 @ nil_Pr6029279252097497404_a_a_a ) ) ) )
       => ( member5679158588528110291_a_a_a @ T @ ( set_Su3113867532901839207_a_a_a @ ( map_Pr1409128909316453038_a_a_a @ F @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_836_single__transition__path,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_Su3173892059090327343_a_b_c] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
     => ( path_S221595587009941175_a_b_c @ M @ ( produc7477144754405272026_a_a_a @ T ) @ ( cons_P6318101215856909548_a_a_a @ T @ nil_Pr6029279252097497404_a_a_a ) ) ) ).

% single_transition_path
thf(fact_837_observable__path__io__target,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a,P4: list_P3520633289758869904_a_a_a] :
      ( ( observ7609480127267726238_a_a_b @ M )
     => ( ( path_P4574188132515142236_a_a_b @ M @ Q @ P4 )
       => ( ( io_tar1879755107597495661_a_a_b @ M
            @ ( map_Pr6873991885683231230_a_a_b
              @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
              @ P4 )
            @ Q )
          = ( insert5166965059073188379_a_a_a @ ( target4721681511209552016_a_a_b @ Q @ P4 ) @ bot_bo8934037479225353397_a_a_a ) ) ) ) ).

% observable_path_io_target
thf(fact_838_observable__path__io__target,axiom,
    ! [M: fsm_Su4683625031830649393_a_a_b,Q: sum_su5154756782089200444_a_a_a,P4: list_P9067677309952966352_a_a_a] :
      ( ( observ8563788498004600205_a_a_b @ M )
     => ( ( path_S4752583732068403787_a_a_b @ M @ Q @ P4 )
       => ( ( io_tar7000603075844232796_a_a_b @ M
            @ ( map_Pr5810527985649246014_a_a_b
              @ ^ [T4: produc8681278163233174848_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3329354246078436523_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) @ ( produc9179341348180701688_a_a_a @ ( produc3812793320627011309_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) ) )
              @ P4 )
            @ Q )
          = ( insert5061175218292437356_a_a_a @ ( target5129991884641307775_a_a_b @ Q @ P4 ) @ bot_bo2676859181106700934_a_a_a ) ) ) ) ).

% observable_path_io_target
thf(fact_839_observable__path__io__target,axiom,
    ! [M: fsm_Su5961347632456545774_a_a_a,Q: sum_su5154756782089200444_a_a_a,P4: list_P4430297574055495993_a_a_a] :
      ( ( observ374133105446117300_a_a_a @ M )
     => ( ( path_S3009051168982617206_a_a_a @ M @ Q @ P4 )
       => ( ( io_tar4514536438336734117_a_a_a @ M
            @ ( map_Pr7525866001273443320od_a_a
              @ ^ [T4: produc1141838204273412275_a_a_a] : ( product_Pair_a_a @ ( produc4618981482445519012_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) @ ( produc2350474654791476471_a_a_a @ ( produc5012511087789358306_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) ) )
              @ P4 )
            @ Q )
          = ( insert5061175218292437356_a_a_a @ ( target7811201204777689922_a_a_a @ Q @ P4 ) @ bot_bo2676859181106700934_a_a_a ) ) ) ) ).

% observable_path_io_target
thf(fact_840_observable__path__io__target,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a] :
      ( ( observ6810049560328217077_a_b_c @ M )
     => ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
       => ( ( io_tar1727080856364058086_a_b_c @ M
            @ ( map_Pr6504563922737449846od_b_c
              @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
              @ P4 )
            @ Q )
          = ( insert5061175218292437356_a_a_a @ ( target5023745622805013891_a_b_c @ Q @ P4 ) @ bot_bo2676859181106700934_a_a_a ) ) ) ) ).

% observable_path_io_target
thf(fact_841_observable__path__io__target,axiom,
    ! [M: fsm_Pr8211287145308421045_a_a_b,Q: produc4770514982935793080_a_a_b,P4: list_P8385316394189796816_a_a_b] :
      ( ( observ4191226443898060305_a_a_b @ M )
     => ( ( path_P1155934449145476303_a_a_b @ M @ Q @ P4 )
       => ( ( io_tar7684873461082605536_a_a_b @ M
            @ ( map_Pr36629913606772286_a_a_b
              @ ^ [T4: produc2838380418727528000_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3595649328014056743_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) @ ( produc8717907853797376116_a_a_b @ ( produc6971105973522642281_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) ) )
              @ P4 )
            @ Q )
          = ( insert8042272945010224104_a_a_b @ ( target1303427827839886083_a_a_b @ Q @ P4 ) @ bot_bo5052542052834138114_a_a_b ) ) ) ) ).

% observable_path_io_target
thf(fact_842_observable__path__io__target,axiom,
    ! [M: fsm_Pr1352325423224000874_b_a_a,Q: produc4770514982935793080_a_a_b,P4: list_P9033668338835616561_a_a_b] :
      ( ( observ7839529218259264560_b_a_a @ M )
     => ( ( path_P447960203046551282_b_a_a @ M @ Q @ P4 )
       => ( ( io_tar7579544333240987425_b_a_a @ M
            @ ( map_Pr2203486926099502080od_a_a
              @ ^ [T4: produc4165969764426093739_a_a_b] : ( product_Pair_a_a @ ( produc2204968541190299680_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) @ ( produc2705278681451453555_a_a_b @ ( produc7577140250099103326_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) ) )
              @ P4 )
            @ Q )
          = ( insert8042272945010224104_a_a_b @ ( target4815919572193323454_b_a_a @ Q @ P4 ) @ bot_bo5052542052834138114_a_a_b ) ) ) ) ).

% observable_path_io_target
thf(fact_843_observable__path__io__target,axiom,
    ! [M: fsm_Pr791368616075801221_a_a_b,Q: produc884577481120224310_a_a_a,P4: list_P1303110702266167372_a_a_a] :
      ( ( observ4404008968300781955_a_a_b @ M )
     => ( ( path_P6406173025370084677_a_a_b @ M @ Q @ P4 )
       => ( ( io_tar7108072963341213940_a_a_b @ M
            @ ( map_Pr2767717211958206138_a_a_b
              @ ^ [T4: produc4009731802749794876_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc1606485967678404637_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) @ ( produc5695092443052942970_a_a_a @ ( produc3123543664576935259_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) ) )
              @ P4 )
            @ Q )
          = ( insert2249508923356641798_a_a_a @ ( target1704919491679417873_a_a_b @ Q @ P4 ) @ bot_bo2898616998271641474_a_a_a ) ) ) ) ).

% observable_path_io_target
thf(fact_844_observable__path__io__target,axiom,
    ! [M: fsm_Pr3307108581279186586_a_a_a,Q: produc884577481120224310_a_a_a,P4: list_P6192864029102241151_a_a_a] :
      ( ( observ1248204241357884222_a_a_a @ M )
     => ( ( path_P3388033272012105212_a_a_a @ M @ Q @ P4 )
       => ( ( io_tar2509673025420183309_a_a_a @ M
            @ ( map_Pr1392725673272416306od_a_a
              @ ^ [T4: produc7205803768117265017_a_a_a] : ( product_Pair_a_a @ ( produc5229955328659460580_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) @ ( produc1756692000304289723_a_a_a @ ( produc30913607518176806_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) ) )
              @ P4 )
            @ Q )
          = ( insert2249508923356641798_a_a_a @ ( target2673679534084926512_a_a_a @ Q @ P4 ) @ bot_bo2898616998271641474_a_a_a ) ) ) ) ).

% observable_path_io_target
thf(fact_845_path__prepend__t,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q6: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,Q: sum_su5154756782089200444_a_a_a,X3: b,Y3: c] :
      ( ( path_S221595587009941175_a_b_c @ M @ Q6 @ P4 )
     => ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q @ ( produc1730628064597884819_a_a_a @ X3 @ ( produc8806894744968565733_a_a_a @ Y3 @ Q6 ) ) ) @ ( transi749694810082594024_a_b_c @ M ) )
       => ( path_S221595587009941175_a_b_c @ M @ Q @ ( cons_P6318101215856909548_a_a_a @ ( produc6863513614254923246_a_a_a @ Q @ ( produc1730628064597884819_a_a_a @ X3 @ ( produc8806894744968565733_a_a_a @ Y3 @ Q6 ) ) ) @ P4 ) ) ) ) ).

% path_prepend_t
thf(fact_846_cons,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Ts: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
     => ( ( path_S221595587009941175_a_b_c @ M @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) @ Ts )
       => ( path_S221595587009941175_a_b_c @ M @ ( produc7477144754405272026_a_a_a @ T ) @ ( cons_P6318101215856909548_a_a_a @ T @ Ts ) ) ) ) ).

% cons
thf(fact_847_cons,axiom,
    ! [T: produc6800189296593107456_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Ts: list_P3520633289758869904_a_a_a] :
      ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ M ) )
     => ( ( path_P4574188132515142236_a_a_b @ M @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) @ Ts )
       => ( path_P4574188132515142236_a_a_b @ M @ ( produc2316789063618113990_a_a_a @ T ) @ ( cons_P5563258090397451722_a_a_a @ T @ Ts ) ) ) ) ).

% cons
thf(fact_848_path__cons__elim,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,T: produc884577481120224310_a_a_a,Ts: list_P8197596098214363964_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ M @ Q @ ( cons_P6318101215856909548_a_a_a @ T @ Ts ) )
     => ~ ( ( Q
            = ( produc7477144754405272026_a_a_a @ T ) )
         => ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
           => ~ ( path_S221595587009941175_a_b_c @ M @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) @ Ts ) ) ) ) ).

% path_cons_elim
thf(fact_849_path__cons__elim,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a,T: produc6800189296593107456_a_a_a,Ts: list_P3520633289758869904_a_a_a] :
      ( ( path_P4574188132515142236_a_a_b @ M @ Q @ ( cons_P5563258090397451722_a_a_a @ T @ Ts ) )
     => ~ ( ( Q
            = ( produc2316789063618113990_a_a_a @ T ) )
         => ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ M ) )
           => ~ ( path_P4574188132515142236_a_a_b @ M @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) @ Ts ) ) ) ) ).

% path_cons_elim
thf(fact_850_canonical__separator__path__initial__ex_I1_J,axiom,
    ! [M: fsm_Su4683625031830649393_a_a_b,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,P4: list_P8614482820184752236_a_a_a] :
      ( ( path_S6747785721213197493_a_a_b @ ( state_2979059454677389811_a_a_b @ M @ Q1 @ Q2 ) @ ( initia8581304622896136204_a_a_b @ ( state_2979059454677389811_a_a_b @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3538144058484115470_a_a_b @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3538144058484115470_a_a_b @ M ) )
         => ( ? [P1: list_P9067677309952966352_a_a_a] :
                ( ( path_S4752583732068403787_a_a_b @ M @ Q1 @ P1 )
                & ( ( map_Pr5810527985649246014_a_a_b
                    @ ^ [T4: produc8681278163233174848_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3329354246078436523_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) @ ( produc9179341348180701688_a_a_a @ ( produc3812793320627011309_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) ) )
                    @ P1 )
                  = ( map_Pr7095168675738074842_a_a_b
                    @ ^ [T4: produc8336027307173600092_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc5719501463807155629_a_a_a @ ( produc2127940979268284968_a_a_a @ T4 ) ) @ ( produc6788288432653770762_a_a_a @ ( produc248128219081304043_a_a_a @ ( produc2127940979268284968_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
            | ? [P22: list_P9067677309952966352_a_a_a] :
                ( ( path_S4752583732068403787_a_a_b @ M @ Q2 @ P22 )
                & ( ( map_Pr5810527985649246014_a_a_b
                    @ ^ [T4: produc8681278163233174848_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3329354246078436523_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) @ ( produc9179341348180701688_a_a_a @ ( produc3812793320627011309_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) ) )
                    @ P22 )
                  = ( map_Pr7095168675738074842_a_a_b
                    @ ^ [T4: produc8336027307173600092_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc5719501463807155629_a_a_a @ ( produc2127940979268284968_a_a_a @ T4 ) ) @ ( produc6788288432653770762_a_a_a @ ( produc248128219081304043_a_a_a @ ( produc2127940979268284968_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(1)
thf(fact_851_canonical__separator__path__initial__ex_I1_J,axiom,
    ! [M: fsm_Pr791368616075801221_a_a_b,Q1: produc884577481120224310_a_a_a,Q2: produc884577481120224310_a_a_a,P4: list_P3072728476344414928_a_a_a] :
      ( ( path_S1304752709593314579_a_a_b @ ( state_3609830002750407581_a_a_b @ M @ Q1 @ Q2 ) @ ( initia2424258989328714876_a_a_b @ ( state_3609830002750407581_a_a_b @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member8362999488266169055_a_a_a @ Q1 @ ( states6021486123142740674_a_a_b @ M ) )
       => ( ( member8362999488266169055_a_a_a @ Q2 @ ( states6021486123142740674_a_a_b @ M ) )
         => ( ? [P1: list_P1303110702266167372_a_a_a] :
                ( ( path_P6406173025370084677_a_a_b @ M @ Q1 @ P1 )
                & ( ( map_Pr2767717211958206138_a_a_b
                    @ ^ [T4: produc4009731802749794876_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc1606485967678404637_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) @ ( produc5695092443052942970_a_a_a @ ( produc3123543664576935259_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) ) )
                    @ P1 )
                  = ( map_Pr1942940760749364030_a_a_b
                    @ ^ [T4: produc4612780081894834240_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3848882253140696291_a_a_a @ ( produc8453961691671531716_a_a_a @ T4 ) ) @ ( produc5739986182704136752_a_a_a @ ( produc8858187255993310245_a_a_a @ ( produc8453961691671531716_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
            | ? [P22: list_P1303110702266167372_a_a_a] :
                ( ( path_P6406173025370084677_a_a_b @ M @ Q2 @ P22 )
                & ( ( map_Pr2767717211958206138_a_a_b
                    @ ^ [T4: produc4009731802749794876_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc1606485967678404637_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) @ ( produc5695092443052942970_a_a_a @ ( produc3123543664576935259_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) ) )
                    @ P22 )
                  = ( map_Pr1942940760749364030_a_a_b
                    @ ^ [T4: produc4612780081894834240_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3848882253140696291_a_a_a @ ( produc8453961691671531716_a_a_a @ T4 ) ) @ ( produc5739986182704136752_a_a_a @ ( produc8858187255993310245_a_a_a @ ( produc8453961691671531716_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(1)
thf(fact_852_canonical__separator__path__initial__ex_I1_J,axiom,
    ! [M: fsm_Pr8211287145308421045_a_a_b,Q1: produc4770514982935793080_a_a_b,Q2: produc4770514982935793080_a_a_b,P4: list_P2242549886795509588_a_a_b] :
      ( ( path_S264780079786346561_a_a_b @ ( state_4270792120431400055_a_a_b @ M @ Q1 @ Q2 ) @ ( initia1897159703354810008_a_a_b @ ( state_4270792120431400055_a_a_b @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member642552543386773071_a_a_b @ Q1 @ ( states5045336508805429138_a_a_b @ M ) )
       => ( ( member642552543386773071_a_a_b @ Q2 @ ( states5045336508805429138_a_a_b @ M ) )
         => ( ? [P1: list_P8385316394189796816_a_a_b] :
                ( ( path_P1155934449145476303_a_a_b @ M @ Q1 @ P1 )
                & ( ( map_Pr36629913606772286_a_a_b
                    @ ^ [T4: produc2838380418727528000_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3595649328014056743_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) @ ( produc8717907853797376116_a_a_b @ ( produc6971105973522642281_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) ) )
                    @ P1 )
                  = ( map_Pr4542665781501985218_a_a_b
                    @ ^ [T4: produc123133041302015044_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3312939215451498273_a_a_b @ ( produc7693017963871686416_a_a_b @ T4 ) ) @ ( produc6995861152403393406_a_a_b @ ( produc3169017415322691423_a_a_b @ ( produc7693017963871686416_a_a_b @ T4 ) ) ) )
                    @ P4 ) ) )
            | ? [P22: list_P8385316394189796816_a_a_b] :
                ( ( path_P1155934449145476303_a_a_b @ M @ Q2 @ P22 )
                & ( ( map_Pr36629913606772286_a_a_b
                    @ ^ [T4: produc2838380418727528000_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3595649328014056743_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) @ ( produc8717907853797376116_a_a_b @ ( produc6971105973522642281_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) ) )
                    @ P22 )
                  = ( map_Pr4542665781501985218_a_a_b
                    @ ^ [T4: produc123133041302015044_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3312939215451498273_a_a_b @ ( produc7693017963871686416_a_a_b @ T4 ) ) @ ( produc6995861152403393406_a_a_b @ ( produc3169017415322691423_a_a_b @ ( produc7693017963871686416_a_a_b @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(1)
thf(fact_853_canonical__separator__path__initial__ex_I1_J,axiom,
    ! [M: fsm_Su5961347632456545774_a_a_a,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,P4: list_P4201227678853489407_a_a_a] :
      ( ( path_S4742588529375693452_a_a_a @ ( state_7218859947466147534_a_a_a @ M @ Q1 @ Q2 ) @ ( initia55471205950526965_a_a_a @ ( state_7218859947466147534_a_a_a @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states5822841456425053299_a_a_a @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states5822841456425053299_a_a_a @ M ) )
         => ( ? [P1: list_P4430297574055495993_a_a_a] :
                ( ( path_S3009051168982617206_a_a_a @ M @ Q1 @ P1 )
                & ( ( map_Pr7525866001273443320od_a_a
                    @ ^ [T4: produc1141838204273412275_a_a_a] : ( product_Pair_a_a @ ( produc4618981482445519012_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) @ ( produc2350474654791476471_a_a_a @ ( produc5012511087789358306_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) ) )
                    @ P1 )
                  = ( map_Pr1037215490889484978od_a_a
                    @ ^ [T4: produc8556884082142916345_a_a_a] : ( product_Pair_a_a @ ( produc5296690695870061044_a_a_a @ ( produc8103174256791760663_a_a_a @ T4 ) ) @ ( produc7776879804106701515_a_a_a @ ( produc1372291505885593910_a_a_a @ ( produc8103174256791760663_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
            | ? [P22: list_P4430297574055495993_a_a_a] :
                ( ( path_S3009051168982617206_a_a_a @ M @ Q2 @ P22 )
                & ( ( map_Pr7525866001273443320od_a_a
                    @ ^ [T4: produc1141838204273412275_a_a_a] : ( product_Pair_a_a @ ( produc4618981482445519012_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) @ ( produc2350474654791476471_a_a_a @ ( produc5012511087789358306_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) ) )
                    @ P22 )
                  = ( map_Pr1037215490889484978od_a_a
                    @ ^ [T4: produc8556884082142916345_a_a_a] : ( product_Pair_a_a @ ( produc5296690695870061044_a_a_a @ ( produc8103174256791760663_a_a_a @ T4 ) ) @ ( produc7776879804106701515_a_a_a @ ( produc1372291505885593910_a_a_a @ ( produc8103174256791760663_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(1)
thf(fact_854_canonical__separator__path__initial__ex_I1_J,axiom,
    ! [M: fsm_Pr3307108581279186586_a_a_a,Q1: produc884577481120224310_a_a_a,Q2: produc884577481120224310_a_a_a,P4: list_P953336824429746729_a_a_a] :
      ( ( path_S4361706828270291886_a_a_a @ ( state_6737487012141364644_a_a_a @ M @ Q1 @ Q2 ) @ ( initia9181111417782089093_a_a_a @ ( state_6737487012141364644_a_a_a @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member8362999488266169055_a_a_a @ Q1 @ ( states7932151110703657663_a_a_a @ M ) )
       => ( ( member8362999488266169055_a_a_a @ Q2 @ ( states7932151110703657663_a_a_a @ M ) )
         => ( ? [P1: list_P6192864029102241151_a_a_a] :
                ( ( path_P3388033272012105212_a_a_a @ M @ Q1 @ P1 )
                & ( ( map_Pr1392725673272416306od_a_a
                    @ ^ [T4: produc7205803768117265017_a_a_a] : ( product_Pair_a_a @ ( produc5229955328659460580_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) @ ( produc1756692000304289723_a_a_a @ ( produc30913607518176806_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) ) )
                    @ P1 )
                  = ( map_Pr7791883050161821448od_a_a
                    @ ^ [T4: produc6752683490396420771_a_a_a] : ( product_Pair_a_a @ ( produc8629031941109950300_a_a_a @ ( produc644117954929008137_a_a_a @ T4 ) ) @ ( produc3231407421338598063_a_a_a @ ( produc8609258968895674010_a_a_a @ ( produc644117954929008137_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
            | ? [P22: list_P6192864029102241151_a_a_a] :
                ( ( path_P3388033272012105212_a_a_a @ M @ Q2 @ P22 )
                & ( ( map_Pr1392725673272416306od_a_a
                    @ ^ [T4: produc7205803768117265017_a_a_a] : ( product_Pair_a_a @ ( produc5229955328659460580_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) @ ( produc1756692000304289723_a_a_a @ ( produc30913607518176806_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) ) )
                    @ P22 )
                  = ( map_Pr7791883050161821448od_a_a
                    @ ^ [T4: produc6752683490396420771_a_a_a] : ( product_Pair_a_a @ ( produc8629031941109950300_a_a_a @ ( produc644117954929008137_a_a_a @ T4 ) ) @ ( produc3231407421338598063_a_a_a @ ( produc8609258968895674010_a_a_a @ ( produc644117954929008137_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(1)
thf(fact_855_canonical__separator__path__initial__ex_I1_J,axiom,
    ! [M: fsm_Pr1352325423224000874_b_a_a,Q1: produc4770514982935793080_a_a_b,Q2: produc4770514982935793080_a_a_b,P4: list_P747566903215087359_a_a_b] :
      ( ( path_S8893950696295478784_b_a_a @ ( state_8792045420005744970_b_a_a @ M @ Q1 @ Q2 ) @ ( initia8327540589931236969_b_a_a @ ( state_8792045420005744970_b_a_a @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member642552543386773071_a_a_b @ Q1 @ ( states6742877326663703023_b_a_a @ M ) )
       => ( ( member642552543386773071_a_a_b @ Q2 @ ( states6742877326663703023_b_a_a @ M ) )
         => ( ? [P1: list_P9033668338835616561_a_a_b] :
                ( ( path_P447960203046551282_b_a_a @ M @ Q1 @ P1 )
                & ( ( map_Pr2203486926099502080od_a_a
                    @ ^ [T4: produc4165969764426093739_a_a_b] : ( product_Pair_a_a @ ( produc2204968541190299680_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) @ ( produc2705278681451453555_a_a_b @ ( produc7577140250099103326_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) ) )
                    @ P1 )
                  = ( map_Pr1120819969589644978od_a_a
                    @ ^ [T4: produc8368057995964161273_a_a_b] : ( product_Pair_a_a @ ( produc7806160504747017832_a_a_b @ ( produc410253929432769815_a_a_b @ T4 ) ) @ ( produc510337100525087551_a_a_b @ ( produc879673915631515562_a_a_b @ ( produc410253929432769815_a_a_b @ T4 ) ) ) )
                    @ P4 ) ) )
            | ? [P22: list_P9033668338835616561_a_a_b] :
                ( ( path_P447960203046551282_b_a_a @ M @ Q2 @ P22 )
                & ( ( map_Pr2203486926099502080od_a_a
                    @ ^ [T4: produc4165969764426093739_a_a_b] : ( product_Pair_a_a @ ( produc2204968541190299680_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) @ ( produc2705278681451453555_a_a_b @ ( produc7577140250099103326_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) ) )
                    @ P22 )
                  = ( map_Pr1120819969589644978od_a_a
                    @ ^ [T4: produc8368057995964161273_a_a_b] : ( product_Pair_a_a @ ( produc7806160504747017832_a_a_b @ ( produc410253929432769815_a_a_b @ T4 ) ) @ ( produc510337100525087551_a_a_b @ ( produc879673915631515562_a_a_b @ ( produc410253929432769815_a_a_b @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(1)
thf(fact_856_canonical__separator__path__initial__ex_I1_J,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a,P4: list_P1661186363840085254_a_a_a] :
      ( ( path_S5554404386621442024_a_a_b @ ( state_7689045803801065988_a_a_b @ M @ Q1 @ Q2 ) @ ( initia7186784010189905471_a_a_b @ ( state_7689045803801065988_a_a_b @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member6990616694304513154_a_a_a @ Q1 @ ( states8463590192175095071_a_a_b @ M ) )
       => ( ( member6990616694304513154_a_a_a @ Q2 @ ( states8463590192175095071_a_a_b @ M ) )
         => ( ? [P1: list_P3520633289758869904_a_a_a] :
                ( ( path_P4574188132515142236_a_a_b @ M @ Q1 @ P1 )
                & ( ( map_Pr6873991885683231230_a_a_b
                    @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                    @ P1 )
                  = ( map_Pr1452853287033278580_a_a_b
                    @ ^ [T4: produc2172198381475533558_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T4 ) ) @ ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
            | ? [P22: list_P3520633289758869904_a_a_a] :
                ( ( path_P4574188132515142236_a_a_b @ M @ Q2 @ P22 )
                & ( ( map_Pr6873991885683231230_a_a_b
                    @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                    @ P22 )
                  = ( map_Pr1452853287033278580_a_a_b
                    @ ^ [T4: produc2172198381475533558_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T4 ) ) @ ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(1)
thf(fact_857_canonical__separator__path__initial__ex_I1_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,P4: list_P1545942051602268092_a_a_a] :
      ( ( path_S1955132947403017421_a_b_c @ ( state_4431404365493471503_a_b_c @ M @ Q1 @ Q2 ) @ ( initia6491387660832626742_a_b_c @ ( state_4431404365493471503_a_b_c @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3035385874452377268_a_b_c @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3035385874452377268_a_b_c @ M ) )
         => ( ? [P1: list_P8197596098214363964_a_a_a] :
                ( ( path_S221595587009941175_a_b_c @ M @ Q1 @ P1 )
                & ( ( map_Pr6504563922737449846od_b_c
                    @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                    @ P1 )
                  = ( map_Pr7462116669582905078od_b_c
                    @ ^ [T4: produc1185401179908872758_a_a_a] : ( product_Pair_b_c @ ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T4 ) ) @ ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
            | ? [P22: list_P8197596098214363964_a_a_a] :
                ( ( path_S221595587009941175_a_b_c @ M @ Q2 @ P22 )
                & ( ( map_Pr6504563922737449846od_b_c
                    @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                    @ P22 )
                  = ( map_Pr7462116669582905078od_b_c
                    @ ^ [T4: produc1185401179908872758_a_a_a] : ( product_Pair_b_c @ ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T4 ) ) @ ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(1)
thf(fact_858_canonical__separator__path__initial__ex_I1_J,axiom,
    ! [M: fsm_a_b_c,Q1: a,Q2: a,P4: list_P8197596098214363964_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( ? [P1: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M @ Q1 @ P1 )
                & ( ( map_Pr776894537227705078od_b_c
                    @ ^ [T4: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) )
                    @ P1 )
                  = ( map_Pr6504563922737449846od_b_c
                    @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
            | ? [P22: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M @ Q2 @ P22 )
                & ( ( map_Pr776894537227705078od_b_c
                    @ ^ [T4: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) )
                    @ P22 )
                  = ( map_Pr6504563922737449846od_b_c
                    @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(1)
thf(fact_859_product__target__split_I1_J,axiom,
    ! [Q1: c,Q2: sum_su5154756782089200444_a_a_a,P4: list_P3520633289758869904_a_a_a,Q17: c,Q24: sum_su5154756782089200444_a_a_a] :
      ( ( ( target4721681511209552016_a_a_b @ ( produc8806894744968565733_a_a_a @ Q1 @ Q2 ) @ P4 )
        = ( produc8806894744968565733_a_a_a @ Q17 @ Q24 ) )
     => ( ( target1000401010099934919_a_a_b @ Q1
          @ ( map_Pr4103997051015666966od_b_c
            @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc9054217211726079306od_b_c @ ( produc6784836004715151097_a_a_a @ ( produc2316789063618113990_a_a_a @ T4 ) ) @ ( produc6554145002957781203od_b_c @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) ) ) ) )
            @ P4 ) )
        = Q17 ) ) ).

% product_target_split(1)
thf(fact_860_product__target__split_I2_J,axiom,
    ! [Q1: c,Q2: sum_su5154756782089200444_a_a_a,P4: list_P3520633289758869904_a_a_a,Q17: c,Q24: sum_su5154756782089200444_a_a_a] :
      ( ( ( target4721681511209552016_a_a_b @ ( produc8806894744968565733_a_a_a @ Q1 @ Q2 ) @ P4 )
        = ( produc8806894744968565733_a_a_a @ Q17 @ Q24 ) )
     => ( ( target5129991884641307775_a_a_b @ Q2
          @ ( map_Pr4032422291525893254_a_a_a
            @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc8935917708474127986_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc2316789063618113990_a_a_a @ T4 ) ) @ ( produc919293830850624703_a_a_a @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc1978028051579340516_a_a_a @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) ) ) ) )
            @ P4 ) )
        = Q24 ) ) ).

% product_target_split(2)
thf(fact_861_paths__for__input__def,axiom,
    ( paths_5737697329168360697_a_a_a
    = ( ^ [M3: fsm_b_2210266817785990191_a_a_a,Q5: b,Xs2: list_c] :
          ( collec5250301242499970682_a_a_b
          @ ^ [P6: list_P7566143005048781605_a_a_b] :
              ( ( path_b563861380003114551_a_a_a @ M3 @ Q5 @ P6 )
              & ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a
                  @ ( map_Pr5613174315646222406_a_a_a
                    @ ^ [T4: produc5464264772254188693_a_a_b] : ( produc8806894744968565733_a_a_a @ ( produc5507164989288522869_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) @ ( produc4064411471574014846_a_a_b @ ( produc5900694594632362163_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) ) )
                    @ P6 ) )
                = Xs2 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_862_paths__for__input__def,axiom,
    ( paths_2436134535752754919_a_a_a
    = ( ^ [M3: fsm_Pr6532079070102214215_a_a_a,Q5: produc7810058533641270553_a_a_a,Xs2: list_c] :
          ( collec2081221580412524894_a_a_a
          @ ^ [P6: list_P7090462827770577289_a_a_a] :
              ( ( path_P4743837418810288169_a_a_a @ M3 @ Q5 @ P6 )
              & ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a
                  @ ( map_Pr7391780466455013546_a_a_a
                    @ ^ [T4: produc8583226035453294713_a_a_a] : ( produc8806894744968565733_a_a_a @ ( produc410120848946820409_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) @ ( produc7477144754405272026_a_a_a @ ( produc1285651877402164091_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) ) )
                    @ P6 ) )
                = Xs2 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_863_paths__for__input__def,axiom,
    ( paths_1915428146606688026_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b,Q5: produc1895207096998757355_a_a_a,Xs2: list_S8072159660131056972_a_a_a] :
          ( collec3075913212816350053_a_a_a
          @ ^ [P6: list_P3520633289758869904_a_a_a] :
              ( ( path_P4574188132515142236_a_a_b @ M3 @ Q5 @ P6 )
              & ( ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b
                  @ ( map_Pr6873991885683231230_a_a_b
                    @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                    @ P6 ) )
                = Xs2 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_864_paths__for__input__def,axiom,
    ( paths_5395431536175187321_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c,Q5: sum_su5154756782089200444_a_a_a,Xs2: list_b] :
          ( collec406916415612872615_a_a_a
          @ ^ [P6: list_P8197596098214363964_a_a_a] :
              ( ( path_S221595587009941175_a_b_c @ M3 @ Q5 @ P6 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr6504563922737449846od_b_c
                    @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                    @ P6 ) )
                = Xs2 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_865_set__conv__nth,axiom,
    ( set_Su3113867532901839207_a_a_a
    = ( ^ [Xs2: list_S8072159660131056972_a_a_a] :
          ( collec299858375640228881_a_a_a
          @ ^ [Uu: sum_su5154756782089200444_a_a_a] :
            ? [I3: nat] :
              ( ( Uu
                = ( nth_Su7097239896299986421_a_a_a @ Xs2 @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_s7186666008910096352_a_a_a @ Xs2 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_866_set__conv__nth,axiom,
    ( set_Pr4578018992751950051_a_a_b
    = ( ^ [Xs2: list_P7155397214832950472_a_a_b] :
          ( collec4985708871760228749_a_a_b
          @ ^ [Uu: produc4770514982935793080_a_a_b] :
            ? [I3: nat] :
              ( ( Uu
                = ( nth_Pr167341066936870769_a_a_b @ Xs2 @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_s1546398270689948508_a_a_b @ Xs2 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_867_io__targets_Osimps,axiom,
    ( io_tar1879755107597495661_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b,Io2: list_P7155397214832950472_a_a_b,Q5: produc1895207096998757355_a_a_a] :
          ( collec2110400985823193024_a_a_a
          @ ^ [Uu: produc1895207096998757355_a_a_a] :
            ? [P6: list_P3520633289758869904_a_a_a] :
              ( ( Uu
                = ( target4721681511209552016_a_a_b @ Q5 @ P6 ) )
              & ( path_P4574188132515142236_a_a_b @ M3 @ Q5 @ P6 )
              & ( ( map_Pr6873991885683231230_a_a_b
                  @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                  @ P6 )
                = Io2 ) ) ) ) ) ).

% io_targets.simps
thf(fact_868_io__targets_Osimps,axiom,
    ( io_tar1727080856364058086_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c,Io2: list_P903359562653991662od_b_c,Q5: sum_su5154756782089200444_a_a_a] :
          ( collec299858375640228881_a_a_a
          @ ^ [Uu: sum_su5154756782089200444_a_a_a] :
            ? [P6: list_P8197596098214363964_a_a_a] :
              ( ( Uu
                = ( target5023745622805013891_a_b_c @ Q5 @ P6 ) )
              & ( path_S221595587009941175_a_b_c @ M3 @ Q5 @ P6 )
              & ( ( map_Pr6504563922737449846od_b_c
                  @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                  @ P6 )
                = Io2 ) ) ) ) ) ).

% io_targets.simps
thf(fact_869_io__targets_Oelims,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b,Xa: list_P7155397214832950472_a_a_b,Xb3: produc1895207096998757355_a_a_a,Y3: set_Pr904891338250078753_a_a_a] :
      ( ( ( io_tar1879755107597495661_a_a_b @ X3 @ Xa @ Xb3 )
        = Y3 )
     => ( Y3
        = ( collec2110400985823193024_a_a_a
          @ ^ [Uu: produc1895207096998757355_a_a_a] :
            ? [P6: list_P3520633289758869904_a_a_a] :
              ( ( Uu
                = ( target4721681511209552016_a_a_b @ Xb3 @ P6 ) )
              & ( path_P4574188132515142236_a_a_b @ X3 @ Xb3 @ P6 )
              & ( ( map_Pr6873991885683231230_a_a_b
                  @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                  @ P6 )
                = Xa ) ) ) ) ) ).

% io_targets.elims
thf(fact_870_io__targets_Oelims,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Xa: list_P903359562653991662od_b_c,Xb3: sum_su5154756782089200444_a_a_a,Y3: set_Su8591296761904255986_a_a_a] :
      ( ( ( io_tar1727080856364058086_a_b_c @ X3 @ Xa @ Xb3 )
        = Y3 )
     => ( Y3
        = ( collec299858375640228881_a_a_a
          @ ^ [Uu: sum_su5154756782089200444_a_a_a] :
            ? [P6: list_P8197596098214363964_a_a_a] :
              ( ( Uu
                = ( target5023745622805013891_a_b_c @ Xb3 @ P6 ) )
              & ( path_S221595587009941175_a_b_c @ X3 @ Xb3 @ P6 )
              & ( ( map_Pr6504563922737449846od_b_c
                  @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                  @ P6 )
                = Xa ) ) ) ) ) ).

% io_targets.elims
thf(fact_871_LS__acyclic__def,axiom,
    ( lS_acy4668073874649717238_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b,Q5: produc1895207096998757355_a_a_a] :
          ( collec5137133736343679517_a_a_b
          @ ^ [Uu: list_P7155397214832950472_a_a_b] :
            ? [P6: list_P3520633289758869904_a_a_a] :
              ( ( Uu
                = ( map_Pr6873991885683231230_a_a_b
                  @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_P4574188132515142236_a_a_b @ M3 @ Q5 @ P6 )
              & ( distin7721000803377255218_a_a_a @ ( visite6172159911691885821_a_a_b @ Q5 @ P6 ) ) ) ) ) ) ).

% LS_acyclic_def
thf(fact_872_LS__acyclic__def,axiom,
    ( lS_acy904398375403819421_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c,Q5: sum_su5154756782089200444_a_a_a] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu: list_P903359562653991662od_b_c] :
            ? [P6: list_P8197596098214363964_a_a_a] :
              ( ( Uu
                = ( map_Pr6504563922737449846od_b_c
                  @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_S221595587009941175_a_b_c @ M3 @ Q5 @ P6 )
              & ( distin5156058890512421251_a_a_a @ ( visite8040902618889214550_a_b_c @ Q5 @ P6 ) ) ) ) ) ) ).

% LS_acyclic_def
thf(fact_873_language__state__for__input_Osimps,axiom,
    ( langua8364342509234237028_a_a_a
    = ( ^ [M3: fsm_b_2210266817785990191_a_a_a,Q5: b,Xs2: list_c] :
          ( collec9018629162734894800_a_a_a
          @ ^ [Uu: list_P1813520604369389947_a_a_a] :
            ? [P6: list_P7566143005048781605_a_a_b] :
              ( ( Uu
                = ( map_Pr5613174315646222406_a_a_a
                  @ ^ [T4: produc5464264772254188693_a_a_b] : ( produc8806894744968565733_a_a_a @ ( produc5507164989288522869_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) @ ( produc4064411471574014846_a_a_b @ ( produc5900694594632362163_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_b563861380003114551_a_a_a @ M3 @ Q5 @ P6 )
              & ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a
                  @ ( map_Pr5613174315646222406_a_a_a
                    @ ^ [T4: produc5464264772254188693_a_a_b] : ( produc8806894744968565733_a_a_a @ ( produc5507164989288522869_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) @ ( produc4064411471574014846_a_a_b @ ( produc5900694594632362163_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) ) )
                    @ P6 ) )
                = Xs2 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_874_language__state__for__input_Osimps,axiom,
    ( langua8128915693617386428_a_a_a
    = ( ^ [M3: fsm_Pr6532079070102214215_a_a_a,Q5: produc7810058533641270553_a_a_a,Xs2: list_c] :
          ( collec9018629162734894800_a_a_a
          @ ^ [Uu: list_P1813520604369389947_a_a_a] :
            ? [P6: list_P7090462827770577289_a_a_a] :
              ( ( Uu
                = ( map_Pr7391780466455013546_a_a_a
                  @ ^ [T4: produc8583226035453294713_a_a_a] : ( produc8806894744968565733_a_a_a @ ( produc410120848946820409_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) @ ( produc7477144754405272026_a_a_a @ ( produc1285651877402164091_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_P4743837418810288169_a_a_a @ M3 @ Q5 @ P6 )
              & ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a
                  @ ( map_Pr7391780466455013546_a_a_a
                    @ ^ [T4: produc8583226035453294713_a_a_a] : ( produc8806894744968565733_a_a_a @ ( produc410120848946820409_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) @ ( produc7477144754405272026_a_a_a @ ( produc1285651877402164091_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) ) )
                    @ P6 ) )
                = Xs2 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_875_language__state__for__input_Osimps,axiom,
    ( langua3023930401053692143_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b,Q5: produc1895207096998757355_a_a_a,Xs2: list_S8072159660131056972_a_a_a] :
          ( collec5137133736343679517_a_a_b
          @ ^ [Uu: list_P7155397214832950472_a_a_b] :
            ? [P6: list_P3520633289758869904_a_a_a] :
              ( ( Uu
                = ( map_Pr6873991885683231230_a_a_b
                  @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_P4574188132515142236_a_a_b @ M3 @ Q5 @ P6 )
              & ( ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b
                  @ ( map_Pr6873991885683231230_a_a_b
                    @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                    @ P6 ) )
                = Xs2 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_876_language__state__for__input_Osimps,axiom,
    ( langua8022076716241063652_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c,Q5: sum_su5154756782089200444_a_a_a,Xs2: list_b] :
          ( collec2280997390073109977od_b_c
          @ ^ [Uu: list_P903359562653991662od_b_c] :
            ? [P6: list_P8197596098214363964_a_a_a] :
              ( ( Uu
                = ( map_Pr6504563922737449846od_b_c
                  @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_S221595587009941175_a_b_c @ M3 @ Q5 @ P6 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr6504563922737449846od_b_c
                    @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                    @ P6 ) )
                = Xs2 ) ) ) ) ) ).

% language_state_for_input.simps
thf(fact_877_language__state__for__input_Oelims,axiom,
    ! [X3: fsm_b_2210266817785990191_a_a_a,Xa: b,Xb3: list_c,Y3: set_li4521133000354268721_a_a_a] :
      ( ( ( langua8364342509234237028_a_a_a @ X3 @ Xa @ Xb3 )
        = Y3 )
     => ( Y3
        = ( collec9018629162734894800_a_a_a
          @ ^ [Uu: list_P1813520604369389947_a_a_a] :
            ? [P6: list_P7566143005048781605_a_a_b] :
              ( ( Uu
                = ( map_Pr5613174315646222406_a_a_a
                  @ ^ [T4: produc5464264772254188693_a_a_b] : ( produc8806894744968565733_a_a_a @ ( produc5507164989288522869_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) @ ( produc4064411471574014846_a_a_b @ ( produc5900694594632362163_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_b563861380003114551_a_a_a @ X3 @ Xa @ P6 )
              & ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a
                  @ ( map_Pr5613174315646222406_a_a_a
                    @ ^ [T4: produc5464264772254188693_a_a_b] : ( produc8806894744968565733_a_a_a @ ( produc5507164989288522869_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) @ ( produc4064411471574014846_a_a_b @ ( produc5900694594632362163_a_a_b @ ( produc6165236293432588641_a_a_b @ T4 ) ) ) )
                    @ P6 ) )
                = Xb3 ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_878_language__state__for__input_Oelims,axiom,
    ! [X3: fsm_Pr6532079070102214215_a_a_a,Xa: produc7810058533641270553_a_a_a,Xb3: list_c,Y3: set_li4521133000354268721_a_a_a] :
      ( ( ( langua8128915693617386428_a_a_a @ X3 @ Xa @ Xb3 )
        = Y3 )
     => ( Y3
        = ( collec9018629162734894800_a_a_a
          @ ^ [Uu: list_P1813520604369389947_a_a_a] :
            ? [P6: list_P7090462827770577289_a_a_a] :
              ( ( Uu
                = ( map_Pr7391780466455013546_a_a_a
                  @ ^ [T4: produc8583226035453294713_a_a_a] : ( produc8806894744968565733_a_a_a @ ( produc410120848946820409_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) @ ( produc7477144754405272026_a_a_a @ ( produc1285651877402164091_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_P4743837418810288169_a_a_a @ X3 @ Xa @ P6 )
              & ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a
                  @ ( map_Pr7391780466455013546_a_a_a
                    @ ^ [T4: produc8583226035453294713_a_a_a] : ( produc8806894744968565733_a_a_a @ ( produc410120848946820409_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) @ ( produc7477144754405272026_a_a_a @ ( produc1285651877402164091_a_a_a @ ( produc4875806743801315069_a_a_a @ T4 ) ) ) )
                    @ P6 ) )
                = Xb3 ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_879_language__state__for__input_Oelims,axiom,
    ! [X3: fsm_Pr6287166851770849218_a_a_b,Xa: produc1895207096998757355_a_a_a,Xb3: list_S8072159660131056972_a_a_a,Y3: set_li1065949846435832574_a_a_b] :
      ( ( ( langua3023930401053692143_a_a_b @ X3 @ Xa @ Xb3 )
        = Y3 )
     => ( Y3
        = ( collec5137133736343679517_a_a_b
          @ ^ [Uu: list_P7155397214832950472_a_a_b] :
            ? [P6: list_P3520633289758869904_a_a_a] :
              ( ( Uu
                = ( map_Pr6873991885683231230_a_a_b
                  @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_P4574188132515142236_a_a_b @ X3 @ Xa @ P6 )
              & ( ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b
                  @ ( map_Pr6873991885683231230_a_a_b
                    @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                    @ P6 ) )
                = Xb3 ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_880_language__state__for__input_Oelims,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Xa: sum_su5154756782089200444_a_a_a,Xb3: list_b,Y3: set_li6436108459499378894od_b_c] :
      ( ( ( langua8022076716241063652_a_b_c @ X3 @ Xa @ Xb3 )
        = Y3 )
     => ( Y3
        = ( collec2280997390073109977od_b_c
          @ ^ [Uu: list_P903359562653991662od_b_c] :
            ? [P6: list_P8197596098214363964_a_a_a] :
              ( ( Uu
                = ( map_Pr6504563922737449846od_b_c
                  @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                  @ P6 ) )
              & ( path_S221595587009941175_a_b_c @ X3 @ Xa @ P6 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr6504563922737449846od_b_c
                    @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                    @ P6 ) )
                = Xb3 ) ) ) ) ) ).

% language_state_for_input.elims
thf(fact_881_paths__for__io__def,axiom,
    ( paths_3868643201761404704_a_a_b
    = ( ^ [M3: fsm_Pr6287166851770849218_a_a_b,Q5: produc1895207096998757355_a_a_a,Io2: list_P7155397214832950472_a_a_b] :
          ( collec3075913212816350053_a_a_a
          @ ^ [P6: list_P3520633289758869904_a_a_a] :
              ( ( path_P4574188132515142236_a_a_b @ M3 @ Q5 @ P6 )
              & ( ( map_Pr6873991885683231230_a_a_b
                  @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                  @ P6 )
                = Io2 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_882_paths__for__io__def,axiom,
    ( paths_5320179807070014195_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c,Q5: sum_su5154756782089200444_a_a_a,Io2: list_P903359562653991662od_b_c] :
          ( collec406916415612872615_a_a_a
          @ ^ [P6: list_P8197596098214363964_a_a_a] :
              ( ( path_S221595587009941175_a_b_c @ M3 @ Q5 @ P6 )
              & ( ( map_Pr6504563922737449846od_b_c
                  @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                  @ P6 )
                = Io2 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_883_canonical__separator__path__initial__ex_I2_J,axiom,
    ! [M: fsm_Su4683625031830649393_a_a_b,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,P4: list_P8614482820184752236_a_a_a] :
      ( ( path_S6747785721213197493_a_a_b @ ( state_2979059454677389811_a_a_b @ M @ Q1 @ Q2 ) @ ( initia8581304622896136204_a_a_b @ ( state_2979059454677389811_a_a_b @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3538144058484115470_a_a_b @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3538144058484115470_a_a_b @ M ) )
         => ? [P1: list_P9067677309952966352_a_a_a,P22: list_P9067677309952966352_a_a_a] :
              ( ( path_S4752583732068403787_a_a_b @ M @ Q1 @ P1 )
              & ( path_S4752583732068403787_a_a_b @ M @ Q2 @ P22 )
              & ( ( map_Pr5810527985649246014_a_a_b
                  @ ^ [T4: produc8681278163233174848_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3329354246078436523_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) @ ( produc9179341348180701688_a_a_a @ ( produc3812793320627011309_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) ) )
                  @ P1 )
                = ( butlas2466425122822928358_a_a_b
                  @ ( map_Pr7095168675738074842_a_a_b
                    @ ^ [T4: produc8336027307173600092_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc5719501463807155629_a_a_a @ ( produc2127940979268284968_a_a_a @ T4 ) ) @ ( produc6788288432653770762_a_a_a @ ( produc248128219081304043_a_a_a @ ( produc2127940979268284968_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
              & ( ( map_Pr5810527985649246014_a_a_b
                  @ ^ [T4: produc8681278163233174848_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3329354246078436523_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) @ ( produc9179341348180701688_a_a_a @ ( produc3812793320627011309_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) ) )
                  @ P22 )
                = ( butlas2466425122822928358_a_a_b
                  @ ( map_Pr7095168675738074842_a_a_b
                    @ ^ [T4: produc8336027307173600092_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc5719501463807155629_a_a_a @ ( produc2127940979268284968_a_a_a @ T4 ) ) @ ( produc6788288432653770762_a_a_a @ ( produc248128219081304043_a_a_a @ ( produc2127940979268284968_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(2)
thf(fact_884_canonical__separator__path__initial__ex_I2_J,axiom,
    ! [M: fsm_Pr791368616075801221_a_a_b,Q1: produc884577481120224310_a_a_a,Q2: produc884577481120224310_a_a_a,P4: list_P3072728476344414928_a_a_a] :
      ( ( path_S1304752709593314579_a_a_b @ ( state_3609830002750407581_a_a_b @ M @ Q1 @ Q2 ) @ ( initia2424258989328714876_a_a_b @ ( state_3609830002750407581_a_a_b @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member8362999488266169055_a_a_a @ Q1 @ ( states6021486123142740674_a_a_b @ M ) )
       => ( ( member8362999488266169055_a_a_a @ Q2 @ ( states6021486123142740674_a_a_b @ M ) )
         => ? [P1: list_P1303110702266167372_a_a_a,P22: list_P1303110702266167372_a_a_a] :
              ( ( path_P6406173025370084677_a_a_b @ M @ Q1 @ P1 )
              & ( path_P6406173025370084677_a_a_b @ M @ Q2 @ P22 )
              & ( ( map_Pr2767717211958206138_a_a_b
                  @ ^ [T4: produc4009731802749794876_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc1606485967678404637_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) @ ( produc5695092443052942970_a_a_a @ ( produc3123543664576935259_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) ) )
                  @ P1 )
                = ( butlas2466425122822928358_a_a_b
                  @ ( map_Pr1942940760749364030_a_a_b
                    @ ^ [T4: produc4612780081894834240_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3848882253140696291_a_a_a @ ( produc8453961691671531716_a_a_a @ T4 ) ) @ ( produc5739986182704136752_a_a_a @ ( produc8858187255993310245_a_a_a @ ( produc8453961691671531716_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
              & ( ( map_Pr2767717211958206138_a_a_b
                  @ ^ [T4: produc4009731802749794876_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc1606485967678404637_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) @ ( produc5695092443052942970_a_a_a @ ( produc3123543664576935259_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) ) )
                  @ P22 )
                = ( butlas2466425122822928358_a_a_b
                  @ ( map_Pr1942940760749364030_a_a_b
                    @ ^ [T4: produc4612780081894834240_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3848882253140696291_a_a_a @ ( produc8453961691671531716_a_a_a @ T4 ) ) @ ( produc5739986182704136752_a_a_a @ ( produc8858187255993310245_a_a_a @ ( produc8453961691671531716_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(2)
thf(fact_885_canonical__separator__path__initial__ex_I2_J,axiom,
    ! [M: fsm_Pr8211287145308421045_a_a_b,Q1: produc4770514982935793080_a_a_b,Q2: produc4770514982935793080_a_a_b,P4: list_P2242549886795509588_a_a_b] :
      ( ( path_S264780079786346561_a_a_b @ ( state_4270792120431400055_a_a_b @ M @ Q1 @ Q2 ) @ ( initia1897159703354810008_a_a_b @ ( state_4270792120431400055_a_a_b @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member642552543386773071_a_a_b @ Q1 @ ( states5045336508805429138_a_a_b @ M ) )
       => ( ( member642552543386773071_a_a_b @ Q2 @ ( states5045336508805429138_a_a_b @ M ) )
         => ? [P1: list_P8385316394189796816_a_a_b,P22: list_P8385316394189796816_a_a_b] :
              ( ( path_P1155934449145476303_a_a_b @ M @ Q1 @ P1 )
              & ( path_P1155934449145476303_a_a_b @ M @ Q2 @ P22 )
              & ( ( map_Pr36629913606772286_a_a_b
                  @ ^ [T4: produc2838380418727528000_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3595649328014056743_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) @ ( produc8717907853797376116_a_a_b @ ( produc6971105973522642281_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) ) )
                  @ P1 )
                = ( butlas2466425122822928358_a_a_b
                  @ ( map_Pr4542665781501985218_a_a_b
                    @ ^ [T4: produc123133041302015044_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3312939215451498273_a_a_b @ ( produc7693017963871686416_a_a_b @ T4 ) ) @ ( produc6995861152403393406_a_a_b @ ( produc3169017415322691423_a_a_b @ ( produc7693017963871686416_a_a_b @ T4 ) ) ) )
                    @ P4 ) ) )
              & ( ( map_Pr36629913606772286_a_a_b
                  @ ^ [T4: produc2838380418727528000_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3595649328014056743_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) @ ( produc8717907853797376116_a_a_b @ ( produc6971105973522642281_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) ) )
                  @ P22 )
                = ( butlas2466425122822928358_a_a_b
                  @ ( map_Pr4542665781501985218_a_a_b
                    @ ^ [T4: produc123133041302015044_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3312939215451498273_a_a_b @ ( produc7693017963871686416_a_a_b @ T4 ) ) @ ( produc6995861152403393406_a_a_b @ ( produc3169017415322691423_a_a_b @ ( produc7693017963871686416_a_a_b @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(2)
thf(fact_886_canonical__separator__path__initial__ex_I2_J,axiom,
    ! [M: fsm_Su5961347632456545774_a_a_a,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,P4: list_P4201227678853489407_a_a_a] :
      ( ( path_S4742588529375693452_a_a_a @ ( state_7218859947466147534_a_a_a @ M @ Q1 @ Q2 ) @ ( initia55471205950526965_a_a_a @ ( state_7218859947466147534_a_a_a @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states5822841456425053299_a_a_a @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states5822841456425053299_a_a_a @ M ) )
         => ? [P1: list_P4430297574055495993_a_a_a,P22: list_P4430297574055495993_a_a_a] :
              ( ( path_S3009051168982617206_a_a_a @ M @ Q1 @ P1 )
              & ( path_S3009051168982617206_a_a_a @ M @ Q2 @ P22 )
              & ( ( map_Pr7525866001273443320od_a_a
                  @ ^ [T4: produc1141838204273412275_a_a_a] : ( product_Pair_a_a @ ( produc4618981482445519012_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) @ ( produc2350474654791476471_a_a_a @ ( produc5012511087789358306_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) ) )
                  @ P1 )
                = ( butlas8142365730073264249od_a_a
                  @ ( map_Pr1037215490889484978od_a_a
                    @ ^ [T4: produc8556884082142916345_a_a_a] : ( product_Pair_a_a @ ( produc5296690695870061044_a_a_a @ ( produc8103174256791760663_a_a_a @ T4 ) ) @ ( produc7776879804106701515_a_a_a @ ( produc1372291505885593910_a_a_a @ ( produc8103174256791760663_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
              & ( ( map_Pr7525866001273443320od_a_a
                  @ ^ [T4: produc1141838204273412275_a_a_a] : ( product_Pair_a_a @ ( produc4618981482445519012_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) @ ( produc2350474654791476471_a_a_a @ ( produc5012511087789358306_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) ) )
                  @ P22 )
                = ( butlas8142365730073264249od_a_a
                  @ ( map_Pr1037215490889484978od_a_a
                    @ ^ [T4: produc8556884082142916345_a_a_a] : ( product_Pair_a_a @ ( produc5296690695870061044_a_a_a @ ( produc8103174256791760663_a_a_a @ T4 ) ) @ ( produc7776879804106701515_a_a_a @ ( produc1372291505885593910_a_a_a @ ( produc8103174256791760663_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(2)
thf(fact_887_canonical__separator__path__initial__ex_I2_J,axiom,
    ! [M: fsm_Pr3307108581279186586_a_a_a,Q1: produc884577481120224310_a_a_a,Q2: produc884577481120224310_a_a_a,P4: list_P953336824429746729_a_a_a] :
      ( ( path_S4361706828270291886_a_a_a @ ( state_6737487012141364644_a_a_a @ M @ Q1 @ Q2 ) @ ( initia9181111417782089093_a_a_a @ ( state_6737487012141364644_a_a_a @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member8362999488266169055_a_a_a @ Q1 @ ( states7932151110703657663_a_a_a @ M ) )
       => ( ( member8362999488266169055_a_a_a @ Q2 @ ( states7932151110703657663_a_a_a @ M ) )
         => ? [P1: list_P6192864029102241151_a_a_a,P22: list_P6192864029102241151_a_a_a] :
              ( ( path_P3388033272012105212_a_a_a @ M @ Q1 @ P1 )
              & ( path_P3388033272012105212_a_a_a @ M @ Q2 @ P22 )
              & ( ( map_Pr1392725673272416306od_a_a
                  @ ^ [T4: produc7205803768117265017_a_a_a] : ( product_Pair_a_a @ ( produc5229955328659460580_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) @ ( produc1756692000304289723_a_a_a @ ( produc30913607518176806_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) ) )
                  @ P1 )
                = ( butlas8142365730073264249od_a_a
                  @ ( map_Pr7791883050161821448od_a_a
                    @ ^ [T4: produc6752683490396420771_a_a_a] : ( product_Pair_a_a @ ( produc8629031941109950300_a_a_a @ ( produc644117954929008137_a_a_a @ T4 ) ) @ ( produc3231407421338598063_a_a_a @ ( produc8609258968895674010_a_a_a @ ( produc644117954929008137_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
              & ( ( map_Pr1392725673272416306od_a_a
                  @ ^ [T4: produc7205803768117265017_a_a_a] : ( product_Pair_a_a @ ( produc5229955328659460580_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) @ ( produc1756692000304289723_a_a_a @ ( produc30913607518176806_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) ) )
                  @ P22 )
                = ( butlas8142365730073264249od_a_a
                  @ ( map_Pr7791883050161821448od_a_a
                    @ ^ [T4: produc6752683490396420771_a_a_a] : ( product_Pair_a_a @ ( produc8629031941109950300_a_a_a @ ( produc644117954929008137_a_a_a @ T4 ) ) @ ( produc3231407421338598063_a_a_a @ ( produc8609258968895674010_a_a_a @ ( produc644117954929008137_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(2)
thf(fact_888_canonical__separator__path__initial__ex_I2_J,axiom,
    ! [M: fsm_Pr1352325423224000874_b_a_a,Q1: produc4770514982935793080_a_a_b,Q2: produc4770514982935793080_a_a_b,P4: list_P747566903215087359_a_a_b] :
      ( ( path_S8893950696295478784_b_a_a @ ( state_8792045420005744970_b_a_a @ M @ Q1 @ Q2 ) @ ( initia8327540589931236969_b_a_a @ ( state_8792045420005744970_b_a_a @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member642552543386773071_a_a_b @ Q1 @ ( states6742877326663703023_b_a_a @ M ) )
       => ( ( member642552543386773071_a_a_b @ Q2 @ ( states6742877326663703023_b_a_a @ M ) )
         => ? [P1: list_P9033668338835616561_a_a_b,P22: list_P9033668338835616561_a_a_b] :
              ( ( path_P447960203046551282_b_a_a @ M @ Q1 @ P1 )
              & ( path_P447960203046551282_b_a_a @ M @ Q2 @ P22 )
              & ( ( map_Pr2203486926099502080od_a_a
                  @ ^ [T4: produc4165969764426093739_a_a_b] : ( product_Pair_a_a @ ( produc2204968541190299680_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) @ ( produc2705278681451453555_a_a_b @ ( produc7577140250099103326_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) ) )
                  @ P1 )
                = ( butlas8142365730073264249od_a_a
                  @ ( map_Pr1120819969589644978od_a_a
                    @ ^ [T4: produc8368057995964161273_a_a_b] : ( product_Pair_a_a @ ( produc7806160504747017832_a_a_b @ ( produc410253929432769815_a_a_b @ T4 ) ) @ ( produc510337100525087551_a_a_b @ ( produc879673915631515562_a_a_b @ ( produc410253929432769815_a_a_b @ T4 ) ) ) )
                    @ P4 ) ) )
              & ( ( map_Pr2203486926099502080od_a_a
                  @ ^ [T4: produc4165969764426093739_a_a_b] : ( product_Pair_a_a @ ( produc2204968541190299680_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) @ ( produc2705278681451453555_a_a_b @ ( produc7577140250099103326_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) ) )
                  @ P22 )
                = ( butlas8142365730073264249od_a_a
                  @ ( map_Pr1120819969589644978od_a_a
                    @ ^ [T4: produc8368057995964161273_a_a_b] : ( product_Pair_a_a @ ( produc7806160504747017832_a_a_b @ ( produc410253929432769815_a_a_b @ T4 ) ) @ ( produc510337100525087551_a_a_b @ ( produc879673915631515562_a_a_b @ ( produc410253929432769815_a_a_b @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(2)
thf(fact_889_canonical__separator__path__initial__ex_I2_J,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a,P4: list_P1661186363840085254_a_a_a] :
      ( ( path_S5554404386621442024_a_a_b @ ( state_7689045803801065988_a_a_b @ M @ Q1 @ Q2 ) @ ( initia7186784010189905471_a_a_b @ ( state_7689045803801065988_a_a_b @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member6990616694304513154_a_a_a @ Q1 @ ( states8463590192175095071_a_a_b @ M ) )
       => ( ( member6990616694304513154_a_a_a @ Q2 @ ( states8463590192175095071_a_a_b @ M ) )
         => ? [P1: list_P3520633289758869904_a_a_a,P22: list_P3520633289758869904_a_a_a] :
              ( ( path_P4574188132515142236_a_a_b @ M @ Q1 @ P1 )
              & ( path_P4574188132515142236_a_a_b @ M @ Q2 @ P22 )
              & ( ( map_Pr6873991885683231230_a_a_b
                  @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                  @ P1 )
                = ( butlas2466425122822928358_a_a_b
                  @ ( map_Pr1452853287033278580_a_a_b
                    @ ^ [T4: produc2172198381475533558_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T4 ) ) @ ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
              & ( ( map_Pr6873991885683231230_a_a_b
                  @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                  @ P22 )
                = ( butlas2466425122822928358_a_a_b
                  @ ( map_Pr1452853287033278580_a_a_b
                    @ ^ [T4: produc2172198381475533558_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc5218696005822594618_a_a_a @ ( produc6971022889549468226_a_a_a @ T4 ) ) @ ( produc3217384781410164503_a_a_a @ ( produc5074774205693787768_a_a_a @ ( produc6971022889549468226_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(2)
thf(fact_890_canonical__separator__path__initial__ex_I2_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,P4: list_P1545942051602268092_a_a_a] :
      ( ( path_S1955132947403017421_a_b_c @ ( state_4431404365493471503_a_b_c @ M @ Q1 @ Q2 ) @ ( initia6491387660832626742_a_b_c @ ( state_4431404365493471503_a_b_c @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3035385874452377268_a_b_c @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3035385874452377268_a_b_c @ M ) )
         => ? [P1: list_P8197596098214363964_a_a_a,P22: list_P8197596098214363964_a_a_a] :
              ( ( path_S221595587009941175_a_b_c @ M @ Q1 @ P1 )
              & ( path_S221595587009941175_a_b_c @ M @ Q2 @ P22 )
              & ( ( map_Pr6504563922737449846od_b_c
                  @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                  @ P1 )
                = ( butlas5354910156707045818od_b_c
                  @ ( map_Pr7462116669582905078od_b_c
                    @ ^ [T4: produc1185401179908872758_a_a_a] : ( product_Pair_b_c @ ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T4 ) ) @ ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
              & ( ( map_Pr6504563922737449846od_b_c
                  @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                  @ P22 )
                = ( butlas5354910156707045818od_b_c
                  @ ( map_Pr7462116669582905078od_b_c
                    @ ^ [T4: produc1185401179908872758_a_a_a] : ( product_Pair_b_c @ ( produc1778722041638253233_a_a_a @ ( produc7507218669542189268_a_a_a @ T4 ) ) @ ( produc5799697061200840009_a_a_a @ ( produc7077694888508561907_a_a_a @ ( produc7507218669542189268_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(2)
thf(fact_891_canonical__separator__path__initial__ex_I2_J,axiom,
    ! [M: fsm_a_b_c,Q1: a,Q2: a,P4: list_P8197596098214363964_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) @ P4 )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ? [P1: list_P6327159017948738492od_c_a,P22: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M @ Q1 @ P1 )
              & ( path_a_b_c @ M @ Q2 @ P22 )
              & ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T4: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) )
                  @ P1 )
                = ( butlas5354910156707045818od_b_c
                  @ ( map_Pr6504563922737449846od_b_c
                    @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) )
              & ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T4: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) )
                  @ P22 )
                = ( butlas5354910156707045818od_b_c
                  @ ( map_Pr6504563922737449846od_b_c
                    @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                    @ P4 ) ) ) ) ) ) ) ).

% canonical_separator_path_initial_ex(2)
thf(fact_892_delete__aux_Ocases,axiom,
    ! [X3: produc1051194571141289317_a_a_b] :
      ( ! [K2: sum_su5154756782089200444_a_a_a] :
          ( X3
         != ( produc421593117374600221_a_a_b @ K2 @ nil_Pr4580708590200510514_a_a_b ) )
     => ~ ! [K2: sum_su5154756782089200444_a_a_a,K3: sum_su5154756782089200444_a_a_a,V: b,Xs3: list_P7155397214832950472_a_a_b] :
            ( X3
           != ( produc421593117374600221_a_a_b @ K2 @ ( cons_P2262690024297812098_a_a_b @ ( produc6086470211827429482_a_a_b @ K3 @ V ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_893_delete__aux_Ocases,axiom,
    ! [X3: produc5160346394017861590od_a_a] :
      ( ! [K2: a] :
          ( X3
         != ( produc6572230313500880070od_a_a @ K2 @ nil_Product_prod_a_a ) )
     => ~ ! [K2: a,K3: a,V: a,Xs3: list_P1396940483166286381od_a_a] :
            ( X3
           != ( produc6572230313500880070od_a_a @ K2 @ ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ K3 @ V ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_894_zip__Cons__Cons,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,Y3: b,Ys: list_b] :
      ( ( zip_Su8251842677396246858_a_a_b @ ( cons_S2140548516868915206_a_a_a @ X3 @ Xs ) @ ( cons_b @ Y3 @ Ys ) )
      = ( cons_P2262690024297812098_a_a_b @ ( produc6086470211827429482_a_a_b @ X3 @ Y3 ) @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_895_zip__Cons__Cons,axiom,
    ! [X3: a,Xs: list_a,Y3: a,Ys: list_a] :
      ( ( zip_a_a @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y3 @ Ys ) )
      = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_896_map__snd__zip,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( map_Pr6814226208063904709_a_a_b @ produc36865705334053431_a_a_b @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_897_map__snd__zip,axiom,
    ! [Xs: list_c,Ys: list_S8072159660131056972_a_a_a] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_s7186666008910096352_a_a_a @ Ys ) )
     => ( ( map_Pr1074589380674644461_a_a_a @ produc1172910294343954743_a_a_a @ ( zip_c_1748895173682607301_a_a_a @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_898_map__snd__zip,axiom,
    ! [Xs: list_b,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s5427893697081163791_a_a_a @ Ys ) )
     => ( ( map_Pr3321722608670633418_a_a_a @ produc6236129573204179685_a_a_a @ ( zip_b_7838609714423136371_a_a_a @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_899_map__snd__zip,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_s3598275489302632765_a_a_a @ Ys ) )
     => ( ( map_Pr3477903750503579787_a_a_a @ produc1629229363059081756_a_a_a @ ( zip_Su467525902673716494_a_a_a @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_900_map__snd__zip,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_b] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( map_Pr8925698157833054977_a_b_b @ produc7675857798057594300_a_a_b @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_901_map__fst__zip,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( map_Pr6814226208063904709_a_a_b @ produc5884781096680243701_a_a_b @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_902_map__fst__zip,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_b] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_903_map__fst__zip,axiom,
    ! [Xs: list_b,Ys: list_P1813520604369389947_a_a_a] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s5427893697081163791_a_a_a @ Ys ) )
     => ( ( map_Pr3622043693041699552_a_a_b @ produc5842599967860340391_a_a_a @ ( zip_b_7838609714423136371_a_a_a @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_904_map__fst__zip,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_s3598275489302632765_a_a_a @ Ys ) )
     => ( ( map_Pr1409128909316453038_a_a_a @ produc7477144754405272026_a_a_a @ ( zip_Su467525902673716494_a_a_a @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_905_map__fst__zip,axiom,
    ! [Xs: list_c,Ys: list_S8072159660131056972_a_a_a] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_s7186666008910096352_a_a_a @ Ys ) )
     => ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a @ ( zip_c_1748895173682607301_a_a_a @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_906_nth__zip,axiom,
    ! [I2: nat,Xs: list_S8072159660131056972_a_a_a,Ys: list_b] :
      ( ( ord_less_nat @ I2 @ ( size_s7186666008910096352_a_a_a @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Ys ) )
       => ( ( nth_Pr167341066936870769_a_a_b @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) @ I2 )
          = ( produc6086470211827429482_a_a_b @ ( nth_Su7097239896299986421_a_a_a @ Xs @ I2 ) @ ( nth_b @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_907_nth__zip,axiom,
    ! [I2: nat,Xs: list_a,Ys: list_a] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
       => ( ( nth_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) @ I2 )
          = ( product_Pair_a_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_908_nth__zip,axiom,
    ! [I2: nat,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ord_less_nat @ I2 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s1546398270689948508_a_a_b @ Ys ) )
       => ( ( nth_Pr2400495744800824280_a_a_b @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Ys ) @ I2 )
          = ( produc5271149956529894921_a_a_b @ ( nth_Pr167341066936870769_a_a_b @ Xs @ I2 ) @ ( nth_Pr167341066936870769_a_a_b @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_909_map__butlast,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b] :
      ( ( map_Pr7354669405448068282_a_a_a @ F @ ( butlas2466425122822928358_a_a_b @ Xs ) )
      = ( butlas3241881294940498794_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_910_map__butlast,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( map_Pr8925698157833054977_a_b_b @ F @ ( butlas2466425122822928358_a_a_b @ Xs ) )
      = ( butlast_b @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_911_map__butlast,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a] :
      ( ( map_Pr1074589380674644461_a_a_a @ F @ ( butlas8814489273740668441_a_a_a @ Xs ) )
      = ( butlas3241881294940498794_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_912_map__butlast,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a] :
      ( ( map_Pr2287363492687751055_a_a_c @ F @ ( butlas8814489273740668441_a_a_a @ Xs ) )
      = ( butlast_c @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_913_map__butlast,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a] :
      ( ( map_Pr3321722608670633418_a_a_a @ F @ ( butlas1772860031844647239_a_a_a @ Xs ) )
      = ( butlas8814489273740668441_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_914_map__butlast,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a] :
      ( ( map_Pr3622043693041699552_a_a_b @ F @ ( butlas1772860031844647239_a_a_a @ Xs ) )
      = ( butlast_b @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_915_in__set__butlastD,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ ( butlas60410705360980104_a_a_a @ Xs ) ) )
     => ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_916_in__set__butlastD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( butlas3241881294940498794_a_a_a @ Xs ) ) )
     => ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_917_in__set__butlastD,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ ( butlas2466425122822928358_a_a_b @ Xs ) ) )
     => ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_918_take__zip,axiom,
    ! [N: nat,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( take_P1600366419461136831_a_a_b @ N @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Ys ) )
      = ( zip_Pr8098534281803463977_a_a_b @ ( take_P6639847873373218378_a_a_b @ N @ Xs ) @ ( take_P6639847873373218378_a_a_b @ N @ Ys ) ) ) ).

% take_zip
thf(fact_919_take__zip,axiom,
    ! [N: nat,Xs: list_S8072159660131056972_a_a_a,Ys: list_b] :
      ( ( take_P6639847873373218378_a_a_b @ N @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) )
      = ( zip_Su8251842677396246858_a_a_b @ ( take_S1842584226280693198_a_a_a @ N @ Xs ) @ ( take_b @ N @ Ys ) ) ) ).

% take_zip
thf(fact_920_zip__same__conv__map,axiom,
    ! [Xs: list_a] :
      ( ( zip_a_a @ Xs @ Xs )
      = ( map_a_7860052162900579309od_a_a
        @ ^ [X: a] : ( product_Pair_a_a @ X @ X )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_921_zip__eq__ConsE,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_b,Xy: produc4770514982935793080_a_a_b,Xys: list_P7155397214832950472_a_a_b] :
      ( ( ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys )
        = ( cons_P2262690024297812098_a_a_b @ Xy @ Xys ) )
     => ~ ! [X4: sum_su5154756782089200444_a_a_a,Xs4: list_S8072159660131056972_a_a_a] :
            ( ( Xs
              = ( cons_S2140548516868915206_a_a_a @ X4 @ Xs4 ) )
           => ! [Y: b,Ys5: list_b] :
                ( ( Ys
                  = ( cons_b @ Y @ Ys5 ) )
               => ( ( Xy
                    = ( produc6086470211827429482_a_a_b @ X4 @ Y ) )
                 => ( Xys
                   != ( zip_Su8251842677396246858_a_a_b @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_922_zip__eq__ConsE,axiom,
    ! [Xs: list_a,Ys: list_a,Xy: product_prod_a_a,Xys: list_P1396940483166286381od_a_a] :
      ( ( ( zip_a_a @ Xs @ Ys )
        = ( cons_P7316939126706565853od_a_a @ Xy @ Xys ) )
     => ~ ! [X4: a,Xs4: list_a] :
            ( ( Xs
              = ( cons_a @ X4 @ Xs4 ) )
           => ! [Y: a,Ys5: list_a] :
                ( ( Ys
                  = ( cons_a @ Y @ Ys5 ) )
               => ( ( Xy
                    = ( product_Pair_a_a @ X4 @ Y ) )
                 => ( Xys
                   != ( zip_a_a @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_923_zip__map__fst__snd,axiom,
    ! [Zs3: list_P1813520604369389947_a_a_a] :
      ( ( zip_c_1748895173682607301_a_a_a @ ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a @ Zs3 ) @ ( map_Pr1074589380674644461_a_a_a @ produc1172910294343954743_a_a_a @ Zs3 ) )
      = Zs3 ) ).

% zip_map_fst_snd
thf(fact_924_zip__map__fst__snd,axiom,
    ! [Zs3: list_P7200181061086074537_a_a_a] :
      ( ( zip_b_7838609714423136371_a_a_a @ ( map_Pr3622043693041699552_a_a_b @ produc5842599967860340391_a_a_a @ Zs3 ) @ ( map_Pr3321722608670633418_a_a_a @ produc6236129573204179685_a_a_a @ Zs3 ) )
      = Zs3 ) ).

% zip_map_fst_snd
thf(fact_925_zip__map__fst__snd,axiom,
    ! [Zs3: list_P8197596098214363964_a_a_a] :
      ( ( zip_Su467525902673716494_a_a_a @ ( map_Pr1409128909316453038_a_a_a @ produc7477144754405272026_a_a_a @ Zs3 ) @ ( map_Pr3477903750503579787_a_a_a @ produc1629229363059081756_a_a_a @ Zs3 ) )
      = Zs3 ) ).

% zip_map_fst_snd
thf(fact_926_zip__map__fst__snd,axiom,
    ! [Zs3: list_P7155397214832950472_a_a_b] :
      ( ( zip_Su8251842677396246858_a_a_b @ ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Zs3 ) @ ( map_Pr8925698157833054977_a_b_b @ produc7675857798057594300_a_a_b @ Zs3 ) )
      = Zs3 ) ).

% zip_map_fst_snd
thf(fact_927_set__zip__rightD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: produc7810058533641270553_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X3 @ Y3 ) @ ( set_Pr8067384617587748683_a_a_a @ ( zip_Su467525902673716494_a_a_a @ Xs @ Ys ) ) )
     => ( member54949653122719408_a_a_a @ Y3 @ ( set_Pr6411044387506519364_a_a_a @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_928_set__zip__rightD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: b,Xs: list_S8072159660131056972_a_a_a,Ys: list_b] :
      ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X3 @ Y3 ) @ ( set_Pr4578018992751950051_a_a_b @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) ) )
     => ( member_b @ Y3 @ ( set_b2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_929_set__zip__rightD,axiom,
    ! [X3: a,Y3: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ( member_a @ Y3 @ ( set_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_930_set__zip__leftD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: produc7810058533641270553_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X3 @ Y3 ) @ ( set_Pr8067384617587748683_a_a_a @ ( zip_Su467525902673716494_a_a_a @ Xs @ Ys ) ) )
     => ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_931_set__zip__leftD,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: b,Xs: list_S8072159660131056972_a_a_a,Ys: list_b] :
      ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X3 @ Y3 ) @ ( set_Pr4578018992751950051_a_a_b @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) ) )
     => ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_932_set__zip__leftD,axiom,
    ! [X3: a,Y3: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_933_in__set__zipE,axiom,
    ! [X3: a,Y3: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member_a @ Y3 @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_934_in__set__zipE,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: b,Xs: list_S8072159660131056972_a_a_a,Ys: list_b] :
      ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X3 @ Y3 ) @ ( set_Pr4578018992751950051_a_a_b @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
         => ~ ( member_b @ Y3 @ ( set_b2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_935_in__set__zipE,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ X3 @ Y3 ) @ ( set_Pr6783383005027252718_a_a_a @ ( zip_Su1361631147075161393_a_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
         => ~ ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_936_in__set__zipE,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: produc4770514982935793080_a_a_b,Xs: list_S8072159660131056972_a_a_a,Ys: list_P7155397214832950472_a_a_b] :
      ( ( member2904354503647999742_a_a_b @ ( produc1400096373322109837_a_a_b @ X3 @ Y3 ) @ ( set_Pr5192501396389873514_a_a_b @ ( zip_Su6597855456904985261_a_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
         => ~ ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_937_in__set__zipE,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Y3: sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b,Ys: list_S8072159660131056972_a_a_a] :
      ( ( member7629133530163241470_a_a_a @ ( produc5636987742475224205_a_a_a @ X3 @ Y3 ) @ ( set_Pr693908386050339434_a_a_a @ ( zip_Pr1611374789203323821_a_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
         => ~ ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_938_in__set__zipE,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: produc7810058533641270553_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X3 @ Y3 ) @ ( set_Pr8067384617587748683_a_a_a @ ( zip_Su467525902673716494_a_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
         => ~ ( member54949653122719408_a_a_a @ Y3 @ ( set_Pr6411044387506519364_a_a_a @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_939_in__set__zipE,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Y3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( member3114646429565181818_a_a_b @ ( produc5271149956529894921_a_a_b @ X3 @ Y3 ) @ ( set_Pr2819031558886761446_a_a_b @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
         => ~ ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_940_in__set__zipE,axiom,
    ! [X3: produc884577481120224310_a_a_a,Y3: sum_su5154756782089200444_a_a_a,Xs: list_P8197596098214363964_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( member8286568445716149432_a_a_a @ ( produc6846751707927469851_a_a_a @ X3 @ Y3 ) @ ( set_Pr5488859925440335180_a_a_a @ ( zip_Pr5410572673177489403_a_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
         => ~ ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_941_in__set__zipE,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Y3: produc884577481120224310_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_P8197596098214363964_a_a_a] :
      ( ( member5834302193316035974_a_a_a @ ( produc7703250942253352993_a_a_a @ X3 @ Y3 ) @ ( set_Pr3036593673040221722_a_a_a @ ( zip_Su6267071907503372545_a_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
         => ~ ( member8362999488266169055_a_a_a @ Y3 @ ( set_Pr8067384617587748683_a_a_a @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_942_in__set__zipE,axiom,
    ! [X3: produc884577481120224310_a_a_a,Y3: produc4770514982935793080_a_a_b,Xs: list_P8197596098214363964_a_a_a,Ys: list_P7155397214832950472_a_a_b] :
      ( ( member136372082342586420_a_a_b @ ( produc7487074570930836119_a_a_b @ X3 @ Y3 ) @ ( set_Pr8252018788724509896_a_a_b @ ( zip_Pr2588242602176979831_a_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
         => ~ ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_943_zip__same,axiom,
    ! [A: produc884577481120224310_a_a_a,B: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member4115638175673106640_a_a_a @ ( produc6861895844393288855_a_a_a @ A @ B ) @ ( set_Pr3788371281935070012_a_a_a @ ( zip_Pr8247141128185945015_a_a_a @ Xs @ Xs ) ) )
      = ( ( member8362999488266169055_a_a_a @ A @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_944_zip__same,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,B: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ A @ B ) @ ( set_Pr6783383005027252718_a_a_a @ ( zip_Su1361631147075161393_a_a_a @ Xs @ Xs ) ) )
      = ( ( member5679158588528110291_a_a_a @ A @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_945_zip__same,axiom,
    ! [A: produc4770514982935793080_a_a_b,B: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member3114646429565181818_a_a_b @ ( produc5271149956529894921_a_a_b @ A @ B ) @ ( set_Pr2819031558886761446_a_a_b @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Xs ) ) )
      = ( ( member642552543386773071_a_a_b @ A @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_946_zip__same,axiom,
    ! [A: a,B: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
      = ( ( member_a @ A @ ( set_a2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_947_nth__butlast,axiom,
    ! [N: nat,Xs: list_P7155397214832950472_a_a_b] :
      ( ( ord_less_nat @ N @ ( size_s1546398270689948508_a_a_b @ ( butlas2466425122822928358_a_a_b @ Xs ) ) )
     => ( ( nth_Pr167341066936870769_a_a_b @ ( butlas2466425122822928358_a_a_b @ Xs ) @ N )
        = ( nth_Pr167341066936870769_a_a_b @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_948_take__butlast,axiom,
    ! [N: nat,Xs: list_P7155397214832950472_a_a_b] :
      ( ( ord_less_nat @ N @ ( size_s1546398270689948508_a_a_b @ Xs ) )
     => ( ( take_P6639847873373218378_a_a_b @ N @ ( butlas2466425122822928358_a_a_b @ Xs ) )
        = ( take_P6639847873373218378_a_a_b @ N @ Xs ) ) ) ).

% take_butlast
thf(fact_949_zip__eq__conv,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,Zs3: list_P1300507108341865047_a_a_b] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( ( zip_Pr8098534281803463977_a_a_b @ Xs @ Ys )
          = Zs3 )
        = ( ( ( map_Pr6814226208063904709_a_a_b @ produc5884781096680243701_a_a_b @ Zs3 )
            = Xs )
          & ( ( map_Pr6814226208063904709_a_a_b @ produc36865705334053431_a_a_b @ Zs3 )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_950_zip__eq__conv,axiom,
    ! [Xs: list_c,Ys: list_S8072159660131056972_a_a_a,Zs3: list_P1813520604369389947_a_a_a] :
      ( ( ( size_size_list_c @ Xs )
        = ( size_s7186666008910096352_a_a_a @ Ys ) )
     => ( ( ( zip_c_1748895173682607301_a_a_a @ Xs @ Ys )
          = Zs3 )
        = ( ( ( map_Pr2287363492687751055_a_a_c @ produc6784836004715151097_a_a_a @ Zs3 )
            = Xs )
          & ( ( map_Pr1074589380674644461_a_a_a @ produc1172910294343954743_a_a_a @ Zs3 )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_951_zip__eq__conv,axiom,
    ! [Xs: list_b,Ys: list_P1813520604369389947_a_a_a,Zs3: list_P7200181061086074537_a_a_a] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_s5427893697081163791_a_a_a @ Ys ) )
     => ( ( ( zip_b_7838609714423136371_a_a_a @ Xs @ Ys )
          = Zs3 )
        = ( ( ( map_Pr3622043693041699552_a_a_b @ produc5842599967860340391_a_a_a @ Zs3 )
            = Xs )
          & ( ( map_Pr3321722608670633418_a_a_a @ produc6236129573204179685_a_a_a @ Zs3 )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_952_zip__eq__conv,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_P7200181061086074537_a_a_a,Zs3: list_P8197596098214363964_a_a_a] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_s3598275489302632765_a_a_a @ Ys ) )
     => ( ( ( zip_Su467525902673716494_a_a_a @ Xs @ Ys )
          = Zs3 )
        = ( ( ( map_Pr1409128909316453038_a_a_a @ produc7477144754405272026_a_a_a @ Zs3 )
            = Xs )
          & ( ( map_Pr3477903750503579787_a_a_a @ produc1629229363059081756_a_a_a @ Zs3 )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_953_zip__eq__conv,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_b,Zs3: list_P7155397214832950472_a_a_b] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys )
          = Zs3 )
        = ( ( ( map_Pr7354669405448068282_a_a_a @ produc4064411471574014846_a_a_b @ Zs3 )
            = Xs )
          & ( ( map_Pr8925698157833054977_a_b_b @ produc7675857798057594300_a_a_b @ Zs3 )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_954_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_P7200181061086074537_a_a_a,Y3: produc7810058533641270553_a_a_a] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_s3598275489302632765_a_a_a @ Ys ) )
     => ( ( member54949653122719408_a_a_a @ Y3 @ ( set_Pr6411044387506519364_a_a_a @ Ys ) )
       => ~ ! [X4: sum_su5154756782089200444_a_a_a] :
              ~ ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X4 @ Y3 ) @ ( set_Pr8067384617587748683_a_a_a @ ( zip_Su467525902673716494_a_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_955_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_b,Y3: b] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( member_b @ Y3 @ ( set_b2 @ Ys ) )
       => ~ ! [X4: sum_su5154756782089200444_a_a_a] :
              ~ ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X4 @ Y3 ) @ ( set_Pr4578018992751950051_a_a_b @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_956_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_a,Y3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ Y3 @ ( set_a2 @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_957_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P8197596098214363964_a_a_a,Y3: produc884577481120224310_a_a_a] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s4783218027424981928_a_a_a @ Ys ) )
     => ( ( member8362999488266169055_a_a_a @ Y3 @ ( set_Pr8067384617587748683_a_a_a @ Ys ) )
       => ~ ! [X4: produc4770514982935793080_a_a_b] :
              ~ ( member8409818790948104714_a_a_a @ ( produc134655035190672037_a_a_a @ X4 @ Y3 ) @ ( set_Pr7302093460475252382_a_a_a @ ( zip_Pr4459195103291591557_a_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_958_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_S8072159660131056972_a_a_a,Y3: sum_su5154756782089200444_a_a_a] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s7186666008910096352_a_a_a @ Ys ) )
     => ( ( member5679158588528110291_a_a_a @ Y3 @ ( set_Su3113867532901839207_a_a_a @ Ys ) )
       => ~ ! [X4: produc4770514982935793080_a_a_b] :
              ~ ( member7629133530163241470_a_a_a @ ( produc5636987742475224205_a_a_a @ X4 @ Y3 ) @ ( set_Pr693908386050339434_a_a_a @ ( zip_Pr1611374789203323821_a_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_959_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,Y3: produc4770514982935793080_a_a_b] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( member642552543386773071_a_a_b @ Y3 @ ( set_Pr4578018992751950051_a_a_b @ Ys ) )
       => ~ ! [X4: produc4770514982935793080_a_a_b] :
              ~ ( member3114646429565181818_a_a_b @ ( produc5271149956529894921_a_a_b @ X4 @ Y3 ) @ ( set_Pr2819031558886761446_a_a_b @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_960_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_P7200181061086074537_a_a_a,X3: sum_su5154756782089200444_a_a_a] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_s3598275489302632765_a_a_a @ Ys ) )
     => ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
       => ~ ! [Y: produc7810058533641270553_a_a_a] :
              ~ ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X3 @ Y ) @ ( set_Pr8067384617587748683_a_a_a @ ( zip_Su467525902673716494_a_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_961_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_b,X3: sum_su5154756782089200444_a_a_a] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_size_list_b @ Ys ) )
     => ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
       => ~ ! [Y: b] :
              ~ ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X3 @ Y ) @ ( set_Pr4578018992751950051_a_a_b @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_962_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_a,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_963_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,Ys: list_P7155397214832950472_a_a_b,X3: produc884577481120224310_a_a_a] :
      ( ( ( size_s4783218027424981928_a_a_a @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
       => ~ ! [Y: produc4770514982935793080_a_a_b] :
              ~ ( member136372082342586420_a_a_b @ ( produc7487074570930836119_a_a_b @ X3 @ Y ) @ ( set_Pr8252018788724509896_a_a_b @ ( zip_Pr2588242602176979831_a_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_964_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_P7155397214832950472_a_a_b,X3: sum_su5154756782089200444_a_a_a] :
      ( ( ( size_s7186666008910096352_a_a_a @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
       => ~ ! [Y: produc4770514982935793080_a_a_b] :
              ~ ( member2904354503647999742_a_a_b @ ( produc1400096373322109837_a_a_b @ X3 @ Y ) @ ( set_Pr5192501396389873514_a_a_b @ ( zip_Su6597855456904985261_a_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_965_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,X3: produc4770514982935793080_a_a_b] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
       => ~ ! [Y: produc4770514982935793080_a_a_b] :
              ~ ( member3114646429565181818_a_a_b @ ( produc5271149956529894921_a_a_b @ X3 @ Y ) @ ( set_Pr2819031558886761446_a_a_b @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_966_in__set__zip,axiom,
    ! [P4: produc4859596459274012881_a_a_b,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( member3114646429565181818_a_a_b @ P4 @ ( set_Pr2819031558886761446_a_a_b @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_Pr167341066936870769_a_a_b @ Xs @ N2 )
              = ( produc5884781096680243701_a_a_b @ P4 ) )
            & ( ( nth_Pr167341066936870769_a_a_b @ Ys @ N2 )
              = ( produc36865705334053431_a_a_b @ P4 ) )
            & ( ord_less_nat @ N2 @ ( size_s1546398270689948508_a_a_b @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s1546398270689948508_a_a_b @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_967_in__set__zip,axiom,
    ! [P4: produc1895207096998757355_a_a_a,Xs: list_c,Ys: list_S8072159660131056972_a_a_a] :
      ( ( member6990616694304513154_a_a_a @ P4 @ ( set_Pr1702711106814914326_a_a_a @ ( zip_c_1748895173682607301_a_a_a @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_c @ Xs @ N2 )
              = ( produc6784836004715151097_a_a_a @ P4 ) )
            & ( ( nth_Su7097239896299986421_a_a_a @ Ys @ N2 )
              = ( produc1172910294343954743_a_a_a @ P4 ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_c @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s7186666008910096352_a_a_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_968_in__set__zip,axiom,
    ! [P4: produc7810058533641270553_a_a_a,Xs: list_b,Ys: list_P1813520604369389947_a_a_a] :
      ( ( member54949653122719408_a_a_a @ P4 @ ( set_Pr6411044387506519364_a_a_a @ ( zip_b_7838609714423136371_a_a_a @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_b @ Xs @ N2 )
              = ( produc5842599967860340391_a_a_a @ P4 ) )
            & ( ( nth_Pr6515405217854610852_a_a_a @ Ys @ N2 )
              = ( produc6236129573204179685_a_a_a @ P4 ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_b @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s5427893697081163791_a_a_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_969_in__set__zip,axiom,
    ! [P4: produc884577481120224310_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ P4 @ ( set_Pr8067384617587748683_a_a_a @ ( zip_Su467525902673716494_a_a_a @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_Su7097239896299986421_a_a_a @ Xs @ N2 )
              = ( produc7477144754405272026_a_a_a @ P4 ) )
            & ( ( nth_Pr6430744503775489490_a_a_a @ Ys @ N2 )
              = ( produc1629229363059081756_a_a_a @ P4 ) )
            & ( ord_less_nat @ N2 @ ( size_s7186666008910096352_a_a_a @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s3598275489302632765_a_a_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_970_in__set__zip,axiom,
    ! [P4: produc4770514982935793080_a_a_b,Xs: list_S8072159660131056972_a_a_a,Ys: list_b] :
      ( ( member642552543386773071_a_a_b @ P4 @ ( set_Pr4578018992751950051_a_a_b @ ( zip_Su8251842677396246858_a_a_b @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_Su7097239896299986421_a_a_a @ Xs @ N2 )
              = ( produc4064411471574014846_a_a_b @ P4 ) )
            & ( ( nth_b @ Ys @ N2 )
              = ( produc7675857798057594300_a_a_b @ P4 ) )
            & ( ord_less_nat @ N2 @ ( size_s7186666008910096352_a_a_a @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_b @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_971_map__append,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( map_Pr7354669405448068282_a_a_a @ F @ ( append157347090090570333_a_a_b @ Xs @ Ys ) )
      = ( append944914365211047137_a_a_a @ ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) @ ( map_Pr7354669405448068282_a_a_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_972_map__append,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( map_Pr8925698157833054977_a_b_b @ F @ ( append157347090090570333_a_a_b @ Xs @ Ys ) )
      = ( append_b @ ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) @ ( map_Pr8925698157833054977_a_b_b @ F @ Ys ) ) ) ).

% map_append
thf(fact_973_map__append,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( map_Pr1074589380674644461_a_a_a @ F @ ( append6505411241008310416_a_a_a @ Xs @ Ys ) )
      = ( append944914365211047137_a_a_a @ ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) @ ( map_Pr1074589380674644461_a_a_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_974_map__append,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,Ys: list_P1813520604369389947_a_a_a] :
      ( ( map_Pr2287363492687751055_a_a_c @ F @ ( append6505411241008310416_a_a_a @ Xs @ Ys ) )
      = ( append_c @ ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) @ ( map_Pr2287363492687751055_a_a_c @ F @ Ys ) ) ) ).

% map_append
thf(fact_975_map__append,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( map_Pr3321722608670633418_a_a_a @ F @ ( append1954881080209075390_a_a_a @ Xs @ Ys ) )
      = ( append6505411241008310416_a_a_a @ ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) @ ( map_Pr3321722608670633418_a_a_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_976_map__append,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,Ys: list_P7200181061086074537_a_a_a] :
      ( ( map_Pr3622043693041699552_a_a_b @ F @ ( append1954881080209075390_a_a_a @ Xs @ Ys ) )
      = ( append_b @ ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) @ ( map_Pr3622043693041699552_a_a_b @ F @ Ys ) ) ) ).

% map_append
thf(fact_977_append__eq__append__conv,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,Us: list_P7155397214832950472_a_a_b,Vs: list_P7155397214832950472_a_a_b] :
      ( ( ( ( size_s1546398270689948508_a_a_b @ Xs )
          = ( size_s1546398270689948508_a_a_b @ Ys ) )
        | ( ( size_s1546398270689948508_a_a_b @ Us )
          = ( size_s1546398270689948508_a_a_b @ Vs ) ) )
     => ( ( ( append157347090090570333_a_a_b @ Xs @ Us )
          = ( append157347090090570333_a_a_b @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_978_set__append,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( set_Pr4578018992751950051_a_a_b @ ( append157347090090570333_a_a_b @ Xs @ Ys ) )
      = ( sup_su2660408029157294746_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ Xs ) @ ( set_Pr4578018992751950051_a_a_b @ Ys ) ) ) ).

% set_append
thf(fact_979_set__append,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( set_Su3113867532901839207_a_a_a @ ( append944914365211047137_a_a_a @ Xs @ Ys ) )
      = ( sup_su7096409449263244574_a_a_a @ ( set_Su3113867532901839207_a_a_a @ Xs ) @ ( set_Su3113867532901839207_a_a_a @ Ys ) ) ) ).

% set_append
thf(fact_980_nth__append__length,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,X3: produc4770514982935793080_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( nth_Pr167341066936870769_a_a_b @ ( append157347090090570333_a_a_b @ Xs @ ( cons_P2262690024297812098_a_a_b @ X3 @ Ys ) ) @ ( size_s1546398270689948508_a_a_b @ Xs ) )
      = X3 ) ).

% nth_append_length
thf(fact_981_zip__append,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Us: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,Vs: list_P7155397214832950472_a_a_b] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Us ) )
     => ( ( zip_Pr8098534281803463977_a_a_b @ ( append157347090090570333_a_a_b @ Xs @ Ys ) @ ( append157347090090570333_a_a_b @ Us @ Vs ) )
        = ( append1962905573249128940_a_a_b @ ( zip_Pr8098534281803463977_a_a_b @ Xs @ Us ) @ ( zip_Pr8098534281803463977_a_a_b @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_982_target__snoc,axiom,
    ! [Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,T: produc884577481120224310_a_a_a] :
      ( ( target5023745622805013891_a_b_c @ Q @ ( append7211258631950116177_a_a_a @ P4 @ ( cons_P6318101215856909548_a_a_a @ T @ nil_Pr6029279252097497404_a_a_a ) ) )
      = ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) ) ).

% target_snoc
thf(fact_983_target__snoc,axiom,
    ! [Q: produc1895207096998757355_a_a_a,P4: list_P3520633289758869904_a_a_a,T: produc6800189296593107456_a_a_a] :
      ( ( target4721681511209552016_a_a_b @ Q @ ( append2262074668660228901_a_a_a @ P4 @ ( cons_P5563258090397451722_a_a_a @ T @ nil_Pr3590249381822989690_a_a_a ) ) )
      = ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) ) ).

% target_snoc
thf(fact_984_path__append__transition,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,T: produc884577481120224310_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
     => ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
       => ( ( ( produc7477144754405272026_a_a_a @ T )
            = ( target5023745622805013891_a_b_c @ Q @ P4 ) )
         => ( path_S221595587009941175_a_b_c @ M @ Q @ ( append7211258631950116177_a_a_a @ P4 @ ( cons_P6318101215856909548_a_a_a @ T @ nil_Pr6029279252097497404_a_a_a ) ) ) ) ) ) ).

% path_append_transition
thf(fact_985_map__eq__append__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b,Ys: list_S8072159660131056972_a_a_a,Zs3: list_S8072159660131056972_a_a_a] :
      ( ( ( map_Pr7354669405448068282_a_a_a @ F @ Xs )
        = ( append944914365211047137_a_a_a @ Ys @ Zs3 ) )
      = ( ? [Us2: list_P7155397214832950472_a_a_b,Vs2: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( append157347090090570333_a_a_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr7354669405448068282_a_a_a @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr7354669405448068282_a_a_a @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_986_map__eq__append__conv,axiom,
    ! [F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b,Ys: list_b,Zs3: list_b] :
      ( ( ( map_Pr8925698157833054977_a_b_b @ F @ Xs )
        = ( append_b @ Ys @ Zs3 ) )
      = ( ? [Us2: list_P7155397214832950472_a_a_b,Vs2: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( append157347090090570333_a_a_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr8925698157833054977_a_b_b @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr8925698157833054977_a_b_b @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_987_map__eq__append__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a,Ys: list_S8072159660131056972_a_a_a,Zs3: list_S8072159660131056972_a_a_a] :
      ( ( ( map_Pr1074589380674644461_a_a_a @ F @ Xs )
        = ( append944914365211047137_a_a_a @ Ys @ Zs3 ) )
      = ( ? [Us2: list_P1813520604369389947_a_a_a,Vs2: list_P1813520604369389947_a_a_a] :
            ( ( Xs
              = ( append6505411241008310416_a_a_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr1074589380674644461_a_a_a @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr1074589380674644461_a_a_a @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_988_map__eq__append__conv,axiom,
    ! [F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a,Ys: list_c,Zs3: list_c] :
      ( ( ( map_Pr2287363492687751055_a_a_c @ F @ Xs )
        = ( append_c @ Ys @ Zs3 ) )
      = ( ? [Us2: list_P1813520604369389947_a_a_a,Vs2: list_P1813520604369389947_a_a_a] :
            ( ( Xs
              = ( append6505411241008310416_a_a_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr2287363492687751055_a_a_c @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr2287363492687751055_a_a_c @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_989_map__eq__append__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a,Ys: list_P1813520604369389947_a_a_a,Zs3: list_P1813520604369389947_a_a_a] :
      ( ( ( map_Pr3321722608670633418_a_a_a @ F @ Xs )
        = ( append6505411241008310416_a_a_a @ Ys @ Zs3 ) )
      = ( ? [Us2: list_P7200181061086074537_a_a_a,Vs2: list_P7200181061086074537_a_a_a] :
            ( ( Xs
              = ( append1954881080209075390_a_a_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr3321722608670633418_a_a_a @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr3321722608670633418_a_a_a @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_990_map__eq__append__conv,axiom,
    ! [F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a,Ys: list_b,Zs3: list_b] :
      ( ( ( map_Pr3622043693041699552_a_a_b @ F @ Xs )
        = ( append_b @ Ys @ Zs3 ) )
      = ( ? [Us2: list_P7200181061086074537_a_a_a,Vs2: list_P7200181061086074537_a_a_a] :
            ( ( Xs
              = ( append1954881080209075390_a_a_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr3622043693041699552_a_a_b @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr3622043693041699552_a_a_b @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_991_append__eq__map__conv,axiom,
    ! [Ys: list_S8072159660131056972_a_a_a,Zs3: list_S8072159660131056972_a_a_a,F: produc4770514982935793080_a_a_b > sum_su5154756782089200444_a_a_a,Xs: list_P7155397214832950472_a_a_b] :
      ( ( ( append944914365211047137_a_a_a @ Ys @ Zs3 )
        = ( map_Pr7354669405448068282_a_a_a @ F @ Xs ) )
      = ( ? [Us2: list_P7155397214832950472_a_a_b,Vs2: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( append157347090090570333_a_a_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr7354669405448068282_a_a_a @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr7354669405448068282_a_a_a @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_992_append__eq__map__conv,axiom,
    ! [Ys: list_b,Zs3: list_b,F: produc4770514982935793080_a_a_b > b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( ( append_b @ Ys @ Zs3 )
        = ( map_Pr8925698157833054977_a_b_b @ F @ Xs ) )
      = ( ? [Us2: list_P7155397214832950472_a_a_b,Vs2: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( append157347090090570333_a_a_b @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr8925698157833054977_a_b_b @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr8925698157833054977_a_b_b @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_993_append__eq__map__conv,axiom,
    ! [Ys: list_S8072159660131056972_a_a_a,Zs3: list_S8072159660131056972_a_a_a,F: produc1895207096998757355_a_a_a > sum_su5154756782089200444_a_a_a,Xs: list_P1813520604369389947_a_a_a] :
      ( ( ( append944914365211047137_a_a_a @ Ys @ Zs3 )
        = ( map_Pr1074589380674644461_a_a_a @ F @ Xs ) )
      = ( ? [Us2: list_P1813520604369389947_a_a_a,Vs2: list_P1813520604369389947_a_a_a] :
            ( ( Xs
              = ( append6505411241008310416_a_a_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr1074589380674644461_a_a_a @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr1074589380674644461_a_a_a @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_994_append__eq__map__conv,axiom,
    ! [Ys: list_c,Zs3: list_c,F: produc1895207096998757355_a_a_a > c,Xs: list_P1813520604369389947_a_a_a] :
      ( ( ( append_c @ Ys @ Zs3 )
        = ( map_Pr2287363492687751055_a_a_c @ F @ Xs ) )
      = ( ? [Us2: list_P1813520604369389947_a_a_a,Vs2: list_P1813520604369389947_a_a_a] :
            ( ( Xs
              = ( append6505411241008310416_a_a_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr2287363492687751055_a_a_c @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr2287363492687751055_a_a_c @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_995_append__eq__map__conv,axiom,
    ! [Ys: list_P1813520604369389947_a_a_a,Zs3: list_P1813520604369389947_a_a_a,F: produc7810058533641270553_a_a_a > produc1895207096998757355_a_a_a,Xs: list_P7200181061086074537_a_a_a] :
      ( ( ( append6505411241008310416_a_a_a @ Ys @ Zs3 )
        = ( map_Pr3321722608670633418_a_a_a @ F @ Xs ) )
      = ( ? [Us2: list_P7200181061086074537_a_a_a,Vs2: list_P7200181061086074537_a_a_a] :
            ( ( Xs
              = ( append1954881080209075390_a_a_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr3321722608670633418_a_a_a @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr3321722608670633418_a_a_a @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_996_append__eq__map__conv,axiom,
    ! [Ys: list_b,Zs3: list_b,F: produc7810058533641270553_a_a_a > b,Xs: list_P7200181061086074537_a_a_a] :
      ( ( ( append_b @ Ys @ Zs3 )
        = ( map_Pr3622043693041699552_a_a_b @ F @ Xs ) )
      = ( ? [Us2: list_P7200181061086074537_a_a_a,Vs2: list_P7200181061086074537_a_a_a] :
            ( ( Xs
              = ( append1954881080209075390_a_a_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr3622043693041699552_a_a_b @ F @ Us2 ) )
            & ( Zs3
              = ( map_Pr3622043693041699552_a_a_b @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_997_list__concat__non__elem,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a,Ys: list_P8197596098214363964_a_a_a] :
      ( ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
     => ( ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Ys ) )
       => ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ ( append7211258631950116177_a_a_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_998_list__concat__non__elem,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
     => ( ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Ys ) )
       => ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( append944914365211047137_a_a_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_999_list__concat__non__elem,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ( ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Ys ) )
       => ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ ( append157347090090570333_a_a_b @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_1000_list__prefix__elem,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a,Ys: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ ( append7211258631950116177_a_a_a @ Xs @ Ys ) ) )
     => ( ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Ys ) )
       => ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_1001_list__prefix__elem,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( append944914365211047137_a_a_a @ Xs @ Ys ) ) )
     => ( ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Ys ) )
       => ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_1002_list__prefix__elem,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ ( append157347090090570333_a_a_b @ Xs @ Ys ) ) )
     => ( ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Ys ) )
       => ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_1003_list__set__sym,axiom,
    ! [X3: list_S8072159660131056972_a_a_a,Y3: list_S8072159660131056972_a_a_a] :
      ( ( set_Su3113867532901839207_a_a_a @ ( append944914365211047137_a_a_a @ X3 @ Y3 ) )
      = ( set_Su3113867532901839207_a_a_a @ ( append944914365211047137_a_a_a @ Y3 @ X3 ) ) ) ).

% list_set_sym
thf(fact_1004_list__set__sym,axiom,
    ! [X3: list_P7155397214832950472_a_a_b,Y3: list_P7155397214832950472_a_a_b] :
      ( ( set_Pr4578018992751950051_a_a_b @ ( append157347090090570333_a_a_b @ X3 @ Y3 ) )
      = ( set_Pr4578018992751950051_a_a_b @ ( append157347090090570333_a_a_b @ Y3 @ X3 ) ) ) ).

% list_set_sym
thf(fact_1005_split__list__first__prop__iff,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( ? [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_S8072159660131056972_a_a_a,X: sum_su5154756782089200444_a_a_a] :
            ( ? [Zs2: list_S8072159660131056972_a_a_a] :
                ( Xs
                = ( append944914365211047137_a_a_a @ Ys3 @ ( cons_S2140548516868915206_a_a_a @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y4: sum_su5154756782089200444_a_a_a] :
                ( ( member5679158588528110291_a_a_a @ Y4 @ ( set_Su3113867532901839207_a_a_a @ Ys3 ) )
               => ~ ( P @ Y4 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1006_split__list__first__prop__iff,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ( ? [X: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_P7155397214832950472_a_a_b,X: produc4770514982935793080_a_a_b] :
            ( ? [Zs2: list_P7155397214832950472_a_a_b] :
                ( Xs
                = ( append157347090090570333_a_a_b @ Ys3 @ ( cons_P2262690024297812098_a_a_b @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y4: produc4770514982935793080_a_a_b] :
                ( ( member642552543386773071_a_a_b @ Y4 @ ( set_Pr4578018992751950051_a_a_b @ Ys3 ) )
               => ~ ( P @ Y4 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1007_split__list__last__prop__iff,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( ? [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_S8072159660131056972_a_a_a,X: sum_su5154756782089200444_a_a_a,Zs2: list_S8072159660131056972_a_a_a] :
            ( ( Xs
              = ( append944914365211047137_a_a_a @ Ys3 @ ( cons_S2140548516868915206_a_a_a @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y4: sum_su5154756782089200444_a_a_a] :
                ( ( member5679158588528110291_a_a_a @ Y4 @ ( set_Su3113867532901839207_a_a_a @ Zs2 ) )
               => ~ ( P @ Y4 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1008_split__list__last__prop__iff,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ( ? [X: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list_P7155397214832950472_a_a_b,X: produc4770514982935793080_a_a_b,Zs2: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( append157347090090570333_a_a_b @ Ys3 @ ( cons_P2262690024297812098_a_a_b @ X @ Zs2 ) ) )
            & ( P @ X )
            & ! [Y4: produc4770514982935793080_a_a_b] :
                ( ( member642552543386773071_a_a_b @ Y4 @ ( set_Pr4578018992751950051_a_a_b @ Zs2 ) )
               => ~ ( P @ Y4 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1009_in__set__conv__decomp__first,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
      = ( ? [Ys3: list_P8197596098214363964_a_a_a,Zs2: list_P8197596098214363964_a_a_a] :
            ( ( Xs
              = ( append7211258631950116177_a_a_a @ Ys3 @ ( cons_P6318101215856909548_a_a_a @ X3 @ Zs2 ) ) )
            & ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1010_in__set__conv__decomp__first,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
      = ( ? [Ys3: list_S8072159660131056972_a_a_a,Zs2: list_S8072159660131056972_a_a_a] :
            ( ( Xs
              = ( append944914365211047137_a_a_a @ Ys3 @ ( cons_S2140548516868915206_a_a_a @ X3 @ Zs2 ) ) )
            & ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1011_in__set__conv__decomp__first,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
      = ( ? [Ys3: list_P7155397214832950472_a_a_b,Zs2: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( append157347090090570333_a_a_b @ Ys3 @ ( cons_P2262690024297812098_a_a_b @ X3 @ Zs2 ) ) )
            & ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1012_in__set__conv__decomp__last,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
      = ( ? [Ys3: list_P8197596098214363964_a_a_a,Zs2: list_P8197596098214363964_a_a_a] :
            ( ( Xs
              = ( append7211258631950116177_a_a_a @ Ys3 @ ( cons_P6318101215856909548_a_a_a @ X3 @ Zs2 ) ) )
            & ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1013_in__set__conv__decomp__last,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
      = ( ? [Ys3: list_S8072159660131056972_a_a_a,Zs2: list_S8072159660131056972_a_a_a] :
            ( ( Xs
              = ( append944914365211047137_a_a_a @ Ys3 @ ( cons_S2140548516868915206_a_a_a @ X3 @ Zs2 ) ) )
            & ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1014_in__set__conv__decomp__last,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
      = ( ? [Ys3: list_P7155397214832950472_a_a_b,Zs2: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( append157347090090570333_a_a_b @ Ys3 @ ( cons_P2262690024297812098_a_a_b @ X3 @ Zs2 ) ) )
            & ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1015_split__list__first__propE,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ? [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
          & ( P @ X7 ) )
     => ~ ! [Ys4: list_S8072159660131056972_a_a_a,X4: sum_su5154756782089200444_a_a_a] :
            ( ? [Zs: list_S8072159660131056972_a_a_a] :
                ( Xs
                = ( append944914365211047137_a_a_a @ Ys4 @ ( cons_S2140548516868915206_a_a_a @ X4 @ Zs ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa2: sum_su5154756782089200444_a_a_a] :
                    ( ( member5679158588528110291_a_a_a @ Xa2 @ ( set_Su3113867532901839207_a_a_a @ Ys4 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1016_split__list__first__propE,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ? [X7: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X7 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
          & ( P @ X7 ) )
     => ~ ! [Ys4: list_P7155397214832950472_a_a_b,X4: produc4770514982935793080_a_a_b] :
            ( ? [Zs: list_P7155397214832950472_a_a_b] :
                ( Xs
                = ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ X4 @ Zs ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa2: produc4770514982935793080_a_a_b] :
                    ( ( member642552543386773071_a_a_b @ Xa2 @ ( set_Pr4578018992751950051_a_a_b @ Ys4 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1017_split__list__last__propE,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ? [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
          & ( P @ X7 ) )
     => ~ ! [Ys4: list_S8072159660131056972_a_a_a,X4: sum_su5154756782089200444_a_a_a,Zs: list_S8072159660131056972_a_a_a] :
            ( ( Xs
              = ( append944914365211047137_a_a_a @ Ys4 @ ( cons_S2140548516868915206_a_a_a @ X4 @ Zs ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa2: sum_su5154756782089200444_a_a_a] :
                    ( ( member5679158588528110291_a_a_a @ Xa2 @ ( set_Su3113867532901839207_a_a_a @ Zs ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1018_split__list__last__propE,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ? [X7: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X7 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
          & ( P @ X7 ) )
     => ~ ! [Ys4: list_P7155397214832950472_a_a_b,X4: produc4770514982935793080_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
            ( ( Xs
              = ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ X4 @ Zs ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa2: produc4770514982935793080_a_a_b] :
                    ( ( member642552543386773071_a_a_b @ Xa2 @ ( set_Pr4578018992751950051_a_a_b @ Zs ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1019_split__list__first__prop,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ? [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
          & ( P @ X7 ) )
     => ? [Ys4: list_S8072159660131056972_a_a_a,X4: sum_su5154756782089200444_a_a_a] :
          ( ? [Zs: list_S8072159660131056972_a_a_a] :
              ( Xs
              = ( append944914365211047137_a_a_a @ Ys4 @ ( cons_S2140548516868915206_a_a_a @ X4 @ Zs ) ) )
          & ( P @ X4 )
          & ! [Xa2: sum_su5154756782089200444_a_a_a] :
              ( ( member5679158588528110291_a_a_a @ Xa2 @ ( set_Su3113867532901839207_a_a_a @ Ys4 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1020_split__list__first__prop,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ? [X7: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X7 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
          & ( P @ X7 ) )
     => ? [Ys4: list_P7155397214832950472_a_a_b,X4: produc4770514982935793080_a_a_b] :
          ( ? [Zs: list_P7155397214832950472_a_a_b] :
              ( Xs
              = ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ X4 @ Zs ) ) )
          & ( P @ X4 )
          & ! [Xa2: produc4770514982935793080_a_a_b] :
              ( ( member642552543386773071_a_a_b @ Xa2 @ ( set_Pr4578018992751950051_a_a_b @ Ys4 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1021_split__list__last__prop,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ? [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
          & ( P @ X7 ) )
     => ? [Ys4: list_S8072159660131056972_a_a_a,X4: sum_su5154756782089200444_a_a_a,Zs: list_S8072159660131056972_a_a_a] :
          ( ( Xs
            = ( append944914365211047137_a_a_a @ Ys4 @ ( cons_S2140548516868915206_a_a_a @ X4 @ Zs ) ) )
          & ( P @ X4 )
          & ! [Xa2: sum_su5154756782089200444_a_a_a] :
              ( ( member5679158588528110291_a_a_a @ Xa2 @ ( set_Su3113867532901839207_a_a_a @ Zs ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_1022_split__list__last__prop,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ? [X7: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X7 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
          & ( P @ X7 ) )
     => ? [Ys4: list_P7155397214832950472_a_a_b,X4: produc4770514982935793080_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
          ( ( Xs
            = ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ X4 @ Zs ) ) )
          & ( P @ X4 )
          & ! [Xa2: produc4770514982935793080_a_a_b] :
              ( ( member642552543386773071_a_a_b @ Xa2 @ ( set_Pr4578018992751950051_a_a_b @ Zs ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_1023_in__set__conv__decomp,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
      = ( ? [Ys3: list_P8197596098214363964_a_a_a,Zs2: list_P8197596098214363964_a_a_a] :
            ( Xs
            = ( append7211258631950116177_a_a_a @ Ys3 @ ( cons_P6318101215856909548_a_a_a @ X3 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1024_in__set__conv__decomp,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
      = ( ? [Ys3: list_S8072159660131056972_a_a_a,Zs2: list_S8072159660131056972_a_a_a] :
            ( Xs
            = ( append944914365211047137_a_a_a @ Ys3 @ ( cons_S2140548516868915206_a_a_a @ X3 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1025_in__set__conv__decomp,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
      = ( ? [Ys3: list_P7155397214832950472_a_a_b,Zs2: list_P7155397214832950472_a_a_b] :
            ( Xs
            = ( append157347090090570333_a_a_b @ Ys3 @ ( cons_P2262690024297812098_a_a_b @ X3 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1026_append__Cons__eq__iff,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a,Ys: list_P8197596098214363964_a_a_a,Xs5: list_P8197596098214363964_a_a_a,Ys6: list_P8197596098214363964_a_a_a] :
      ( ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
     => ( ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Ys ) )
       => ( ( ( append7211258631950116177_a_a_a @ Xs @ ( cons_P6318101215856909548_a_a_a @ X3 @ Ys ) )
            = ( append7211258631950116177_a_a_a @ Xs5 @ ( cons_P6318101215856909548_a_a_a @ X3 @ Ys6 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1027_append__Cons__eq__iff,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_S8072159660131056972_a_a_a,Xs5: list_S8072159660131056972_a_a_a,Ys6: list_S8072159660131056972_a_a_a] :
      ( ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
     => ( ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Ys ) )
       => ( ( ( append944914365211047137_a_a_a @ Xs @ ( cons_S2140548516868915206_a_a_a @ X3 @ Ys ) )
            = ( append944914365211047137_a_a_a @ Xs5 @ ( cons_S2140548516868915206_a_a_a @ X3 @ Ys6 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1028_append__Cons__eq__iff,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,Xs5: list_P7155397214832950472_a_a_b,Ys6: list_P7155397214832950472_a_a_b] :
      ( ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ( ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Ys ) )
       => ( ( ( append157347090090570333_a_a_b @ Xs @ ( cons_P2262690024297812098_a_a_b @ X3 @ Ys ) )
            = ( append157347090090570333_a_a_b @ Xs5 @ ( cons_P2262690024297812098_a_a_b @ X3 @ Ys6 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1029_split__list__propE,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ? [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
          & ( P @ X7 ) )
     => ~ ! [Ys4: list_S8072159660131056972_a_a_a,X4: sum_su5154756782089200444_a_a_a] :
            ( ? [Zs: list_S8072159660131056972_a_a_a] :
                ( Xs
                = ( append944914365211047137_a_a_a @ Ys4 @ ( cons_S2140548516868915206_a_a_a @ X4 @ Zs ) ) )
           => ~ ( P @ X4 ) ) ) ).

% split_list_propE
thf(fact_1030_split__list__propE,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ? [X7: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X7 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
          & ( P @ X7 ) )
     => ~ ! [Ys4: list_P7155397214832950472_a_a_b,X4: produc4770514982935793080_a_a_b] :
            ( ? [Zs: list_P7155397214832950472_a_a_b] :
                ( Xs
                = ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ X4 @ Zs ) ) )
           => ~ ( P @ X4 ) ) ) ).

% split_list_propE
thf(fact_1031_split__list__first,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
     => ? [Ys4: list_P8197596098214363964_a_a_a,Zs: list_P8197596098214363964_a_a_a] :
          ( ( Xs
            = ( append7211258631950116177_a_a_a @ Ys4 @ ( cons_P6318101215856909548_a_a_a @ X3 @ Zs ) ) )
          & ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1032_split__list__first,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
     => ? [Ys4: list_S8072159660131056972_a_a_a,Zs: list_S8072159660131056972_a_a_a] :
          ( ( Xs
            = ( append944914365211047137_a_a_a @ Ys4 @ ( cons_S2140548516868915206_a_a_a @ X3 @ Zs ) ) )
          & ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1033_split__list__first,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ? [Ys4: list_P7155397214832950472_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
          ( ( Xs
            = ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ X3 @ Zs ) ) )
          & ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1034_split__list__prop,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ? [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
          & ( P @ X7 ) )
     => ? [Ys4: list_S8072159660131056972_a_a_a,X4: sum_su5154756782089200444_a_a_a] :
          ( ? [Zs: list_S8072159660131056972_a_a_a] :
              ( Xs
              = ( append944914365211047137_a_a_a @ Ys4 @ ( cons_S2140548516868915206_a_a_a @ X4 @ Zs ) ) )
          & ( P @ X4 ) ) ) ).

% split_list_prop
thf(fact_1035_split__list__prop,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ? [X7: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X7 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
          & ( P @ X7 ) )
     => ? [Ys4: list_P7155397214832950472_a_a_b,X4: produc4770514982935793080_a_a_b] :
          ( ? [Zs: list_P7155397214832950472_a_a_b] :
              ( Xs
              = ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ X4 @ Zs ) ) )
          & ( P @ X4 ) ) ) ).

% split_list_prop
thf(fact_1036_split__list__last,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
     => ? [Ys4: list_P8197596098214363964_a_a_a,Zs: list_P8197596098214363964_a_a_a] :
          ( ( Xs
            = ( append7211258631950116177_a_a_a @ Ys4 @ ( cons_P6318101215856909548_a_a_a @ X3 @ Zs ) ) )
          & ~ ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Zs ) ) ) ) ).

% split_list_last
thf(fact_1037_split__list__last,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
     => ? [Ys4: list_S8072159660131056972_a_a_a,Zs: list_S8072159660131056972_a_a_a] :
          ( ( Xs
            = ( append944914365211047137_a_a_a @ Ys4 @ ( cons_S2140548516868915206_a_a_a @ X3 @ Zs ) ) )
          & ~ ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Zs ) ) ) ) ).

% split_list_last
thf(fact_1038_split__list__last,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ? [Ys4: list_P7155397214832950472_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
          ( ( Xs
            = ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ X3 @ Zs ) ) )
          & ~ ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Zs ) ) ) ) ).

% split_list_last
thf(fact_1039_split__list,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
     => ? [Ys4: list_P8197596098214363964_a_a_a,Zs: list_P8197596098214363964_a_a_a] :
          ( Xs
          = ( append7211258631950116177_a_a_a @ Ys4 @ ( cons_P6318101215856909548_a_a_a @ X3 @ Zs ) ) ) ) ).

% split_list
thf(fact_1040_split__list,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
     => ? [Ys4: list_S8072159660131056972_a_a_a,Zs: list_S8072159660131056972_a_a_a] :
          ( Xs
          = ( append944914365211047137_a_a_a @ Ys4 @ ( cons_S2140548516868915206_a_a_a @ X3 @ Zs ) ) ) ) ).

% split_list
thf(fact_1041_split__list,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
     => ? [Ys4: list_P7155397214832950472_a_a_b,Zs: list_P7155397214832950472_a_a_b] :
          ( Xs
          = ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ X3 @ Zs ) ) ) ) ).

% split_list
thf(fact_1042_in__set__butlast__appendI,axiom,
    ! [X3: produc884577481120224310_a_a_a,Xs: list_P8197596098214363964_a_a_a,Ys: list_P8197596098214363964_a_a_a] :
      ( ( ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ ( butlas60410705360980104_a_a_a @ Xs ) ) )
        | ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ ( butlas60410705360980104_a_a_a @ Ys ) ) ) )
     => ( member8362999488266169055_a_a_a @ X3 @ ( set_Pr8067384617587748683_a_a_a @ ( butlas60410705360980104_a_a_a @ ( append7211258631950116177_a_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_1043_in__set__butlast__appendI,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,Xs: list_S8072159660131056972_a_a_a,Ys: list_S8072159660131056972_a_a_a] :
      ( ( ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( butlas3241881294940498794_a_a_a @ Xs ) ) )
        | ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( butlas3241881294940498794_a_a_a @ Ys ) ) ) )
     => ( member5679158588528110291_a_a_a @ X3 @ ( set_Su3113867532901839207_a_a_a @ ( butlas3241881294940498794_a_a_a @ ( append944914365211047137_a_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_1044_in__set__butlast__appendI,axiom,
    ! [X3: produc4770514982935793080_a_a_b,Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ ( butlas2466425122822928358_a_a_b @ Xs ) ) )
        | ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ ( butlas2466425122822928358_a_a_b @ Ys ) ) ) )
     => ( member642552543386773071_a_a_b @ X3 @ ( set_Pr4578018992751950051_a_a_b @ ( butlas2466425122822928358_a_a_b @ ( append157347090090570333_a_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_1045_same__length__different,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b] :
      ( ( Xs != Ys )
     => ( ( ( size_s1546398270689948508_a_a_b @ Xs )
          = ( size_s1546398270689948508_a_a_b @ Ys ) )
       => ? [Pre: list_P7155397214832950472_a_a_b,X4: produc4770514982935793080_a_a_b,Xs4: list_P7155397214832950472_a_a_b,Y: produc4770514982935793080_a_a_b,Ys5: list_P7155397214832950472_a_a_b] :
            ( ( X4 != Y )
            & ( Xs
              = ( append157347090090570333_a_a_b @ Pre @ ( append157347090090570333_a_a_b @ ( cons_P2262690024297812098_a_a_b @ X4 @ nil_Pr4580708590200510514_a_a_b ) @ Xs4 ) ) )
            & ( Ys
              = ( append157347090090570333_a_a_b @ Pre @ ( append157347090090570333_a_a_b @ ( cons_P2262690024297812098_a_a_b @ Y @ nil_Pr4580708590200510514_a_a_b ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1046_rev__induct2,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,Ys: list_P7155397214832950472_a_a_b,P: list_P7155397214832950472_a_a_b > list_P7155397214832950472_a_a_b > $o] :
      ( ( ( size_s1546398270689948508_a_a_b @ Xs )
        = ( size_s1546398270689948508_a_a_b @ Ys ) )
     => ( ( P @ nil_Pr4580708590200510514_a_a_b @ nil_Pr4580708590200510514_a_a_b )
       => ( ! [X4: produc4770514982935793080_a_a_b,Xs3: list_P7155397214832950472_a_a_b,Y: produc4770514982935793080_a_a_b,Ys4: list_P7155397214832950472_a_a_b] :
              ( ( ( size_s1546398270689948508_a_a_b @ Xs3 )
                = ( size_s1546398270689948508_a_a_b @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( append157347090090570333_a_a_b @ Xs3 @ ( cons_P2262690024297812098_a_a_b @ X4 @ nil_Pr4580708590200510514_a_a_b ) ) @ ( append157347090090570333_a_a_b @ Ys4 @ ( cons_P2262690024297812098_a_a_b @ Y @ nil_Pr4580708590200510514_a_a_b ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1047_not__distinct__conv__prefix,axiom,
    ! [As: list_P8197596098214363964_a_a_a] :
      ( ( ~ ( distin5648706335258269743_a_a_a @ As ) )
      = ( ? [Xs2: list_P8197596098214363964_a_a_a,Y4: produc884577481120224310_a_a_a,Ys3: list_P8197596098214363964_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ Y4 @ ( set_Pr8067384617587748683_a_a_a @ Xs2 ) )
            & ( distin5648706335258269743_a_a_a @ Xs2 )
            & ( As
              = ( append7211258631950116177_a_a_a @ Xs2 @ ( cons_P6318101215856909548_a_a_a @ Y4 @ Ys3 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1048_not__distinct__conv__prefix,axiom,
    ! [As: list_S8072159660131056972_a_a_a] :
      ( ( ~ ( distin5156058890512421251_a_a_a @ As ) )
      = ( ? [Xs2: list_S8072159660131056972_a_a_a,Y4: sum_su5154756782089200444_a_a_a,Ys3: list_S8072159660131056972_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ Y4 @ ( set_Su3113867532901839207_a_a_a @ Xs2 ) )
            & ( distin5156058890512421251_a_a_a @ Xs2 )
            & ( As
              = ( append944914365211047137_a_a_a @ Xs2 @ ( cons_S2140548516868915206_a_a_a @ Y4 @ Ys3 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1049_not__distinct__conv__prefix,axiom,
    ! [As: list_P7155397214832950472_a_a_b] :
      ( ( ~ ( distin1372936652459515135_a_a_b @ As ) )
      = ( ? [Xs2: list_P7155397214832950472_a_a_b,Y4: produc4770514982935793080_a_a_b,Ys3: list_P7155397214832950472_a_a_b] :
            ( ( member642552543386773071_a_a_b @ Y4 @ ( set_Pr4578018992751950051_a_a_b @ Xs2 ) )
            & ( distin1372936652459515135_a_a_b @ Xs2 )
            & ( As
              = ( append157347090090570333_a_a_b @ Xs2 @ ( cons_P2262690024297812098_a_a_b @ Y4 @ Ys3 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1050_non__sym__dist__pairs_H_Osimps_I2_J,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( non_sym_dist_pairs_a @ ( cons_a @ X3 @ Xs ) )
      = ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X3 ) @ Xs ) @ ( non_sym_dist_pairs_a @ Xs ) ) ) ).

% non_sym_dist_pairs'.simps(2)
thf(fact_1051_path__append__transition__elim_I2_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,T: produc884577481120224310_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ M @ Q @ ( append7211258631950116177_a_a_a @ P4 @ ( cons_P6318101215856909548_a_a_a @ T @ nil_Pr6029279252097497404_a_a_a ) ) )
     => ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) ) ) ).

% path_append_transition_elim(2)
thf(fact_1052_path__append__transition__elim_I3_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,T: produc884577481120224310_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ M @ Q @ ( append7211258631950116177_a_a_a @ P4 @ ( cons_P6318101215856909548_a_a_a @ T @ nil_Pr6029279252097497404_a_a_a ) ) )
     => ( ( produc7477144754405272026_a_a_a @ T )
        = ( target5023745622805013891_a_b_c @ Q @ P4 ) ) ) ).

% path_append_transition_elim(3)
thf(fact_1053_product__transitions__def,axiom,
    ! [A2: fsm_Su3173892059090327343_a_b_c,B3: fsm_Su3173892059090327343_a_b_c] :
      ( ( transi549026741542983407_a_b_c @ ( produc8615301468332162573_a_a_a @ A2 @ B3 ) )
      = ( collec181093262231583073_a_a_a
        @ ^ [Uu: produc7598642544577812982_a_a_a] :
          ? [QA: sum_su5154756782089200444_a_a_a,QB: sum_su5154756782089200444_a_a_a,X: b,Y4: c,QA2: sum_su5154756782089200444_a_a_a,QB2: sum_su5154756782089200444_a_a_a] :
            ( ( Uu
              = ( produc2666597461487561190_a_a_a @ ( produc5672884958730733073_a_a_a @ QA @ QB ) @ ( produc8193768035335866968_a_a_a @ X @ ( produc2664588836397278192_a_a_a @ Y4 @ ( produc5672884958730733073_a_a_a @ QA2 @ QB2 ) ) ) ) )
            & ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ QA @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ QA2 ) ) ) @ ( transi749694810082594024_a_b_c @ A2 ) )
            & ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ QB @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ QB2 ) ) ) @ ( transi749694810082594024_a_b_c @ B3 ) ) ) ) ) ).

% product_transitions_def
thf(fact_1054_product__transitions__def,axiom,
    ! [A2: fsm_Su3173892059090327343_a_b_c,B3: fsm_b_b_c] :
      ( ( transi9045946477028714084_b_b_c @ ( produc4284470041628015470_b_c_b @ A2 @ B3 ) )
      = ( collec2650623390879315225_a_a_b
        @ ^ [Uu: produc6221990862682875182_a_a_b] :
          ? [QA: sum_su5154756782089200444_a_a_a,QB: b,X: b,Y4: c,QA2: sum_su5154756782089200444_a_a_a,QB2: b] :
            ( ( Uu
              = ( produc1195494797170244326_a_a_b @ ( produc6086470211827429482_a_a_b @ QA @ QB ) @ ( produc3453035827950477327_a_a_b @ X @ ( produc1395193086026067297_a_a_b @ Y4 @ ( produc6086470211827429482_a_a_b @ QA2 @ QB2 ) ) ) ) )
            & ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ QA @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ QA2 ) ) ) @ ( transi749694810082594024_a_b_c @ A2 ) )
            & ( member8152910379392791327od_c_b @ ( produc1782518189699212006od_c_b @ QB @ ( produc3440865918304110462od_c_b @ X @ ( product_Pair_c_b @ Y4 @ QB2 ) ) ) @ ( transitions_b_b_c @ B3 ) ) ) ) ) ).

% product_transitions_def
thf(fact_1055_non__sym__dist__pairs_H_Oelims,axiom,
    ! [X3: list_a,Y3: list_P1396940483166286381od_a_a] :
      ( ( ( non_sym_dist_pairs_a @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_a )
         => ( Y3 != nil_Product_prod_a_a ) )
       => ~ ! [X4: a,Xs3: list_a] :
              ( ( X3
                = ( cons_a @ X4 @ Xs3 ) )
             => ( Y3
               != ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X4 ) @ Xs3 ) @ ( non_sym_dist_pairs_a @ Xs3 ) ) ) ) ) ) ).

% non_sym_dist_pairs'.elims
thf(fact_1056_canonical__separator__path__split__target__isl,axiom,
    ! [M: fsm_a_b_c,Q1: a,Q2: a,P4: list_P8197596098214363964_a_a_a,T: produc884577481120224310_a_a_a] :
      ( ( path_S221595587009941175_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) @ ( append7211258631950116177_a_a_a @ P4 @ ( cons_P6318101215856909548_a_a_a @ T @ nil_Pr6029279252097497404_a_a_a ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( sum_is4240621810064316983_a_a_a @ ( target5023745622805013891_a_b_c @ ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) @ P4 ) ) ) ) ) ).

% canonical_separator_path_split_target_isl
thf(fact_1057_path__io__split_I1_J,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a,P4: list_P3520633289758869904_a_a_a,Io1: list_P7155397214832950472_a_a_b,Io22: list_P7155397214832950472_a_a_b] :
      ( ( path_P4574188132515142236_a_a_b @ M @ Q @ P4 )
     => ( ( ( map_Pr6873991885683231230_a_a_b
            @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
            @ P4 )
          = ( append157347090090570333_a_a_b @ Io1 @ Io22 ) )
       => ( path_P4574188132515142236_a_a_b @ M @ Q @ ( take_P6207148901550968722_a_a_a @ ( size_s1546398270689948508_a_a_b @ Io1 ) @ P4 ) ) ) ) ).

% path_io_split(1)
thf(fact_1058_path__io__split_I1_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,Io1: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c] :
      ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
     => ( ( ( map_Pr6504563922737449846od_b_c
            @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
            @ P4 )
          = ( append2547753245680614915od_b_c @ Io1 @ Io22 ) )
       => ( path_S221595587009941175_a_b_c @ M @ Q @ ( take_P6848719478162124068_a_a_a @ ( size_s3392097710323735898od_b_c @ Io1 ) @ P4 ) ) ) ) ).

% path_io_split(1)
thf(fact_1059_path__io__split_I2_J,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a,P4: list_P3520633289758869904_a_a_a,Io1: list_P7155397214832950472_a_a_b,Io22: list_P7155397214832950472_a_a_b] :
      ( ( path_P4574188132515142236_a_a_b @ M @ Q @ P4 )
     => ( ( ( map_Pr6873991885683231230_a_a_b
            @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
            @ P4 )
          = ( append157347090090570333_a_a_b @ Io1 @ Io22 ) )
       => ( ( map_Pr6873991885683231230_a_a_b
            @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
            @ ( take_P6207148901550968722_a_a_a @ ( size_s1546398270689948508_a_a_b @ Io1 ) @ P4 ) )
          = Io1 ) ) ) ).

% path_io_split(2)
thf(fact_1060_path__io__split_I2_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,P4: list_P8197596098214363964_a_a_a,Io1: list_P903359562653991662od_b_c,Io22: list_P903359562653991662od_b_c] :
      ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
     => ( ( ( map_Pr6504563922737449846od_b_c
            @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
            @ P4 )
          = ( append2547753245680614915od_b_c @ Io1 @ Io22 ) )
       => ( ( map_Pr6504563922737449846od_b_c
            @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
            @ ( take_P6848719478162124068_a_a_a @ ( size_s3392097710323735898od_b_c @ Io1 ) @ P4 ) )
          = Io1 ) ) ) ).

% path_io_split(2)
thf(fact_1061_product__transition__split_I2_J,axiom,
    ! [T: produc5452756132721879828_a_a_a,A2: fsm_c_b_c,B3: fsm_Su3173892059090327343_a_b_c] :
      ( ( member4956778701725638589_a_a_a @ T @ ( transi4612001117653690135_a_b_c @ ( produc5612601397720187563_a_a_a @ A2 @ B3 ) ) )
     => ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc3788472483471646264_a_a_a @ T ) ) @ ( produc1730628064597884819_a_a_a @ ( produc4674560010915000278_a_a_a @ ( produc7812802799185138298_a_a_a @ T ) ) @ ( produc8806894744968565733_a_a_a @ ( produc2631857103351487144_a_a_a @ ( produc823359682969028116_a_a_a @ ( produc7812802799185138298_a_a_a @ T ) ) ) @ ( produc1172910294343954743_a_a_a @ ( produc3025386708695326438_a_a_a @ ( produc823359682969028116_a_a_a @ ( produc7812802799185138298_a_a_a @ T ) ) ) ) ) ) ) @ ( transi749694810082594024_a_b_c @ B3 ) ) ) ).

% product_transition_split(2)
thf(fact_1062_product__transition__split_I2_J,axiom,
    ! [T: produc6800189296593107456_a_a_a,A2: fsm_c_8682073512053177019_a_a_b,B3: fsm_Su4683625031830649393_a_a_b] :
      ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) ) )
     => ( member6311545829279235415_a_a_a @ ( produc8935917708474127986_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc2316789063618113990_a_a_a @ T ) ) @ ( produc919293830850624703_a_a_a @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) @ ( produc1978028051579340516_a_a_a @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) ) ) ) ) @ ( transi473620957736112346_a_a_b @ B3 ) ) ) ).

% product_transition_split(2)
thf(fact_1063_product__transition__split_I1_J,axiom,
    ! [T: produc6221990862682875182_a_a_b,A2: fsm_Su3173892059090327343_a_b_c,B3: fsm_b_b_c] :
      ( ( member5726013431686633943_a_a_b @ T @ ( transi9045946477028714084_b_b_c @ ( produc4284470041628015470_b_c_b @ A2 @ B3 ) ) )
     => ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ ( produc4064411471574014846_a_a_b @ ( produc8830533715806318290_a_a_b @ T ) ) @ ( produc1730628064597884819_a_a_a @ ( produc793064584523784995_a_a_b @ ( produc3631491994665034516_a_a_b @ T ) ) @ ( produc8806894744968565733_a_a_a @ ( produc5507164989288522869_a_a_b @ ( produc6165236293432588641_a_a_b @ ( produc3631491994665034516_a_a_b @ T ) ) ) @ ( produc4064411471574014846_a_a_b @ ( produc5900694594632362163_a_a_b @ ( produc6165236293432588641_a_a_b @ ( produc3631491994665034516_a_a_b @ T ) ) ) ) ) ) ) @ ( transi749694810082594024_a_b_c @ A2 ) ) ) ).

% product_transition_split(1)
thf(fact_1064_product__transition__split_I1_J,axiom,
    ! [T: produc7062490817812474422_a_a_a,A2: fsm_Su3173892059090327343_a_b_c,B3: fsm_Pr6383261684945925900_a_b_c] :
      ( ( member1936903858404289887_a_a_a @ T @ ( transi5606176809373972940_a_b_c @ ( produc331288983406632426_a_a_a @ A2 @ B3 ) ) )
     => ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ ( produc7477144754405272026_a_a_a @ ( produc6245311756135949970_a_a_a @ T ) ) @ ( produc1730628064597884819_a_a_a @ ( produc3382922180580566817_a_a_a @ ( produc1688335080334495700_a_a_a @ T ) ) @ ( produc8806894744968565733_a_a_a @ ( produc410120848946820409_a_a_a @ ( produc7407252496294058851_a_a_a @ ( produc1688335080334495700_a_a_a @ T ) ) ) @ ( produc7477144754405272026_a_a_a @ ( produc1285651877402164091_a_a_a @ ( produc7407252496294058851_a_a_a @ ( produc1688335080334495700_a_a_a @ T ) ) ) ) ) ) ) @ ( transi749694810082594024_a_b_c @ A2 ) ) ) ).

% product_transition_split(1)
thf(fact_1065_product__transition__split_I1_J,axiom,
    ! [T: produc6800189296593107456_a_a_a,A2: fsm_c_8682073512053177019_a_a_b,B3: fsm_Su4683625031830649393_a_a_b] :
      ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) ) )
     => ( member5460219220941260647od_b_c @ ( produc9054217211726079306od_b_c @ ( produc6784836004715151097_a_a_a @ ( produc2316789063618113990_a_a_a @ T ) ) @ ( produc6554145002957781203od_b_c @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) @ ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) ) ) ) ) @ ( transi5949722234232290860_a_a_b @ A2 ) ) ) ).

% product_transition_split(1)
thf(fact_1066_paths__from__product__path_I2_J,axiom,
    ! [A2: fsm_c_8682073512053177019_a_a_b,B3: fsm_Su4683625031830649393_a_a_b,P4: list_P3520633289758869904_a_a_a] :
      ( ( path_P4574188132515142236_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) @ ( initia5013972905137889861_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) ) @ P4 )
     => ( path_S4752583732068403787_a_a_b @ B3 @ ( initia3760672355826311988_a_a_b @ B3 )
        @ ( map_Pr4032422291525893254_a_a_a
          @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc8935917708474127986_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc2316789063618113990_a_a_a @ T4 ) ) @ ( produc919293830850624703_a_a_a @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc1978028051579340516_a_a_a @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) ) ) ) )
          @ P4 ) ) ) ).

% paths_from_product_path(2)
thf(fact_1067_paths__from__product__path_I1_J,axiom,
    ! [A2: fsm_c_8682073512053177019_a_a_b,B3: fsm_Su4683625031830649393_a_a_b,P4: list_P3520633289758869904_a_a_a] :
      ( ( path_P4574188132515142236_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) @ ( initia5013972905137889861_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) ) @ P4 )
     => ( path_c5421623011159638011_a_a_b @ A2 @ ( initia6039622272215124050_a_a_b @ A2 )
        @ ( map_Pr4103997051015666966od_b_c
          @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc9054217211726079306od_b_c @ ( produc6784836004715151097_a_a_a @ ( produc2316789063618113990_a_a_a @ T4 ) ) @ ( produc6554145002957781203od_b_c @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) ) ) ) )
          @ P4 ) ) ) ).

% paths_from_product_path(1)
thf(fact_1068_observable__io__targets__next,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,T: produc6800189296593107456_a_a_a,Io: list_P7155397214832950472_a_a_b] :
      ( ( observ7609480127267726238_a_a_b @ M )
     => ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ M ) )
       => ( ( io_tar1879755107597495661_a_a_b @ M
            @ ( append157347090090570333_a_a_b
              @ ( map_Pr6873991885683231230_a_a_b
                @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                @ ( cons_P5563258090397451722_a_a_a @ T @ nil_Pr3590249381822989690_a_a_a ) )
              @ Io )
            @ ( produc2316789063618113990_a_a_a @ T ) )
          = ( io_tar1879755107597495661_a_a_b @ M @ Io @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) ) ) ) ) ).

% observable_io_targets_next
thf(fact_1069_observable__io__targets__next,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,T: produc884577481120224310_a_a_a,Io: list_P903359562653991662od_b_c] :
      ( ( observ6810049560328217077_a_b_c @ M )
     => ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
       => ( ( io_tar1727080856364058086_a_b_c @ M
            @ ( append2547753245680614915od_b_c
              @ ( map_Pr6504563922737449846od_b_c
                @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                @ ( cons_P6318101215856909548_a_a_a @ T @ nil_Pr6029279252097497404_a_a_a ) )
              @ Io )
            @ ( produc7477144754405272026_a_a_a @ T ) )
          = ( io_tar1727080856364058086_a_b_c @ M @ Io @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) ) ) ) ) ).

% observable_io_targets_next
thf(fact_1070_paths__from__product__path_I4_J,axiom,
    ! [A2: fsm_c_8682073512053177019_a_a_b,B3: fsm_Su4683625031830649393_a_a_b,P4: list_P3520633289758869904_a_a_a] :
      ( ( path_P4574188132515142236_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) @ ( initia5013972905137889861_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) ) @ P4 )
     => ( ( target5129991884641307775_a_a_b @ ( initia3760672355826311988_a_a_b @ B3 )
          @ ( map_Pr4032422291525893254_a_a_a
            @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc8935917708474127986_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc2316789063618113990_a_a_a @ T4 ) ) @ ( produc919293830850624703_a_a_a @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc1978028051579340516_a_a_a @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) ) ) ) )
            @ P4 ) )
        = ( produc1172910294343954743_a_a_a @ ( target4721681511209552016_a_a_b @ ( initia5013972905137889861_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) ) @ P4 ) ) ) ) ).

% paths_from_product_path(4)
thf(fact_1071_paths__from__product__path_I3_J,axiom,
    ! [A2: fsm_c_8682073512053177019_a_a_b,B3: fsm_Su4683625031830649393_a_a_b,P4: list_P3520633289758869904_a_a_a] :
      ( ( path_P4574188132515142236_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) @ ( initia5013972905137889861_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) ) @ P4 )
     => ( ( target1000401010099934919_a_a_b @ ( initia6039622272215124050_a_a_b @ A2 )
          @ ( map_Pr4103997051015666966od_b_c
            @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc9054217211726079306od_b_c @ ( produc6784836004715151097_a_a_a @ ( produc2316789063618113990_a_a_a @ T4 ) ) @ ( produc6554145002957781203od_b_c @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) ) ) ) )
            @ P4 ) )
        = ( produc6784836004715151097_a_a_a @ ( target4721681511209552016_a_a_b @ ( initia5013972905137889861_a_a_b @ ( produc6121419909112718161_a_a_a @ A2 @ B3 ) ) @ P4 ) ) ) ) ).

% paths_from_product_path(3)
thf(fact_1072_from__FSM__simps_I4_J,axiom,
    ! [Q: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c] :
      ( ( member5679158588528110291_a_a_a @ Q @ ( states3035385874452377268_a_b_c @ M ) )
     => ( ( transi749694810082594024_a_b_c @ ( from_F8539169374725194592_a_b_c @ M @ Q ) )
        = ( transi749694810082594024_a_b_c @ M ) ) ) ).

% from_FSM_simps(4)
thf(fact_1073_reachable__states__path,axiom,
    ! [Q: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c,P4: list_P8197596098214363964_a_a_a,T: produc884577481120224310_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ Q @ ( reacha2263492793340247097_a_b_c @ M ) )
     => ( ( path_S221595587009941175_a_b_c @ M @ Q @ P4 )
       => ( ( member8362999488266169055_a_a_a @ T @ ( set_Pr8067384617587748683_a_a_a @ P4 ) )
         => ( member5679158588528110291_a_a_a @ ( produc7477144754405272026_a_a_a @ T ) @ ( reacha2263492793340247097_a_b_c @ M ) ) ) ) ) ).

% reachable_states_path
thf(fact_1074_reachable__states__next,axiom,
    ! [Q: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c,T: produc884577481120224310_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ Q @ ( reacha2263492793340247097_a_b_c @ M ) )
     => ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
       => ( ( ( produc7477144754405272026_a_a_a @ T )
            = Q )
         => ( member5679158588528110291_a_a_a @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) @ ( reacha2263492793340247097_a_b_c @ M ) ) ) ) ) ).

% reachable_states_next
thf(fact_1075_reachable__states__next,axiom,
    ! [Q: produc1895207096998757355_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,T: produc6800189296593107456_a_a_a] :
      ( ( member6990616694304513154_a_a_a @ Q @ ( reacha7147495264601923674_a_a_b @ M ) )
     => ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ M ) )
       => ( ( ( produc2316789063618113990_a_a_a @ T )
            = Q )
         => ( member6990616694304513154_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) @ ( reacha7147495264601923674_a_a_b @ M ) ) ) ) ) ).

% reachable_states_next
thf(fact_1076_acyclic__no__self__loop,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a] :
      ( ( acycli6334973534855091236_a_b_c @ M )
     => ( ( member5679158588528110291_a_a_a @ Q @ ( reacha2263492793340247097_a_b_c @ M ) )
       => ~ ? [X7: b,Y5: c] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q @ ( produc1730628064597884819_a_a_a @ X7 @ ( produc8806894744968565733_a_a_a @ Y5 @ Q ) ) ) @ ( transi749694810082594024_a_b_c @ M ) ) ) ) ).

% acyclic_no_self_loop
thf(fact_1077_reachable__states__initial__or__target,axiom,
    ! [Q: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c] :
      ( ( member5679158588528110291_a_a_a @ Q @ ( reacha2263492793340247097_a_b_c @ M ) )
     => ( ( Q
          = ( initia839594848065427214_a_b_c @ M ) )
        | ? [X4: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ M ) )
            & ( member5679158588528110291_a_a_a @ ( produc7477144754405272026_a_a_a @ X4 ) @ ( reacha2263492793340247097_a_b_c @ M ) )
            & ( ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
              = Q ) ) ) ) ).

% reachable_states_initial_or_target
thf(fact_1078_reachable__states__initial__or__target,axiom,
    ! [Q: produc1895207096998757355_a_a_a,M: fsm_Pr6287166851770849218_a_a_b] :
      ( ( member6990616694304513154_a_a_a @ Q @ ( reacha7147495264601923674_a_a_b @ M ) )
     => ( ( Q
          = ( initia5013972905137889861_a_a_b @ M ) )
        | ? [X4: produc6800189296593107456_a_a_a] :
            ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ M ) )
            & ( member6990616694304513154_a_a_a @ ( produc2316789063618113990_a_a_a @ X4 ) @ ( reacha7147495264601923674_a_a_b @ M ) )
            & ( ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
              = Q ) ) ) ) ).

% reachable_states_initial_or_target
thf(fact_1079_reachable__states__induct,axiom,
    ! [Q: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( member5679158588528110291_a_a_a @ Q @ ( reacha2263492793340247097_a_b_c @ M ) )
     => ( ( P @ ( initia839594848065427214_a_b_c @ M ) )
       => ( ! [T5: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ T5 @ ( transi749694810082594024_a_b_c @ M ) )
             => ( ( member5679158588528110291_a_a_a @ ( produc7477144754405272026_a_a_a @ T5 ) @ ( reacha2263492793340247097_a_b_c @ M ) )
               => ( ( P @ ( produc7477144754405272026_a_a_a @ T5 ) )
                 => ( P @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T5 ) ) ) ) ) ) )
         => ( P @ Q ) ) ) ) ).

% reachable_states_induct
thf(fact_1080_reachable__states__induct,axiom,
    ! [Q: produc1895207096998757355_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,P: produc1895207096998757355_a_a_a > $o] :
      ( ( member6990616694304513154_a_a_a @ Q @ ( reacha7147495264601923674_a_a_b @ M ) )
     => ( ( P @ ( initia5013972905137889861_a_a_b @ M ) )
       => ( ! [T5: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ T5 @ ( transi7010299678732630507_a_a_b @ M ) )
             => ( ( member6990616694304513154_a_a_a @ ( produc2316789063618113990_a_a_a @ T5 ) @ ( reacha7147495264601923674_a_a_b @ M ) )
               => ( ( P @ ( produc2316789063618113990_a_a_a @ T5 ) )
                 => ( P @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T5 ) ) ) ) ) ) )
         => ( P @ Q ) ) ) ) ).

% reachable_states_induct
thf(fact_1081_reachable__states__cases,axiom,
    ! [Q: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( member5679158588528110291_a_a_a @ Q @ ( reacha2263492793340247097_a_b_c @ M ) )
     => ( ( P @ ( initia839594848065427214_a_b_c @ M ) )
       => ( ! [T5: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ T5 @ ( transi749694810082594024_a_b_c @ M ) )
             => ( ( member5679158588528110291_a_a_a @ ( produc7477144754405272026_a_a_a @ T5 ) @ ( reacha2263492793340247097_a_b_c @ M ) )
               => ( P @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T5 ) ) ) ) ) )
         => ( P @ Q ) ) ) ) ).

% reachable_states_cases
thf(fact_1082_reachable__states__cases,axiom,
    ! [Q: produc1895207096998757355_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,P: produc1895207096998757355_a_a_a > $o] :
      ( ( member6990616694304513154_a_a_a @ Q @ ( reacha7147495264601923674_a_a_b @ M ) )
     => ( ( P @ ( initia5013972905137889861_a_a_b @ M ) )
       => ( ! [T5: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ T5 @ ( transi7010299678732630507_a_a_b @ M ) )
             => ( ( member6990616694304513154_a_a_a @ ( produc2316789063618113990_a_a_a @ T5 ) @ ( reacha7147495264601923674_a_a_b @ M ) )
               => ( P @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T5 ) ) ) ) ) )
         => ( P @ Q ) ) ) ) ).

% reachable_states_cases
thf(fact_1083_canonical__separator__states,axiom,
    ! [S1: a,S2: a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member5679158588528110291_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ S1 @ S2 ) ) @ ( states3035385874452377268_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ S1 @ S2 ) @ ( states3827391290251485425_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) ) ) ) ) ) ).

% canonical_separator_states
thf(fact_1084_acyclic__induction,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ( acycli6334973534855091236_a_b_c @ M )
     => ( ! [Q7: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ Q7 @ ( reacha2263492793340247097_a_b_c @ M ) )
           => ( ! [T6: produc884577481120224310_a_a_a] :
                  ( ( member8362999488266169055_a_a_a @ T6 @ ( transi749694810082594024_a_b_c @ M ) )
                 => ( ( ( produc7477144754405272026_a_a_a @ T6 )
                      = Q7 )
                   => ( P @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T6 ) ) ) ) ) )
             => ( P @ Q7 ) ) )
       => ! [X7: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X7 @ ( reacha2263492793340247097_a_b_c @ M ) )
           => ( P @ X7 ) ) ) ) ).

% acyclic_induction
thf(fact_1085_acyclic__induction,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,P: produc1895207096998757355_a_a_a > $o] :
      ( ( acycli3019483576907578287_a_a_b @ M )
     => ( ! [Q7: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ Q7 @ ( reacha7147495264601923674_a_a_b @ M ) )
           => ( ! [T6: produc6800189296593107456_a_a_a] :
                  ( ( member4643618012521571351_a_a_a @ T6 @ ( transi7010299678732630507_a_a_b @ M ) )
                 => ( ( ( produc2316789063618113990_a_a_a @ T6 )
                      = Q7 )
                   => ( P @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T6 ) ) ) ) ) )
             => ( P @ Q7 ) ) )
       => ! [X7: produc1895207096998757355_a_a_a] :
            ( ( member6990616694304513154_a_a_a @ X7 @ ( reacha7147495264601923674_a_a_b @ M ) )
           => ( P @ X7 ) ) ) ) ).

% acyclic_induction
thf(fact_1086_canonical__separator__transition__source,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_a_b_c,Q1: a,Q2: a] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) ) )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
         => ~ ! [Q12: a,Q22: a] :
                ( ( ( produc7477144754405272026_a_a_a @ T )
                  = ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q12 @ Q22 ) ) )
               => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Q12 @ Q22 ) @ ( states3827391290251485425_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) ) ) ) ) ) ) ).

% canonical_separator_transition_source
thf(fact_1087_product__from__transition__split_I1_J,axiom,
    ! [T: produc7669896877331299902_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a] :
      ( ( member5238278507536988629_a_a_a @ T @ ( transi4297421501747413173_a_a_b @ ( produc2190053497452919855_a_a_a @ ( from_F6245563941867181427_a_a_b @ M @ Q1 ) @ ( from_F6245563941867181427_a_a_b @ M @ Q2 ) ) ) )
     => ( ( member6990616694304513154_a_a_a @ Q1 @ ( states8463590192175095071_a_a_b @ M ) )
       => ( ( member6990616694304513154_a_a_a @ Q2 @ ( states8463590192175095071_a_a_b @ M ) )
         => ? [X4: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ M ) )
              & ( ( produc2316789063618113990_a_a_a @ X4 )
                = ( produc196831967674047067_a_a_a @ ( produc302551054129649228_a_a_a @ T ) ) )
              & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
                = ( produc1943859948161608478_a_a_a @ ( produc1023885589438848650_a_a_a @ T ) ) )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                = ( produc1515593567674010107_a_a_a @ ( produc3460917645060139100_a_a_a @ ( produc1023885589438848650_a_a_a @ T ) ) ) ) ) ) ) ) ).

% product_from_transition_split(1)
thf(fact_1088_product__from__transition__split_I1_J,axiom,
    ! [T: produc7598642544577812982_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a] :
      ( ( member3147200746921337119_a_a_a @ T @ ( transi549026741542983407_a_b_c @ ( produc8615301468332162573_a_a_a @ ( from_F8539169374725194592_a_b_c @ M @ Q1 ) @ ( from_F8539169374725194592_a_b_c @ M @ Q2 ) ) ) )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3035385874452377268_a_b_c @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3035385874452377268_a_b_c @ M ) )
         => ? [X4: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ M ) )
              & ( ( produc7477144754405272026_a_a_a @ X4 )
                = ( produc3656708485065107965_a_a_a @ ( produc5035679817875094482_a_a_a @ T ) ) )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
                = ( produc8807399175486215748_a_a_a @ ( produc7332565265614368020_a_a_a @ T ) ) )
              & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                = ( produc5074649251625090012_a_a_a @ ( produc2959483784140025478_a_a_a @ ( produc7332565265614368020_a_a_a @ T ) ) ) ) ) ) ) ) ).

% product_from_transition_split(1)
thf(fact_1089_from__product__initial__paths__ex,axiom,
    ! [Q1: sum_su5154756782089200444_a_a_a,M: fsm_Su4683625031830649393_a_a_b,Q2: sum_su5154756782089200444_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3538144058484115470_a_a_b @ M ) )
     => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3538144058484115470_a_a_b @ M ) )
       => ? [P1: list_P9067677309952966352_a_a_a,P22: list_P9067677309952966352_a_a_a] :
            ( ( path_S4752583732068403787_a_a_b @ ( from_F7700525888172323938_a_a_b @ M @ Q1 ) @ ( initia3760672355826311988_a_a_b @ ( from_F7700525888172323938_a_a_b @ M @ Q1 ) ) @ P1 )
            & ( path_S4752583732068403787_a_a_b @ ( from_F7700525888172323938_a_a_b @ M @ Q2 ) @ ( initia3760672355826311988_a_a_b @ ( from_F7700525888172323938_a_a_b @ M @ Q2 ) ) @ P22 )
            & ( ( target5129991884641307775_a_a_b @ ( initia3760672355826311988_a_a_b @ ( from_F7700525888172323938_a_a_b @ M @ Q1 ) ) @ P1 )
              = Q1 )
            & ( ( target5129991884641307775_a_a_b @ ( initia3760672355826311988_a_a_b @ ( from_F7700525888172323938_a_a_b @ M @ Q2 ) ) @ P22 )
              = Q2 )
            & ( ( map_Pr5810527985649246014_a_a_b
                @ ^ [T4: produc8681278163233174848_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3329354246078436523_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) @ ( produc9179341348180701688_a_a_a @ ( produc3812793320627011309_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) ) )
                @ P1 )
              = ( map_Pr5810527985649246014_a_a_b
                @ ^ [T4: produc8681278163233174848_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc3329354246078436523_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) @ ( produc9179341348180701688_a_a_a @ ( produc3812793320627011309_a_a_a @ ( produc5843923735430010052_a_a_a @ T4 ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_1090_from__product__initial__paths__ex,axiom,
    ! [Q1: produc884577481120224310_a_a_a,M: fsm_Pr791368616075801221_a_a_b,Q2: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ Q1 @ ( states6021486123142740674_a_a_b @ M ) )
     => ( ( member8362999488266169055_a_a_a @ Q2 @ ( states6021486123142740674_a_a_b @ M ) )
       => ? [P1: list_P1303110702266167372_a_a_a,P22: list_P1303110702266167372_a_a_a] :
            ( ( path_P6406173025370084677_a_a_b @ ( from_F1976116041108351342_a_a_b @ M @ Q1 ) @ ( initia6928045674339925532_a_a_b @ ( from_F1976116041108351342_a_a_b @ M @ Q1 ) ) @ P1 )
            & ( path_P6406173025370084677_a_a_b @ ( from_F1976116041108351342_a_a_b @ M @ Q2 ) @ ( initia6928045674339925532_a_a_b @ ( from_F1976116041108351342_a_a_b @ M @ Q2 ) ) @ P22 )
            & ( ( target1704919491679417873_a_a_b @ ( initia6928045674339925532_a_a_b @ ( from_F1976116041108351342_a_a_b @ M @ Q1 ) ) @ P1 )
              = Q1 )
            & ( ( target1704919491679417873_a_a_b @ ( initia6928045674339925532_a_a_b @ ( from_F1976116041108351342_a_a_b @ M @ Q2 ) ) @ P22 )
              = Q2 )
            & ( ( map_Pr2767717211958206138_a_a_b
                @ ^ [T4: produc4009731802749794876_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc1606485967678404637_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) @ ( produc5695092443052942970_a_a_a @ ( produc3123543664576935259_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) ) )
                @ P1 )
              = ( map_Pr2767717211958206138_a_a_b
                @ ^ [T4: produc4009731802749794876_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc1606485967678404637_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) @ ( produc5695092443052942970_a_a_a @ ( produc3123543664576935259_a_a_a @ ( produc6588247977250318856_a_a_a @ T4 ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_1091_from__product__initial__paths__ex,axiom,
    ! [Q1: produc4770514982935793080_a_a_b,M: fsm_Pr8211287145308421045_a_a_b,Q2: produc4770514982935793080_a_a_b] :
      ( ( member642552543386773071_a_a_b @ Q1 @ ( states5045336508805429138_a_a_b @ M ) )
     => ( ( member642552543386773071_a_a_b @ Q2 @ ( states5045336508805429138_a_a_b @ M ) )
       => ? [P1: list_P8385316394189796816_a_a_b,P22: list_P8385316394189796816_a_a_b] :
            ( ( path_P1155934449145476303_a_a_b @ ( from_F2827310258497515494_a_a_b @ M @ Q1 ) @ ( initia1595719221768223928_a_a_b @ ( from_F2827310258497515494_a_a_b @ M @ Q1 ) ) @ P1 )
            & ( path_P1155934449145476303_a_a_b @ ( from_F2827310258497515494_a_a_b @ M @ Q2 ) @ ( initia1595719221768223928_a_a_b @ ( from_F2827310258497515494_a_a_b @ M @ Q2 ) ) @ P22 )
            & ( ( target1303427827839886083_a_a_b @ ( initia1595719221768223928_a_a_b @ ( from_F2827310258497515494_a_a_b @ M @ Q1 ) ) @ P1 )
              = Q1 )
            & ( ( target1303427827839886083_a_a_b @ ( initia1595719221768223928_a_a_b @ ( from_F2827310258497515494_a_a_b @ M @ Q2 ) ) @ P22 )
              = Q2 )
            & ( ( map_Pr36629913606772286_a_a_b
                @ ^ [T4: produc2838380418727528000_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3595649328014056743_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) @ ( produc8717907853797376116_a_a_b @ ( produc6971105973522642281_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) ) )
                @ P1 )
              = ( map_Pr36629913606772286_a_a_b
                @ ^ [T4: produc2838380418727528000_a_a_b] : ( produc6086470211827429482_a_a_b @ ( produc3595649328014056743_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) @ ( produc8717907853797376116_a_a_b @ ( produc6971105973522642281_a_a_b @ ( produc4144864385024741828_a_a_b @ T4 ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_1092_from__product__initial__paths__ex,axiom,
    ! [Q1: sum_su5154756782089200444_a_a_a,M: fsm_Su5961347632456545774_a_a_a,Q2: sum_su5154756782089200444_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ Q1 @ ( states5822841456425053299_a_a_a @ M ) )
     => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states5822841456425053299_a_a_a @ M ) )
       => ? [P1: list_P4430297574055495993_a_a_a,P22: list_P4430297574055495993_a_a_a] :
            ( ( path_S3009051168982617206_a_a_a @ ( from_F2103252919843094815_a_a_a @ M @ Q1 ) @ ( initia3627050430038103245_a_a_a @ ( from_F2103252919843094815_a_a_a @ M @ Q1 ) ) @ P1 )
            & ( path_S3009051168982617206_a_a_a @ ( from_F2103252919843094815_a_a_a @ M @ Q2 ) @ ( initia3627050430038103245_a_a_a @ ( from_F2103252919843094815_a_a_a @ M @ Q2 ) ) @ P22 )
            & ( ( target7811201204777689922_a_a_a @ ( initia3627050430038103245_a_a_a @ ( from_F2103252919843094815_a_a_a @ M @ Q1 ) ) @ P1 )
              = Q1 )
            & ( ( target7811201204777689922_a_a_a @ ( initia3627050430038103245_a_a_a @ ( from_F2103252919843094815_a_a_a @ M @ Q2 ) ) @ P22 )
              = Q2 )
            & ( ( map_Pr7525866001273443320od_a_a
                @ ^ [T4: produc1141838204273412275_a_a_a] : ( product_Pair_a_a @ ( produc4618981482445519012_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) @ ( produc2350474654791476471_a_a_a @ ( produc5012511087789358306_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) ) )
                @ P1 )
              = ( map_Pr7525866001273443320od_a_a
                @ ^ [T4: produc1141838204273412275_a_a_a] : ( product_Pair_a_a @ ( produc4618981482445519012_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) @ ( produc2350474654791476471_a_a_a @ ( produc5012511087789358306_a_a_a @ ( produc5988991054768383257_a_a_a @ T4 ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_1093_from__product__initial__paths__ex,axiom,
    ! [Q1: produc884577481120224310_a_a_a,M: fsm_Pr3307108581279186586_a_a_a,Q2: produc884577481120224310_a_a_a] :
      ( ( member8362999488266169055_a_a_a @ Q1 @ ( states7932151110703657663_a_a_a @ M ) )
     => ( ( member8362999488266169055_a_a_a @ Q2 @ ( states7932151110703657663_a_a_a @ M ) )
       => ? [P1: list_P6192864029102241151_a_a_a,P22: list_P6192864029102241151_a_a_a] :
            ( ( path_P3388033272012105212_a_a_a @ ( from_F8247771635465257747_a_a_a @ M @ Q1 ) @ ( initia4901907407836924901_a_a_a @ ( from_F8247771635465257747_a_a_a @ M @ Q1 ) ) @ P1 )
            & ( path_P3388033272012105212_a_a_a @ ( from_F8247771635465257747_a_a_a @ M @ Q2 ) @ ( initia4901907407836924901_a_a_a @ ( from_F8247771635465257747_a_a_a @ M @ Q2 ) ) @ P22 )
            & ( ( target2673679534084926512_a_a_a @ ( initia4901907407836924901_a_a_a @ ( from_F8247771635465257747_a_a_a @ M @ Q1 ) ) @ P1 )
              = Q1 )
            & ( ( target2673679534084926512_a_a_a @ ( initia4901907407836924901_a_a_a @ ( from_F8247771635465257747_a_a_a @ M @ Q2 ) ) @ P22 )
              = Q2 )
            & ( ( map_Pr1392725673272416306od_a_a
                @ ^ [T4: produc7205803768117265017_a_a_a] : ( product_Pair_a_a @ ( produc5229955328659460580_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) @ ( produc1756692000304289723_a_a_a @ ( produc30913607518176806_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) ) )
                @ P1 )
              = ( map_Pr1392725673272416306od_a_a
                @ ^ [T4: produc7205803768117265017_a_a_a] : ( product_Pair_a_a @ ( produc5229955328659460580_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) @ ( produc1756692000304289723_a_a_a @ ( produc30913607518176806_a_a_a @ ( produc8180239367361206679_a_a_a @ T4 ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_1094_from__product__initial__paths__ex,axiom,
    ! [Q1: produc4770514982935793080_a_a_b,M: fsm_Pr1352325423224000874_b_a_a,Q2: produc4770514982935793080_a_a_b] :
      ( ( member642552543386773071_a_a_b @ Q1 @ ( states6742877326663703023_b_a_a @ M ) )
     => ( ( member642552543386773071_a_a_b @ Q2 @ ( states6742877326663703023_b_a_a @ M ) )
       => ? [P1: list_P9033668338835616561_a_a_b,P22: list_P9033668338835616561_a_a_b] :
            ( ( path_P447960203046551282_b_a_a @ ( from_F7986001506485183131_b_a_a @ M @ Q1 ) @ ( initia5419579234797426761_b_a_a @ ( from_F7986001506485183131_b_a_a @ M @ Q1 ) ) @ P1 )
            & ( path_P447960203046551282_b_a_a @ ( from_F7986001506485183131_b_a_a @ M @ Q2 ) @ ( initia5419579234797426761_b_a_a @ ( from_F7986001506485183131_b_a_a @ M @ Q2 ) ) @ P22 )
            & ( ( target4815919572193323454_b_a_a @ ( initia5419579234797426761_b_a_a @ ( from_F7986001506485183131_b_a_a @ M @ Q1 ) ) @ P1 )
              = Q1 )
            & ( ( target4815919572193323454_b_a_a @ ( initia5419579234797426761_b_a_a @ ( from_F7986001506485183131_b_a_a @ M @ Q2 ) ) @ P22 )
              = Q2 )
            & ( ( map_Pr2203486926099502080od_a_a
                @ ^ [T4: produc4165969764426093739_a_a_b] : ( product_Pair_a_a @ ( produc2204968541190299680_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) @ ( produc2705278681451453555_a_a_b @ ( produc7577140250099103326_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) ) )
                @ P1 )
              = ( map_Pr2203486926099502080od_a_a
                @ ^ [T4: produc4165969764426093739_a_a_b] : ( product_Pair_a_a @ ( produc2204968541190299680_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) @ ( produc2705278681451453555_a_a_b @ ( produc7577140250099103326_a_a_b @ ( produc5472006684323512593_a_a_b @ T4 ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_1095_from__product__initial__paths__ex,axiom,
    ! [Q1: produc1895207096998757355_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Q2: produc1895207096998757355_a_a_a] :
      ( ( member6990616694304513154_a_a_a @ Q1 @ ( states8463590192175095071_a_a_b @ M ) )
     => ( ( member6990616694304513154_a_a_a @ Q2 @ ( states8463590192175095071_a_a_b @ M ) )
       => ? [P1: list_P3520633289758869904_a_a_a,P22: list_P3520633289758869904_a_a_a] :
            ( ( path_P4574188132515142236_a_a_b @ ( from_F6245563941867181427_a_a_b @ M @ Q1 ) @ ( initia5013972905137889861_a_a_b @ ( from_F6245563941867181427_a_a_b @ M @ Q1 ) ) @ P1 )
            & ( path_P4574188132515142236_a_a_b @ ( from_F6245563941867181427_a_a_b @ M @ Q2 ) @ ( initia5013972905137889861_a_a_b @ ( from_F6245563941867181427_a_a_b @ M @ Q2 ) ) @ P22 )
            & ( ( target4721681511209552016_a_a_b @ ( initia5013972905137889861_a_a_b @ ( from_F6245563941867181427_a_a_b @ M @ Q1 ) ) @ P1 )
              = Q1 )
            & ( ( target4721681511209552016_a_a_b @ ( initia5013972905137889861_a_a_b @ ( from_F6245563941867181427_a_a_b @ M @ Q2 ) ) @ P22 )
              = Q2 )
            & ( ( map_Pr6873991885683231230_a_a_b
                @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                @ P1 )
              = ( map_Pr6873991885683231230_a_a_b
                @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_1096_from__product__initial__paths__ex,axiom,
    ! [Q1: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Q2: sum_su5154756782089200444_a_a_a] :
      ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3035385874452377268_a_b_c @ M ) )
     => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3035385874452377268_a_b_c @ M ) )
       => ? [P1: list_P8197596098214363964_a_a_a,P22: list_P8197596098214363964_a_a_a] :
            ( ( path_S221595587009941175_a_b_c @ ( from_F8539169374725194592_a_b_c @ M @ Q1 ) @ ( initia839594848065427214_a_b_c @ ( from_F8539169374725194592_a_b_c @ M @ Q1 ) ) @ P1 )
            & ( path_S221595587009941175_a_b_c @ ( from_F8539169374725194592_a_b_c @ M @ Q2 ) @ ( initia839594848065427214_a_b_c @ ( from_F8539169374725194592_a_b_c @ M @ Q2 ) ) @ P22 )
            & ( ( target5023745622805013891_a_b_c @ ( initia839594848065427214_a_b_c @ ( from_F8539169374725194592_a_b_c @ M @ Q1 ) ) @ P1 )
              = Q1 )
            & ( ( target5023745622805013891_a_b_c @ ( initia839594848065427214_a_b_c @ ( from_F8539169374725194592_a_b_c @ M @ Q2 ) ) @ P22 )
              = Q2 )
            & ( ( map_Pr6504563922737449846od_b_c
                @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                @ P1 )
              = ( map_Pr6504563922737449846od_b_c
                @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
                @ P22 ) ) ) ) ) ).

% from_product_initial_paths_ex
thf(fact_1097_product__undefined__input,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a,Qq2: produc5928450642596067255_a_a_a,X3: sum_su5154756782089200444_a_a_a] :
      ( ~ ? [X4: produc7669896877331299902_a_a_a] :
            ( ( member5238278507536988629_a_a_a @ X4 @ ( transi4297421501747413173_a_a_b @ ( produc2190053497452919855_a_a_a @ ( from_F6245563941867181427_a_a_b @ M @ Q1 ) @ ( from_F6245563941867181427_a_a_b @ M @ Q2 ) ) ) )
            & ( ( produc302551054129649228_a_a_a @ X4 )
              = Qq2 )
            & ( ( produc1943859948161608478_a_a_a @ ( produc1023885589438848650_a_a_a @ X4 ) )
              = X3 ) )
     => ( ( member6990616694304513154_a_a_a @ Q1 @ ( states8463590192175095071_a_a_b @ M ) )
       => ( ( member6990616694304513154_a_a_a @ Q2 @ ( states8463590192175095071_a_a_b @ M ) )
         => ~ ? [X7: produc6800189296593107456_a_a_a] :
                ( ( member4643618012521571351_a_a_a @ X7 @ ( transi7010299678732630507_a_a_b @ M ) )
                & ? [Xa2: produc6800189296593107456_a_a_a] :
                    ( ( member4643618012521571351_a_a_a @ Xa2 @ ( transi7010299678732630507_a_a_b @ M ) )
                    & ( ( produc2316789063618113990_a_a_a @ X7 )
                      = ( produc196831967674047067_a_a_a @ Qq2 ) )
                    & ( ( produc2316789063618113990_a_a_a @ Xa2 )
                      = ( produc3572288613182632605_a_a_a @ Qq2 ) )
                    & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) )
                      = X3 )
                    & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) )
                      = X3 )
                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
                      = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% product_undefined_input
thf(fact_1098_product__undefined__input,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,Qq2: produc2766379329844667609_a_a_a,X3: b] :
      ( ~ ? [X4: produc7598642544577812982_a_a_a] :
            ( ( member3147200746921337119_a_a_a @ X4 @ ( transi549026741542983407_a_b_c @ ( produc8615301468332162573_a_a_a @ ( from_F8539169374725194592_a_b_c @ M @ Q1 ) @ ( from_F8539169374725194592_a_b_c @ M @ Q2 ) ) ) )
            & ( ( produc5035679817875094482_a_a_a @ X4 )
              = Qq2 )
            & ( ( produc8807399175486215748_a_a_a @ ( produc7332565265614368020_a_a_a @ X4 ) )
              = X3 ) )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3035385874452377268_a_b_c @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3035385874452377268_a_b_c @ M ) )
         => ~ ? [X7: produc884577481120224310_a_a_a] :
                ( ( member8362999488266169055_a_a_a @ X7 @ ( transi749694810082594024_a_b_c @ M ) )
                & ? [Xa2: produc884577481120224310_a_a_a] :
                    ( ( member8362999488266169055_a_a_a @ Xa2 @ ( transi749694810082594024_a_b_c @ M ) )
                    & ( ( produc7477144754405272026_a_a_a @ X7 )
                      = ( produc3656708485065107965_a_a_a @ Qq2 ) )
                    & ( ( produc7477144754405272026_a_a_a @ Xa2 )
                      = ( produc9018290274662147647_a_a_a @ Qq2 ) )
                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) )
                      = X3 )
                    & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) )
                      = X3 )
                    & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
                      = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% product_undefined_input
thf(fact_1099_product__from__transition__split_I2_J,axiom,
    ! [T: produc7669896877331299902_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a] :
      ( ( member5238278507536988629_a_a_a @ T @ ( transi4297421501747413173_a_a_b @ ( produc2190053497452919855_a_a_a @ ( from_F6245563941867181427_a_a_b @ M @ Q1 ) @ ( from_F6245563941867181427_a_a_b @ M @ Q2 ) ) ) )
     => ( ( member6990616694304513154_a_a_a @ Q1 @ ( states8463590192175095071_a_a_b @ M ) )
       => ( ( member6990616694304513154_a_a_a @ Q2 @ ( states8463590192175095071_a_a_b @ M ) )
         => ? [X4: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ M ) )
              & ( ( produc2316789063618113990_a_a_a @ X4 )
                = ( produc3572288613182632605_a_a_a @ ( produc302551054129649228_a_a_a @ T ) ) )
              & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
                = ( produc1943859948161608478_a_a_a @ ( produc1023885589438848650_a_a_a @ T ) ) )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                = ( produc1515593567674010107_a_a_a @ ( produc3460917645060139100_a_a_a @ ( produc1023885589438848650_a_a_a @ T ) ) ) ) ) ) ) ) ).

% product_from_transition_split(2)
thf(fact_1100_product__from__transition__split_I2_J,axiom,
    ! [T: produc7598642544577812982_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a] :
      ( ( member3147200746921337119_a_a_a @ T @ ( transi549026741542983407_a_b_c @ ( produc8615301468332162573_a_a_a @ ( from_F8539169374725194592_a_b_c @ M @ Q1 ) @ ( from_F8539169374725194592_a_b_c @ M @ Q2 ) ) ) )
     => ( ( member5679158588528110291_a_a_a @ Q1 @ ( states3035385874452377268_a_b_c @ M ) )
       => ( ( member5679158588528110291_a_a_a @ Q2 @ ( states3035385874452377268_a_b_c @ M ) )
         => ? [X4: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ M ) )
              & ( ( produc7477144754405272026_a_a_a @ X4 )
                = ( produc9018290274662147647_a_a_a @ ( produc5035679817875094482_a_a_a @ T ) ) )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
                = ( produc8807399175486215748_a_a_a @ ( produc7332565265614368020_a_a_a @ T ) ) )
              & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                = ( produc5074649251625090012_a_a_a @ ( produc2959483784140025478_a_a_a @ ( produc7332565265614368020_a_a_a @ T ) ) ) ) ) ) ) ) ).

% product_from_transition_split(2)
thf(fact_1101_distinguishing__transitions__right__def,axiom,
    ( state_9218246414040145401_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c,Q13: sum_su5154756782089200444_a_a_a,Q25: sum_su5154756782089200444_a_a_a] :
          ( collec7563953601204149921_a_a_a
          @ ^ [Uu: produc1185401179908872758_a_a_a] :
            ? [Q14: sum_su5154756782089200444_a_a_a,Q26: sum_su5154756782089200444_a_a_a,X: b,Y4: c] :
              ( ( Uu
                = ( produc7266650933148349350_a_a_a @ ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ Q14 @ Q26 ) ) @ ( produc4152729215829753029_a_a_a @ X @ ( produc1982557662236492637_a_a_a @ Y4 @ ( sum_In5332043013833219623_a_a_a @ Q25 ) ) ) ) )
              & ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ Q14 @ Q26 ) @ ( states2002317254985447971_a_b_c @ ( produc8615301468332162573_a_a_a @ ( from_F8539169374725194592_a_b_c @ M3 @ Q13 ) @ ( from_F8539169374725194592_a_b_c @ M3 @ Q25 ) ) ) )
              & ~ ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q14 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) )
              & ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q26 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) ) ) ) ) ) ).

% distinguishing_transitions_right_def
thf(fact_1102_distinguishing__transitions__left__def,axiom,
    ( state_676393888658453318_a_b_c
    = ( ^ [M3: fsm_Su3173892059090327343_a_b_c,Q13: sum_su5154756782089200444_a_a_a,Q25: sum_su5154756782089200444_a_a_a] :
          ( collec7563953601204149921_a_a_a
          @ ^ [Uu: produc1185401179908872758_a_a_a] :
            ? [Q14: sum_su5154756782089200444_a_a_a,Q26: sum_su5154756782089200444_a_a_a,X: b,Y4: c] :
              ( ( Uu
                = ( produc7266650933148349350_a_a_a @ ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ Q14 @ Q26 ) ) @ ( produc4152729215829753029_a_a_a @ X @ ( produc1982557662236492637_a_a_a @ Y4 @ ( sum_In5332043013833219623_a_a_a @ Q13 ) ) ) ) )
              & ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ Q14 @ Q26 ) @ ( states2002317254985447971_a_b_c @ ( produc8615301468332162573_a_a_a @ ( from_F8539169374725194592_a_b_c @ M3 @ Q13 ) @ ( from_F8539169374725194592_a_b_c @ M3 @ Q25 ) ) ) )
              & ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q14 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) )
              & ~ ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q26 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M3 ) ) ) ) ) ) ).

% distinguishing_transitions_left_def
thf(fact_1103_distinguishing__transitions__left__right__intro,axiom,
    ! [S1: a,S2: a,M: fsm_a_b_c,Q1: a,Q2: a,X3: b,Y3: c] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ S1 @ S2 ) @ ( states3827391290251485425_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) ) )
     => ( ~ ? [X4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M ) )
              & ( ( produc4225943634317630866od_c_a @ X4 )
                = S1 )
              & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                = X3 )
              & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                = Y3 ) )
       => ( ? [X7: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X7 @ ( transitions_a_b_c @ M ) )
              & ( ( produc4225943634317630866od_c_a @ X7 )
                = S2 )
              & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X7 ) )
                = X3 )
              & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X7 ) ) )
                = Y3 ) )
         => ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ S1 @ S2 ) ) @ ( produc1730628064597884819_a_a_a @ X3 @ ( produc8806894744968565733_a_a_a @ Y3 @ ( sum_In6345099547827038749od_a_a @ Q2 ) ) ) ) @ ( state_2881338517818858960_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ).

% distinguishing_transitions_left_right_intro
thf(fact_1104_distinguishing__transitions__left__right__intro,axiom,
    ! [S1: produc1895207096998757355_a_a_a,S2: produc1895207096998757355_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a,X3: sum_su5154756782089200444_a_a_a,Y3: b] :
      ( ( member4183500612887236192_a_a_a @ ( produc8806572864378474095_a_a_a @ S1 @ S2 ) @ ( states7450307986677394561_a_a_b @ ( produc2190053497452919855_a_a_a @ ( from_F6245563941867181427_a_a_b @ M @ Q1 ) @ ( from_F6245563941867181427_a_a_b @ M @ Q2 ) ) ) )
     => ( ~ ? [X4: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ M ) )
              & ( ( produc2316789063618113990_a_a_a @ X4 )
                = S1 )
              & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
                = X3 )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                = Y3 ) )
       => ( ? [X7: produc6800189296593107456_a_a_a] :
              ( ( member4643618012521571351_a_a_a @ X7 @ ( transi7010299678732630507_a_a_b @ M ) )
              & ( ( produc2316789063618113990_a_a_a @ X7 )
                = S2 )
              & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) )
                = X3 )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
                = Y3 ) )
         => ( member2160747886658840973_a_a_a @ ( produc4893933519386964208_a_a_a @ ( sum_In122109773348503092_a_a_a @ ( produc8806572864378474095_a_a_a @ S1 @ S2 ) ) @ ( produc9113862492967280422_a_a_a @ X3 @ ( produc2736063840373621035_a_a_a @ Y3 @ ( sum_In4154768905285699222_a_a_a @ Q2 ) ) ) ) @ ( state_234223814407059111_a_a_b @ M @ Q1 @ Q2 ) ) ) ) ) ).

% distinguishing_transitions_left_right_intro
thf(fact_1105_distinguishing__transitions__left__right__intro,axiom,
    ! [S1: sum_su5154756782089200444_a_a_a,S2: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,X3: b,Y3: c] :
      ( ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ S1 @ S2 ) @ ( states2002317254985447971_a_b_c @ ( produc8615301468332162573_a_a_a @ ( from_F8539169374725194592_a_b_c @ M @ Q1 ) @ ( from_F8539169374725194592_a_b_c @ M @ Q2 ) ) ) )
     => ( ~ ? [X4: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ M ) )
              & ( ( produc7477144754405272026_a_a_a @ X4 )
                = S1 )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
                = X3 )
              & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                = Y3 ) )
       => ( ? [X7: produc884577481120224310_a_a_a] :
              ( ( member8362999488266169055_a_a_a @ X7 @ ( transi749694810082594024_a_b_c @ M ) )
              & ( ( produc7477144754405272026_a_a_a @ X7 )
                = S2 )
              & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) )
                = X3 )
              & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
                = Y3 ) )
         => ( member3530913115812046943_a_a_a @ ( produc7266650933148349350_a_a_a @ ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ S1 @ S2 ) ) @ ( produc4152729215829753029_a_a_a @ X3 @ ( produc1982557662236492637_a_a_a @ Y3 @ ( sum_In5332043013833219623_a_a_a @ Q2 ) ) ) ) @ ( state_5647776831386003244_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ).

% distinguishing_transitions_left_right_intro
thf(fact_1106_distinguishing__transitions__left__alt__intro,axiom,
    ! [S1: a,S2: a,M: fsm_a_b_c,Q1: a,Q2: a,X3: b,Y3: c] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ S1 @ S2 ) @ ( states3827391290251485425_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) ) )
     => ( ? [X7: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X7 @ ( transitions_a_b_c @ M ) )
            & ( ( produc4225943634317630866od_c_a @ X7 )
              = S1 )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X7 ) )
              = X3 )
            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X7 ) ) )
              = Y3 ) )
       => ( ~ ? [X4: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ M ) )
                & ( ( produc4225943634317630866od_c_a @ X4 )
                  = S2 )
                & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                  = X3 )
                & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                  = Y3 ) )
         => ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ S1 @ S2 ) ) @ ( produc1730628064597884819_a_a_a @ X3 @ ( produc8806894744968565733_a_a_a @ Y3 @ ( sum_In6345099547827038749od_a_a @ Q1 ) ) ) ) @ ( state_2046487007905918109_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ).

% distinguishing_transitions_left_alt_intro
thf(fact_1107_distinguishing__transitions__left__alt__intro,axiom,
    ! [S1: produc1895207096998757355_a_a_a,S2: produc1895207096998757355_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Q1: produc1895207096998757355_a_a_a,Q2: produc1895207096998757355_a_a_a,X3: sum_su5154756782089200444_a_a_a,Y3: b] :
      ( ( member4183500612887236192_a_a_a @ ( produc8806572864378474095_a_a_a @ S1 @ S2 ) @ ( states7450307986677394561_a_a_b @ ( produc2190053497452919855_a_a_a @ ( from_F6245563941867181427_a_a_b @ M @ Q1 ) @ ( from_F6245563941867181427_a_a_b @ M @ Q2 ) ) ) )
     => ( ? [X7: produc6800189296593107456_a_a_a] :
            ( ( member4643618012521571351_a_a_a @ X7 @ ( transi7010299678732630507_a_a_b @ M ) )
            & ( ( produc2316789063618113990_a_a_a @ X7 )
              = S1 )
            & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) )
              = X3 )
            & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X7 ) ) )
              = Y3 ) )
       => ( ~ ? [X4: produc6800189296593107456_a_a_a] :
                ( ( member4643618012521571351_a_a_a @ X4 @ ( transi7010299678732630507_a_a_b @ M ) )
                & ( ( produc2316789063618113990_a_a_a @ X4 )
                  = S2 )
                & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) )
                  = X3 )
                & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ X4 ) ) )
                  = Y3 ) )
         => ( member2160747886658840973_a_a_a @ ( produc4893933519386964208_a_a_a @ ( sum_In122109773348503092_a_a_a @ ( produc8806572864378474095_a_a_a @ S1 @ S2 ) ) @ ( produc9113862492967280422_a_a_a @ X3 @ ( produc2736063840373621035_a_a_a @ Y3 @ ( sum_In4154768905285699222_a_a_a @ Q1 ) ) ) ) @ ( state_8712764720510717172_a_a_b @ M @ Q1 @ Q2 ) ) ) ) ) ).

% distinguishing_transitions_left_alt_intro
thf(fact_1108_distinguishing__transitions__left__alt__intro,axiom,
    ! [S1: sum_su5154756782089200444_a_a_a,S2: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,X3: b,Y3: c] :
      ( ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ S1 @ S2 ) @ ( states2002317254985447971_a_b_c @ ( produc8615301468332162573_a_a_a @ ( from_F8539169374725194592_a_b_c @ M @ Q1 ) @ ( from_F8539169374725194592_a_b_c @ M @ Q2 ) ) ) )
     => ( ? [X7: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X7 @ ( transi749694810082594024_a_b_c @ M ) )
            & ( ( produc7477144754405272026_a_a_a @ X7 )
              = S1 )
            & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) )
              = X3 )
            & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X7 ) ) )
              = Y3 ) )
       => ( ~ ? [X4: produc884577481120224310_a_a_a] :
                ( ( member8362999488266169055_a_a_a @ X4 @ ( transi749694810082594024_a_b_c @ M ) )
                & ( ( produc7477144754405272026_a_a_a @ X4 )
                  = S2 )
                & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) )
                  = X3 )
                & ( ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ X4 ) ) )
                  = Y3 ) )
         => ( member3530913115812046943_a_a_a @ ( produc7266650933148349350_a_a_a @ ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ S1 @ S2 ) ) @ ( produc4152729215829753029_a_a_a @ X3 @ ( produc1982557662236492637_a_a_a @ Y3 @ ( sum_In5332043013833219623_a_a_a @ Q1 ) ) ) ) @ ( state_1106402805630587423_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ).

% distinguishing_transitions_left_alt_intro
thf(fact_1109_canonical__separator__transitions__def,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) )
          = ( sup_su9011528761648865002_a_a_a
            @ ( sup_su9011528761648865002_a_a_a
              @ ( collec3562233214436895265_a_a_a
                @ ^ [Uu: produc884577481120224310_a_a_a] :
                  ? [Q14: a,Q26: a,X: b,Y4: c,Q15: a,Q27: a] :
                    ( ( Uu
                      = ( produc6863513614254923246_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q14 @ Q26 ) ) @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ ( sum_In6412843608951811077_a_a_a @ ( product_Pair_a_a @ Q15 @ Q27 ) ) ) ) ) )
                    & ( member6611408146729689119od_a_a @ ( produc2632091562447838182od_a_a @ ( product_Pair_a_a @ Q14 @ Q26 ) @ ( produc2343341850531023270od_a_a @ X @ ( produc8376336777614417598od_a_a @ Y4 @ ( product_Pair_a_a @ Q15 @ Q27 ) ) ) ) @ ( transi6401751653757136829_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) ) ) ) )
              @ ( state_251499256903769654_a_b_c @ M @ Q1 @ Q2 ) )
            @ ( state_581478702265048387_a_b_c @ M @ Q1 @ Q2 ) ) ) ) ) ).

% canonical_separator_transitions_def
thf(fact_1110_canonical__separator__path__shift,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a,P4: list_P558552684728352636od_a_a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ( path_P4567602787516311982_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) @ ( initia4141967198046469911_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) ) @ P4 )
          = ( path_S221595587009941175_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) @ ( initia839594848065427214_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) )
            @ ( map_Pr8315520957595816964_a_a_a
              @ ^ [T4: produc3308035872450592246od_a_a] : ( produc6863513614254923246_a_a_a @ ( sum_In6412843608951811077_a_a_a @ ( produc2892452523101172178od_a_a @ T4 ) ) @ ( produc1730628064597884819_a_a_a @ ( produc806483927813978002od_a_a @ ( produc8587744923287169300od_a_a @ T4 ) ) @ ( produc8806894744968565733_a_a_a @ ( produc5237419938360121514od_a_a @ ( produc4668670682878874324od_a_a @ ( produc8587744923287169300od_a_a @ T4 ) ) ) @ ( sum_In6412843608951811077_a_a_a @ ( produc933545150919345644od_a_a @ ( produc4668670682878874324od_a_a @ ( produc8587744923287169300od_a_a @ T4 ) ) ) ) ) ) )
              @ P4 ) ) ) ) ) ).

% canonical_separator_path_shift
thf(fact_1111_h__out__alt__def,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,X3: b] :
      ( ( h_out_2422034477872479448_a_b_c @ M @ ( produc6086470211827429482_a_a_b @ Q @ X3 ) )
      = ( collect_c
        @ ^ [Uu: c] :
          ? [T4: produc884577481120224310_a_a_a] :
            ( ( Uu
              = ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
            & ( member8362999488266169055_a_a_a @ T4 @ ( transi749694810082594024_a_b_c @ M ) )
            & ( ( produc7477144754405272026_a_a_a @ T4 )
              = Q )
            & ( ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) )
              = X3 ) ) ) ) ).

% h_out_alt_def
thf(fact_1112_h__out__alt__def,axiom,
    ! [M: fsm_Pr6287166851770849218_a_a_b,Q: produc1895207096998757355_a_a_a,X3: sum_su5154756782089200444_a_a_a] :
      ( ( h_out_6310775740478713339_a_a_b @ M @ ( produc8580279754556576192_a_a_a @ Q @ X3 ) )
      = ( collect_b
        @ ^ [Uu: b] :
          ? [T4: produc6800189296593107456_a_a_a] :
            ( ( Uu
              = ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
            & ( member4643618012521571351_a_a_a @ T4 @ ( transi7010299678732630507_a_a_b @ M ) )
            & ( ( produc2316789063618113990_a_a_a @ T4 )
              = Q )
            & ( ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) )
              = X3 ) ) ) ) ).

% h_out_alt_def
thf(fact_1113_bex__empty,axiom,
    ! [P: sum_su5154756782089200444_a_a_a > $o] :
      ~ ? [X7: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X7 @ bot_bo2676859181106700934_a_a_a )
          & ( P @ X7 ) ) ).

% bex_empty
thf(fact_1114_bex__empty,axiom,
    ! [P: produc4770514982935793080_a_a_b > $o] :
      ~ ? [X7: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X7 @ bot_bo5052542052834138114_a_a_b )
          & ( P @ X7 ) ) ).

% bex_empty
thf(fact_1115_bex__empty,axiom,
    ! [P: produc884577481120224310_a_a_a > $o] :
      ~ ? [X7: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X7 @ bot_bo2898616998271641474_a_a_a )
          & ( P @ X7 ) ) ).

% bex_empty
thf(fact_1116_Bex__def,axiom,
    ( bex_Su2623642379066622940_a_a_a
    = ( ^ [A3: set_Su8591296761904255986_a_a_a,P3: sum_su5154756782089200444_a_a_a > $o] :
        ? [X: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X @ A3 )
          & ( P3 @ X ) ) ) ) ).

% Bex_def
thf(fact_1117_Bex__def,axiom,
    ( bex_Pr583515590951275414_a_a_a
    = ( ^ [A3: set_Pr4618682941230328598_a_a_a,P3: produc884577481120224310_a_a_a > $o] :
        ? [X: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X @ A3 )
          & ( P3 @ X ) ) ) ) ).

% Bex_def
thf(fact_1118_Bex__def,axiom,
    ( bex_Pr62707341681643096_a_a_b
    = ( ^ [A3: set_Pr6246767948713639278_a_a_b,P3: produc4770514982935793080_a_a_b > $o] :
        ? [X: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X @ A3 )
          & ( P3 @ X ) ) ) ) ).

% Bex_def
thf(fact_1119_h__out_Osimps,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a,X3: b] :
      ( ( h_out_2422034477872479448_a_b_c @ M @ ( produc6086470211827429482_a_a_b @ Q @ X3 ) )
      = ( collect_c
        @ ^ [Y4: c] :
          ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q @ ( produc1730628064597884819_a_a_a @ X3 @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M ) ) ) ) ).

% h_out.simps
thf(fact_1120_h__out_Osimps,axiom,
    ! [M: fsm_a_a_a,Q: a,X3: a] :
      ( ( h_out_a_a_a @ M @ ( product_Pair_a_a @ Q @ X3 ) )
      = ( collect_a
        @ ^ [Y4: a] :
          ? [Q4: a] : ( member1536750027164364322od_a_a @ ( produc7673539858576465641od_a_a @ Q @ ( produc431845341423274048od_a_a @ X3 @ ( product_Pair_a_a @ Y4 @ Q4 ) ) ) @ ( transitions_a_a_a @ M ) ) ) ) ).

% h_out.simps
thf(fact_1121_h__out_Oelims,axiom,
    ! [X3: fsm_Su3173892059090327343_a_b_c,Xa: produc4770514982935793080_a_a_b,Y3: set_c] :
      ( ( ( h_out_2422034477872479448_a_b_c @ X3 @ Xa )
        = Y3 )
     => ~ ! [Q7: sum_su5154756782089200444_a_a_a,X4: b] :
            ( ( Xa
              = ( produc6086470211827429482_a_a_b @ Q7 @ X4 ) )
           => ( Y3
             != ( collect_c
                @ ^ [Y4: c] :
                  ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q7 @ ( produc1730628064597884819_a_a_a @ X4 @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ X3 ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1122_h__out_Oelims,axiom,
    ! [X3: fsm_a_a_a,Xa: product_prod_a_a,Y3: set_a] :
      ( ( ( h_out_a_a_a @ X3 @ Xa )
        = Y3 )
     => ~ ! [Q7: a,X4: a] :
            ( ( Xa
              = ( product_Pair_a_a @ Q7 @ X4 ) )
           => ( Y3
             != ( collect_a
                @ ^ [Y4: a] :
                  ? [Q4: a] : ( member1536750027164364322od_a_a @ ( produc7673539858576465641od_a_a @ Q7 @ ( produc431845341423274048od_a_a @ X4 @ ( product_Pair_a_a @ Y4 @ Q4 ) ) ) @ ( transitions_a_a_a @ X3 ) ) ) ) ) ) ).

% h_out.elims
thf(fact_1123_distinguishing__transitions__alt__def,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q1: sum_su5154756782089200444_a_a_a,Q2: sum_su5154756782089200444_a_a_a,PS: set_Pr5813531400943588409_a_a_a] :
      ( ( fSM_di7339306060024166443_a_b_c @ ( h_out_2422034477872479448_a_b_c @ M ) @ Q1 @ Q2 @ PS @ ( inputs3694848887974842907_a_b_c @ M ) )
      = ( sup_su7948560395287882858_a_a_a
        @ ( collec7563953601204149921_a_a_a
          @ ^ [Uu: produc1185401179908872758_a_a_a] :
            ? [Q14: sum_su5154756782089200444_a_a_a,Q26: sum_su5154756782089200444_a_a_a,X: b,Y4: c] :
              ( ( Uu
                = ( produc7266650933148349350_a_a_a @ ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ Q14 @ Q26 ) ) @ ( produc4152729215829753029_a_a_a @ X @ ( produc1982557662236492637_a_a_a @ Y4 @ ( sum_In5332043013833219623_a_a_a @ Q1 ) ) ) ) )
              & ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ Q14 @ Q26 ) @ PS )
              & ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q14 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M ) )
              & ~ ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q26 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M ) ) ) )
        @ ( collec7563953601204149921_a_a_a
          @ ^ [Uu: produc1185401179908872758_a_a_a] :
            ? [Q14: sum_su5154756782089200444_a_a_a,Q26: sum_su5154756782089200444_a_a_a,X: b,Y4: c] :
              ( ( Uu
                = ( produc7266650933148349350_a_a_a @ ( sum_In3471042157297199015_a_a_a @ ( produc5672884958730733073_a_a_a @ Q14 @ Q26 ) ) @ ( produc4152729215829753029_a_a_a @ X @ ( produc1982557662236492637_a_a_a @ Y4 @ ( sum_In5332043013833219623_a_a_a @ Q2 ) ) ) ) )
              & ( member6836137003679210882_a_a_a @ ( produc5672884958730733073_a_a_a @ Q14 @ Q26 ) @ PS )
              & ~ ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q14 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M ) )
              & ? [Q4: sum_su5154756782089200444_a_a_a] : ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ Q26 @ ( produc1730628064597884819_a_a_a @ X @ ( produc8806894744968565733_a_a_a @ Y4 @ Q4 ) ) ) @ ( transi749694810082594024_a_b_c @ M ) ) ) ) ) ) ).

% distinguishing_transitions_alt_def
thf(fact_1124_canonical__separator__simps_I5_J,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q2: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q2 @ ( states_a_b_c @ M ) )
       => ( ( transi749694810082594024_a_b_c @ ( state_7524743879100055981_a_b_c @ M @ Q1 @ Q2 ) )
          = ( sup_su9011528761648865002_a_a_a @ ( fSM_sh2270962973017757008_b_c_a @ ( transi6401751653757136829_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) ) ) @ ( fSM_di5981764419675225489_a_b_c @ ( h_out_a_b_c @ M ) @ Q1 @ Q2 @ ( states3827391290251485425_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) ) @ ( inputs2873161410287034186_a_b_c @ ( produc2510185919789642449_b_c_a @ ( from_FSM_a_b_c @ M @ Q1 ) @ ( from_FSM_a_b_c @ M @ Q2 ) ) ) ) ) ) ) ) ).

% canonical_separator_simps(5)
thf(fact_1125_FSM_Ocanonical__separator_H__simps_I5_J,axiom,
    ! [P: fsm_Pr5633030146480774220_a_b_c,Q1: a,Q2: a,M: fsm_a_b_c] :
      ( ( ( initia4141967198046469911_a_b_c @ P )
        = ( product_Pair_a_a @ Q1 @ Q2 ) )
     => ( ( transi749694810082594024_a_b_c @ ( canoni1795827711669416976_a_b_c @ M @ P @ Q1 @ Q2 ) )
        = ( sup_su9011528761648865002_a_a_a @ ( fSM_sh2270962973017757008_b_c_a @ ( transi6401751653757136829_a_b_c @ P ) ) @ ( fSM_di5981764419675225489_a_b_c @ ( h_out_a_b_c @ M ) @ Q1 @ Q2 @ ( states3827391290251485425_a_b_c @ P ) @ ( inputs2873161410287034186_a_b_c @ P ) ) ) ) ) ).

% FSM.canonical_separator'_simps(5)
thf(fact_1126_canonical__separator_H__simps__without__assm_I5_J,axiom,
    ! [P: fsm_Pr5633030146480774220_a_b_c,Q1: a,Q2: a,M: fsm_a_b_c] :
      ( ( ( ( initia4141967198046469911_a_b_c @ P )
          = ( product_Pair_a_a @ Q1 @ Q2 ) )
       => ( ( transi749694810082594024_a_b_c @ ( canoni1795827711669416976_a_b_c @ M @ P @ Q1 @ Q2 ) )
          = ( sup_su9011528761648865002_a_a_a @ ( fSM_sh2270962973017757008_b_c_a @ ( transi6401751653757136829_a_b_c @ P ) ) @ ( fSM_di5981764419675225489_a_b_c @ ( h_out_a_b_c @ M ) @ Q1 @ Q2 @ ( states3827391290251485425_a_b_c @ P ) @ ( inputs2873161410287034186_a_b_c @ P ) ) ) ) )
      & ( ( ( initia4141967198046469911_a_b_c @ P )
         != ( product_Pair_a_a @ Q1 @ Q2 ) )
       => ( ( transi749694810082594024_a_b_c @ ( canoni1795827711669416976_a_b_c @ M @ P @ Q1 @ Q2 ) )
          = bot_bo2898616998271641474_a_a_a ) ) ) ).

% canonical_separator'_simps_without_assm(5)
thf(fact_1127_io__targets__next,axiom,
    ! [T: produc884577481120224310_a_a_a,M: fsm_Su3173892059090327343_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( member8362999488266169055_a_a_a @ T @ ( transi749694810082594024_a_b_c @ M ) )
     => ( ord_le4476889455585342546_a_a_a @ ( io_tar1727080856364058086_a_b_c @ M @ Io @ ( produc1172910294343954743_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T ) ) ) )
        @ ( io_tar1727080856364058086_a_b_c @ M
          @ ( append2547753245680614915od_b_c
            @ ( map_Pr6504563922737449846od_b_c
              @ ^ [T4: produc884577481120224310_a_a_a] : ( product_Pair_b_c @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) @ ( produc6784836004715151097_a_a_a @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ T4 ) ) ) )
              @ ( cons_P6318101215856909548_a_a_a @ T @ nil_Pr6029279252097497404_a_a_a ) )
            @ Io )
          @ ( produc7477144754405272026_a_a_a @ T ) ) ) ) ).

% io_targets_next
thf(fact_1128_io__targets__next,axiom,
    ! [T: produc6800189296593107456_a_a_a,M: fsm_Pr6287166851770849218_a_a_b,Io: list_P7155397214832950472_a_a_b] :
      ( ( member4643618012521571351_a_a_a @ T @ ( transi7010299678732630507_a_a_b @ M ) )
     => ( ord_le6734046778807601793_a_a_a @ ( io_tar1879755107597495661_a_a_b @ M @ Io @ ( produc6236129573204179685_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T ) ) ) )
        @ ( io_tar1879755107597495661_a_a_b @ M
          @ ( append157347090090570333_a_a_b
            @ ( map_Pr6873991885683231230_a_a_b
              @ ^ [T4: produc6800189296593107456_a_a_a] : ( produc6086470211827429482_a_a_b @ ( produc7477144754405272026_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) @ ( produc5842599967860340391_a_a_a @ ( produc1629229363059081756_a_a_a @ ( produc3833846760516644612_a_a_a @ T4 ) ) ) )
              @ ( cons_P5563258090397451722_a_a_a @ T @ nil_Pr3590249381822989690_a_a_a ) )
            @ Io )
          @ ( produc2316789063618113990_a_a_a @ T ) ) ) ) ).

% io_targets_next
thf(fact_1129_subsetI,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ! [X4: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X4 @ A2 )
         => ( member5679158588528110291_a_a_a @ X4 @ B3 ) )
     => ( ord_le4476889455585342546_a_a_a @ A2 @ B3 ) ) ).

% subsetI
thf(fact_1130_subsetI,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ! [X4: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X4 @ A2 )
         => ( member8362999488266169055_a_a_a @ X4 @ B3 ) )
     => ( ord_le8158965334049305270_a_a_a @ A2 @ B3 ) ) ).

% subsetI
thf(fact_1131_subsetI,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ! [X4: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X4 @ A2 )
         => ( member642552543386773071_a_a_b @ X4 @ B3 ) )
     => ( ord_le2852551352416386510_a_a_b @ A2 @ B3 ) ) ).

% subsetI
thf(fact_1132_le__sup__iff,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a,Z2: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) @ Z2 )
      = ( ( ord_le4476889455585342546_a_a_a @ X3 @ Z2 )
        & ( ord_le4476889455585342546_a_a_a @ Y3 @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_1133_sup_Obounded__iff,axiom,
    ! [B: set_Su8591296761904255986_a_a_a,C: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ B @ C ) @ A )
      = ( ( ord_le4476889455585342546_a_a_a @ B @ A )
        & ( ord_le4476889455585342546_a_a_a @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1134_empty__subsetI,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ bot_bo2676859181106700934_a_a_a @ A2 ) ).

% empty_subsetI
thf(fact_1135_empty__subsetI,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] : ( ord_le2852551352416386510_a_a_b @ bot_bo5052542052834138114_a_a_b @ A2 ) ).

% empty_subsetI
thf(fact_1136_empty__subsetI,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] : ( ord_le8158965334049305270_a_a_a @ bot_bo2898616998271641474_a_a_a @ A2 ) ).

% empty_subsetI
thf(fact_1137_subset__empty,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A2 @ bot_bo2676859181106700934_a_a_a )
      = ( A2 = bot_bo2676859181106700934_a_a_a ) ) ).

% subset_empty
thf(fact_1138_subset__empty,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ A2 @ bot_bo5052542052834138114_a_a_b )
      = ( A2 = bot_bo5052542052834138114_a_a_b ) ) ).

% subset_empty
thf(fact_1139_subset__empty,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ A2 @ bot_bo2898616998271641474_a_a_a )
      = ( A2 = bot_bo2898616998271641474_a_a_a ) ) ).

% subset_empty
thf(fact_1140_insert__subset,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ ( insert5061175218292437356_a_a_a @ X3 @ A2 ) @ B3 )
      = ( ( member5679158588528110291_a_a_a @ X3 @ B3 )
        & ( ord_le4476889455585342546_a_a_a @ A2 @ B3 ) ) ) ).

% insert_subset
thf(fact_1141_insert__subset,axiom,
    ! [X3: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ ( insert2249508923356641798_a_a_a @ X3 @ A2 ) @ B3 )
      = ( ( member8362999488266169055_a_a_a @ X3 @ B3 )
        & ( ord_le8158965334049305270_a_a_a @ A2 @ B3 ) ) ) ).

% insert_subset
thf(fact_1142_insert__subset,axiom,
    ! [X3: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ ( insert8042272945010224104_a_a_b @ X3 @ A2 ) @ B3 )
      = ( ( member642552543386773071_a_a_b @ X3 @ B3 )
        & ( ord_le2852551352416386510_a_a_b @ A2 @ B3 ) ) ) ).

% insert_subset
thf(fact_1143_Un__subset__iff,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,C2: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) @ C2 )
      = ( ( ord_le4476889455585342546_a_a_a @ A2 @ C2 )
        & ( ord_le4476889455585342546_a_a_a @ B3 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_1144_singleton__insert__inj__eq,axiom,
    ! [B: sum_su5154756782089200444_a_a_a,A: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( ( insert5061175218292437356_a_a_a @ B @ bot_bo2676859181106700934_a_a_a )
        = ( insert5061175218292437356_a_a_a @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_le4476889455585342546_a_a_a @ A2 @ ( insert5061175218292437356_a_a_a @ B @ bot_bo2676859181106700934_a_a_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1145_singleton__insert__inj__eq,axiom,
    ! [B: produc4770514982935793080_a_a_b,A: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b] :
      ( ( ( insert8042272945010224104_a_a_b @ B @ bot_bo5052542052834138114_a_a_b )
        = ( insert8042272945010224104_a_a_b @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_le2852551352416386510_a_a_b @ A2 @ ( insert8042272945010224104_a_a_b @ B @ bot_bo5052542052834138114_a_a_b ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1146_singleton__insert__inj__eq,axiom,
    ! [B: produc884577481120224310_a_a_a,A: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a] :
      ( ( ( insert2249508923356641798_a_a_a @ B @ bot_bo2898616998271641474_a_a_a )
        = ( insert2249508923356641798_a_a_a @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_le8158965334049305270_a_a_a @ A2 @ ( insert2249508923356641798_a_a_a @ B @ bot_bo2898616998271641474_a_a_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1147_singleton__insert__inj__eq_H,axiom,
    ! [A: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a,B: sum_su5154756782089200444_a_a_a] :
      ( ( ( insert5061175218292437356_a_a_a @ A @ A2 )
        = ( insert5061175218292437356_a_a_a @ B @ bot_bo2676859181106700934_a_a_a ) )
      = ( ( A = B )
        & ( ord_le4476889455585342546_a_a_a @ A2 @ ( insert5061175218292437356_a_a_a @ B @ bot_bo2676859181106700934_a_a_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1148_singleton__insert__inj__eq_H,axiom,
    ! [A: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b,B: produc4770514982935793080_a_a_b] :
      ( ( ( insert8042272945010224104_a_a_b @ A @ A2 )
        = ( insert8042272945010224104_a_a_b @ B @ bot_bo5052542052834138114_a_a_b ) )
      = ( ( A = B )
        & ( ord_le2852551352416386510_a_a_b @ A2 @ ( insert8042272945010224104_a_a_b @ B @ bot_bo5052542052834138114_a_a_b ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1149_singleton__insert__inj__eq_H,axiom,
    ! [A: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a,B: produc884577481120224310_a_a_a] :
      ( ( ( insert2249508923356641798_a_a_a @ A @ A2 )
        = ( insert2249508923356641798_a_a_a @ B @ bot_bo2898616998271641474_a_a_a ) )
      = ( ( A = B )
        & ( ord_le8158965334049305270_a_a_a @ A2 @ ( insert2249508923356641798_a_a_a @ B @ bot_bo2898616998271641474_a_a_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1150_less__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
      ( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_1151_leD,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).

% leD
thf(fact_1152_leI,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% leI
thf(fact_1153_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1154_antisym__conv1,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1155_antisym__conv2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1156_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1157_not__le__imp__less,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ord_less_nat @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1158_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B6: nat] :
          ( ( ord_less_nat @ A4 @ B6 )
          | ( A4 = B6 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1159_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A4 @ B6 )
          & ( A4 != B6 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1160_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1161_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1162_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A4 @ B6 )
          & ~ ( ord_less_eq_nat @ B6 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1163_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B6: nat,A4: nat] :
          ( ( ord_less_nat @ B6 @ A4 )
          | ( A4 = B6 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1164_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B6: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B6 @ A4 )
          & ( A4 != B6 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1165_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1166_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1167_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B6: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B6 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B6 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1168_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1169_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1170_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_nat @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_1171_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_1172_linorder__not__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_1173_linorder__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_1174_order__less__imp__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1175_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1176_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1177_order__le__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1178_order__less__le__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1179_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y: nat] :
              ( ( ord_less_nat @ X4 @ Y )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1180_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y: nat] :
              ( ( ord_less_nat @ X4 @ Y )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1181_linorder__le__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1182_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1183_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A @ bot_bo2676859181106700934_a_a_a )
     => ( A = bot_bo2676859181106700934_a_a_a ) ) ).

% bot.extremum_uniqueI
thf(fact_1184_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr6246767948713639278_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ A @ bot_bo5052542052834138114_a_a_b )
     => ( A = bot_bo5052542052834138114_a_a_b ) ) ).

% bot.extremum_uniqueI
thf(fact_1185_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr4618682941230328598_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ A @ bot_bo2898616998271641474_a_a_a )
     => ( A = bot_bo2898616998271641474_a_a_a ) ) ).

% bot.extremum_uniqueI
thf(fact_1186_bot_Oextremum__unique,axiom,
    ! [A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A @ bot_bo2676859181106700934_a_a_a )
      = ( A = bot_bo2676859181106700934_a_a_a ) ) ).

% bot.extremum_unique
thf(fact_1187_bot_Oextremum__unique,axiom,
    ! [A: set_Pr6246767948713639278_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ A @ bot_bo5052542052834138114_a_a_b )
      = ( A = bot_bo5052542052834138114_a_a_b ) ) ).

% bot.extremum_unique
thf(fact_1188_bot_Oextremum__unique,axiom,
    ! [A: set_Pr4618682941230328598_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ A @ bot_bo2898616998271641474_a_a_a )
      = ( A = bot_bo2898616998271641474_a_a_a ) ) ).

% bot.extremum_unique
thf(fact_1189_bot_Oextremum,axiom,
    ! [A: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ bot_bo2676859181106700934_a_a_a @ A ) ).

% bot.extremum
thf(fact_1190_bot_Oextremum,axiom,
    ! [A: set_Pr6246767948713639278_a_a_b] : ( ord_le2852551352416386510_a_a_b @ bot_bo5052542052834138114_a_a_b @ A ) ).

% bot.extremum
thf(fact_1191_bot_Oextremum,axiom,
    ! [A: set_Pr4618682941230328598_a_a_a] : ( ord_le8158965334049305270_a_a_a @ bot_bo2898616998271641474_a_a_a @ A ) ).

% bot.extremum
thf(fact_1192_inf__sup__ord_I4_J,axiom,
    ! [Y3: set_Su8591296761904255986_a_a_a,X3: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ Y3 @ ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_1193_inf__sup__ord_I3_J,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_1194_le__supE,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a,X3: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ A @ B ) @ X3 )
     => ~ ( ( ord_le4476889455585342546_a_a_a @ A @ X3 )
         => ~ ( ord_le4476889455585342546_a_a_a @ B @ X3 ) ) ) ).

% le_supE
thf(fact_1195_le__supI,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,X3: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A @ X3 )
     => ( ( ord_le4476889455585342546_a_a_a @ B @ X3 )
       => ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_1196_sup__ge1,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) ) ).

% sup_ge1
thf(fact_1197_sup__ge2,axiom,
    ! [Y3: set_Su8591296761904255986_a_a_a,X3: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ Y3 @ ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 ) ) ).

% sup_ge2
thf(fact_1198_le__supI1,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ X3 @ A )
     => ( ord_le4476889455585342546_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ).

% le_supI1
thf(fact_1199_le__supI2,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ X3 @ B )
     => ( ord_le4476889455585342546_a_a_a @ X3 @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ).

% le_supI2
thf(fact_1200_sup_Omono,axiom,
    ! [C: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a,D: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ C @ A )
     => ( ( ord_le4476889455585342546_a_a_a @ D @ B )
       => ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ C @ D ) @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1201_sup__mono,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,C: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a,D: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A @ C )
     => ( ( ord_le4476889455585342546_a_a_a @ B @ D )
       => ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ A @ B ) @ ( sup_su7096409449263244574_a_a_a @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_1202_sup__least,axiom,
    ! [Y3: set_Su8591296761904255986_a_a_a,X3: set_Su8591296761904255986_a_a_a,Z2: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ Y3 @ X3 )
     => ( ( ord_le4476889455585342546_a_a_a @ Z2 @ X3 )
       => ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ Y3 @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_1203_le__iff__sup,axiom,
    ( ord_le4476889455585342546_a_a_a
    = ( ^ [X: set_Su8591296761904255986_a_a_a,Y4: set_Su8591296761904255986_a_a_a] :
          ( ( sup_su7096409449263244574_a_a_a @ X @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_1204_sup_OorderE,axiom,
    ! [B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ B @ A )
     => ( A
        = ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1205_sup_OorderI,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( A
        = ( sup_su7096409449263244574_a_a_a @ A @ B ) )
     => ( ord_le4476889455585342546_a_a_a @ B @ A ) ) ).

% sup.orderI
thf(fact_1206_sup__unique,axiom,
    ! [F: set_Su8591296761904255986_a_a_a > set_Su8591296761904255986_a_a_a > set_Su8591296761904255986_a_a_a,X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a] :
      ( ! [X4: set_Su8591296761904255986_a_a_a,Y: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ X4 @ ( F @ X4 @ Y ) )
     => ( ! [X4: set_Su8591296761904255986_a_a_a,Y: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ Y @ ( F @ X4 @ Y ) )
       => ( ! [X4: set_Su8591296761904255986_a_a_a,Y: set_Su8591296761904255986_a_a_a,Z: set_Su8591296761904255986_a_a_a] :
              ( ( ord_le4476889455585342546_a_a_a @ Y @ X4 )
             => ( ( ord_le4476889455585342546_a_a_a @ Z @ X4 )
               => ( ord_le4476889455585342546_a_a_a @ ( F @ Y @ Z ) @ X4 ) ) )
         => ( ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 )
            = ( F @ X3 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_1207_sup_Oabsorb1,axiom,
    ! [B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ B @ A )
     => ( ( sup_su7096409449263244574_a_a_a @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1208_sup_Oabsorb2,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A @ B )
     => ( ( sup_su7096409449263244574_a_a_a @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1209_sup__absorb1,axiom,
    ! [Y3: set_Su8591296761904255986_a_a_a,X3: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ Y3 @ X3 )
     => ( ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 )
        = X3 ) ) ).

% sup_absorb1
thf(fact_1210_sup__absorb2,axiom,
    ! [X3: set_Su8591296761904255986_a_a_a,Y3: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ X3 @ Y3 )
     => ( ( sup_su7096409449263244574_a_a_a @ X3 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_1211_sup_OboundedE,axiom,
    ! [B: set_Su8591296761904255986_a_a_a,C: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ B @ C ) @ A )
     => ~ ( ( ord_le4476889455585342546_a_a_a @ B @ A )
         => ~ ( ord_le4476889455585342546_a_a_a @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_1212_sup_OboundedI,axiom,
    ! [B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a,C: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ B @ A )
     => ( ( ord_le4476889455585342546_a_a_a @ C @ A )
       => ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1213_sup_Oorder__iff,axiom,
    ( ord_le4476889455585342546_a_a_a
    = ( ^ [B6: set_Su8591296761904255986_a_a_a,A4: set_Su8591296761904255986_a_a_a] :
          ( A4
          = ( sup_su7096409449263244574_a_a_a @ A4 @ B6 ) ) ) ) ).

% sup.order_iff
thf(fact_1214_sup_Ocobounded1,axiom,
    ! [A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ A @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1215_sup_Ocobounded2,axiom,
    ! [B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ B @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1216_sup_Oabsorb__iff1,axiom,
    ( ord_le4476889455585342546_a_a_a
    = ( ^ [B6: set_Su8591296761904255986_a_a_a,A4: set_Su8591296761904255986_a_a_a] :
          ( ( sup_su7096409449263244574_a_a_a @ A4 @ B6 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_1217_sup_Oabsorb__iff2,axiom,
    ( ord_le4476889455585342546_a_a_a
    = ( ^ [A4: set_Su8591296761904255986_a_a_a,B6: set_Su8591296761904255986_a_a_a] :
          ( ( sup_su7096409449263244574_a_a_a @ A4 @ B6 )
          = B6 ) ) ) ).

% sup.absorb_iff2
thf(fact_1218_sup_OcoboundedI1,axiom,
    ! [C: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ C @ A )
     => ( ord_le4476889455585342546_a_a_a @ C @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1219_sup_OcoboundedI2,axiom,
    ! [C: set_Su8591296761904255986_a_a_a,B: set_Su8591296761904255986_a_a_a,A: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ C @ B )
     => ( ord_le4476889455585342546_a_a_a @ C @ ( sup_su7096409449263244574_a_a_a @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1220_subset__code_I1_J,axiom,
    ! [Xs: list_P8197596098214363964_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ ( set_Pr8067384617587748683_a_a_a @ Xs ) @ B3 )
      = ( ! [X: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X @ ( set_Pr8067384617587748683_a_a_a @ Xs ) )
           => ( member8362999488266169055_a_a_a @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1221_subset__code_I1_J,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ ( set_Su3113867532901839207_a_a_a @ Xs ) @ B3 )
      = ( ! [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ ( set_Su3113867532901839207_a_a_a @ Xs ) )
           => ( member5679158588528110291_a_a_a @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1222_subset__code_I1_J,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ Xs ) @ B3 )
      = ( ! [X: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X @ ( set_Pr4578018992751950051_a_a_b @ Xs ) )
           => ( member642552543386773071_a_a_b @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1223_subset__insertI2,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,B: sum_su5154756782089200444_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A2 @ B3 )
     => ( ord_le4476889455585342546_a_a_a @ A2 @ ( insert5061175218292437356_a_a_a @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_1224_subset__insertI2,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b,B: produc4770514982935793080_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ A2 @ B3 )
     => ( ord_le2852551352416386510_a_a_b @ A2 @ ( insert8042272945010224104_a_a_b @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_1225_subset__insertI2,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a,B: produc884577481120224310_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ A2 @ B3 )
     => ( ord_le8158965334049305270_a_a_a @ A2 @ ( insert2249508923356641798_a_a_a @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_1226_subset__insertI,axiom,
    ! [B3: set_Su8591296761904255986_a_a_a,A: sum_su5154756782089200444_a_a_a] : ( ord_le4476889455585342546_a_a_a @ B3 @ ( insert5061175218292437356_a_a_a @ A @ B3 ) ) ).

% subset_insertI
thf(fact_1227_subset__insertI,axiom,
    ! [B3: set_Pr6246767948713639278_a_a_b,A: produc4770514982935793080_a_a_b] : ( ord_le2852551352416386510_a_a_b @ B3 @ ( insert8042272945010224104_a_a_b @ A @ B3 ) ) ).

% subset_insertI
thf(fact_1228_subset__insertI,axiom,
    ! [B3: set_Pr4618682941230328598_a_a_a,A: produc884577481120224310_a_a_a] : ( ord_le8158965334049305270_a_a_a @ B3 @ ( insert2249508923356641798_a_a_a @ A @ B3 ) ) ).

% subset_insertI
thf(fact_1229_subset__insert,axiom,
    ! [X3: sum_su5154756782089200444_a_a_a,A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ~ ( member5679158588528110291_a_a_a @ X3 @ A2 )
     => ( ( ord_le4476889455585342546_a_a_a @ A2 @ ( insert5061175218292437356_a_a_a @ X3 @ B3 ) )
        = ( ord_le4476889455585342546_a_a_a @ A2 @ B3 ) ) ) ).

% subset_insert
thf(fact_1230_subset__insert,axiom,
    ! [X3: produc884577481120224310_a_a_a,A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a] :
      ( ~ ( member8362999488266169055_a_a_a @ X3 @ A2 )
     => ( ( ord_le8158965334049305270_a_a_a @ A2 @ ( insert2249508923356641798_a_a_a @ X3 @ B3 ) )
        = ( ord_le8158965334049305270_a_a_a @ A2 @ B3 ) ) ) ).

% subset_insert
thf(fact_1231_subset__insert,axiom,
    ! [X3: produc4770514982935793080_a_a_b,A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b] :
      ( ~ ( member642552543386773071_a_a_b @ X3 @ A2 )
     => ( ( ord_le2852551352416386510_a_a_b @ A2 @ ( insert8042272945010224104_a_a_b @ X3 @ B3 ) )
        = ( ord_le2852551352416386510_a_a_b @ A2 @ B3 ) ) ) ).

% subset_insert
thf(fact_1232_insert__mono,axiom,
    ! [C2: set_Su8591296761904255986_a_a_a,D2: set_Su8591296761904255986_a_a_a,A: sum_su5154756782089200444_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ C2 @ D2 )
     => ( ord_le4476889455585342546_a_a_a @ ( insert5061175218292437356_a_a_a @ A @ C2 ) @ ( insert5061175218292437356_a_a_a @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_1233_insert__mono,axiom,
    ! [C2: set_Pr6246767948713639278_a_a_b,D2: set_Pr6246767948713639278_a_a_b,A: produc4770514982935793080_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ C2 @ D2 )
     => ( ord_le2852551352416386510_a_a_b @ ( insert8042272945010224104_a_a_b @ A @ C2 ) @ ( insert8042272945010224104_a_a_b @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_1234_insert__mono,axiom,
    ! [C2: set_Pr4618682941230328598_a_a_a,D2: set_Pr4618682941230328598_a_a_a,A: produc884577481120224310_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ C2 @ D2 )
     => ( ord_le8158965334049305270_a_a_a @ ( insert2249508923356641798_a_a_a @ A @ C2 ) @ ( insert2249508923356641798_a_a_a @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_1235_Un__mono,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,C2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,D2: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A2 @ C2 )
     => ( ( ord_le4476889455585342546_a_a_a @ B3 @ D2 )
       => ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) @ ( sup_su7096409449263244574_a_a_a @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_1236_Un__least,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,C2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A2 @ C2 )
     => ( ( ord_le4476889455585342546_a_a_a @ B3 @ C2 )
       => ( ord_le4476889455585342546_a_a_a @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) @ C2 ) ) ) ).

% Un_least
thf(fact_1237_Un__upper1,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ A2 @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) ) ).

% Un_upper1
thf(fact_1238_Un__upper2,axiom,
    ! [B3: set_Su8591296761904255986_a_a_a,A2: set_Su8591296761904255986_a_a_a] : ( ord_le4476889455585342546_a_a_a @ B3 @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) ) ).

% Un_upper2
thf(fact_1239_Un__absorb1,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A2 @ B3 )
     => ( ( sup_su7096409449263244574_a_a_a @ A2 @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_1240_Un__absorb2,axiom,
    ! [B3: set_Su8591296761904255986_a_a_a,A2: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ B3 @ A2 )
     => ( ( sup_su7096409449263244574_a_a_a @ A2 @ B3 )
        = A2 ) ) ).

% Un_absorb2
thf(fact_1241_subset__UnE,axiom,
    ! [C2: set_Su8591296761904255986_a_a_a,A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ C2 @ ( sup_su7096409449263244574_a_a_a @ A2 @ B3 ) )
     => ~ ! [A7: set_Su8591296761904255986_a_a_a] :
            ( ( ord_le4476889455585342546_a_a_a @ A7 @ A2 )
           => ! [B8: set_Su8591296761904255986_a_a_a] :
                ( ( ord_le4476889455585342546_a_a_a @ B8 @ B3 )
               => ( C2
                 != ( sup_su7096409449263244574_a_a_a @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_1242_subset__Un__eq,axiom,
    ( ord_le4476889455585342546_a_a_a
    = ( ^ [A3: set_Su8591296761904255986_a_a_a,B4: set_Su8591296761904255986_a_a_a] :
          ( ( sup_su7096409449263244574_a_a_a @ A3 @ B4 )
          = B4 ) ) ) ).

% subset_Un_eq
thf(fact_1243_in__mono,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,X3: sum_su5154756782089200444_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A2 @ B3 )
     => ( ( member5679158588528110291_a_a_a @ X3 @ A2 )
       => ( member5679158588528110291_a_a_a @ X3 @ B3 ) ) ) ).

% in_mono
thf(fact_1244_in__mono,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a,X3: produc884577481120224310_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ A2 @ B3 )
     => ( ( member8362999488266169055_a_a_a @ X3 @ A2 )
       => ( member8362999488266169055_a_a_a @ X3 @ B3 ) ) ) ).

% in_mono
thf(fact_1245_in__mono,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b,X3: produc4770514982935793080_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ A2 @ B3 )
     => ( ( member642552543386773071_a_a_b @ X3 @ A2 )
       => ( member642552543386773071_a_a_b @ X3 @ B3 ) ) ) ).

% in_mono
thf(fact_1246_subsetD,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,B3: set_Su8591296761904255986_a_a_a,C: sum_su5154756782089200444_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A2 @ B3 )
     => ( ( member5679158588528110291_a_a_a @ C @ A2 )
       => ( member5679158588528110291_a_a_a @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1247_subsetD,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,B3: set_Pr4618682941230328598_a_a_a,C: produc884577481120224310_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ A2 @ B3 )
     => ( ( member8362999488266169055_a_a_a @ C @ A2 )
       => ( member8362999488266169055_a_a_a @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1248_subsetD,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,B3: set_Pr6246767948713639278_a_a_b,C: produc4770514982935793080_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ A2 @ B3 )
     => ( ( member642552543386773071_a_a_b @ C @ A2 )
       => ( member642552543386773071_a_a_b @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1249_subset__eq,axiom,
    ( ord_le4476889455585342546_a_a_a
    = ( ^ [A3: set_Su8591296761904255986_a_a_a,B4: set_Su8591296761904255986_a_a_a] :
        ! [X: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ X @ A3 )
         => ( member5679158588528110291_a_a_a @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_1250_subset__eq,axiom,
    ( ord_le8158965334049305270_a_a_a
    = ( ^ [A3: set_Pr4618682941230328598_a_a_a,B4: set_Pr4618682941230328598_a_a_a] :
        ! [X: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ X @ A3 )
         => ( member8362999488266169055_a_a_a @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_1251_subset__eq,axiom,
    ( ord_le2852551352416386510_a_a_b
    = ( ^ [A3: set_Pr6246767948713639278_a_a_b,B4: set_Pr6246767948713639278_a_a_b] :
        ! [X: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ X @ A3 )
         => ( member642552543386773071_a_a_b @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_1252_subset__iff,axiom,
    ( ord_le4476889455585342546_a_a_a
    = ( ^ [A3: set_Su8591296761904255986_a_a_a,B4: set_Su8591296761904255986_a_a_a] :
        ! [T4: sum_su5154756782089200444_a_a_a] :
          ( ( member5679158588528110291_a_a_a @ T4 @ A3 )
         => ( member5679158588528110291_a_a_a @ T4 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_1253_subset__iff,axiom,
    ( ord_le8158965334049305270_a_a_a
    = ( ^ [A3: set_Pr4618682941230328598_a_a_a,B4: set_Pr4618682941230328598_a_a_a] :
        ! [T4: produc884577481120224310_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ T4 @ A3 )
         => ( member8362999488266169055_a_a_a @ T4 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_1254_subset__iff,axiom,
    ( ord_le2852551352416386510_a_a_b
    = ( ^ [A3: set_Pr6246767948713639278_a_a_b,B4: set_Pr6246767948713639278_a_a_b] :
        ! [T4: produc4770514982935793080_a_a_b] :
          ( ( member642552543386773071_a_a_b @ T4 @ A3 )
         => ( member642552543386773071_a_a_b @ T4 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_1255_Collect__subset,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,P: sum_su5154756782089200444_a_a_a > $o] :
      ( ord_le4476889455585342546_a_a_a
      @ ( collec299858375640228881_a_a_a
        @ ^ [X: sum_su5154756782089200444_a_a_a] :
            ( ( member5679158588528110291_a_a_a @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_1256_Collect__subset,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,P: produc884577481120224310_a_a_a > $o] :
      ( ord_le8158965334049305270_a_a_a
      @ ( collec3562233214436895265_a_a_a
        @ ^ [X: produc884577481120224310_a_a_a] :
            ( ( member8362999488266169055_a_a_a @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_1257_Collect__subset,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,P: produc4770514982935793080_a_a_b > $o] :
      ( ord_le2852551352416386510_a_a_b
      @ ( collec4985708871760228749_a_a_b
        @ ^ [X: produc4770514982935793080_a_a_b] :
            ( ( member642552543386773071_a_a_b @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_1258_subrelI,axiom,
    ! [R2: set_Pr4618682941230328598_a_a_a,S3: set_Pr4618682941230328598_a_a_a] :
      ( ! [X4: sum_su5154756782089200444_a_a_a,Y: produc7810058533641270553_a_a_a] :
          ( ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X4 @ Y ) @ R2 )
         => ( member8362999488266169055_a_a_a @ ( produc6863513614254923246_a_a_a @ X4 @ Y ) @ S3 ) )
     => ( ord_le8158965334049305270_a_a_a @ R2 @ S3 ) ) ).

% subrelI
thf(fact_1259_subrelI,axiom,
    ! [R2: set_Pr6246767948713639278_a_a_b,S3: set_Pr6246767948713639278_a_a_b] :
      ( ! [X4: sum_su5154756782089200444_a_a_a,Y: b] :
          ( ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X4 @ Y ) @ R2 )
         => ( member642552543386773071_a_a_b @ ( produc6086470211827429482_a_a_b @ X4 @ Y ) @ S3 ) )
     => ( ord_le2852551352416386510_a_a_b @ R2 @ S3 ) ) ).

% subrelI
thf(fact_1260_subrelI,axiom,
    ! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
      ( ! [X4: a,Y: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y ) @ R2 )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y ) @ S3 ) )
     => ( ord_le746702958409616551od_a_a @ R2 @ S3 ) ) ).

% subrelI
thf(fact_1261_set__subset__Cons,axiom,
    ! [Xs: list_S8072159660131056972_a_a_a,X3: sum_su5154756782089200444_a_a_a] : ( ord_le4476889455585342546_a_a_a @ ( set_Su3113867532901839207_a_a_a @ Xs ) @ ( set_Su3113867532901839207_a_a_a @ ( cons_S2140548516868915206_a_a_a @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_1262_set__subset__Cons,axiom,
    ! [Xs: list_P7155397214832950472_a_a_b,X3: produc4770514982935793080_a_a_b] : ( ord_le2852551352416386510_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ Xs ) @ ( set_Pr4578018992751950051_a_a_b @ ( cons_P2262690024297812098_a_a_b @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_1263_subset__singletonD,axiom,
    ! [A2: set_Su8591296761904255986_a_a_a,X3: sum_su5154756782089200444_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ A2 @ ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) )
     => ( ( A2 = bot_bo2676859181106700934_a_a_a )
        | ( A2
          = ( insert5061175218292437356_a_a_a @ X3 @ bot_bo2676859181106700934_a_a_a ) ) ) ) ).

% subset_singletonD
thf(fact_1264_subset__singletonD,axiom,
    ! [A2: set_Pr6246767948713639278_a_a_b,X3: produc4770514982935793080_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ A2 @ ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) )
     => ( ( A2 = bot_bo5052542052834138114_a_a_b )
        | ( A2
          = ( insert8042272945010224104_a_a_b @ X3 @ bot_bo5052542052834138114_a_a_b ) ) ) ) ).

% subset_singletonD
thf(fact_1265_subset__singletonD,axiom,
    ! [A2: set_Pr4618682941230328598_a_a_a,X3: produc884577481120224310_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ A2 @ ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) )
     => ( ( A2 = bot_bo2898616998271641474_a_a_a )
        | ( A2
          = ( insert2249508923356641798_a_a_a @ X3 @ bot_bo2898616998271641474_a_a_a ) ) ) ) ).

% subset_singletonD
thf(fact_1266_subset__singleton__iff,axiom,
    ! [X9: set_Su8591296761904255986_a_a_a,A: sum_su5154756782089200444_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ X9 @ ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) )
      = ( ( X9 = bot_bo2676859181106700934_a_a_a )
        | ( X9
          = ( insert5061175218292437356_a_a_a @ A @ bot_bo2676859181106700934_a_a_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_1267_subset__singleton__iff,axiom,
    ! [X9: set_Pr6246767948713639278_a_a_b,A: produc4770514982935793080_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ X9 @ ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) )
      = ( ( X9 = bot_bo5052542052834138114_a_a_b )
        | ( X9
          = ( insert8042272945010224104_a_a_b @ A @ bot_bo5052542052834138114_a_a_b ) ) ) ) ).

% subset_singleton_iff
thf(fact_1268_subset__singleton__iff,axiom,
    ! [X9: set_Pr4618682941230328598_a_a_a,A: produc884577481120224310_a_a_a] :
      ( ( ord_le8158965334049305270_a_a_a @ X9 @ ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) )
      = ( ( X9 = bot_bo2898616998271641474_a_a_a )
        | ( X9
          = ( insert2249508923356641798_a_a_a @ A @ bot_bo2898616998271641474_a_a_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_1269_list__append__subset3,axiom,
    ! [Xs1: list_S8072159660131056972_a_a_a,Ys1: list_S8072159660131056972_a_a_a,Xs22: list_S8072159660131056972_a_a_a,Ys22: list_S8072159660131056972_a_a_a,Xs32: list_S8072159660131056972_a_a_a,Ys32: list_S8072159660131056972_a_a_a] :
      ( ( ord_le4476889455585342546_a_a_a @ ( set_Su3113867532901839207_a_a_a @ Xs1 ) @ ( set_Su3113867532901839207_a_a_a @ Ys1 ) )
     => ( ( ord_le4476889455585342546_a_a_a @ ( set_Su3113867532901839207_a_a_a @ Xs22 ) @ ( set_Su3113867532901839207_a_a_a @ Ys22 ) )
       => ( ( ord_le4476889455585342546_a_a_a @ ( set_Su3113867532901839207_a_a_a @ Xs32 ) @ ( set_Su3113867532901839207_a_a_a @ Ys32 ) )
         => ( ord_le4476889455585342546_a_a_a @ ( set_Su3113867532901839207_a_a_a @ ( append944914365211047137_a_a_a @ Xs1 @ ( append944914365211047137_a_a_a @ Xs22 @ Xs32 ) ) ) @ ( set_Su3113867532901839207_a_a_a @ ( append944914365211047137_a_a_a @ Ys1 @ ( append944914365211047137_a_a_a @ Ys22 @ Ys32 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_1270_list__append__subset3,axiom,
    ! [Xs1: list_P7155397214832950472_a_a_b,Ys1: list_P7155397214832950472_a_a_b,Xs22: list_P7155397214832950472_a_a_b,Ys22: list_P7155397214832950472_a_a_b,Xs32: list_P7155397214832950472_a_a_b,Ys32: list_P7155397214832950472_a_a_b] :
      ( ( ord_le2852551352416386510_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ Xs1 ) @ ( set_Pr4578018992751950051_a_a_b @ Ys1 ) )
     => ( ( ord_le2852551352416386510_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ Xs22 ) @ ( set_Pr4578018992751950051_a_a_b @ Ys22 ) )
       => ( ( ord_le2852551352416386510_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ Xs32 ) @ ( set_Pr4578018992751950051_a_a_b @ Ys32 ) )
         => ( ord_le2852551352416386510_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ ( append157347090090570333_a_a_b @ Xs1 @ ( append157347090090570333_a_a_b @ Xs22 @ Xs32 ) ) ) @ ( set_Pr4578018992751950051_a_a_b @ ( append157347090090570333_a_a_b @ Ys1 @ ( append157347090090570333_a_a_b @ Ys22 @ Ys32 ) ) ) ) ) ) ) ).

% list_append_subset3
thf(fact_1271_list__prefix__subset,axiom,
    ! [Ts: list_S8072159660131056972_a_a_a,Xs: list_S8072159660131056972_a_a_a] :
      ( ? [Ys2: list_S8072159660131056972_a_a_a] :
          ( Ts
          = ( append944914365211047137_a_a_a @ Xs @ Ys2 ) )
     => ( ord_le4476889455585342546_a_a_a @ ( set_Su3113867532901839207_a_a_a @ Xs ) @ ( set_Su3113867532901839207_a_a_a @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_1272_list__prefix__subset,axiom,
    ! [Ts: list_P7155397214832950472_a_a_b,Xs: list_P7155397214832950472_a_a_b] :
      ( ? [Ys2: list_P7155397214832950472_a_a_b] :
          ( Ts
          = ( append157347090090570333_a_a_b @ Xs @ Ys2 ) )
     => ( ord_le2852551352416386510_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ Xs ) @ ( set_Pr4578018992751950051_a_a_b @ Ts ) ) ) ).

% list_prefix_subset
thf(fact_1273_set__take__subset,axiom,
    ! [N: nat,Xs: list_P7155397214832950472_a_a_b] : ( ord_le2852551352416386510_a_a_b @ ( set_Pr4578018992751950051_a_a_b @ ( take_P6639847873373218378_a_a_b @ N @ Xs ) ) @ ( set_Pr4578018992751950051_a_a_b @ Xs ) ) ).

% set_take_subset

% Conjectures (3)
thf(conj_0,hypothesis,
    $true ).

thf(conj_1,hypothesis,
    $true ).

thf(conj_2,conjecture,
    thesis ).

%------------------------------------------------------------------------------