TPTP Problem File: SLH0990^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FSM_Tests/0042_Observability/prob_00679_032998__19612256_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1925 ( 659 unt; 643 typ;   0 def)
%            Number of atoms       : 3576 (2031 equ;   0 cnn)
%            Maximal formula atoms :   23 (   2 avg)
%            Number of connectives : 14130 ( 566   ~;  37   |; 475   &;11592   @)
%                                         (   0 <=>;1460  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (   6 avg)
%            Number of types       :  138 ( 137 usr)
%            Number of type conns  : 1655 (1655   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  509 ( 506 usr;  48 con; 0-4 aty)
%            Number of variables   : 4114 ( 507   ^;3480   !; 127   ?;4114   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:28:15.135
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__c,type,
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thf(ty_n_tf__b,type,
    b: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (506)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
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thf(sy_c_FSM_Ofrom__FSM_001t__FSet__Ofset_Itf__a_J_001tf__b_001tf__c,type,
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thf(sy_c_FSM_Ofrom__FSM_001tf__a_001tf__b_001tf__c,type,
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thf(sy_c_FSM_Oftransitions_001tf__a_001tf__b_001tf__c,type,
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thf(sy_c_FSM_Oinitial_001t__FSet__Ofset_Itf__a_J_001tf__b_001tf__c,type,
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    image_1461573630340395376od_c_a: ( produc4838349196543436939fset_a > produc2669932201312201782od_c_a ) > set_Pr1098063973736579051fset_a > set_Pr8357624050790843030od_c_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_001tf__a,type,
    image_5808782396235892280et_a_a: ( produc4838349196543436939fset_a > a ) > set_Pr1098063973736579051fset_a > set_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__FSet__Ofset_Itf__a_J,type,
    image_a_fset_a: ( a > fset_a ) > set_a > set_fset_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_J,type,
    image_3174156142289807421fset_a: ( a > produc21584292878202806fset_a ) > set_a > set_Pr4695589342295920918fset_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    image_2663017512022304511od_c_a: ( a > produc3205353004677322104od_c_a ) > set_a > set_Pr2640428967771533528od_c_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    image_4184489845059386301od_c_a: ( a > produc2669932201312201782od_c_a ) > set_a > set_Pr8357624050790843030od_c_a ).

thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
    image_a_a: ( a > a ) > set_a > set_a ).

thf(sy_c_member_001t__FSet__Ofset_Itf__a_J,type,
    member_fset_a: fset_a > set_fset_a > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mtf__b_J_J,type,
    member6660424755668143345et_a_b: list_P7633400287818446792et_a_b > set_li5025376902699835944et_a_b > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mtf__a_J_J,type,
    member5697445122164201327_c_a_a: list_P8814837271076367046_c_a_a > set_li2844990292994971558_c_a_a > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    member6895035109809442519od_a_b: list_P1467974523212632366od_a_b > set_li6306622547289818894od_a_b > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    member2650335117833904444od_c_a: list_P5767727266746070163od_c_a > set_li4839836742003378931od_c_a > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    member6330420149250801815od_b_c: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J,type,
    member2596827339850278714fset_a: list_P3569802872000582161fset_a > set_li191631857062832113fset_a > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__c_Mtf__a_J_J,type,
    member5552703068553123156od_c_a: list_P125642481956313003od_c_a > set_li4905775889721270027od_c_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_J,type,
    member4899939873214261983fset_a: produc21584292878202806fset_a > set_Pr4695589342295920918fset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J_J,type,
    member5709309679808970746od_c_a: produc4887275193591018193od_c_a > set_Pr5541715923002146993od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_J_J_J,type,
    member9109638507678586746fset_a: produc7303452399287242961fset_a > set_Pr22987033768835633fset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    member6699836568822684667od_c_a: produc1135898663509404882od_c_a > set_Pr6020917337727550514od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__c_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    member5500486793640445664od_c_a: produc3729687299602777783od_c_a > set_Pr7664180988061153431od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_Mt__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_J_J,type,
    member808877655204049083fset_a: produc7245291065743820818fset_a > set_Pr4691815135995353842fset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    member6508725278956418977od_c_a: produc3205353004677322104od_c_a > set_Pr2640428967771533528od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_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__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J,type,
    member7243057332918018972fset_a: produc1137077444975408883fset_a > set_Pr7381868494853627219fset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    member7862447941013992593od_b_c: product_prod_b_c > set_Product_prod_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J,type,
    member1042037746285471284fset_a: produc4838349196543436939fset_a > set_Pr1098063973736579051fset_a > $o ).

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

thf(sy_v_M,type,
    m: fsm_a_b_c ).

thf(sy_v_dones,type,
    dones: fset_fset_a ).

thf(sy_v_donesa____,type,
    donesa: fset_fset_a ).

thf(sy_v_ioPa____,type,
    ioPa: list_P903359562653991662od_b_c ).

thf(sy_v_ioTa____,type,
    ioTa: product_prod_b_c ).

thf(sy_v_pM____,type,
    pM: list_P6327159017948738492od_c_a ).

thf(sy_v_pP____,type,
    pP: list_P1058411382900698172fset_a ).

thf(sy_v_p_H____,type,
    p: list_P1058411382900698172fset_a ).

thf(sy_v_q,type,
    q: fset_a ).

thf(sy_v_q_H_H____,type,
    q2: a ).

thf(sy_v_qa____,type,
    qa: fset_a ).

thf(sy_v_qb____,type,
    qb: fset_a ).

thf(sy_v_qc____,type,
    qc: fset_a ).

thf(sy_v_qtrans____,type,
    qtrans: fset_P2596122609397385116fset_a ).

thf(sy_v_tP____,type,
    tP: produc21584292878202806fset_a ).

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

thf(sy_v_ts,type,
    ts: fset_P2596122609397385116fset_a ).

thf(sy_v_tsa____,type,
    tsa: fset_P2596122609397385116fset_a ).

% Relevant facts (1265)
thf(fact_0__092_060open_062q_H_H_A_092_060in_062_Afset_A_It__target_AtP_J_092_060close_062,axiom,
    member_a @ q2 @ ( fset_a2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ tP ) ) ) ) ).

% \<open>q'' \<in> fset (t_target tP)\<close>
thf(fact_1__092_060open_062q_H_H_A_124_092_060in_062_124_At__target_AtP_092_060close_062,axiom,
    fmember_a @ q2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ tP ) ) ) ).

% \<open>q'' |\<in>| t_target tP\<close>
thf(fact_2_Cons_Oprems_I2_J,axiom,
    qc != bot_bot_fset_a ).

% Cons.prems(2)
thf(fact_3__092_060open_062path_AM_Aq_H_H_ApM_092_060close_062,axiom,
    path_a_b_c @ m @ q2 @ pM ).

% \<open>path M q'' pM\<close>
thf(fact_4__092_060open_062t__source_AtP_A_061_Aq_092_060close_062,axiom,
    ( ( produc1739398754328533778fset_a @ tP )
    = qc ) ).

% \<open>t_source tP = q\<close>
thf(fact_5_retains__outputs__for__states__and__inputs__def,axiom,
    ( retain7919859912583161496_c_a_a
    = ( ^ [M: fsm_Pr7813689363856283786_c_a_a,S: fsm_Pr7813689363856283786_c_a_a] :
        ! [X: produc1135898663509404882od_c_a] :
          ( ( member6699836568822684667od_c_a @ X @ ( transi5048145386435558395_c_a_a @ S ) )
         => ! [Y: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ Y @ ( transi5048145386435558395_c_a_a @ M ) )
             => ( ( ( ( produc5475348670448042158od_c_a @ X )
                    = ( produc5475348670448042158od_c_a @ Y ) )
                  & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X ) )
                    = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) )
               => ( member6699836568822684667od_c_a @ Y @ ( transi5048145386435558395_c_a_a @ S ) ) ) ) ) ) ) ).

% retains_outputs_for_states_and_inputs_def
thf(fact_6_retains__outputs__for__states__and__inputs__def,axiom,
    ( retain1579183450281877784et_a_b
    = ( ^ [M: fsm_Pr2398083450103406218et_a_b,S: fsm_Pr2398083450103406218et_a_b] :
        ! [X: produc7245291065743820818fset_a] :
          ( ( member808877655204049083fset_a @ X @ ( transi3322184091305251195et_a_b @ S ) )
         => ! [Y: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ Y @ ( transi3322184091305251195et_a_b @ M ) )
             => ( ( ( ( produc8095675384246233966fset_a @ X )
                    = ( produc8095675384246233966fset_a @ Y ) )
                  & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X ) )
                    = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) )
               => ( member808877655204049083fset_a @ Y @ ( transi3322184091305251195et_a_b @ S ) ) ) ) ) ) ) ).

% retains_outputs_for_states_and_inputs_def
thf(fact_7_retains__outputs__for__states__and__inputs__def,axiom,
    ( retain1438601208673325207_a_b_c
    = ( ^ [M: fsm_fset_a_b_c,S: fsm_fset_a_b_c] :
        ! [X: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ X @ ( transi6427852111903025530_a_b_c @ S ) )
         => ! [Y: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ Y @ ( transi6427852111903025530_a_b_c @ M ) )
             => ( ( ( ( produc1739398754328533778fset_a @ X )
                    = ( produc1739398754328533778fset_a @ Y ) )
                  & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) )
               => ( member4899939873214261983fset_a @ Y @ ( transi6427852111903025530_a_b_c @ S ) ) ) ) ) ) ) ).

% retains_outputs_for_states_and_inputs_def
thf(fact_8_retains__outputs__for__states__and__inputs__def,axiom,
    ( retain6283235197096808344_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b,S: fsm_Pr1886043760914421514_a_a_b] :
        ! [X: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X @ ( transi2941759273464634107_a_a_b @ S ) )
         => ! [Y: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Y @ ( transi2941759273464634107_a_a_b @ M ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X )
                    = ( produc8171714036265584980od_c_a @ Y ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) )
               => ( member6508725278956418977od_c_a @ Y @ ( transi2941759273464634107_a_a_b @ S ) ) ) ) ) ) ) ).

% retains_outputs_for_states_and_inputs_def
thf(fact_9_retains__outputs__for__states__and__inputs__def,axiom,
    ( retain2430384024318543025_a_b_c
    = ( ^ [M: fsm_a_b_c,S: fsm_a_b_c] :
        ! [X: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ S ) )
         => ! [Y: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y @ ( transitions_a_b_c @ M ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X )
                    = ( produc4225943634317630866od_c_a @ Y ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) )
               => ( member5502040268208861279od_c_a @ Y @ ( transitions_a_b_c @ S ) ) ) ) ) ) ) ).

% retains_outputs_for_states_and_inputs_def
thf(fact_10_deterministic_Oelims_I3_J,axiom,
    ! [X2: fsm_Pr7813689363856283786_c_a_a] :
      ( ~ ( determ3842564420232101959_c_a_a @ X2 )
     => ~ ! [X3: produc1135898663509404882od_c_a] :
            ( ( member6699836568822684667od_c_a @ X3 @ ( transi5048145386435558395_c_a_a @ X2 ) )
           => ! [Xa: produc1135898663509404882od_c_a] :
                ( ( member6699836568822684667od_c_a @ Xa @ ( transi5048145386435558395_c_a_a @ X2 ) )
               => ( ( ( ( produc5475348670448042158od_c_a @ X3 )
                      = ( produc5475348670448042158od_c_a @ Xa ) )
                    & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X3 ) )
                      = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Xa ) ) ) )
                 => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X3 ) ) )
                      = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Xa ) ) ) )
                    & ( ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X3 ) ) )
                      = ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Xa ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_11_deterministic_Oelims_I3_J,axiom,
    ! [X2: fsm_Pr2398083450103406218et_a_b] :
      ( ~ ( determ7969653133781418439et_a_b @ X2 )
     => ~ ! [X3: produc7245291065743820818fset_a] :
            ( ( member808877655204049083fset_a @ X3 @ ( transi3322184091305251195et_a_b @ X2 ) )
           => ! [Xa: produc7245291065743820818fset_a] :
                ( ( member808877655204049083fset_a @ Xa @ ( transi3322184091305251195et_a_b @ X2 ) )
               => ( ( ( ( produc8095675384246233966fset_a @ X3 )
                      = ( produc8095675384246233966fset_a @ Xa ) )
                    & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X3 ) )
                      = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Xa ) ) ) )
                 => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X3 ) ) )
                      = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Xa ) ) ) )
                    & ( ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X3 ) ) )
                      = ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Xa ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_12_deterministic_Oelims_I3_J,axiom,
    ! [X2: fsm_fset_a_b_c] :
      ( ~ ( determ4918356626253405126_a_b_c @ X2 )
     => ~ ! [X3: produc21584292878202806fset_a] :
            ( ( member4899939873214261983fset_a @ X3 @ ( transi6427852111903025530_a_b_c @ X2 ) )
           => ! [Xa: produc21584292878202806fset_a] :
                ( ( member4899939873214261983fset_a @ Xa @ ( transi6427852111903025530_a_b_c @ X2 ) )
               => ( ( ( ( produc1739398754328533778fset_a @ X3 )
                      = ( produc1739398754328533778fset_a @ Xa ) )
                    & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X3 ) )
                      = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Xa ) ) ) )
                 => ( ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X3 ) ) )
                      = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Xa ) ) ) )
                    & ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X3 ) ) )
                      = ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Xa ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_13_deterministic_Oelims_I3_J,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b] :
      ( ~ ( determ4061018006534944583_a_a_b @ X2 )
     => ~ ! [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ X2 ) )
           => ! [Xa: produc3205353004677322104od_c_a] :
                ( ( member6508725278956418977od_c_a @ Xa @ ( transi2941759273464634107_a_a_b @ X2 ) )
               => ( ( ( ( produc8171714036265584980od_c_a @ X3 )
                      = ( produc8171714036265584980od_c_a @ Xa ) )
                    & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) )
                      = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) )
                 => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) )
                    & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
                      = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_14_deterministic_Oelims_I3_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ~ ( deterministic_a_b_c @ X2 )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X2 ) )
           => ! [Xa: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa @ ( transitions_a_b_c @ X2 ) )
               => ( ( ( ( produc4225943634317630866od_c_a @ X3 )
                      = ( produc4225943634317630866od_c_a @ Xa ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) )
                 => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                      = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) )
                    & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                      = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(3)
thf(fact_15_deterministic_Oelims_I2_J,axiom,
    ! [X2: fsm_Pr7813689363856283786_c_a_a] :
      ( ( determ3842564420232101959_c_a_a @ X2 )
     => ! [X4: produc1135898663509404882od_c_a] :
          ( ( member6699836568822684667od_c_a @ X4 @ ( transi5048145386435558395_c_a_a @ X2 ) )
         => ! [Xa2: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ Xa2 @ ( transi5048145386435558395_c_a_a @ X2 ) )
             => ( ( ( ( produc5475348670448042158od_c_a @ X4 )
                    = ( produc5475348670448042158od_c_a @ Xa2 ) )
                  & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X4 ) )
                    = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Xa2 ) ) ) )
               => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X4 ) ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Xa2 ) ) ) )
                  & ( ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X4 ) ) )
                    = ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_16_deterministic_Oelims_I2_J,axiom,
    ! [X2: fsm_Pr2398083450103406218et_a_b] :
      ( ( determ7969653133781418439et_a_b @ X2 )
     => ! [X4: produc7245291065743820818fset_a] :
          ( ( member808877655204049083fset_a @ X4 @ ( transi3322184091305251195et_a_b @ X2 ) )
         => ! [Xa2: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ Xa2 @ ( transi3322184091305251195et_a_b @ X2 ) )
             => ( ( ( ( produc8095675384246233966fset_a @ X4 )
                    = ( produc8095675384246233966fset_a @ Xa2 ) )
                  & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X4 ) )
                    = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Xa2 ) ) ) )
               => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X4 ) ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Xa2 ) ) ) )
                  & ( ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X4 ) ) )
                    = ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_17_deterministic_Oelims_I2_J,axiom,
    ! [X2: fsm_fset_a_b_c] :
      ( ( determ4918356626253405126_a_b_c @ X2 )
     => ! [X4: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ X4 @ ( transi6427852111903025530_a_b_c @ X2 ) )
         => ! [Xa2: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ Xa2 @ ( transi6427852111903025530_a_b_c @ X2 ) )
             => ( ( ( ( produc1739398754328533778fset_a @ X4 )
                    = ( produc1739398754328533778fset_a @ Xa2 ) )
                  & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X4 ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Xa2 ) ) ) )
               => ( ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X4 ) ) )
                    = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Xa2 ) ) ) )
                  & ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X4 ) ) )
                    = ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_18_deterministic_Oelims_I2_J,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( determ4061018006534944583_a_a_b @ X2 )
     => ! [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ X2 ) )
         => ! [Xa2: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Xa2 @ ( transi2941759273464634107_a_a_b @ X2 ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X4 )
                    = ( produc8171714036265584980od_c_a @ Xa2 ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) )
               => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) )
                  & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                    = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_19_deterministic_Oelims_I2_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ( deterministic_a_b_c @ X2 )
     => ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X2 ) )
         => ! [Xa2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X2 ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                    = ( produc4225943634317630866od_c_a @ Xa2 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) )
               => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) )
                  & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(2)
thf(fact_20_deterministic_Oelims_I1_J,axiom,
    ! [X2: fsm_Pr7813689363856283786_c_a_a,Y2: $o] :
      ( ( ( determ3842564420232101959_c_a_a @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ X @ ( transi5048145386435558395_c_a_a @ X2 ) )
             => ! [Y: produc1135898663509404882od_c_a] :
                  ( ( member6699836568822684667od_c_a @ Y @ ( transi5048145386435558395_c_a_a @ X2 ) )
                 => ( ( ( ( produc5475348670448042158od_c_a @ X )
                        = ( produc5475348670448042158od_c_a @ Y ) )
                      & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X ) )
                        = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) )
                   => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X ) ) )
                        = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) )
                      & ( ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X ) ) )
                        = ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_21_deterministic_Oelims_I1_J,axiom,
    ! [X2: fsm_Pr2398083450103406218et_a_b,Y2: $o] :
      ( ( ( determ7969653133781418439et_a_b @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ X @ ( transi3322184091305251195et_a_b @ X2 ) )
             => ! [Y: produc7245291065743820818fset_a] :
                  ( ( member808877655204049083fset_a @ Y @ ( transi3322184091305251195et_a_b @ X2 ) )
                 => ( ( ( ( produc8095675384246233966fset_a @ X )
                        = ( produc8095675384246233966fset_a @ Y ) )
                      & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X ) )
                        = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) )
                   => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X ) ) )
                        = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) )
                      & ( ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X ) ) )
                        = ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_22_deterministic_Oelims_I1_J,axiom,
    ! [X2: fsm_fset_a_b_c,Y2: $o] :
      ( ( ( determ4918356626253405126_a_b_c @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ X @ ( transi6427852111903025530_a_b_c @ X2 ) )
             => ! [Y: produc21584292878202806fset_a] :
                  ( ( member4899939873214261983fset_a @ Y @ ( transi6427852111903025530_a_b_c @ X2 ) )
                 => ( ( ( ( produc1739398754328533778fset_a @ X )
                        = ( produc1739398754328533778fset_a @ Y ) )
                      & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X ) )
                        = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) )
                   => ( ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X ) ) )
                        = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) )
                      & ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X ) ) )
                        = ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_23_deterministic_Oelims_I1_J,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b,Y2: $o] :
      ( ( ( determ4061018006534944583_a_a_b @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X @ ( transi2941759273464634107_a_a_b @ X2 ) )
             => ! [Y: produc3205353004677322104od_c_a] :
                  ( ( member6508725278956418977od_c_a @ Y @ ( transi2941759273464634107_a_a_b @ X2 ) )
                 => ( ( ( ( produc8171714036265584980od_c_a @ X )
                        = ( produc8171714036265584980od_c_a @ Y ) )
                      & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X ) )
                        = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) )
                   => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) )
                      & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
                        = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_24_deterministic_Oelims_I1_J,axiom,
    ! [X2: fsm_a_b_c,Y2: $o] :
      ( ( ( deterministic_a_b_c @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X2 ) )
             => ! [Y: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y @ ( transitions_a_b_c @ X2 ) )
                 => ( ( ( ( produc4225943634317630866od_c_a @ X )
                        = ( produc4225943634317630866od_c_a @ Y ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) )
                   => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                        = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) )
                      & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                        = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) ) ) ) ) ) ) ) ) ).

% deterministic.elims(1)
thf(fact_25_deterministic_Osimps,axiom,
    ( determ3842564420232101959_c_a_a
    = ( ^ [M: fsm_Pr7813689363856283786_c_a_a] :
        ! [X: produc1135898663509404882od_c_a] :
          ( ( member6699836568822684667od_c_a @ X @ ( transi5048145386435558395_c_a_a @ M ) )
         => ! [Y: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ Y @ ( transi5048145386435558395_c_a_a @ M ) )
             => ( ( ( ( produc5475348670448042158od_c_a @ X )
                    = ( produc5475348670448042158od_c_a @ Y ) )
                  & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X ) )
                    = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) )
               => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X ) ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) )
                  & ( ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X ) ) )
                    = ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_26_deterministic_Osimps,axiom,
    ( determ7969653133781418439et_a_b
    = ( ^ [M: fsm_Pr2398083450103406218et_a_b] :
        ! [X: produc7245291065743820818fset_a] :
          ( ( member808877655204049083fset_a @ X @ ( transi3322184091305251195et_a_b @ M ) )
         => ! [Y: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ Y @ ( transi3322184091305251195et_a_b @ M ) )
             => ( ( ( ( produc8095675384246233966fset_a @ X )
                    = ( produc8095675384246233966fset_a @ Y ) )
                  & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X ) )
                    = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) )
               => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X ) ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) )
                  & ( ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X ) ) )
                    = ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_27_deterministic_Osimps,axiom,
    ( determ4918356626253405126_a_b_c
    = ( ^ [M: fsm_fset_a_b_c] :
        ! [X: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ X @ ( transi6427852111903025530_a_b_c @ M ) )
         => ! [Y: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ Y @ ( transi6427852111903025530_a_b_c @ M ) )
             => ( ( ( ( produc1739398754328533778fset_a @ X )
                    = ( produc1739398754328533778fset_a @ Y ) )
                  & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) )
               => ( ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X ) ) )
                    = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) )
                  & ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X ) ) )
                    = ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_28_deterministic_Osimps,axiom,
    ( determ4061018006534944583_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [X: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X @ ( transi2941759273464634107_a_a_b @ M ) )
         => ! [Y: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Y @ ( transi2941759273464634107_a_a_b @ M ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X )
                    = ( produc8171714036265584980od_c_a @ Y ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) )
               => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) )
                  & ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
                    = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_29_deterministic_Osimps,axiom,
    ( deterministic_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [X: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ M ) )
         => ! [Y: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y @ ( transitions_a_b_c @ M ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X )
                    = ( produc4225943634317630866od_c_a @ Y ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) )
               => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) )
                  & ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% deterministic.simps
thf(fact_30__092_060open_062t__target_AtP_A_092_060noteq_062_A_123_124_124_125_092_060close_062,axiom,
    ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ tP ) ) )
   != bot_bot_fset_a ) ).

% \<open>t_target tP \<noteq> {||}\<close>
thf(fact_31_prod_Oexpand,axiom,
    ! [Prod: produc4838349196543436939fset_a,Prod2: produc4838349196543436939fset_a] :
      ( ( ( ( product_fst_c_fset_a @ Prod )
          = ( product_fst_c_fset_a @ Prod2 ) )
        & ( ( product_snd_c_fset_a @ Prod )
          = ( product_snd_c_fset_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_32_prod_Oexpand,axiom,
    ! [Prod: product_prod_c_a,Prod2: product_prod_c_a] :
      ( ( ( ( product_fst_c_a @ Prod )
          = ( product_fst_c_a @ Prod2 ) )
        & ( ( product_snd_c_a @ Prod )
          = ( product_snd_c_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_33_prod_Oexpand,axiom,
    ! [Prod: produc1137077444975408883fset_a,Prod2: produc1137077444975408883fset_a] :
      ( ( ( ( produc5616988955220391631fset_a @ Prod )
          = ( produc5616988955220391631fset_a @ Prod2 ) )
        & ( ( produc8704244169652099857fset_a @ Prod )
          = ( produc8704244169652099857fset_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_34_prod_Oexpand,axiom,
    ! [Prod: produc2133359284458116493od_c_a,Prod2: produc2133359284458116493od_c_a] :
      ( ( ( ( produc301949074746585577od_c_a @ Prod )
          = ( produc301949074746585577od_c_a @ Prod2 ) )
        & ( ( produc5221446324160585515od_c_a @ Prod )
          = ( produc5221446324160585515od_c_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_35_prod_Oexpand,axiom,
    ! [Prod: produc2669932201312201782od_c_a,Prod2: produc2669932201312201782od_c_a] :
      ( ( ( ( produc4225943634317630866od_c_a @ Prod )
          = ( produc4225943634317630866od_c_a @ Prod2 ) )
        & ( ( produc8088130389382527188od_c_a @ Prod )
          = ( produc8088130389382527188od_c_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_36_prod_Oexpand,axiom,
    ! [Prod: produc3205353004677322104od_c_a,Prod2: produc3205353004677322104od_c_a] :
      ( ( ( ( produc8171714036265584980od_c_a @ Prod )
          = ( produc8171714036265584980od_c_a @ Prod2 ) )
        & ( ( produc4643634399596806294od_c_a @ Prod )
          = ( produc4643634399596806294od_c_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_37_prod_Oexpand,axiom,
    ! [Prod: produc21584292878202806fset_a,Prod2: produc21584292878202806fset_a] :
      ( ( ( ( produc1739398754328533778fset_a @ Prod )
          = ( produc1739398754328533778fset_a @ Prod2 ) )
        & ( ( produc8188425890649629780fset_a @ Prod )
          = ( produc8188425890649629780fset_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_38_prod__eqI,axiom,
    ! [P: produc4838349196543436939fset_a,Q: produc4838349196543436939fset_a] :
      ( ( ( product_fst_c_fset_a @ P )
        = ( product_fst_c_fset_a @ Q ) )
     => ( ( ( product_snd_c_fset_a @ P )
          = ( product_snd_c_fset_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_39_prod__eqI,axiom,
    ! [P: product_prod_c_a,Q: product_prod_c_a] :
      ( ( ( product_fst_c_a @ P )
        = ( product_fst_c_a @ Q ) )
     => ( ( ( product_snd_c_a @ P )
          = ( product_snd_c_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_40_prod__eqI,axiom,
    ! [P: produc1137077444975408883fset_a,Q: produc1137077444975408883fset_a] :
      ( ( ( produc5616988955220391631fset_a @ P )
        = ( produc5616988955220391631fset_a @ Q ) )
     => ( ( ( produc8704244169652099857fset_a @ P )
          = ( produc8704244169652099857fset_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_41_prod__eqI,axiom,
    ! [P: produc2133359284458116493od_c_a,Q: produc2133359284458116493od_c_a] :
      ( ( ( produc301949074746585577od_c_a @ P )
        = ( produc301949074746585577od_c_a @ Q ) )
     => ( ( ( produc5221446324160585515od_c_a @ P )
          = ( produc5221446324160585515od_c_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_42_prod__eqI,axiom,
    ! [P: produc2669932201312201782od_c_a,Q: produc2669932201312201782od_c_a] :
      ( ( ( produc4225943634317630866od_c_a @ P )
        = ( produc4225943634317630866od_c_a @ Q ) )
     => ( ( ( produc8088130389382527188od_c_a @ P )
          = ( produc8088130389382527188od_c_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_43_prod__eqI,axiom,
    ! [P: produc3205353004677322104od_c_a,Q: produc3205353004677322104od_c_a] :
      ( ( ( produc8171714036265584980od_c_a @ P )
        = ( produc8171714036265584980od_c_a @ Q ) )
     => ( ( ( produc4643634399596806294od_c_a @ P )
          = ( produc4643634399596806294od_c_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_44_prod__eqI,axiom,
    ! [P: produc21584292878202806fset_a,Q: produc21584292878202806fset_a] :
      ( ( ( produc1739398754328533778fset_a @ P )
        = ( produc1739398754328533778fset_a @ Q ) )
     => ( ( ( produc8188425890649629780fset_a @ P )
          = ( produc8188425890649629780fset_a @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_45_exE__realizer_H,axiom,
    ! [P2: fset_a > c > $o,P: produc4838349196543436939fset_a] :
      ( ( P2 @ ( product_snd_c_fset_a @ P ) @ ( product_fst_c_fset_a @ P ) )
     => ~ ! [X3: c,Y3: fset_a] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_46_exE__realizer_H,axiom,
    ! [P2: a > c > $o,P: product_prod_c_a] :
      ( ( P2 @ ( product_snd_c_a @ P ) @ ( product_fst_c_a @ P ) )
     => ~ ! [X3: c,Y3: a] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_47_exE__realizer_H,axiom,
    ! [P2: produc4838349196543436939fset_a > b > $o,P: produc1137077444975408883fset_a] :
      ( ( P2 @ ( produc8704244169652099857fset_a @ P ) @ ( produc5616988955220391631fset_a @ P ) )
     => ~ ! [X3: b,Y3: produc4838349196543436939fset_a] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_48_exE__realizer_H,axiom,
    ! [P2: product_prod_c_a > b > $o,P: produc2133359284458116493od_c_a] :
      ( ( P2 @ ( produc5221446324160585515od_c_a @ P ) @ ( produc301949074746585577od_c_a @ P ) )
     => ~ ! [X3: b,Y3: product_prod_c_a] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_49_exE__realizer_H,axiom,
    ! [P2: produc2133359284458116493od_c_a > a > $o,P: produc2669932201312201782od_c_a] :
      ( ( P2 @ ( produc8088130389382527188od_c_a @ P ) @ ( produc4225943634317630866od_c_a @ P ) )
     => ~ ! [X3: a,Y3: produc2133359284458116493od_c_a] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_50_exE__realizer_H,axiom,
    ! [P2: produc2669932201312201782od_c_a > product_prod_c_a > $o,P: produc3205353004677322104od_c_a] :
      ( ( P2 @ ( produc4643634399596806294od_c_a @ P ) @ ( produc8171714036265584980od_c_a @ P ) )
     => ~ ! [X3: product_prod_c_a,Y3: produc2669932201312201782od_c_a] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_51_exE__realizer_H,axiom,
    ! [P2: produc1137077444975408883fset_a > fset_a > $o,P: produc21584292878202806fset_a] :
      ( ( P2 @ ( produc8188425890649629780fset_a @ P ) @ ( produc1739398754328533778fset_a @ P ) )
     => ~ ! [X3: fset_a,Y3: produc1137077444975408883fset_a] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_52_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc4838349196543436939fset_a,Z: produc4838349196543436939fset_a] : ( Y4 = Z ) )
    = ( ^ [S2: produc4838349196543436939fset_a,T: produc4838349196543436939fset_a] :
          ( ( ( product_fst_c_fset_a @ S2 )
            = ( product_fst_c_fset_a @ T ) )
          & ( ( product_snd_c_fset_a @ S2 )
            = ( product_snd_c_fset_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_53_prod__eq__iff,axiom,
    ( ( ^ [Y4: product_prod_c_a,Z: product_prod_c_a] : ( Y4 = Z ) )
    = ( ^ [S2: product_prod_c_a,T: product_prod_c_a] :
          ( ( ( product_fst_c_a @ S2 )
            = ( product_fst_c_a @ T ) )
          & ( ( product_snd_c_a @ S2 )
            = ( product_snd_c_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_54_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc1137077444975408883fset_a,Z: produc1137077444975408883fset_a] : ( Y4 = Z ) )
    = ( ^ [S2: produc1137077444975408883fset_a,T: produc1137077444975408883fset_a] :
          ( ( ( produc5616988955220391631fset_a @ S2 )
            = ( produc5616988955220391631fset_a @ T ) )
          & ( ( produc8704244169652099857fset_a @ S2 )
            = ( produc8704244169652099857fset_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_55_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc2133359284458116493od_c_a,Z: produc2133359284458116493od_c_a] : ( Y4 = Z ) )
    = ( ^ [S2: produc2133359284458116493od_c_a,T: produc2133359284458116493od_c_a] :
          ( ( ( produc301949074746585577od_c_a @ S2 )
            = ( produc301949074746585577od_c_a @ T ) )
          & ( ( produc5221446324160585515od_c_a @ S2 )
            = ( produc5221446324160585515od_c_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_56_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc2669932201312201782od_c_a,Z: produc2669932201312201782od_c_a] : ( Y4 = Z ) )
    = ( ^ [S2: produc2669932201312201782od_c_a,T: produc2669932201312201782od_c_a] :
          ( ( ( produc4225943634317630866od_c_a @ S2 )
            = ( produc4225943634317630866od_c_a @ T ) )
          & ( ( produc8088130389382527188od_c_a @ S2 )
            = ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_57_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc3205353004677322104od_c_a,Z: produc3205353004677322104od_c_a] : ( Y4 = Z ) )
    = ( ^ [S2: produc3205353004677322104od_c_a,T: produc3205353004677322104od_c_a] :
          ( ( ( produc8171714036265584980od_c_a @ S2 )
            = ( produc8171714036265584980od_c_a @ T ) )
          & ( ( produc4643634399596806294od_c_a @ S2 )
            = ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_58_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc21584292878202806fset_a,Z: produc21584292878202806fset_a] : ( Y4 = Z ) )
    = ( ^ [S2: produc21584292878202806fset_a,T: produc21584292878202806fset_a] :
          ( ( ( produc1739398754328533778fset_a @ S2 )
            = ( produc1739398754328533778fset_a @ T ) )
          & ( ( produc8188425890649629780fset_a @ S2 )
            = ( produc8188425890649629780fset_a @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_59__092_060open_062qb_____A_092_060noteq_062_A_123_124_124_125_092_060close_062,axiom,
    qb != bot_bot_fset_a ).

% \<open>qb__ \<noteq> {||}\<close>
thf(fact_60_all__not__fin__conv,axiom,
    ! [A: fset_P2596122609397385116fset_a] :
      ( ( ! [X: produc21584292878202806fset_a] :
            ~ ( fmembe5403795861166450045fset_a @ X @ A ) )
      = ( A = bot_bo2316247384970961928fset_a ) ) ).

% all_not_fin_conv
thf(fact_61_all__not__fin__conv,axiom,
    ! [A: fset_P8043821951902805788od_c_a] :
      ( ( ! [X: produc2669932201312201782od_c_a] :
            ~ ( fmembe4375279799106287869od_c_a @ X @ A ) )
      = ( A = bot_bo1116342619594069896od_c_a ) ) ).

% all_not_fin_conv
thf(fact_62_all__not__fin__conv,axiom,
    ! [A: fset_a] :
      ( ( ! [X: a] :
            ~ ( fmember_a @ X @ A ) )
      = ( A = bot_bot_fset_a ) ) ).

% all_not_fin_conv
thf(fact_63_all__not__fin__conv,axiom,
    ! [A: fset_fset_a] :
      ( ( ! [X: fset_a] :
            ~ ( fmember_fset_a @ X @ A ) )
      = ( A = bot_bot_fset_fset_a ) ) ).

% all_not_fin_conv
thf(fact_64_fempty__iff,axiom,
    ! [C: produc21584292878202806fset_a] :
      ~ ( fmembe5403795861166450045fset_a @ C @ bot_bo2316247384970961928fset_a ) ).

% fempty_iff
thf(fact_65_fempty__iff,axiom,
    ! [C: produc2669932201312201782od_c_a] :
      ~ ( fmembe4375279799106287869od_c_a @ C @ bot_bo1116342619594069896od_c_a ) ).

% fempty_iff
thf(fact_66_fempty__iff,axiom,
    ! [C: a] :
      ~ ( fmember_a @ C @ bot_bot_fset_a ) ).

% fempty_iff
thf(fact_67_fempty__iff,axiom,
    ! [C: fset_a] :
      ~ ( fmember_fset_a @ C @ bot_bot_fset_fset_a ) ).

% fempty_iff
thf(fact_68_equalsffemptyI,axiom,
    ! [A: fset_P2596122609397385116fset_a] :
      ( ! [Y3: produc21584292878202806fset_a] :
          ~ ( fmembe5403795861166450045fset_a @ Y3 @ A )
     => ( A = bot_bo2316247384970961928fset_a ) ) ).

% equalsffemptyI
thf(fact_69_equalsffemptyI,axiom,
    ! [A: fset_P8043821951902805788od_c_a] :
      ( ! [Y3: produc2669932201312201782od_c_a] :
          ~ ( fmembe4375279799106287869od_c_a @ Y3 @ A )
     => ( A = bot_bo1116342619594069896od_c_a ) ) ).

% equalsffemptyI
thf(fact_70_equalsffemptyI,axiom,
    ! [A: fset_a] :
      ( ! [Y3: a] :
          ~ ( fmember_a @ Y3 @ A )
     => ( A = bot_bot_fset_a ) ) ).

% equalsffemptyI
thf(fact_71_equalsffemptyI,axiom,
    ! [A: fset_fset_a] :
      ( ! [Y3: fset_a] :
          ~ ( fmember_fset_a @ Y3 @ A )
     => ( A = bot_bot_fset_fset_a ) ) ).

% equalsffemptyI
thf(fact_72_equalsffemptyD,axiom,
    ! [A: fset_P2596122609397385116fset_a,A2: produc21584292878202806fset_a] :
      ( ( A = bot_bo2316247384970961928fset_a )
     => ~ ( fmembe5403795861166450045fset_a @ A2 @ A ) ) ).

% equalsffemptyD
thf(fact_73_equalsffemptyD,axiom,
    ! [A: fset_P8043821951902805788od_c_a,A2: produc2669932201312201782od_c_a] :
      ( ( A = bot_bo1116342619594069896od_c_a )
     => ~ ( fmembe4375279799106287869od_c_a @ A2 @ A ) ) ).

% equalsffemptyD
thf(fact_74_equalsffemptyD,axiom,
    ! [A: fset_a,A2: a] :
      ( ( A = bot_bot_fset_a )
     => ~ ( fmember_a @ A2 @ A ) ) ).

% equalsffemptyD
thf(fact_75_equalsffemptyD,axiom,
    ! [A: fset_fset_a,A2: fset_a] :
      ( ( A = bot_bot_fset_fset_a )
     => ~ ( fmember_fset_a @ A2 @ A ) ) ).

% equalsffemptyD
thf(fact_76_ex__fin__conv,axiom,
    ! [A: fset_P2596122609397385116fset_a] :
      ( ( ? [X: produc21584292878202806fset_a] : ( fmembe5403795861166450045fset_a @ X @ A ) )
      = ( A != bot_bo2316247384970961928fset_a ) ) ).

% ex_fin_conv
thf(fact_77_ex__fin__conv,axiom,
    ! [A: fset_P8043821951902805788od_c_a] :
      ( ( ? [X: produc2669932201312201782od_c_a] : ( fmembe4375279799106287869od_c_a @ X @ A ) )
      = ( A != bot_bo1116342619594069896od_c_a ) ) ).

% ex_fin_conv
thf(fact_78_ex__fin__conv,axiom,
    ! [A: fset_a] :
      ( ( ? [X: a] : ( fmember_a @ X @ A ) )
      = ( A != bot_bot_fset_a ) ) ).

% ex_fin_conv
thf(fact_79_ex__fin__conv,axiom,
    ! [A: fset_fset_a] :
      ( ( ? [X: fset_a] : ( fmember_fset_a @ X @ A ) )
      = ( A != bot_bot_fset_fset_a ) ) ).

% ex_fin_conv
thf(fact_80_femptyE,axiom,
    ! [A2: produc21584292878202806fset_a] :
      ~ ( fmembe5403795861166450045fset_a @ A2 @ bot_bo2316247384970961928fset_a ) ).

% femptyE
thf(fact_81_femptyE,axiom,
    ! [A2: produc2669932201312201782od_c_a] :
      ~ ( fmembe4375279799106287869od_c_a @ A2 @ bot_bo1116342619594069896od_c_a ) ).

% femptyE
thf(fact_82_femptyE,axiom,
    ! [A2: a] :
      ~ ( fmember_a @ A2 @ bot_bot_fset_a ) ).

% femptyE
thf(fact_83_femptyE,axiom,
    ! [A2: fset_a] :
      ~ ( fmember_fset_a @ A2 @ bot_bot_fset_fset_a ) ).

% femptyE
thf(fact_84_fmember__iff__member__fset,axiom,
    ( fmembe4248414558865794111od_c_a
    = ( ^ [X: produc3205353004677322104od_c_a,A3: fset_P9065897409413591390od_c_a] : ( member6508725278956418977od_c_a @ X @ ( fset_P4425921658469021927od_c_a @ A3 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_85_fmember__iff__member__fset,axiom,
    ( fmember_fset_a
    = ( ^ [X: fset_a,A3: fset_fset_a] : ( member_fset_a @ X @ ( fset_fset_a2 @ A3 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_86_fmember__iff__member__fset,axiom,
    ( fmembe5403795861166450045fset_a
    = ( ^ [X: produc21584292878202806fset_a,A3: fset_P2596122609397385116fset_a] : ( member4899939873214261983fset_a @ X @ ( fset_P7830090094348214821fset_a @ A3 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_87_fmember__iff__member__fset,axiom,
    ( fmembe4375279799106287869od_c_a
    = ( ^ [X: produc2669932201312201782od_c_a,A3: fset_P8043821951902805788od_c_a] : ( member5502040268208861279od_c_a @ X @ ( fset_P7369495421554515365od_c_a @ A3 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_88_fmember__iff__member__fset,axiom,
    ( fmember_a
    = ( ^ [X: a,A3: fset_a] : ( member_a @ X @ ( fset_a2 @ A3 ) ) ) ) ).

% fmember_iff_member_fset
thf(fact_89_notin__fset,axiom,
    ! [X2: produc3205353004677322104od_c_a,S3: fset_P9065897409413591390od_c_a] :
      ( ( ~ ( fmembe4248414558865794111od_c_a @ X2 @ S3 ) )
      = ( ~ ( member6508725278956418977od_c_a @ X2 @ ( fset_P4425921658469021927od_c_a @ S3 ) ) ) ) ).

% notin_fset
thf(fact_90_notin__fset,axiom,
    ! [X2: fset_a,S3: fset_fset_a] :
      ( ( ~ ( fmember_fset_a @ X2 @ S3 ) )
      = ( ~ ( member_fset_a @ X2 @ ( fset_fset_a2 @ S3 ) ) ) ) ).

% notin_fset
thf(fact_91_notin__fset,axiom,
    ! [X2: produc21584292878202806fset_a,S3: fset_P2596122609397385116fset_a] :
      ( ( ~ ( fmembe5403795861166450045fset_a @ X2 @ S3 ) )
      = ( ~ ( member4899939873214261983fset_a @ X2 @ ( fset_P7830090094348214821fset_a @ S3 ) ) ) ) ).

% notin_fset
thf(fact_92_notin__fset,axiom,
    ! [X2: produc2669932201312201782od_c_a,S3: fset_P8043821951902805788od_c_a] :
      ( ( ~ ( fmembe4375279799106287869od_c_a @ X2 @ S3 ) )
      = ( ~ ( member5502040268208861279od_c_a @ X2 @ ( fset_P7369495421554515365od_c_a @ S3 ) ) ) ) ).

% notin_fset
thf(fact_93_notin__fset,axiom,
    ! [X2: a,S3: fset_a] :
      ( ( ~ ( fmember_a @ X2 @ S3 ) )
      = ( ~ ( member_a @ X2 @ ( fset_a2 @ S3 ) ) ) ) ).

% notin_fset
thf(fact_94_fmember_Orep__eq,axiom,
    ( fmembe4248414558865794111od_c_a
    = ( ^ [X: produc3205353004677322104od_c_a,Xa3: fset_P9065897409413591390od_c_a] : ( member6508725278956418977od_c_a @ X @ ( fset_P4425921658469021927od_c_a @ Xa3 ) ) ) ) ).

% fmember.rep_eq
thf(fact_95_fmember_Orep__eq,axiom,
    ( fmember_fset_a
    = ( ^ [X: fset_a,Xa3: fset_fset_a] : ( member_fset_a @ X @ ( fset_fset_a2 @ Xa3 ) ) ) ) ).

% fmember.rep_eq
thf(fact_96_fmember_Orep__eq,axiom,
    ( fmembe5403795861166450045fset_a
    = ( ^ [X: produc21584292878202806fset_a,Xa3: fset_P2596122609397385116fset_a] : ( member4899939873214261983fset_a @ X @ ( fset_P7830090094348214821fset_a @ Xa3 ) ) ) ) ).

% fmember.rep_eq
thf(fact_97_fmember_Orep__eq,axiom,
    ( fmembe4375279799106287869od_c_a
    = ( ^ [X: produc2669932201312201782od_c_a,Xa3: fset_P8043821951902805788od_c_a] : ( member5502040268208861279od_c_a @ X @ ( fset_P7369495421554515365od_c_a @ Xa3 ) ) ) ) ).

% fmember.rep_eq
thf(fact_98_fmember_Orep__eq,axiom,
    ( fmember_a
    = ( ^ [X: a,Xa3: fset_a] : ( member_a @ X @ ( fset_a2 @ Xa3 ) ) ) ) ).

% fmember.rep_eq
thf(fact_99_single__input_Osimps,axiom,
    ( single8385926175230902071_c_a_a
    = ( ^ [M: fsm_Pr7813689363856283786_c_a_a] :
        ! [X: produc1135898663509404882od_c_a] :
          ( ( member6699836568822684667od_c_a @ X @ ( transi5048145386435558395_c_a_a @ M ) )
         => ! [Y: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ Y @ ( transi5048145386435558395_c_a_a @ M ) )
             => ( ( ( produc5475348670448042158od_c_a @ X )
                  = ( produc5475348670448042158od_c_a @ Y ) )
               => ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X ) )
                  = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_100_single__input_Osimps,axiom,
    ( single5089827247979212983et_a_b
    = ( ^ [M: fsm_Pr2398083450103406218et_a_b] :
        ! [X: produc7245291065743820818fset_a] :
          ( ( member808877655204049083fset_a @ X @ ( transi3322184091305251195et_a_b @ M ) )
         => ! [Y: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ Y @ ( transi3322184091305251195et_a_b @ M ) )
             => ( ( ( produc8095675384246233966fset_a @ X )
                  = ( produc8095675384246233966fset_a @ Y ) )
               => ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X ) )
                  = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_101_single__input_Osimps,axiom,
    ( single_input_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [X: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ M ) )
         => ! [Y: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y @ ( transitions_a_b_c @ M ) )
             => ( ( ( produc4225943634317630866od_c_a @ X )
                  = ( produc4225943634317630866od_c_a @ Y ) )
               => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                  = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_102_single__input_Osimps,axiom,
    ( single999786218259502647_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [X: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X @ ( transi2941759273464634107_a_a_b @ M ) )
         => ! [Y: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Y @ ( transi2941759273464634107_a_a_b @ M ) )
             => ( ( ( produc8171714036265584980od_c_a @ X )
                  = ( produc8171714036265584980od_c_a @ Y ) )
               => ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X ) )
                  = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_103_single__input_Osimps,axiom,
    ( single3763647048569287350_a_b_c
    = ( ^ [M: fsm_fset_a_b_c] :
        ! [X: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ X @ ( transi6427852111903025530_a_b_c @ M ) )
         => ! [Y: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ Y @ ( transi6427852111903025530_a_b_c @ M ) )
             => ( ( ( produc1739398754328533778fset_a @ X )
                  = ( produc1739398754328533778fset_a @ Y ) )
               => ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X ) )
                  = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) ) ) ) ) ) ).

% single_input.simps
thf(fact_104_single__input_Oelims_I1_J,axiom,
    ! [X2: fsm_Pr7813689363856283786_c_a_a,Y2: $o] :
      ( ( ( single8385926175230902071_c_a_a @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ X @ ( transi5048145386435558395_c_a_a @ X2 ) )
             => ! [Y: produc1135898663509404882od_c_a] :
                  ( ( member6699836568822684667od_c_a @ Y @ ( transi5048145386435558395_c_a_a @ X2 ) )
                 => ( ( ( produc5475348670448042158od_c_a @ X )
                      = ( produc5475348670448042158od_c_a @ Y ) )
                   => ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X ) )
                      = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_105_single__input_Oelims_I1_J,axiom,
    ! [X2: fsm_Pr2398083450103406218et_a_b,Y2: $o] :
      ( ( ( single5089827247979212983et_a_b @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ X @ ( transi3322184091305251195et_a_b @ X2 ) )
             => ! [Y: produc7245291065743820818fset_a] :
                  ( ( member808877655204049083fset_a @ Y @ ( transi3322184091305251195et_a_b @ X2 ) )
                 => ( ( ( produc8095675384246233966fset_a @ X )
                      = ( produc8095675384246233966fset_a @ Y ) )
                   => ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X ) )
                      = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_106_single__input_Oelims_I1_J,axiom,
    ! [X2: fsm_a_b_c,Y2: $o] :
      ( ( ( single_input_a_b_c @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X2 ) )
             => ! [Y: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y @ ( transitions_a_b_c @ X2 ) )
                 => ( ( ( produc4225943634317630866od_c_a @ X )
                      = ( produc4225943634317630866od_c_a @ Y ) )
                   => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_107_single__input_Oelims_I1_J,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b,Y2: $o] :
      ( ( ( single999786218259502647_a_a_b @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X @ ( transi2941759273464634107_a_a_b @ X2 ) )
             => ! [Y: produc3205353004677322104od_c_a] :
                  ( ( member6508725278956418977od_c_a @ Y @ ( transi2941759273464634107_a_a_b @ X2 ) )
                 => ( ( ( produc8171714036265584980od_c_a @ X )
                      = ( produc8171714036265584980od_c_a @ Y ) )
                   => ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X ) )
                      = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_108_single__input_Oelims_I1_J,axiom,
    ! [X2: fsm_fset_a_b_c,Y2: $o] :
      ( ( ( single3763647048569287350_a_b_c @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ X @ ( transi6427852111903025530_a_b_c @ X2 ) )
             => ! [Y: produc21584292878202806fset_a] :
                  ( ( member4899939873214261983fset_a @ Y @ ( transi6427852111903025530_a_b_c @ X2 ) )
                 => ( ( ( produc1739398754328533778fset_a @ X )
                      = ( produc1739398754328533778fset_a @ Y ) )
                   => ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X ) )
                      = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) ) ) ) ) ) ) ).

% single_input.elims(1)
thf(fact_109_assms_I6_J,axiom,
    fmember_fset_a @ q @ dones ).

% assms(6)
thf(fact_110_assms_I5_J,axiom,
    ~ ( fmember_fset_a @ bot_bot_fset_a @ dones ) ).

% assms(5)
thf(fact_111_bot__fset_Orep__eq,axiom,
    ( ( fset_nat2 @ bot_bot_fset_nat )
    = bot_bot_set_nat ) ).

% bot_fset.rep_eq
thf(fact_112_bot__fset_Orep__eq,axiom,
    ( ( fset_a2 @ bot_bot_fset_a )
    = bot_bot_set_a ) ).

% bot_fset.rep_eq
thf(fact_113_bot__fset_Orep__eq,axiom,
    ( ( fset_fset_a2 @ bot_bot_fset_fset_a )
    = bot_bot_set_fset_a ) ).

% bot_fset.rep_eq
thf(fact_114_fset__eqI,axiom,
    ! [A: fset_a,B: fset_a] :
      ( ! [X3: a] :
          ( ( fmember_a @ X3 @ A )
          = ( fmember_a @ X3 @ B ) )
     => ( A = B ) ) ).

% fset_eqI
thf(fact_115_fset__eqI,axiom,
    ! [A: fset_fset_a,B: fset_fset_a] :
      ( ! [X3: fset_a] :
          ( ( fmember_fset_a @ X3 @ A )
          = ( fmember_fset_a @ X3 @ B ) )
     => ( A = B ) ) ).

% fset_eqI
thf(fact_116_fset__eqI,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ! [X3: produc21584292878202806fset_a] :
          ( ( fmembe5403795861166450045fset_a @ X3 @ A )
          = ( fmembe5403795861166450045fset_a @ X3 @ B ) )
     => ( A = B ) ) ).

% fset_eqI
thf(fact_117_fset__eqI,axiom,
    ! [A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( fmembe4375279799106287869od_c_a @ X3 @ A )
          = ( fmembe4375279799106287869od_c_a @ X3 @ B ) )
     => ( A = B ) ) ).

% fset_eqI
thf(fact_118_fequalityCE,axiom,
    ! [A: fset_a,B: fset_a,C: a] :
      ( ( A = B )
     => ( ( ( fmember_a @ C @ A )
         => ~ ( fmember_a @ C @ B ) )
       => ~ ( ~ ( fmember_a @ C @ A )
           => ( fmember_a @ C @ B ) ) ) ) ).

% fequalityCE
thf(fact_119_fequalityCE,axiom,
    ! [A: fset_fset_a,B: fset_fset_a,C: fset_a] :
      ( ( A = B )
     => ( ( ( fmember_fset_a @ C @ A )
         => ~ ( fmember_fset_a @ C @ B ) )
       => ~ ( ~ ( fmember_fset_a @ C @ A )
           => ( fmember_fset_a @ C @ B ) ) ) ) ).

% fequalityCE
thf(fact_120_fequalityCE,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a,C: produc21584292878202806fset_a] :
      ( ( A = B )
     => ( ( ( fmembe5403795861166450045fset_a @ C @ A )
         => ~ ( fmembe5403795861166450045fset_a @ C @ B ) )
       => ~ ( ~ ( fmembe5403795861166450045fset_a @ C @ A )
           => ( fmembe5403795861166450045fset_a @ C @ B ) ) ) ) ).

% fequalityCE
thf(fact_121_fequalityCE,axiom,
    ! [A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a,C: produc2669932201312201782od_c_a] :
      ( ( A = B )
     => ( ( ( fmembe4375279799106287869od_c_a @ C @ A )
         => ~ ( fmembe4375279799106287869od_c_a @ C @ B ) )
       => ~ ( ~ ( fmembe4375279799106287869od_c_a @ C @ A )
           => ( fmembe4375279799106287869od_c_a @ C @ B ) ) ) ) ).

% fequalityCE
thf(fact_122_eq__fmem__trans,axiom,
    ! [A2: a,B2: a,A: fset_a] :
      ( ( A2 = B2 )
     => ( ( fmember_a @ B2 @ A )
       => ( fmember_a @ A2 @ A ) ) ) ).

% eq_fmem_trans
thf(fact_123_eq__fmem__trans,axiom,
    ! [A2: fset_a,B2: fset_a,A: fset_fset_a] :
      ( ( A2 = B2 )
     => ( ( fmember_fset_a @ B2 @ A )
       => ( fmember_fset_a @ A2 @ A ) ) ) ).

% eq_fmem_trans
thf(fact_124_eq__fmem__trans,axiom,
    ! [A2: produc21584292878202806fset_a,B2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( A2 = B2 )
     => ( ( fmembe5403795861166450045fset_a @ B2 @ A )
       => ( fmembe5403795861166450045fset_a @ A2 @ A ) ) ) ).

% eq_fmem_trans
thf(fact_125_eq__fmem__trans,axiom,
    ! [A2: produc2669932201312201782od_c_a,B2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( A2 = B2 )
     => ( ( fmembe4375279799106287869od_c_a @ B2 @ A )
       => ( fmembe4375279799106287869od_c_a @ A2 @ A ) ) ) ).

% eq_fmem_trans
thf(fact_126_eqfset__imp__iff,axiom,
    ! [A: fset_a,B: fset_a,X2: a] :
      ( ( A = B )
     => ( ( fmember_a @ X2 @ A )
        = ( fmember_a @ X2 @ B ) ) ) ).

% eqfset_imp_iff
thf(fact_127_eqfset__imp__iff,axiom,
    ! [A: fset_fset_a,B: fset_fset_a,X2: fset_a] :
      ( ( A = B )
     => ( ( fmember_fset_a @ X2 @ A )
        = ( fmember_fset_a @ X2 @ B ) ) ) ).

% eqfset_imp_iff
thf(fact_128_eqfset__imp__iff,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a,X2: produc21584292878202806fset_a] :
      ( ( A = B )
     => ( ( fmembe5403795861166450045fset_a @ X2 @ A )
        = ( fmembe5403795861166450045fset_a @ X2 @ B ) ) ) ).

% eqfset_imp_iff
thf(fact_129_eqfset__imp__iff,axiom,
    ! [A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a,X2: produc2669932201312201782od_c_a] :
      ( ( A = B )
     => ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
        = ( fmembe4375279799106287869od_c_a @ X2 @ B ) ) ) ).

% eqfset_imp_iff
thf(fact_130_if__split__fmem1,axiom,
    ! [Q2: $o,X2: a,Y2: a,B2: fset_a] :
      ( ( fmember_a @ ( if_a @ Q2 @ X2 @ Y2 ) @ B2 )
      = ( ( Q2
         => ( fmember_a @ X2 @ B2 ) )
        & ( ~ Q2
         => ( fmember_a @ Y2 @ B2 ) ) ) ) ).

% if_split_fmem1
thf(fact_131_if__split__fmem1,axiom,
    ! [Q2: $o,X2: fset_a,Y2: fset_a,B2: fset_fset_a] :
      ( ( fmember_fset_a @ ( if_fset_a @ Q2 @ X2 @ Y2 ) @ B2 )
      = ( ( Q2
         => ( fmember_fset_a @ X2 @ B2 ) )
        & ( ~ Q2
         => ( fmember_fset_a @ Y2 @ B2 ) ) ) ) ).

% if_split_fmem1
thf(fact_132_if__split__fmem1,axiom,
    ! [Q2: $o,X2: produc21584292878202806fset_a,Y2: produc21584292878202806fset_a,B2: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ ( if_Pro8211175460565361788fset_a @ Q2 @ X2 @ Y2 ) @ B2 )
      = ( ( Q2
         => ( fmembe5403795861166450045fset_a @ X2 @ B2 ) )
        & ( ~ Q2
         => ( fmembe5403795861166450045fset_a @ Y2 @ B2 ) ) ) ) ).

% if_split_fmem1
thf(fact_133_if__split__fmem1,axiom,
    ! [Q2: $o,X2: produc2669932201312201782od_c_a,Y2: produc2669932201312201782od_c_a,B2: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ ( if_Pro5723405847749414652od_c_a @ Q2 @ X2 @ Y2 ) @ B2 )
      = ( ( Q2
         => ( fmembe4375279799106287869od_c_a @ X2 @ B2 ) )
        & ( ~ Q2
         => ( fmembe4375279799106287869od_c_a @ Y2 @ B2 ) ) ) ) ).

% if_split_fmem1
thf(fact_134_if__split__fmem2,axiom,
    ! [A2: a,Q2: $o,X2: fset_a,Y2: fset_a] :
      ( ( fmember_a @ A2 @ ( if_fset_a @ Q2 @ X2 @ Y2 ) )
      = ( ( Q2
         => ( fmember_a @ A2 @ X2 ) )
        & ( ~ Q2
         => ( fmember_a @ A2 @ Y2 ) ) ) ) ).

% if_split_fmem2
thf(fact_135_if__split__fmem2,axiom,
    ! [A2: fset_a,Q2: $o,X2: fset_fset_a,Y2: fset_fset_a] :
      ( ( fmember_fset_a @ A2 @ ( if_fset_fset_a @ Q2 @ X2 @ Y2 ) )
      = ( ( Q2
         => ( fmember_fset_a @ A2 @ X2 ) )
        & ( ~ Q2
         => ( fmember_fset_a @ A2 @ Y2 ) ) ) ) ).

% if_split_fmem2
thf(fact_136_if__split__fmem2,axiom,
    ! [A2: produc21584292878202806fset_a,Q2: $o,X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ A2 @ ( if_fse1100752710543153506fset_a @ Q2 @ X2 @ Y2 ) )
      = ( ( Q2
         => ( fmembe5403795861166450045fset_a @ A2 @ X2 ) )
        & ( ~ Q2
         => ( fmembe5403795861166450045fset_a @ A2 @ Y2 ) ) ) ) ).

% if_split_fmem2
thf(fact_137_if__split__fmem2,axiom,
    ! [A2: produc2669932201312201782od_c_a,Q2: $o,X2: fset_P8043821951902805788od_c_a,Y2: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ A2 @ ( if_fse3511714763770754786od_c_a @ Q2 @ X2 @ Y2 ) )
      = ( ( Q2
         => ( fmembe4375279799106287869od_c_a @ A2 @ X2 ) )
        & ( ~ Q2
         => ( fmembe4375279799106287869od_c_a @ A2 @ Y2 ) ) ) ) ).

% if_split_fmem2
thf(fact_138_eqfelem__imp__iff,axiom,
    ! [X2: a,Y2: a,A: fset_a] :
      ( ( X2 = Y2 )
     => ( ( fmember_a @ X2 @ A )
        = ( fmember_a @ Y2 @ A ) ) ) ).

% eqfelem_imp_iff
thf(fact_139_eqfelem__imp__iff,axiom,
    ! [X2: fset_a,Y2: fset_a,A: fset_fset_a] :
      ( ( X2 = Y2 )
     => ( ( fmember_fset_a @ X2 @ A )
        = ( fmember_fset_a @ Y2 @ A ) ) ) ).

% eqfelem_imp_iff
thf(fact_140_eqfelem__imp__iff,axiom,
    ! [X2: produc21584292878202806fset_a,Y2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( X2 = Y2 )
     => ( ( fmembe5403795861166450045fset_a @ X2 @ A )
        = ( fmembe5403795861166450045fset_a @ Y2 @ A ) ) ) ).

% eqfelem_imp_iff
thf(fact_141_eqfelem__imp__iff,axiom,
    ! [X2: produc2669932201312201782od_c_a,Y2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( X2 = Y2 )
     => ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
        = ( fmembe4375279799106287869od_c_a @ Y2 @ A ) ) ) ).

% eqfelem_imp_iff
thf(fact_142_fset__cong,axiom,
    ! [X2: fset_a,Y2: fset_a] :
      ( ( ( fset_a2 @ X2 )
        = ( fset_a2 @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% fset_cong
thf(fact_143_single__input_Oelims_I3_J,axiom,
    ! [X2: fsm_Pr7813689363856283786_c_a_a] :
      ( ~ ( single8385926175230902071_c_a_a @ X2 )
     => ~ ! [X3: produc1135898663509404882od_c_a] :
            ( ( member6699836568822684667od_c_a @ X3 @ ( transi5048145386435558395_c_a_a @ X2 ) )
           => ! [Xa: produc1135898663509404882od_c_a] :
                ( ( member6699836568822684667od_c_a @ Xa @ ( transi5048145386435558395_c_a_a @ X2 ) )
               => ( ( ( produc5475348670448042158od_c_a @ X3 )
                    = ( produc5475348670448042158od_c_a @ Xa ) )
                 => ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X3 ) )
                    = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Xa ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_144_single__input_Oelims_I3_J,axiom,
    ! [X2: fsm_Pr2398083450103406218et_a_b] :
      ( ~ ( single5089827247979212983et_a_b @ X2 )
     => ~ ! [X3: produc7245291065743820818fset_a] :
            ( ( member808877655204049083fset_a @ X3 @ ( transi3322184091305251195et_a_b @ X2 ) )
           => ! [Xa: produc7245291065743820818fset_a] :
                ( ( member808877655204049083fset_a @ Xa @ ( transi3322184091305251195et_a_b @ X2 ) )
               => ( ( ( produc8095675384246233966fset_a @ X3 )
                    = ( produc8095675384246233966fset_a @ Xa ) )
                 => ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X3 ) )
                    = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Xa ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_145_single__input_Oelims_I3_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ~ ( single_input_a_b_c @ X2 )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X2 ) )
           => ! [Xa: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa @ ( transitions_a_b_c @ X2 ) )
               => ( ( ( produc4225943634317630866od_c_a @ X3 )
                    = ( produc4225943634317630866od_c_a @ Xa ) )
                 => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_146_single__input_Oelims_I3_J,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b] :
      ( ~ ( single999786218259502647_a_a_b @ X2 )
     => ~ ! [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ X2 ) )
           => ! [Xa: produc3205353004677322104od_c_a] :
                ( ( member6508725278956418977od_c_a @ Xa @ ( transi2941759273464634107_a_a_b @ X2 ) )
               => ( ( ( produc8171714036265584980od_c_a @ X3 )
                    = ( produc8171714036265584980od_c_a @ Xa ) )
                 => ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_147_single__input_Oelims_I3_J,axiom,
    ! [X2: fsm_fset_a_b_c] :
      ( ~ ( single3763647048569287350_a_b_c @ X2 )
     => ~ ! [X3: produc21584292878202806fset_a] :
            ( ( member4899939873214261983fset_a @ X3 @ ( transi6427852111903025530_a_b_c @ X2 ) )
           => ! [Xa: produc21584292878202806fset_a] :
                ( ( member4899939873214261983fset_a @ Xa @ ( transi6427852111903025530_a_b_c @ X2 ) )
               => ( ( ( produc1739398754328533778fset_a @ X3 )
                    = ( produc1739398754328533778fset_a @ Xa ) )
                 => ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X3 ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Xa ) ) ) ) ) ) ) ).

% single_input.elims(3)
thf(fact_148_single__input_Oelims_I2_J,axiom,
    ! [X2: fsm_Pr7813689363856283786_c_a_a] :
      ( ( single8385926175230902071_c_a_a @ X2 )
     => ! [X4: produc1135898663509404882od_c_a] :
          ( ( member6699836568822684667od_c_a @ X4 @ ( transi5048145386435558395_c_a_a @ X2 ) )
         => ! [Xa2: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ Xa2 @ ( transi5048145386435558395_c_a_a @ X2 ) )
             => ( ( ( produc5475348670448042158od_c_a @ X4 )
                  = ( produc5475348670448042158od_c_a @ Xa2 ) )
               => ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X4 ) )
                  = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_149_single__input_Oelims_I2_J,axiom,
    ! [X2: fsm_Pr2398083450103406218et_a_b] :
      ( ( single5089827247979212983et_a_b @ X2 )
     => ! [X4: produc7245291065743820818fset_a] :
          ( ( member808877655204049083fset_a @ X4 @ ( transi3322184091305251195et_a_b @ X2 ) )
         => ! [Xa2: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ Xa2 @ ( transi3322184091305251195et_a_b @ X2 ) )
             => ( ( ( produc8095675384246233966fset_a @ X4 )
                  = ( produc8095675384246233966fset_a @ Xa2 ) )
               => ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X4 ) )
                  = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_150_single__input_Oelims_I2_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ( single_input_a_b_c @ X2 )
     => ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X2 ) )
         => ! [Xa2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X2 ) )
             => ( ( ( produc4225943634317630866od_c_a @ X4 )
                  = ( produc4225943634317630866od_c_a @ Xa2 ) )
               => ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                  = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_151_single__input_Oelims_I2_J,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( single999786218259502647_a_a_b @ X2 )
     => ! [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ X2 ) )
         => ! [Xa2: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Xa2 @ ( transi2941759273464634107_a_a_b @ X2 ) )
             => ( ( ( produc8171714036265584980od_c_a @ X4 )
                  = ( produc8171714036265584980od_c_a @ Xa2 ) )
               => ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                  = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_152_single__input_Oelims_I2_J,axiom,
    ! [X2: fsm_fset_a_b_c] :
      ( ( single3763647048569287350_a_b_c @ X2 )
     => ! [X4: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ X4 @ ( transi6427852111903025530_a_b_c @ X2 ) )
         => ! [Xa2: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ Xa2 @ ( transi6427852111903025530_a_b_c @ X2 ) )
             => ( ( ( produc1739398754328533778fset_a @ X4 )
                  = ( produc1739398754328533778fset_a @ Xa2 ) )
               => ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X4 ) )
                  = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Xa2 ) ) ) ) ) ) ) ).

% single_input.elims(2)
thf(fact_153_assms_I2_J,axiom,
    ! [Q: fset_a,T2: produc2669932201312201782od_c_a] :
      ( ( fmember_fset_a @ Q @ dones )
     => ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ m ) )
       => ( ( fmember_a @ ( produc4225943634317630866od_c_a @ T2 ) @ Q )
         => ? [T3: produc21584292878202806fset_a] :
              ( ( fmembe5403795861166450045fset_a @ T3 @ ts )
              & ( ( produc1739398754328533778fset_a @ T3 )
                = Q )
              & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T3 ) )
                = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) )
              & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) )
                = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) ) ) ) ).

% assms(2)
thf(fact_154__C1_Oprems_C_I5_J,axiom,
    ~ ( fmember_fset_a @ bot_bot_fset_a @ donesa ) ).

% "1.prems"(5)
thf(fact_155_mem__Collect__eq,axiom,
    ! [A2: a,P2: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_156_mem__Collect__eq,axiom,
    ! [A2: produc3205353004677322104od_c_a,P2: produc3205353004677322104od_c_a > $o] :
      ( ( member6508725278956418977od_c_a @ A2 @ ( collec7519840507084740067od_c_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_157_mem__Collect__eq,axiom,
    ! [A2: produc21584292878202806fset_a,P2: produc21584292878202806fset_a > $o] :
      ( ( member4899939873214261983fset_a @ A2 @ ( collec8021744632519760673fset_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
    ! [A2: produc2669932201312201782od_c_a,P2: produc2669932201312201782od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ A2 @ ( collec8405073525749625505od_c_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_159_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_160_Collect__mem__eq,axiom,
    ! [A: set_Pr2640428967771533528od_c_a] :
      ( ( collec7519840507084740067od_c_a
        @ ^ [X: produc3205353004677322104od_c_a] : ( member6508725278956418977od_c_a @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_161_Collect__mem__eq,axiom,
    ! [A: set_Pr4695589342295920918fset_a] :
      ( ( collec8021744632519760673fset_a
        @ ^ [X: produc21584292878202806fset_a] : ( member4899939873214261983fset_a @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_162_Collect__mem__eq,axiom,
    ! [A: set_Pr8357624050790843030od_c_a] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_163_Collect__cong,axiom,
    ! [P2: produc2669932201312201782od_c_a > $o,Q2: produc2669932201312201782od_c_a > $o] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec8405073525749625505od_c_a @ P2 )
        = ( collec8405073525749625505od_c_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_164__C1_Oprems_C_I2_J,axiom,
    ! [Q: fset_a,T2: produc2669932201312201782od_c_a] :
      ( ( fmember_fset_a @ Q @ donesa )
     => ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ m ) )
       => ( ( fmember_a @ ( produc4225943634317630866od_c_a @ T2 ) @ Q )
         => ? [T3: produc21584292878202806fset_a] :
              ( ( fmembe5403795861166450045fset_a @ T3 @ tsa )
              & ( ( produc1739398754328533778fset_a @ T3 )
                = Q )
              & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T3 ) )
                = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) )
              & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) )
                = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) ) ) ) ).

% "1.prems"(2)
thf(fact_165_observable_Oelims_I3_J,axiom,
    ! [X2: fsm_Pr7813689363856283786_c_a_a] :
      ( ~ ( observ5792423948907539502_c_a_a @ X2 )
     => ~ ! [X3: produc1135898663509404882od_c_a] :
            ( ( member6699836568822684667od_c_a @ X3 @ ( transi5048145386435558395_c_a_a @ X2 ) )
           => ! [Xa: produc1135898663509404882od_c_a] :
                ( ( member6699836568822684667od_c_a @ Xa @ ( transi5048145386435558395_c_a_a @ X2 ) )
               => ( ( ( ( produc5475348670448042158od_c_a @ X3 )
                      = ( produc5475348670448042158od_c_a @ Xa ) )
                    & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X3 ) )
                      = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Xa ) ) )
                    & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X3 ) ) )
                      = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Xa ) ) ) ) )
                 => ( ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X3 ) ) )
                    = ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Xa ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_166_observable_Oelims_I3_J,axiom,
    ! [X2: fsm_Pr2398083450103406218et_a_b] :
      ( ~ ( observ1610022369204726958et_a_b @ X2 )
     => ~ ! [X3: produc7245291065743820818fset_a] :
            ( ( member808877655204049083fset_a @ X3 @ ( transi3322184091305251195et_a_b @ X2 ) )
           => ! [Xa: produc7245291065743820818fset_a] :
                ( ( member808877655204049083fset_a @ Xa @ ( transi3322184091305251195et_a_b @ X2 ) )
               => ( ( ( ( produc8095675384246233966fset_a @ X3 )
                      = ( produc8095675384246233966fset_a @ Xa ) )
                    & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X3 ) )
                      = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Xa ) ) )
                    & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X3 ) ) )
                      = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Xa ) ) ) ) )
                 => ( ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X3 ) ) )
                    = ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Xa ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_167_observable_Oelims_I3_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ~ ( observable_a_b_c @ X2 )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X2 ) )
           => ! [Xa: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa @ ( transitions_a_b_c @ X2 ) )
               => ( ( ( ( produc4225943634317630866od_c_a @ X3 )
                      = ( produc4225943634317630866od_c_a @ Xa ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) )
                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                      = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) ) )
                 => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_168_observable_Oelims_I3_J,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b] :
      ( ~ ( observ2964502264774598958_a_a_b @ X2 )
     => ~ ! [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ X2 ) )
           => ! [Xa: produc3205353004677322104od_c_a] :
                ( ( member6508725278956418977od_c_a @ Xa @ ( transi2941759273464634107_a_a_b @ X2 ) )
               => ( ( ( ( produc8171714036265584980od_c_a @ X3 )
                      = ( produc8171714036265584980od_c_a @ Xa ) )
                    & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) )
                      = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) ) )
                 => ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
                    = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_169_observable_Oelims_I3_J,axiom,
    ! [X2: fsm_fset_a_b_c] :
      ( ~ ( observ4396094731789362477_a_b_c @ X2 )
     => ~ ! [X3: produc21584292878202806fset_a] :
            ( ( member4899939873214261983fset_a @ X3 @ ( transi6427852111903025530_a_b_c @ X2 ) )
           => ! [Xa: produc21584292878202806fset_a] :
                ( ( member4899939873214261983fset_a @ Xa @ ( transi6427852111903025530_a_b_c @ X2 ) )
               => ( ( ( ( produc1739398754328533778fset_a @ X3 )
                      = ( produc1739398754328533778fset_a @ Xa ) )
                    & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X3 ) )
                      = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Xa ) ) )
                    & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X3 ) ) )
                      = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Xa ) ) ) ) )
                 => ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X3 ) ) )
                    = ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Xa ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_170_observable_Oelims_I2_J,axiom,
    ! [X2: fsm_Pr7813689363856283786_c_a_a] :
      ( ( observ5792423948907539502_c_a_a @ X2 )
     => ! [X4: produc1135898663509404882od_c_a] :
          ( ( member6699836568822684667od_c_a @ X4 @ ( transi5048145386435558395_c_a_a @ X2 ) )
         => ! [Xa2: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ Xa2 @ ( transi5048145386435558395_c_a_a @ X2 ) )
             => ( ( ( ( produc5475348670448042158od_c_a @ X4 )
                    = ( produc5475348670448042158od_c_a @ Xa2 ) )
                  & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X4 ) )
                    = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Xa2 ) ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X4 ) ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Xa2 ) ) ) ) )
               => ( ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X4 ) ) )
                  = ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_171_observable_Oelims_I2_J,axiom,
    ! [X2: fsm_Pr2398083450103406218et_a_b] :
      ( ( observ1610022369204726958et_a_b @ X2 )
     => ! [X4: produc7245291065743820818fset_a] :
          ( ( member808877655204049083fset_a @ X4 @ ( transi3322184091305251195et_a_b @ X2 ) )
         => ! [Xa2: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ Xa2 @ ( transi3322184091305251195et_a_b @ X2 ) )
             => ( ( ( ( produc8095675384246233966fset_a @ X4 )
                    = ( produc8095675384246233966fset_a @ Xa2 ) )
                  & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X4 ) )
                    = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Xa2 ) ) )
                  & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X4 ) ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Xa2 ) ) ) ) )
               => ( ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X4 ) ) )
                  = ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_172_observable_Oelims_I2_J,axiom,
    ! [X2: fsm_a_b_c] :
      ( ( observable_a_b_c @ X2 )
     => ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ ( transitions_a_b_c @ X2 ) )
         => ! [Xa2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X2 ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X4 )
                    = ( produc4225943634317630866od_c_a @ Xa2 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X4 ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_173_observable_Oelims_I2_J,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( observ2964502264774598958_a_a_b @ X2 )
     => ! [X4: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X4 @ ( transi2941759273464634107_a_a_b @ X2 ) )
         => ! [Xa2: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Xa2 @ ( transi2941759273464634107_a_a_b @ X2 ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X4 )
                    = ( produc8171714036265584980od_c_a @ Xa2 ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) ) )
               => ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X4 ) ) )
                  = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_174_observable_Oelims_I2_J,axiom,
    ! [X2: fsm_fset_a_b_c] :
      ( ( observ4396094731789362477_a_b_c @ X2 )
     => ! [X4: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ X4 @ ( transi6427852111903025530_a_b_c @ X2 ) )
         => ! [Xa2: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ Xa2 @ ( transi6427852111903025530_a_b_c @ X2 ) )
             => ( ( ( ( produc1739398754328533778fset_a @ X4 )
                    = ( produc1739398754328533778fset_a @ Xa2 ) )
                  & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X4 ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Xa2 ) ) )
                  & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X4 ) ) )
                    = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Xa2 ) ) ) ) )
               => ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X4 ) ) )
                  = ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_175_observable_Oelims_I1_J,axiom,
    ! [X2: fsm_Pr7813689363856283786_c_a_a,Y2: $o] :
      ( ( ( observ5792423948907539502_c_a_a @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ X @ ( transi5048145386435558395_c_a_a @ X2 ) )
             => ! [Y: produc1135898663509404882od_c_a] :
                  ( ( member6699836568822684667od_c_a @ Y @ ( transi5048145386435558395_c_a_a @ X2 ) )
                 => ( ( ( ( produc5475348670448042158od_c_a @ X )
                        = ( produc5475348670448042158od_c_a @ Y ) )
                      & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X ) )
                        = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) )
                      & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X ) ) )
                        = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) ) )
                   => ( ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X ) ) )
                      = ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_176_observable_Oelims_I1_J,axiom,
    ! [X2: fsm_Pr2398083450103406218et_a_b,Y2: $o] :
      ( ( ( observ1610022369204726958et_a_b @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ X @ ( transi3322184091305251195et_a_b @ X2 ) )
             => ! [Y: produc7245291065743820818fset_a] :
                  ( ( member808877655204049083fset_a @ Y @ ( transi3322184091305251195et_a_b @ X2 ) )
                 => ( ( ( ( produc8095675384246233966fset_a @ X )
                        = ( produc8095675384246233966fset_a @ Y ) )
                      & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X ) )
                        = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Y ) ) )
                      & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X ) ) )
                        = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) ) )
                   => ( ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X ) ) )
                      = ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_177_observable_Oelims_I1_J,axiom,
    ! [X2: fsm_a_b_c,Y2: $o] :
      ( ( ( observable_a_b_c @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ X2 ) )
             => ! [Y: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y @ ( transitions_a_b_c @ X2 ) )
                 => ( ( ( ( produc4225943634317630866od_c_a @ X )
                        = ( produc4225943634317630866od_c_a @ Y ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) )
                      & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                        = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) ) )
                   => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                      = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_178_observable_Oelims_I1_J,axiom,
    ! [X2: fsm_Pr1886043760914421514_a_a_b,Y2: $o] :
      ( ( ( observ2964502264774598958_a_a_b @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X @ ( transi2941759273464634107_a_a_b @ X2 ) )
             => ! [Y: produc3205353004677322104od_c_a] :
                  ( ( member6508725278956418977od_c_a @ Y @ ( transi2941759273464634107_a_a_b @ X2 ) )
                 => ( ( ( ( produc8171714036265584980od_c_a @ X )
                        = ( produc8171714036265584980od_c_a @ Y ) )
                      & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X ) )
                        = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) ) )
                   => ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
                      = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_179_observable_Oelims_I1_J,axiom,
    ! [X2: fsm_fset_a_b_c,Y2: $o] :
      ( ( ( observ4396094731789362477_a_b_c @ X2 )
        = Y2 )
     => ( Y2
        = ( ! [X: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ X @ ( transi6427852111903025530_a_b_c @ X2 ) )
             => ! [Y: produc21584292878202806fset_a] :
                  ( ( member4899939873214261983fset_a @ Y @ ( transi6427852111903025530_a_b_c @ X2 ) )
                 => ( ( ( ( produc1739398754328533778fset_a @ X )
                        = ( produc1739398754328533778fset_a @ Y ) )
                      & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X ) )
                        = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Y ) ) )
                      & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X ) ) )
                        = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) ) )
                   => ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X ) ) )
                      = ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_180_observable_Osimps,axiom,
    ( observ5792423948907539502_c_a_a
    = ( ^ [M: fsm_Pr7813689363856283786_c_a_a] :
        ! [X: produc1135898663509404882od_c_a] :
          ( ( member6699836568822684667od_c_a @ X @ ( transi5048145386435558395_c_a_a @ M ) )
         => ! [Y: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ Y @ ( transi5048145386435558395_c_a_a @ M ) )
             => ( ( ( ( produc5475348670448042158od_c_a @ X )
                    = ( produc5475348670448042158od_c_a @ Y ) )
                  & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X ) )
                    = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X ) ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) ) )
               => ( ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X ) ) )
                  = ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Y ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_181_observable_Osimps,axiom,
    ( observ1610022369204726958et_a_b
    = ( ^ [M: fsm_Pr2398083450103406218et_a_b] :
        ! [X: produc7245291065743820818fset_a] :
          ( ( member808877655204049083fset_a @ X @ ( transi3322184091305251195et_a_b @ M ) )
         => ! [Y: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ Y @ ( transi3322184091305251195et_a_b @ M ) )
             => ( ( ( ( produc8095675384246233966fset_a @ X )
                    = ( produc8095675384246233966fset_a @ Y ) )
                  & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X ) )
                    = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Y ) ) )
                  & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X ) ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) ) )
               => ( ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X ) ) )
                  = ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Y ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_182_observable_Osimps,axiom,
    ( observable_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [X: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ M ) )
         => ! [Y: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y @ ( transitions_a_b_c @ M ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X )
                    = ( produc4225943634317630866od_c_a @ Y ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_183_observable_Osimps,axiom,
    ( observ2964502264774598958_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [X: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ X @ ( transi2941759273464634107_a_a_b @ M ) )
         => ! [Y: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ Y @ ( transi2941759273464634107_a_a_b @ M ) )
             => ( ( ( ( produc8171714036265584980od_c_a @ X )
                    = ( produc8171714036265584980od_c_a @ Y ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) ) )
               => ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
                  = ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_184_observable_Osimps,axiom,
    ( observ4396094731789362477_a_b_c
    = ( ^ [M: fsm_fset_a_b_c] :
        ! [X: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ X @ ( transi6427852111903025530_a_b_c @ M ) )
         => ! [Y: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ Y @ ( transi6427852111903025530_a_b_c @ M ) )
             => ( ( ( ( produc1739398754328533778fset_a @ X )
                    = ( produc1739398754328533778fset_a @ Y ) )
                  & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Y ) ) )
                  & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X ) ) )
                    = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) ) )
               => ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X ) ) )
                  = ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Y ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_185_path_Ocons,axiom,
    ! [T4: produc21584292878202806fset_a,M2: fsm_fset_a_b_c,Ts: list_P1058411382900698172fset_a] :
      ( ( member4899939873214261983fset_a @ T4 @ ( transi6427852111903025530_a_b_c @ M2 ) )
     => ( ( path_fset_a_b_c @ M2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) @ Ts )
       => ( path_fset_a_b_c @ M2 @ ( produc1739398754328533778fset_a @ T4 ) @ ( cons_P3377334492504755052fset_a @ T4 @ Ts ) ) ) ) ).

% path.cons
thf(fact_186_path_Ocons,axiom,
    ! [T4: produc2669932201312201782od_c_a,M2: fsm_a_b_c,Ts: list_P6327159017948738492od_c_a] :
      ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
     => ( ( path_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) @ Ts )
       => ( path_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T4 ) @ ( cons_P8659537020926305772od_c_a @ T4 @ Ts ) ) ) ) ).

% path.cons
thf(fact_187_path_Ocons,axiom,
    ! [T4: produc7245291065743820818fset_a,M2: fsm_Pr2398083450103406218et_a_b,Ts: list_P5281568020286996248fset_a] :
      ( ( member808877655204049083fset_a @ T4 @ ( transi3322184091305251195et_a_b @ M2 ) )
     => ( ( path_P379120284129538668et_a_b @ M2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) @ Ts )
       => ( path_P379120284129538668et_a_b @ M2 @ ( produc8095675384246233966fset_a @ T4 ) @ ( cons_P8607030675745668296fset_a @ T4 @ Ts ) ) ) ) ).

% path.cons
thf(fact_188_path_Ocons,axiom,
    ! [T4: produc3205353004677322104od_c_a,M2: fsm_Pr1886043760914421514_a_a_b,Ts: list_P1562850763667305982od_c_a] :
      ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
     => ( ( path_P1107610407223809260_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) @ Ts )
       => ( path_P1107610407223809260_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T4 ) @ ( cons_P5525787621418763054od_c_a @ T4 @ Ts ) ) ) ) ).

% path.cons
thf(fact_189_path_Ocons,axiom,
    ! [T4: produc1135898663509404882od_c_a,M2: fsm_Pr7813689363856283786_c_a_a,Ts: list_P1143992737879525208od_c_a] :
      ( ( member6699836568822684667od_c_a @ T4 @ ( transi5048145386435558395_c_a_a @ M2 ) )
     => ( ( path_P5541858805596636140_c_a_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) @ Ts )
       => ( path_P5541858805596636140_c_a_a @ M2 @ ( produc5475348670448042158od_c_a @ T4 ) @ ( cons_P3341831520629876872od_c_a @ T4 @ Ts ) ) ) ) ).

% path.cons
thf(fact_190_path__cons__elim,axiom,
    ! [M2: fsm_fset_a_b_c,Q: fset_a,T4: produc21584292878202806fset_a,Ts: list_P1058411382900698172fset_a] :
      ( ( path_fset_a_b_c @ M2 @ Q @ ( cons_P3377334492504755052fset_a @ T4 @ Ts ) )
     => ~ ( ( Q
            = ( produc1739398754328533778fset_a @ T4 ) )
         => ( ( member4899939873214261983fset_a @ T4 @ ( transi6427852111903025530_a_b_c @ M2 ) )
           => ~ ( path_fset_a_b_c @ M2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) @ Ts ) ) ) ) ).

% path_cons_elim
thf(fact_191_path__cons__elim,axiom,
    ! [M2: fsm_a_b_c,Q: a,T4: produc2669932201312201782od_c_a,Ts: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M2 @ Q @ ( cons_P8659537020926305772od_c_a @ T4 @ Ts ) )
     => ~ ( ( Q
            = ( produc4225943634317630866od_c_a @ T4 ) )
         => ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
           => ~ ( path_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) @ Ts ) ) ) ) ).

% path_cons_elim
thf(fact_192_path__cons__elim,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,Q: produc4838349196543436939fset_a,T4: produc7245291065743820818fset_a,Ts: list_P5281568020286996248fset_a] :
      ( ( path_P379120284129538668et_a_b @ M2 @ Q @ ( cons_P8607030675745668296fset_a @ T4 @ Ts ) )
     => ~ ( ( Q
            = ( produc8095675384246233966fset_a @ T4 ) )
         => ( ( member808877655204049083fset_a @ T4 @ ( transi3322184091305251195et_a_b @ M2 ) )
           => ~ ( path_P379120284129538668et_a_b @ M2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) @ Ts ) ) ) ) ).

% path_cons_elim
thf(fact_193_path__cons__elim,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,T4: produc3205353004677322104od_c_a,Ts: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ M2 @ Q @ ( cons_P5525787621418763054od_c_a @ T4 @ Ts ) )
     => ~ ( ( Q
            = ( produc8171714036265584980od_c_a @ T4 ) )
         => ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
           => ~ ( path_P1107610407223809260_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) @ Ts ) ) ) ) ).

% path_cons_elim
thf(fact_194_path__cons__elim,axiom,
    ! [M2: fsm_Pr7813689363856283786_c_a_a,Q: produc2133359284458116493od_c_a,T4: produc1135898663509404882od_c_a,Ts: list_P1143992737879525208od_c_a] :
      ( ( path_P5541858805596636140_c_a_a @ M2 @ Q @ ( cons_P3341831520629876872od_c_a @ T4 @ Ts ) )
     => ~ ( ( Q
            = ( produc5475348670448042158od_c_a @ T4 ) )
         => ( ( member6699836568822684667od_c_a @ T4 @ ( transi5048145386435558395_c_a_a @ M2 ) )
           => ~ ( path_P5541858805596636140_c_a_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) @ Ts ) ) ) ) ).

% path_cons_elim
thf(fact_195_observable__transition__unique,axiom,
    ! [M2: fsm_Pr7813689363856283786_c_a_a,T4: produc1135898663509404882od_c_a] :
      ( ( observ5792423948907539502_c_a_a @ M2 )
     => ( ( member6699836568822684667od_c_a @ T4 @ ( transi5048145386435558395_c_a_a @ M2 ) )
       => ? [X3: produc1135898663509404882od_c_a] :
            ( ( member6699836568822684667od_c_a @ X3 @ ( transi5048145386435558395_c_a_a @ M2 ) )
            & ( ( produc5475348670448042158od_c_a @ X3 )
              = ( produc5475348670448042158od_c_a @ T4 ) )
            & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X3 ) )
              = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) )
            & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X3 ) ) )
              = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) )
            & ! [Y5: produc1135898663509404882od_c_a] :
                ( ( ( member6699836568822684667od_c_a @ Y5 @ ( transi5048145386435558395_c_a_a @ M2 ) )
                  & ( ( produc5475348670448042158od_c_a @ Y5 )
                    = ( produc5475348670448042158od_c_a @ T4 ) )
                  & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ Y5 ) )
                    = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ Y5 ) ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_196_observable__transition__unique,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,T4: produc7245291065743820818fset_a] :
      ( ( observ1610022369204726958et_a_b @ M2 )
     => ( ( member808877655204049083fset_a @ T4 @ ( transi3322184091305251195et_a_b @ M2 ) )
       => ? [X3: produc7245291065743820818fset_a] :
            ( ( member808877655204049083fset_a @ X3 @ ( transi3322184091305251195et_a_b @ M2 ) )
            & ( ( produc8095675384246233966fset_a @ X3 )
              = ( produc8095675384246233966fset_a @ T4 ) )
            & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X3 ) )
              = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) )
            & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X3 ) ) )
              = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) )
            & ! [Y5: produc7245291065743820818fset_a] :
                ( ( ( member808877655204049083fset_a @ Y5 @ ( transi3322184091305251195et_a_b @ M2 ) )
                  & ( ( produc8095675384246233966fset_a @ Y5 )
                    = ( produc8095675384246233966fset_a @ T4 ) )
                  & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ Y5 ) )
                    = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) )
                  & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ Y5 ) ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_197_observable__transition__unique,axiom,
    ! [M2: fsm_a_b_c,T4: produc2669932201312201782od_c_a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
       => ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ M2 ) )
            & ( ( produc4225943634317630866od_c_a @ X3 )
              = ( produc4225943634317630866od_c_a @ T4 ) )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) )
            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) )
            & ! [Y5: produc2669932201312201782od_c_a] :
                ( ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ M2 ) )
                  & ( ( produc4225943634317630866od_c_a @ Y5 )
                    = ( produc4225943634317630866od_c_a @ T4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_198_observable__transition__unique,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T4: produc3205353004677322104od_c_a] :
      ( ( observ2964502264774598958_a_a_b @ M2 )
     => ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ? [X3: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X3 @ ( transi2941759273464634107_a_a_b @ M2 ) )
            & ( ( produc8171714036265584980od_c_a @ X3 )
              = ( produc8171714036265584980od_c_a @ T4 ) )
            & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) )
              = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X3 ) ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) )
            & ! [Y5: produc3205353004677322104od_c_a] :
                ( ( ( member6508725278956418977od_c_a @ Y5 @ ( transi2941759273464634107_a_a_b @ M2 ) )
                  & ( ( produc8171714036265584980od_c_a @ Y5 )
                    = ( produc8171714036265584980od_c_a @ T4 ) )
                  & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) )
                    = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ Y5 ) ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_199_observable__transition__unique,axiom,
    ! [M2: fsm_fset_a_b_c,T4: produc21584292878202806fset_a] :
      ( ( observ4396094731789362477_a_b_c @ M2 )
     => ( ( member4899939873214261983fset_a @ T4 @ ( transi6427852111903025530_a_b_c @ M2 ) )
       => ? [X3: produc21584292878202806fset_a] :
            ( ( member4899939873214261983fset_a @ X3 @ ( transi6427852111903025530_a_b_c @ M2 ) )
            & ( ( produc1739398754328533778fset_a @ X3 )
              = ( produc1739398754328533778fset_a @ T4 ) )
            & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X3 ) )
              = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
            & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X3 ) ) )
              = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) )
            & ! [Y5: produc21584292878202806fset_a] :
                ( ( ( member4899939873214261983fset_a @ Y5 @ ( transi6427852111903025530_a_b_c @ M2 ) )
                  & ( ( produc1739398754328533778fset_a @ Y5 )
                    = ( produc1739398754328533778fset_a @ T4 ) )
                  & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ Y5 ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
                  & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ Y5 ) ) )
                    = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_200_assms_I1_J,axiom,
    ! [T4: produc21584292878202806fset_a] :
      ( ( fmembe5403795861166450045fset_a @ T4 @ ts )
     => ( ( fmember_fset_a @ ( produc1739398754328533778fset_a @ T4 ) @ dones )
        & ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
         != bot_bot_fset_a )
        & ( ( fset_a2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) )
          = ( image_9171994905208062541_c_a_a
            @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
            @ ( collec8405073525749625505od_c_a
              @ ^ [T5: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ T5 @ ( transitions_a_b_c @ m ) )
                  & ( fmember_a @ ( produc4225943634317630866od_c_a @ T5 ) @ ( produc1739398754328533778fset_a @ T4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) )
                    = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) ) ) ) ) ) ) ).

% assms(1)
thf(fact_201__C1_Oprems_C_I6_J,axiom,
    fmember_fset_a @ qa @ donesa ).

% "1.prems"(6)
thf(fact_202__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062tP_ApP_O_Ap_H_A_061_AtP_A_D_ApP_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [TP: produc21584292878202806fset_a,PP: list_P1058411382900698172fset_a] :
        ( p
       != ( cons_P3377334492504755052fset_a @ TP @ PP ) ) ).

% \<open>\<And>thesis. (\<And>tP pP. p' = tP # pP \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_203__092_060open_062p_H_A_061_AtP_A_D_ApP_092_060close_062,axiom,
    ( p
    = ( cons_P3377334492504755052fset_a @ tP @ pP ) ) ).

% \<open>p' = tP # pP\<close>
thf(fact_204__C1_Oprems_C_I1_J,axiom,
    ! [T4: produc21584292878202806fset_a] :
      ( ( fmembe5403795861166450045fset_a @ T4 @ tsa )
     => ( ( fmember_fset_a @ ( produc1739398754328533778fset_a @ T4 ) @ donesa )
        & ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
         != bot_bot_fset_a )
        & ( ( fset_a2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) )
          = ( image_9171994905208062541_c_a_a
            @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
            @ ( collec8405073525749625505od_c_a
              @ ^ [T5: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ T5 @ ( transitions_a_b_c @ m ) )
                  & ( fmember_a @ ( produc4225943634317630866od_c_a @ T5 ) @ ( produc1739398754328533778fset_a @ T4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) )
                    = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) ) ) ) ) ) ) ).

% "1.prems"(1)
thf(fact_205_image__empty,axiom,
    ! [F: produc2669932201312201782od_c_a > a] :
      ( ( image_9171994905208062541_c_a_a @ F @ bot_bo267491471446728962od_c_a )
      = bot_bot_set_a ) ).

% image_empty
thf(fact_206_image__empty,axiom,
    ! [F: nat > nat] :
      ( ( image_nat_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_207_empty__is__image,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: set_Pr8357624050790843030od_c_a] :
      ( ( bot_bot_set_a
        = ( image_9171994905208062541_c_a_a @ F @ A ) )
      = ( A = bot_bo267491471446728962od_c_a ) ) ).

% empty_is_image
thf(fact_208_empty__is__image,axiom,
    ! [F: nat > nat,A: set_nat] :
      ( ( bot_bot_set_nat
        = ( image_nat_nat @ F @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_209_image__is__empty,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: set_Pr8357624050790843030od_c_a] :
      ( ( ( image_9171994905208062541_c_a_a @ F @ A )
        = bot_bot_set_a )
      = ( A = bot_bo267491471446728962od_c_a ) ) ).

% image_is_empty
thf(fact_210_image__is__empty,axiom,
    ! [F: nat > nat,A: set_nat] :
      ( ( ( image_nat_nat @ F @ A )
        = bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_211_pathlike__cons__elim,axiom,
    ! [Ts: fset_P8043821951902805788od_c_a,Q: a,T4: produc2669932201312201782od_c_a,P: list_P6327159017948738492od_c_a] :
      ( ( pathlike_a_b_c @ Ts @ Q @ ( cons_P8659537020926305772od_c_a @ T4 @ P ) )
     => ~ ( ( Q
            = ( produc4225943634317630866od_c_a @ T4 ) )
         => ( ( fmembe4375279799106287869od_c_a @ T4 @ Ts )
           => ~ ( pathlike_a_b_c @ Ts @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) @ P ) ) ) ) ).

% pathlike_cons_elim
thf(fact_212_pathlike__cons__elim,axiom,
    ! [Ts: fset_P1491459140212632184fset_a,Q: produc4838349196543436939fset_a,T4: produc7245291065743820818fset_a,P: list_P5281568020286996248fset_a] :
      ( ( pathli6897467406352051220et_a_b @ Ts @ Q @ ( cons_P8607030675745668296fset_a @ T4 @ P ) )
     => ~ ( ( Q
            = ( produc8095675384246233966fset_a @ T4 ) )
         => ( ( fmembe4210165678437738585fset_a @ T4 @ Ts )
           => ~ ( pathli6897467406352051220et_a_b @ Ts @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) @ P ) ) ) ) ).

% pathlike_cons_elim
thf(fact_213_pathlike__cons__elim,axiom,
    ! [Ts: fset_P9065897409413591390od_c_a,Q: product_prod_c_a,T4: produc3205353004677322104od_c_a,P: list_P1562850763667305982od_c_a] :
      ( ( pathli3518824893434361492_a_a_b @ Ts @ Q @ ( cons_P5525787621418763054od_c_a @ T4 @ P ) )
     => ~ ( ( Q
            = ( produc8171714036265584980od_c_a @ T4 ) )
         => ( ( fmembe4248414558865794111od_c_a @ T4 @ Ts )
           => ~ ( pathli3518824893434361492_a_a_b @ Ts @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) @ P ) ) ) ) ).

% pathlike_cons_elim
thf(fact_214_pathlike__cons__elim,axiom,
    ! [Ts: fset_P5322974059806139064od_c_a,Q: produc2133359284458116493od_c_a,T4: produc1135898663509404882od_c_a,P: list_P1143992737879525208od_c_a] :
      ( ( pathli3873806031239288212_c_a_a @ Ts @ Q @ ( cons_P3341831520629876872od_c_a @ T4 @ P ) )
     => ~ ( ( Q
            = ( produc5475348670448042158od_c_a @ T4 ) )
         => ( ( fmembe6414739546088136345od_c_a @ T4 @ Ts )
           => ~ ( pathli3873806031239288212_c_a_a @ Ts @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) @ P ) ) ) ) ).

% pathlike_cons_elim
thf(fact_215_pathlike__cons__elim,axiom,
    ! [Ts: fset_P2596122609397385116fset_a,Q: fset_a,T4: produc21584292878202806fset_a,P: list_P1058411382900698172fset_a] :
      ( ( pathlike_fset_a_b_c @ Ts @ Q @ ( cons_P3377334492504755052fset_a @ T4 @ P ) )
     => ~ ( ( Q
            = ( produc1739398754328533778fset_a @ T4 ) )
         => ( ( fmembe5403795861166450045fset_a @ T4 @ Ts )
           => ~ ( pathlike_fset_a_b_c @ Ts @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) @ P ) ) ) ) ).

% pathlike_cons_elim
thf(fact_216_pathlike_Ocons,axiom,
    ! [T4: produc2669932201312201782od_c_a,Ts: fset_P8043821951902805788od_c_a,P: list_P6327159017948738492od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ T4 @ Ts )
     => ( ( pathlike_a_b_c @ Ts @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) @ P )
       => ( pathlike_a_b_c @ Ts @ ( produc4225943634317630866od_c_a @ T4 ) @ ( cons_P8659537020926305772od_c_a @ T4 @ P ) ) ) ) ).

% pathlike.cons
thf(fact_217_pathlike_Ocons,axiom,
    ! [T4: produc7245291065743820818fset_a,Ts: fset_P1491459140212632184fset_a,P: list_P5281568020286996248fset_a] :
      ( ( fmembe4210165678437738585fset_a @ T4 @ Ts )
     => ( ( pathli6897467406352051220et_a_b @ Ts @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) @ P )
       => ( pathli6897467406352051220et_a_b @ Ts @ ( produc8095675384246233966fset_a @ T4 ) @ ( cons_P8607030675745668296fset_a @ T4 @ P ) ) ) ) ).

% pathlike.cons
thf(fact_218_pathlike_Ocons,axiom,
    ! [T4: produc3205353004677322104od_c_a,Ts: fset_P9065897409413591390od_c_a,P: list_P1562850763667305982od_c_a] :
      ( ( fmembe4248414558865794111od_c_a @ T4 @ Ts )
     => ( ( pathli3518824893434361492_a_a_b @ Ts @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) @ P )
       => ( pathli3518824893434361492_a_a_b @ Ts @ ( produc8171714036265584980od_c_a @ T4 ) @ ( cons_P5525787621418763054od_c_a @ T4 @ P ) ) ) ) ).

% pathlike.cons
thf(fact_219_pathlike_Ocons,axiom,
    ! [T4: produc1135898663509404882od_c_a,Ts: fset_P5322974059806139064od_c_a,P: list_P1143992737879525208od_c_a] :
      ( ( fmembe6414739546088136345od_c_a @ T4 @ Ts )
     => ( ( pathli3873806031239288212_c_a_a @ Ts @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) @ P )
       => ( pathli3873806031239288212_c_a_a @ Ts @ ( produc5475348670448042158od_c_a @ T4 ) @ ( cons_P3341831520629876872od_c_a @ T4 @ P ) ) ) ) ).

% pathlike.cons
thf(fact_220_pathlike_Ocons,axiom,
    ! [T4: produc21584292878202806fset_a,Ts: fset_P2596122609397385116fset_a,P: list_P1058411382900698172fset_a] :
      ( ( fmembe5403795861166450045fset_a @ T4 @ Ts )
     => ( ( pathlike_fset_a_b_c @ Ts @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) @ P )
       => ( pathlike_fset_a_b_c @ Ts @ ( produc1739398754328533778fset_a @ T4 ) @ ( cons_P3377334492504755052fset_a @ T4 @ P ) ) ) ) ).

% pathlike.cons
thf(fact_221_empty__iff,axiom,
    ! [C: produc2669932201312201782od_c_a] :
      ~ ( member5502040268208861279od_c_a @ C @ bot_bo267491471446728962od_c_a ) ).

% empty_iff
thf(fact_222_empty__iff,axiom,
    ! [C: a] :
      ~ ( member_a @ C @ bot_bot_set_a ) ).

% empty_iff
thf(fact_223_empty__iff,axiom,
    ! [C: produc3205353004677322104od_c_a] :
      ~ ( member6508725278956418977od_c_a @ C @ bot_bo5472487333715441476od_c_a ) ).

% empty_iff
thf(fact_224_empty__iff,axiom,
    ! [C: produc21584292878202806fset_a] :
      ~ ( member4899939873214261983fset_a @ C @ bot_bo6764261144281768834fset_a ) ).

% empty_iff
thf(fact_225_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_226_all__not__in__conv,axiom,
    ! [A: set_Pr8357624050790843030od_c_a] :
      ( ( ! [X: produc2669932201312201782od_c_a] :
            ~ ( member5502040268208861279od_c_a @ X @ A ) )
      = ( A = bot_bo267491471446728962od_c_a ) ) ).

% all_not_in_conv
thf(fact_227_all__not__in__conv,axiom,
    ! [A: set_a] :
      ( ( ! [X: a] :
            ~ ( member_a @ X @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_228_all__not__in__conv,axiom,
    ! [A: set_Pr2640428967771533528od_c_a] :
      ( ( ! [X: produc3205353004677322104od_c_a] :
            ~ ( member6508725278956418977od_c_a @ X @ A ) )
      = ( A = bot_bo5472487333715441476od_c_a ) ) ).

% all_not_in_conv
thf(fact_229_all__not__in__conv,axiom,
    ! [A: set_Pr4695589342295920918fset_a] :
      ( ( ! [X: produc21584292878202806fset_a] :
            ~ ( member4899939873214261983fset_a @ X @ A ) )
      = ( A = bot_bo6764261144281768834fset_a ) ) ).

% all_not_in_conv
thf(fact_230_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_231_Collect__empty__eq,axiom,
    ! [P2: produc2669932201312201782od_c_a > $o] :
      ( ( ( collec8405073525749625505od_c_a @ P2 )
        = bot_bo267491471446728962od_c_a )
      = ( ! [X: produc2669932201312201782od_c_a] :
            ~ ( P2 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_232_Collect__empty__eq,axiom,
    ! [P2: nat > $o] :
      ( ( ( collect_nat @ P2 )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P2 @ X ) ) ) ).

% Collect_empty_eq
thf(fact_233_empty__Collect__eq,axiom,
    ! [P2: produc2669932201312201782od_c_a > $o] :
      ( ( bot_bo267491471446728962od_c_a
        = ( collec8405073525749625505od_c_a @ P2 ) )
      = ( ! [X: produc2669932201312201782od_c_a] :
            ~ ( P2 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_234_empty__Collect__eq,axiom,
    ! [P2: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P2 ) )
      = ( ! [X: nat] :
            ~ ( P2 @ X ) ) ) ).

% empty_Collect_eq
thf(fact_235_image__eqI,axiom,
    ! [B2: a,F: a > a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_236_image__eqI,axiom,
    ! [B2: a,F: produc2669932201312201782od_c_a > a,X2: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member5502040268208861279od_c_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_9171994905208062541_c_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_237_image__eqI,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: a > produc2669932201312201782od_c_a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member5502040268208861279od_c_a @ B2 @ ( image_4184489845059386301od_c_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_238_image__eqI,axiom,
    ! [B2: produc21584292878202806fset_a,F: a > produc21584292878202806fset_a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member4899939873214261983fset_a @ B2 @ ( image_3174156142289807421fset_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_239_image__eqI,axiom,
    ! [B2: a,F: produc21584292878202806fset_a > a,X2: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member4899939873214261983fset_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_3466649715130532045et_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_240_image__eqI,axiom,
    ! [B2: produc3205353004677322104od_c_a,F: a > produc3205353004677322104od_c_a,X2: a,A: set_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member_a @ X2 @ A )
       => ( member6508725278956418977od_c_a @ B2 @ ( image_2663017512022304511od_c_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_241_image__eqI,axiom,
    ! [B2: a,F: produc3205353004677322104od_c_a > a,X2: produc3205353004677322104od_c_a,A: set_Pr2640428967771533528od_c_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member6508725278956418977od_c_a @ X2 @ A )
       => ( member_a @ B2 @ ( image_7535793878625517963_c_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_242_image__eqI,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,X2: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member5502040268208861279od_c_a @ X2 @ A )
       => ( member5502040268208861279od_c_a @ B2 @ ( image_2169818278710580869od_c_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_243_image__eqI,axiom,
    ! [B2: produc21584292878202806fset_a,F: produc2669932201312201782od_c_a > produc21584292878202806fset_a,X2: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member5502040268208861279od_c_a @ X2 @ A )
       => ( member4899939873214261983fset_a @ B2 @ ( image_3290014519244618501fset_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_244_image__eqI,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: produc21584292878202806fset_a > produc2669932201312201782od_c_a,X2: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( member4899939873214261983fset_a @ X2 @ A )
       => ( member5502040268208861279od_c_a @ B2 @ ( image_9201318158128649477od_c_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_245_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: a,F: a > a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_246_rev__image__eqI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,B2: a,F: produc2669932201312201782od_c_a > a] :
      ( ( member5502040268208861279od_c_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_9171994905208062541_c_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_247_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: produc2669932201312201782od_c_a,F: a > produc2669932201312201782od_c_a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member5502040268208861279od_c_a @ B2 @ ( image_4184489845059386301od_c_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_248_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: produc21584292878202806fset_a,F: a > produc21584292878202806fset_a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member4899939873214261983fset_a @ B2 @ ( image_3174156142289807421fset_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_249_rev__image__eqI,axiom,
    ! [X2: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a,B2: a,F: produc21584292878202806fset_a > a] :
      ( ( member4899939873214261983fset_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_3466649715130532045et_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_250_rev__image__eqI,axiom,
    ! [X2: a,A: set_a,B2: produc3205353004677322104od_c_a,F: a > produc3205353004677322104od_c_a] :
      ( ( member_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member6508725278956418977od_c_a @ B2 @ ( image_2663017512022304511od_c_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_251_rev__image__eqI,axiom,
    ! [X2: produc3205353004677322104od_c_a,A: set_Pr2640428967771533528od_c_a,B2: a,F: produc3205353004677322104od_c_a > a] :
      ( ( member6508725278956418977od_c_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member_a @ B2 @ ( image_7535793878625517963_c_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_252_rev__image__eqI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,B2: produc2669932201312201782od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member5502040268208861279od_c_a @ B2 @ ( image_2169818278710580869od_c_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_253_rev__image__eqI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,B2: produc21584292878202806fset_a,F: produc2669932201312201782od_c_a > produc21584292878202806fset_a] :
      ( ( member5502040268208861279od_c_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member4899939873214261983fset_a @ B2 @ ( image_3290014519244618501fset_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_254_rev__image__eqI,axiom,
    ! [X2: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a,B2: produc2669932201312201782od_c_a,F: produc21584292878202806fset_a > produc2669932201312201782od_c_a] :
      ( ( member4899939873214261983fset_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( member5502040268208861279od_c_a @ B2 @ ( image_9201318158128649477od_c_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_255_ball__imageD,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: set_Pr8357624050790843030od_c_a,P2: a > $o] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( image_9171994905208062541_c_a_a @ F @ A ) )
         => ( P2 @ X3 ) )
     => ! [X4: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X4 @ A )
         => ( P2 @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_256_image__cong,axiom,
    ! [M2: set_Pr8357624050790843030od_c_a,N: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > a,G: produc2669932201312201782od_c_a > a] :
      ( ( M2 = N )
     => ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_9171994905208062541_c_a_a @ F @ M2 )
          = ( image_9171994905208062541_c_a_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_257_bex__imageD,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: set_Pr8357624050790843030od_c_a,P2: a > $o] :
      ( ? [X4: a] :
          ( ( member_a @ X4 @ ( image_9171994905208062541_c_a_a @ F @ A ) )
          & ( P2 @ X4 ) )
     => ? [X3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X3 @ A )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_258_image__iff,axiom,
    ! [Z2: a,F: produc2669932201312201782od_c_a > a,A: set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ Z2 @ ( image_9171994905208062541_c_a_a @ F @ A ) )
      = ( ? [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ A )
            & ( Z2
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_259_imageI,axiom,
    ! [X2: a,A: set_a,F: a > a] :
      ( ( member_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_260_imageI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > a] :
      ( ( member5502040268208861279od_c_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_9171994905208062541_c_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_261_imageI,axiom,
    ! [X2: a,A: set_a,F: a > produc2669932201312201782od_c_a] :
      ( ( member_a @ X2 @ A )
     => ( member5502040268208861279od_c_a @ ( F @ X2 ) @ ( image_4184489845059386301od_c_a @ F @ A ) ) ) ).

% imageI
thf(fact_262_imageI,axiom,
    ! [X2: a,A: set_a,F: a > produc21584292878202806fset_a] :
      ( ( member_a @ X2 @ A )
     => ( member4899939873214261983fset_a @ ( F @ X2 ) @ ( image_3174156142289807421fset_a @ F @ A ) ) ) ).

% imageI
thf(fact_263_imageI,axiom,
    ! [X2: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a,F: produc21584292878202806fset_a > a] :
      ( ( member4899939873214261983fset_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_3466649715130532045et_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_264_imageI,axiom,
    ! [X2: a,A: set_a,F: a > produc3205353004677322104od_c_a] :
      ( ( member_a @ X2 @ A )
     => ( member6508725278956418977od_c_a @ ( F @ X2 ) @ ( image_2663017512022304511od_c_a @ F @ A ) ) ) ).

% imageI
thf(fact_265_imageI,axiom,
    ! [X2: produc3205353004677322104od_c_a,A: set_Pr2640428967771533528od_c_a,F: produc3205353004677322104od_c_a > a] :
      ( ( member6508725278956418977od_c_a @ X2 @ A )
     => ( member_a @ ( F @ X2 ) @ ( image_7535793878625517963_c_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_266_imageI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ X2 @ A )
     => ( member5502040268208861279od_c_a @ ( F @ X2 ) @ ( image_2169818278710580869od_c_a @ F @ A ) ) ) ).

% imageI
thf(fact_267_imageI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > produc21584292878202806fset_a] :
      ( ( member5502040268208861279od_c_a @ X2 @ A )
     => ( member4899939873214261983fset_a @ ( F @ X2 ) @ ( image_3290014519244618501fset_a @ F @ A ) ) ) ).

% imageI
thf(fact_268_imageI,axiom,
    ! [X2: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a,F: produc21584292878202806fset_a > produc2669932201312201782od_c_a] :
      ( ( member4899939873214261983fset_a @ X2 @ A )
     => ( member5502040268208861279od_c_a @ ( F @ X2 ) @ ( image_9201318158128649477od_c_a @ F @ A ) ) ) ).

% imageI
thf(fact_269_bot__set__def,axiom,
    ( bot_bo267491471446728962od_c_a
    = ( collec8405073525749625505od_c_a @ bot_bo4987066436690517083_c_a_o ) ) ).

% bot_set_def
thf(fact_270_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_271_ex__in__conv,axiom,
    ! [A: set_Pr8357624050790843030od_c_a] :
      ( ( ? [X: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X @ A ) )
      = ( A != bot_bo267491471446728962od_c_a ) ) ).

% ex_in_conv
thf(fact_272_ex__in__conv,axiom,
    ! [A: set_a] :
      ( ( ? [X: a] : ( member_a @ X @ A ) )
      = ( A != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_273_ex__in__conv,axiom,
    ! [A: set_Pr2640428967771533528od_c_a] :
      ( ( ? [X: produc3205353004677322104od_c_a] : ( member6508725278956418977od_c_a @ X @ A ) )
      = ( A != bot_bo5472487333715441476od_c_a ) ) ).

% ex_in_conv
thf(fact_274_ex__in__conv,axiom,
    ! [A: set_Pr4695589342295920918fset_a] :
      ( ( ? [X: produc21584292878202806fset_a] : ( member4899939873214261983fset_a @ X @ A ) )
      = ( A != bot_bo6764261144281768834fset_a ) ) ).

% ex_in_conv
thf(fact_275_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_276_equals0I,axiom,
    ! [A: set_Pr8357624050790843030od_c_a] :
      ( ! [Y3: produc2669932201312201782od_c_a] :
          ~ ( member5502040268208861279od_c_a @ Y3 @ A )
     => ( A = bot_bo267491471446728962od_c_a ) ) ).

% equals0I
thf(fact_277_equals0I,axiom,
    ! [A: set_a] :
      ( ! [Y3: a] :
          ~ ( member_a @ Y3 @ A )
     => ( A = bot_bot_set_a ) ) ).

% equals0I
thf(fact_278_equals0I,axiom,
    ! [A: set_Pr2640428967771533528od_c_a] :
      ( ! [Y3: produc3205353004677322104od_c_a] :
          ~ ( member6508725278956418977od_c_a @ Y3 @ A )
     => ( A = bot_bo5472487333715441476od_c_a ) ) ).

% equals0I
thf(fact_279_equals0I,axiom,
    ! [A: set_Pr4695589342295920918fset_a] :
      ( ! [Y3: produc21584292878202806fset_a] :
          ~ ( member4899939873214261983fset_a @ Y3 @ A )
     => ( A = bot_bo6764261144281768834fset_a ) ) ).

% equals0I
thf(fact_280_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_281_equals0D,axiom,
    ! [A: set_Pr8357624050790843030od_c_a,A2: produc2669932201312201782od_c_a] :
      ( ( A = bot_bo267491471446728962od_c_a )
     => ~ ( member5502040268208861279od_c_a @ A2 @ A ) ) ).

% equals0D
thf(fact_282_equals0D,axiom,
    ! [A: set_a,A2: a] :
      ( ( A = bot_bot_set_a )
     => ~ ( member_a @ A2 @ A ) ) ).

% equals0D
thf(fact_283_equals0D,axiom,
    ! [A: set_Pr2640428967771533528od_c_a,A2: produc3205353004677322104od_c_a] :
      ( ( A = bot_bo5472487333715441476od_c_a )
     => ~ ( member6508725278956418977od_c_a @ A2 @ A ) ) ).

% equals0D
thf(fact_284_equals0D,axiom,
    ! [A: set_Pr4695589342295920918fset_a,A2: produc21584292878202806fset_a] :
      ( ( A = bot_bo6764261144281768834fset_a )
     => ~ ( member4899939873214261983fset_a @ A2 @ A ) ) ).

% equals0D
thf(fact_285_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_286_emptyE,axiom,
    ! [A2: produc2669932201312201782od_c_a] :
      ~ ( member5502040268208861279od_c_a @ A2 @ bot_bo267491471446728962od_c_a ) ).

% emptyE
thf(fact_287_emptyE,axiom,
    ! [A2: a] :
      ~ ( member_a @ A2 @ bot_bot_set_a ) ).

% emptyE
thf(fact_288_emptyE,axiom,
    ! [A2: produc3205353004677322104od_c_a] :
      ~ ( member6508725278956418977od_c_a @ A2 @ bot_bo5472487333715441476od_c_a ) ).

% emptyE
thf(fact_289_emptyE,axiom,
    ! [A2: produc21584292878202806fset_a] :
      ~ ( member4899939873214261983fset_a @ A2 @ bot_bo6764261144281768834fset_a ) ).

% emptyE
thf(fact_290_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_291_Compr__image__eq,axiom,
    ! [F: a > a,A: set_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ ( image_a_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_a_a @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_292_Compr__image__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: set_Pr8357624050790843030od_c_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ ( image_9171994905208062541_c_a_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_9171994905208062541_c_a_a @ F
        @ ( collec8405073525749625505od_c_a
          @ ^ [X: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_293_Compr__image__eq,axiom,
    ! [F: a > produc2669932201312201782od_c_a,A: set_a,P2: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( image_4184489845059386301od_c_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_4184489845059386301od_c_a @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_294_Compr__image__eq,axiom,
    ! [F: produc21584292878202806fset_a > a,A: set_Pr4695589342295920918fset_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ ( image_3466649715130532045et_a_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_3466649715130532045et_a_a @ F
        @ ( collec8021744632519760673fset_a
          @ ^ [X: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_295_Compr__image__eq,axiom,
    ! [F: a > produc21584292878202806fset_a,A: set_a,P2: produc21584292878202806fset_a > $o] :
      ( ( collec8021744632519760673fset_a
        @ ^ [X: produc21584292878202806fset_a] :
            ( ( member4899939873214261983fset_a @ X @ ( image_3174156142289807421fset_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_3174156142289807421fset_a @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_296_Compr__image__eq,axiom,
    ! [F: produc3205353004677322104od_c_a > a,A: set_Pr2640428967771533528od_c_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a @ X @ ( image_7535793878625517963_c_a_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_7535793878625517963_c_a_a @ F
        @ ( collec7519840507084740067od_c_a
          @ ^ [X: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_297_Compr__image__eq,axiom,
    ! [F: a > produc3205353004677322104od_c_a,A: set_a,P2: produc3205353004677322104od_c_a > $o] :
      ( ( collec7519840507084740067od_c_a
        @ ^ [X: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X @ ( image_2663017512022304511od_c_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_2663017512022304511od_c_a @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_298_Compr__image__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,P2: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( image_2169818278710580869od_c_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_2169818278710580869od_c_a @ F
        @ ( collec8405073525749625505od_c_a
          @ ^ [X: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_299_Compr__image__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > produc21584292878202806fset_a,A: set_Pr8357624050790843030od_c_a,P2: produc21584292878202806fset_a > $o] :
      ( ( collec8021744632519760673fset_a
        @ ^ [X: produc21584292878202806fset_a] :
            ( ( member4899939873214261983fset_a @ X @ ( image_3290014519244618501fset_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_3290014519244618501fset_a @ F
        @ ( collec8405073525749625505od_c_a
          @ ^ [X: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_300_Compr__image__eq,axiom,
    ! [F: produc21584292878202806fset_a > produc2669932201312201782od_c_a,A: set_Pr4695589342295920918fset_a,P2: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( image_9201318158128649477od_c_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_9201318158128649477od_c_a @ F
        @ ( collec8021744632519760673fset_a
          @ ^ [X: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_301_image__image,axiom,
    ! [F: a > a,G: produc2669932201312201782od_c_a > a,A: set_Pr8357624050790843030od_c_a] :
      ( ( image_a_a @ F @ ( image_9171994905208062541_c_a_a @ G @ A ) )
      = ( image_9171994905208062541_c_a_a
        @ ^ [X: produc2669932201312201782od_c_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_302_image__image,axiom,
    ! [F: produc2669932201312201782od_c_a > a,G: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a] :
      ( ( image_9171994905208062541_c_a_a @ F @ ( image_2169818278710580869od_c_a @ G @ A ) )
      = ( image_9171994905208062541_c_a_a
        @ ^ [X: produc2669932201312201782od_c_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_303_imageE,axiom,
    ! [B2: a,F: a > a,A: set_a] :
      ( ( member_a @ B2 @ ( image_a_a @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_304_imageE,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: a > produc2669932201312201782od_c_a,A: set_a] :
      ( ( member5502040268208861279od_c_a @ B2 @ ( image_4184489845059386301od_c_a @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_305_imageE,axiom,
    ! [B2: a,F: produc2669932201312201782od_c_a > a,A: set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ B2 @ ( image_9171994905208062541_c_a_a @ F @ A ) )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member5502040268208861279od_c_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_306_imageE,axiom,
    ! [B2: a,F: produc21584292878202806fset_a > a,A: set_Pr4695589342295920918fset_a] :
      ( ( member_a @ B2 @ ( image_3466649715130532045et_a_a @ F @ A ) )
     => ~ ! [X3: produc21584292878202806fset_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member4899939873214261983fset_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_307_imageE,axiom,
    ! [B2: produc21584292878202806fset_a,F: a > produc21584292878202806fset_a,A: set_a] :
      ( ( member4899939873214261983fset_a @ B2 @ ( image_3174156142289807421fset_a @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_308_imageE,axiom,
    ! [B2: a,F: produc3205353004677322104od_c_a > a,A: set_Pr2640428967771533528od_c_a] :
      ( ( member_a @ B2 @ ( image_7535793878625517963_c_a_a @ F @ A ) )
     => ~ ! [X3: produc3205353004677322104od_c_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member6508725278956418977od_c_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_309_imageE,axiom,
    ! [B2: produc3205353004677322104od_c_a,F: a > produc3205353004677322104od_c_a,A: set_a] :
      ( ( member6508725278956418977od_c_a @ B2 @ ( image_2663017512022304511od_c_a @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_310_imageE,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ B2 @ ( image_2169818278710580869od_c_a @ F @ A ) )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member5502040268208861279od_c_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_311_imageE,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: produc21584292878202806fset_a > produc2669932201312201782od_c_a,A: set_Pr4695589342295920918fset_a] :
      ( ( member5502040268208861279od_c_a @ B2 @ ( image_9201318158128649477od_c_a @ F @ A ) )
     => ~ ! [X3: produc21584292878202806fset_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member4899939873214261983fset_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_312_imageE,axiom,
    ! [B2: produc21584292878202806fset_a,F: produc2669932201312201782od_c_a > produc21584292878202806fset_a,A: set_Pr8357624050790843030od_c_a] :
      ( ( member4899939873214261983fset_a @ B2 @ ( image_3290014519244618501fset_a @ F @ A ) )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member5502040268208861279od_c_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_313_Set_Oempty__def,axiom,
    ( bot_bo267491471446728962od_c_a
    = ( collec8405073525749625505od_c_a
      @ ^ [X: produc2669932201312201782od_c_a] : $false ) ) ).

% Set.empty_def
thf(fact_314_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_315_list_Oinject,axiom,
    ! [X21: produc21584292878202806fset_a,X22: list_P1058411382900698172fset_a,Y21: produc21584292878202806fset_a,Y22: list_P1058411382900698172fset_a] :
      ( ( ( cons_P3377334492504755052fset_a @ X21 @ X22 )
        = ( cons_P3377334492504755052fset_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_316_list_Oinject,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c,Y21: product_prod_b_c,Y22: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X21 @ X22 )
        = ( cons_P4529483553340347422od_b_c @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_317_qtrans__prop,axiom,
    ! [T4: produc21584292878202806fset_a] :
      ( ( fmembe5403795861166450045fset_a @ T4 @ qtrans )
      = ( ( fmember_fset_a @ ( produc1739398754328533778fset_a @ T4 )
          @ ( minus_5856361552470604977fset_a
            @ ( fimage5315711910825859755fset_a
              @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
              @ tsa )
            @ donesa ) )
        & ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
         != bot_bot_fset_a )
        & ( ( fset_a2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) )
          = ( image_9171994905208062541_c_a_a
            @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
            @ ( collec8405073525749625505od_c_a
              @ ^ [T5: produc2669932201312201782od_c_a] :
                  ( ( fmembe4375279799106287869od_c_a @ T5 @ ( ftransitions_a_b_c @ m ) )
                  & ( fmember_a @ ( produc4225943634317630866od_c_a @ T5 ) @ ( produc1739398754328533778fset_a @ T4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) )
                    = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) ) ) ) ) ) ) ).

% qtrans_prop
thf(fact_318_pathlike_Ocases,axiom,
    ! [A1: fset_P8043821951902805788od_c_a,A22: a,A32: list_P6327159017948738492od_c_a] :
      ( ( pathlike_a_b_c @ A1 @ A22 @ A32 )
     => ( ( A32 != nil_Pr1342775757158464060od_c_a )
       => ~ ! [T3: produc2669932201312201782od_c_a] :
              ( ( A22
                = ( produc4225943634317630866od_c_a @ T3 ) )
             => ! [P3: list_P6327159017948738492od_c_a] :
                  ( ( A32
                    = ( cons_P8659537020926305772od_c_a @ T3 @ P3 ) )
                 => ( ( fmembe4375279799106287869od_c_a @ T3 @ A1 )
                   => ~ ( pathlike_a_b_c @ A1 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) @ P3 ) ) ) ) ) ) ).

% pathlike.cases
thf(fact_319_pathlike_Ocases,axiom,
    ! [A1: fset_P1491459140212632184fset_a,A22: produc4838349196543436939fset_a,A32: list_P5281568020286996248fset_a] :
      ( ( pathli6897467406352051220et_a_b @ A1 @ A22 @ A32 )
     => ( ( A32 != nil_Pr4003949106000869656fset_a )
       => ~ ! [T3: produc7245291065743820818fset_a] :
              ( ( A22
                = ( produc8095675384246233966fset_a @ T3 ) )
             => ! [P3: list_P5281568020286996248fset_a] :
                  ( ( A32
                    = ( cons_P8607030675745668296fset_a @ T3 @ P3 ) )
                 => ( ( fmembe4210165678437738585fset_a @ T3 @ A1 )
                   => ~ ( pathli6897467406352051220et_a_b @ A1 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T3 ) ) ) @ P3 ) ) ) ) ) ) ).

% pathlike.cases
thf(fact_320_pathlike_Ocases,axiom,
    ! [A1: fset_P9065897409413591390od_c_a,A22: product_prod_c_a,A32: list_P1562850763667305982od_c_a] :
      ( ( pathli3518824893434361492_a_a_b @ A1 @ A22 @ A32 )
     => ( ( A32 != nil_Pr7913020553331505022od_c_a )
       => ~ ! [T3: produc3205353004677322104od_c_a] :
              ( ( A22
                = ( produc8171714036265584980od_c_a @ T3 ) )
             => ! [P3: list_P1562850763667305982od_c_a] :
                  ( ( A32
                    = ( cons_P5525787621418763054od_c_a @ T3 @ P3 ) )
                 => ( ( fmembe4248414558865794111od_c_a @ T3 @ A1 )
                   => ~ ( pathli3518824893434361492_a_a_b @ A1 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T3 ) ) ) @ P3 ) ) ) ) ) ) ).

% pathlike.cases
thf(fact_321_pathlike_Ocases,axiom,
    ! [A1: fset_P5322974059806139064od_c_a,A22: produc2133359284458116493od_c_a,A32: list_P1143992737879525208od_c_a] :
      ( ( pathli3873806031239288212_c_a_a @ A1 @ A22 @ A32 )
     => ( ( A32 != nil_Pr260931953701973208od_c_a )
       => ~ ! [T3: produc1135898663509404882od_c_a] :
              ( ( A22
                = ( produc5475348670448042158od_c_a @ T3 ) )
             => ! [P3: list_P1143992737879525208od_c_a] :
                  ( ( A32
                    = ( cons_P3341831520629876872od_c_a @ T3 @ P3 ) )
                 => ( ( fmembe6414739546088136345od_c_a @ T3 @ A1 )
                   => ~ ( pathli3873806031239288212_c_a_a @ A1 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T3 ) ) ) @ P3 ) ) ) ) ) ) ).

% pathlike.cases
thf(fact_322_pathlike_Ocases,axiom,
    ! [A1: fset_P2596122609397385116fset_a,A22: fset_a,A32: list_P1058411382900698172fset_a] :
      ( ( pathlike_fset_a_b_c @ A1 @ A22 @ A32 )
     => ( ( A32 != nil_Pr4260649278601056188fset_a )
       => ~ ! [T3: produc21584292878202806fset_a] :
              ( ( A22
                = ( produc1739398754328533778fset_a @ T3 ) )
             => ! [P3: list_P1058411382900698172fset_a] :
                  ( ( A32
                    = ( cons_P3377334492504755052fset_a @ T3 @ P3 ) )
                 => ( ( fmembe5403795861166450045fset_a @ T3 @ A1 )
                   => ~ ( pathlike_fset_a_b_c @ A1 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) ) @ P3 ) ) ) ) ) ) ).

% pathlike.cases
thf(fact_323_pathlike_Osimps,axiom,
    ( pathlike_a_b_c
    = ( ^ [A12: fset_P8043821951902805788od_c_a,A23: a,A33: list_P6327159017948738492od_c_a] :
          ( ? [Ts2: fset_P8043821951902805788od_c_a,Q3: a] :
              ( ( A12 = Ts2 )
              & ( A23 = Q3 )
              & ( A33 = nil_Pr1342775757158464060od_c_a ) )
          | ? [T: produc2669932201312201782od_c_a,Ts2: fset_P8043821951902805788od_c_a,P4: list_P6327159017948738492od_c_a] :
              ( ( A12 = Ts2 )
              & ( A23
                = ( produc4225943634317630866od_c_a @ T ) )
              & ( A33
                = ( cons_P8659537020926305772od_c_a @ T @ P4 ) )
              & ( fmembe4375279799106287869od_c_a @ T @ Ts2 )
              & ( pathlike_a_b_c @ Ts2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ P4 ) ) ) ) ) ).

% pathlike.simps
thf(fact_324_pathlike_Osimps,axiom,
    ( pathli6897467406352051220et_a_b
    = ( ^ [A12: fset_P1491459140212632184fset_a,A23: produc4838349196543436939fset_a,A33: list_P5281568020286996248fset_a] :
          ( ? [Ts2: fset_P1491459140212632184fset_a,Q3: produc4838349196543436939fset_a] :
              ( ( A12 = Ts2 )
              & ( A23 = Q3 )
              & ( A33 = nil_Pr4003949106000869656fset_a ) )
          | ? [T: produc7245291065743820818fset_a,Ts2: fset_P1491459140212632184fset_a,P4: list_P5281568020286996248fset_a] :
              ( ( A12 = Ts2 )
              & ( A23
                = ( produc8095675384246233966fset_a @ T ) )
              & ( A33
                = ( cons_P8607030675745668296fset_a @ T @ P4 ) )
              & ( fmembe4210165678437738585fset_a @ T @ Ts2 )
              & ( pathli6897467406352051220et_a_b @ Ts2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) @ P4 ) ) ) ) ) ).

% pathlike.simps
thf(fact_325_pathlike_Osimps,axiom,
    ( pathli3518824893434361492_a_a_b
    = ( ^ [A12: fset_P9065897409413591390od_c_a,A23: product_prod_c_a,A33: list_P1562850763667305982od_c_a] :
          ( ? [Ts2: fset_P9065897409413591390od_c_a,Q3: product_prod_c_a] :
              ( ( A12 = Ts2 )
              & ( A23 = Q3 )
              & ( A33 = nil_Pr7913020553331505022od_c_a ) )
          | ? [T: produc3205353004677322104od_c_a,Ts2: fset_P9065897409413591390od_c_a,P4: list_P1562850763667305982od_c_a] :
              ( ( A12 = Ts2 )
              & ( A23
                = ( produc8171714036265584980od_c_a @ T ) )
              & ( A33
                = ( cons_P5525787621418763054od_c_a @ T @ P4 ) )
              & ( fmembe4248414558865794111od_c_a @ T @ Ts2 )
              & ( pathli3518824893434361492_a_a_b @ Ts2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ P4 ) ) ) ) ) ).

% pathlike.simps
thf(fact_326_pathlike_Osimps,axiom,
    ( pathli3873806031239288212_c_a_a
    = ( ^ [A12: fset_P5322974059806139064od_c_a,A23: produc2133359284458116493od_c_a,A33: list_P1143992737879525208od_c_a] :
          ( ? [Ts2: fset_P5322974059806139064od_c_a,Q3: produc2133359284458116493od_c_a] :
              ( ( A12 = Ts2 )
              & ( A23 = Q3 )
              & ( A33 = nil_Pr260931953701973208od_c_a ) )
          | ? [T: produc1135898663509404882od_c_a,Ts2: fset_P5322974059806139064od_c_a,P4: list_P1143992737879525208od_c_a] :
              ( ( A12 = Ts2 )
              & ( A23
                = ( produc5475348670448042158od_c_a @ T ) )
              & ( A33
                = ( cons_P3341831520629876872od_c_a @ T @ P4 ) )
              & ( fmembe6414739546088136345od_c_a @ T @ Ts2 )
              & ( pathli3873806031239288212_c_a_a @ Ts2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) ) @ P4 ) ) ) ) ) ).

% pathlike.simps
thf(fact_327_pathlike_Osimps,axiom,
    ( pathlike_fset_a_b_c
    = ( ^ [A12: fset_P2596122609397385116fset_a,A23: fset_a,A33: list_P1058411382900698172fset_a] :
          ( ? [Ts2: fset_P2596122609397385116fset_a,Q3: fset_a] :
              ( ( A12 = Ts2 )
              & ( A23 = Q3 )
              & ( A33 = nil_Pr4260649278601056188fset_a ) )
          | ? [T: produc21584292878202806fset_a,Ts2: fset_P2596122609397385116fset_a,P4: list_P1058411382900698172fset_a] :
              ( ( A12 = Ts2 )
              & ( A23
                = ( produc1739398754328533778fset_a @ T ) )
              & ( A33
                = ( cons_P3377334492504755052fset_a @ T @ P4 ) )
              & ( fmembe5403795861166450045fset_a @ T @ Ts2 )
              & ( pathlike_fset_a_b_c @ Ts2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) @ P4 ) ) ) ) ) ).

% pathlike.simps
thf(fact_328_observable__transition__target__language__eq,axiom,
    ! [M2: fsm_Pr7813689363856283786_c_a_a,T1: produc1135898663509404882od_c_a,T22: produc1135898663509404882od_c_a] :
      ( ( ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc5475348670448042158od_c_a @ T1 ) )
        = ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc5475348670448042158od_c_a @ T22 ) ) )
     => ( ( member6699836568822684667od_c_a @ T1 @ ( transi5048145386435558395_c_a_a @ M2 ) )
       => ( ( member6699836568822684667od_c_a @ T22 @ ( transi5048145386435558395_c_a_a @ M2 ) )
         => ( ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T1 ) )
              = ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T22 ) ) )
           => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T1 ) ) )
                = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T22 ) ) ) )
             => ( ( observ5792423948907539502_c_a_a @ M2 )
               => ( ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T1 ) ) ) )
                  = ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_329_observable__transition__target__language__eq,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,T1: produc7245291065743820818fset_a,T22: produc7245291065743820818fset_a] :
      ( ( ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8095675384246233966fset_a @ T1 ) )
        = ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8095675384246233966fset_a @ T22 ) ) )
     => ( ( member808877655204049083fset_a @ T1 @ ( transi3322184091305251195et_a_b @ M2 ) )
       => ( ( member808877655204049083fset_a @ T22 @ ( transi3322184091305251195et_a_b @ M2 ) )
         => ( ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T1 ) )
              = ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T22 ) ) )
           => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T1 ) ) )
                = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T22 ) ) ) )
             => ( ( observ1610022369204726958et_a_b @ M2 )
               => ( ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T1 ) ) ) )
                  = ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_330_observable__transition__target__language__eq,axiom,
    ! [M2: fsm_a_b_c,T1: produc2669932201312201782od_c_a,T22: produc2669932201312201782od_c_a] :
      ( ( ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T1 ) )
        = ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T22 ) ) )
     => ( ( member5502040268208861279od_c_a @ T1 @ ( transitions_a_b_c @ M2 ) )
       => ( ( member5502040268208861279od_c_a @ T22 @ ( transitions_a_b_c @ M2 ) )
         => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) )
           => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) )
                = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) )
             => ( ( observable_a_b_c @ M2 )
               => ( ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) ) )
                  = ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_331_observable__transition__target__language__eq,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T1: produc3205353004677322104od_c_a,T22: produc3205353004677322104od_c_a] :
      ( ( ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T1 ) )
        = ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T22 ) ) )
     => ( ( member6508725278956418977od_c_a @ T1 @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ( ( member6508725278956418977od_c_a @ T22 @ ( transi2941759273464634107_a_a_b @ M2 ) )
         => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T1 ) )
              = ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T22 ) ) )
           => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T1 ) ) )
                = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T22 ) ) ) )
             => ( ( observ2964502264774598958_a_a_b @ M2 )
               => ( ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T1 ) ) ) )
                  = ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_332_observable__transition__target__language__eq,axiom,
    ! [M2: fsm_fset_a_b_c,T1: produc21584292878202806fset_a,T22: produc21584292878202806fset_a] :
      ( ( ( lS_fset_a_b_c @ M2 @ ( produc1739398754328533778fset_a @ T1 ) )
        = ( lS_fset_a_b_c @ M2 @ ( produc1739398754328533778fset_a @ T22 ) ) )
     => ( ( member4899939873214261983fset_a @ T1 @ ( transi6427852111903025530_a_b_c @ M2 ) )
       => ( ( member4899939873214261983fset_a @ T22 @ ( transi6427852111903025530_a_b_c @ M2 ) )
         => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T1 ) )
              = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T22 ) ) )
           => ( ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T1 ) ) )
                = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T22 ) ) ) )
             => ( ( observ4396094731789362477_a_b_c @ M2 )
               => ( ( lS_fset_a_b_c @ M2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T1 ) ) ) )
                  = ( lS_fset_a_b_c @ M2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_333__092_060open_062pathlike_A_Its_A_124_092_060union_062_124_Aqtrans_J_Aq_Ap_H_092_060close_062,axiom,
    pathlike_fset_a_b_c @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) @ qc @ p ).

% \<open>pathlike (ts |\<union>| qtrans) q p'\<close>
thf(fact_334__092_060open_062pathlike_A_Its_A_124_092_060union_062_124_Aqtrans_J_A_It__target_AtP_J_ApP_092_060close_062,axiom,
    pathlike_fset_a_b_c @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ tP ) ) ) @ pP ).

% \<open>pathlike (ts |\<union>| qtrans) (t_target tP) pP\<close>
thf(fact_335__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_H_H_ApM_O_A_092_060lbrakk_062q_H_H_A_124_092_060in_062_124_At__target_AtP_059_Apath_AM_Aq_H_H_ApM_059_Ap__io_ApM_A_061_AioP_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Q4: a] :
        ( ( fmember_a @ Q4 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ tP ) ) ) )
       => ! [PM: list_P6327159017948738492od_c_a] :
            ( ( path_a_b_c @ m @ Q4 @ PM )
           => ( ( map_Pr776894537227705078od_b_c
                @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                @ PM )
             != ioPa ) ) ) ).

% \<open>\<And>thesis. (\<And>q'' pM. \<lbrakk>q'' |\<in>| t_target tP; path M q'' pM; p_io pM = ioP\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_336_old_Oprod_Oinject,axiom,
    ! [A2: b,B2: c,A5: b,B3: c] :
      ( ( ( product_Pair_b_c @ A2 @ B2 )
        = ( product_Pair_b_c @ A5 @ B3 ) )
      = ( ( A2 = A5 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_337_old_Oprod_Oinject,axiom,
    ! [A2: fset_a,B2: produc1137077444975408883fset_a,A5: fset_a,B3: produc1137077444975408883fset_a] :
      ( ( ( produc3008349866153638182fset_a @ A2 @ B2 )
        = ( produc3008349866153638182fset_a @ A5 @ B3 ) )
      = ( ( A2 = A5 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_338_old_Oprod_Oinject,axiom,
    ! [A2: b,B2: produc4838349196543436939fset_a,A5: b,B3: produc4838349196543436939fset_a] :
      ( ( ( produc4496741132823077603fset_a @ A2 @ B2 )
        = ( produc4496741132823077603fset_a @ A5 @ B3 ) )
      = ( ( A2 = A5 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_339_old_Oprod_Oinject,axiom,
    ! [A2: c,B2: fset_a,A5: c,B3: fset_a] :
      ( ( ( produc1535188672328598523fset_a @ A2 @ B2 )
        = ( produc1535188672328598523fset_a @ A5 @ B3 ) )
      = ( ( A2 = A5 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_340_prod_Oinject,axiom,
    ! [X1: b,X23: c,Y1: b,Y23: c] :
      ( ( ( product_Pair_b_c @ X1 @ X23 )
        = ( product_Pair_b_c @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_341_prod_Oinject,axiom,
    ! [X1: fset_a,X23: produc1137077444975408883fset_a,Y1: fset_a,Y23: produc1137077444975408883fset_a] :
      ( ( ( produc3008349866153638182fset_a @ X1 @ X23 )
        = ( produc3008349866153638182fset_a @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_342_prod_Oinject,axiom,
    ! [X1: b,X23: produc4838349196543436939fset_a,Y1: b,Y23: produc4838349196543436939fset_a] :
      ( ( ( produc4496741132823077603fset_a @ X1 @ X23 )
        = ( produc4496741132823077603fset_a @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_343_prod_Oinject,axiom,
    ! [X1: c,X23: fset_a,Y1: c,Y23: fset_a] :
      ( ( ( produc1535188672328598523fset_a @ X1 @ X23 )
        = ( produc1535188672328598523fset_a @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_344_fminus__idemp,axiom,
    ! [A: fset_fset_a,B: fset_fset_a] :
      ( ( minus_5856361552470604977fset_a @ ( minus_5856361552470604977fset_a @ A @ B ) @ B )
      = ( minus_5856361552470604977fset_a @ A @ B ) ) ).

% fminus_idemp
thf(fact_345__092_060open_062p__io_ApM_A_061_AioP_092_060close_062,axiom,
    ( ( map_Pr776894537227705078od_b_c
      @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
      @ pM )
    = ioPa ) ).

% \<open>p_io pM = ioP\<close>
thf(fact_346_list_Omap__disc__iff,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,A2: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr776894537227705078od_b_c @ F @ A2 )
        = nil_Product_prod_b_c )
      = ( A2 = nil_Pr1342775757158464060od_c_a ) ) ).

% list.map_disc_iff
thf(fact_347_list_Omap__disc__iff,axiom,
    ! [F: produc21584292878202806fset_a > product_prod_b_c,A2: list_P1058411382900698172fset_a] :
      ( ( ( map_Pr7697975548294988406od_b_c @ F @ A2 )
        = nil_Product_prod_b_c )
      = ( A2 = nil_Pr4260649278601056188fset_a ) ) ).

% list.map_disc_iff
thf(fact_348_Nil__is__map__conv,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a] :
      ( ( nil_Product_prod_b_c
        = ( map_Pr776894537227705078od_b_c @ F @ Xs ) )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% Nil_is_map_conv
thf(fact_349_Nil__is__map__conv,axiom,
    ! [F: produc21584292878202806fset_a > product_prod_b_c,Xs: list_P1058411382900698172fset_a] :
      ( ( nil_Product_prod_b_c
        = ( map_Pr7697975548294988406od_b_c @ F @ Xs ) )
      = ( Xs = nil_Pr4260649278601056188fset_a ) ) ).

% Nil_is_map_conv
thf(fact_350_map__is__Nil__conv,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr776894537227705078od_b_c @ F @ Xs )
        = nil_Product_prod_b_c )
      = ( Xs = nil_Pr1342775757158464060od_c_a ) ) ).

% map_is_Nil_conv
thf(fact_351_map__is__Nil__conv,axiom,
    ! [F: produc21584292878202806fset_a > product_prod_b_c,Xs: list_P1058411382900698172fset_a] :
      ( ( ( map_Pr7697975548294988406od_b_c @ F @ Xs )
        = nil_Product_prod_b_c )
      = ( Xs = nil_Pr4260649278601056188fset_a ) ) ).

% map_is_Nil_conv
thf(fact_352__092_060open_062tP_A_124_092_060in_062_124_Ats_A_124_092_060union_062_124_Aqtrans_092_060close_062,axiom,
    fmembe5403795861166450045fset_a @ tP @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) ).

% \<open>tP |\<in>| ts |\<union>| qtrans\<close>
thf(fact_353_fimage__eqI,axiom,
    ! [B2: a,F: a > a,X2: a,A: fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmember_a @ X2 @ A )
       => ( fmember_a @ B2 @ ( fimage_a_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_354_fimage__eqI,axiom,
    ! [B2: fset_a,F: a > fset_a,X2: a,A: fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmember_a @ X2 @ A )
       => ( fmember_fset_a @ B2 @ ( fimage_a_fset_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_355_fimage__eqI,axiom,
    ! [B2: a,F: fset_a > a,X2: fset_a,A: fset_fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmember_fset_a @ X2 @ A )
       => ( fmember_a @ B2 @ ( fimage_fset_a_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_356_fimage__eqI,axiom,
    ! [B2: fset_a,F: fset_a > fset_a,X2: fset_a,A: fset_fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmember_fset_a @ X2 @ A )
       => ( fmember_fset_a @ B2 @ ( fimage_fset_a_fset_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_357_fimage__eqI,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: a > produc2669932201312201782od_c_a,X2: a,A: fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmember_a @ X2 @ A )
       => ( fmembe4375279799106287869od_c_a @ B2 @ ( fimage4212853655448708405od_c_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_358_fimage__eqI,axiom,
    ! [B2: a,F: produc2669932201312201782od_c_a > a,X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
       => ( fmember_a @ B2 @ ( fimage9200358715597384645_c_a_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_359_fimage__eqI,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: fset_a > produc2669932201312201782od_c_a,X2: fset_a,A: fset_fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmember_fset_a @ X2 @ A )
       => ( fmembe4375279799106287869od_c_a @ B2 @ ( fimage3543014039228378575od_c_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_360_fimage__eqI,axiom,
    ! [B2: fset_a,F: produc2669932201312201782od_c_a > fset_a,X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
       => ( fmember_fset_a @ B2 @ ( fimage2540242761017857067fset_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_361_fimage__eqI,axiom,
    ! [B2: produc21584292878202806fset_a,F: a > produc21584292878202806fset_a,X2: a,A: fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmember_a @ X2 @ A )
       => ( fmembe5403795861166450045fset_a @ B2 @ ( fimage5428608421149153717fset_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_362_fimage__eqI,axiom,
    ! [B2: a,F: produc21584292878202806fset_a > a,X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( B2
        = ( F @ X2 ) )
     => ( ( fmembe5403795861166450045fset_a @ X2 @ A )
       => ( fmember_a @ B2 @ ( fimage5721101993989878341et_a_a @ F @ A ) ) ) ) ).

% fimage_eqI
thf(fact_363_funion__iff,axiom,
    ! [C: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ C @ ( sup_su7793179031141480176od_c_a @ A @ B ) )
      = ( ( fmembe4375279799106287869od_c_a @ C @ A )
        | ( fmembe4375279799106287869od_c_a @ C @ B ) ) ) ).

% funion_iff
thf(fact_364_funion__iff,axiom,
    ! [C: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ C @ ( sup_su6818703939759823728fset_a @ A @ B ) )
      = ( ( fmembe5403795861166450045fset_a @ C @ A )
        | ( fmembe5403795861166450045fset_a @ C @ B ) ) ) ).

% funion_iff
thf(fact_365_funion__iff,axiom,
    ! [C: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fmember_fset_a @ C @ ( sup_sup_fset_fset_a @ A @ B ) )
      = ( ( fmember_fset_a @ C @ A )
        | ( fmember_fset_a @ C @ B ) ) ) ).

% funion_iff
thf(fact_366_funion__iff,axiom,
    ! [C: a,A: fset_a,B: fset_a] :
      ( ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) )
      = ( ( fmember_a @ C @ A )
        | ( fmember_a @ C @ B ) ) ) ).

% funion_iff
thf(fact_367_funionCI,axiom,
    ! [C: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( ~ ( fmembe4375279799106287869od_c_a @ C @ B )
       => ( fmembe4375279799106287869od_c_a @ C @ A ) )
     => ( fmembe4375279799106287869od_c_a @ C @ ( sup_su7793179031141480176od_c_a @ A @ B ) ) ) ).

% funionCI
thf(fact_368_funionCI,axiom,
    ! [C: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( ~ ( fmembe5403795861166450045fset_a @ C @ B )
       => ( fmembe5403795861166450045fset_a @ C @ A ) )
     => ( fmembe5403795861166450045fset_a @ C @ ( sup_su6818703939759823728fset_a @ A @ B ) ) ) ).

% funionCI
thf(fact_369_funionCI,axiom,
    ! [C: fset_a,B: fset_fset_a,A: fset_fset_a] :
      ( ( ~ ( fmember_fset_a @ C @ B )
       => ( fmember_fset_a @ C @ A ) )
     => ( fmember_fset_a @ C @ ( sup_sup_fset_fset_a @ A @ B ) ) ) ).

% funionCI
thf(fact_370_funionCI,axiom,
    ! [C: a,B: fset_a,A: fset_a] :
      ( ( ~ ( fmember_a @ C @ B )
       => ( fmember_a @ C @ A ) )
     => ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) ) ) ).

% funionCI
thf(fact_371_fminus__iff,axiom,
    ! [C: a,A: fset_a,B: fset_a] :
      ( ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) )
      = ( ( fmember_a @ C @ A )
        & ~ ( fmember_a @ C @ B ) ) ) ).

% fminus_iff
thf(fact_372_fminus__iff,axiom,
    ! [C: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ C @ ( minus_7225097421573603715fset_a @ A @ B ) )
      = ( ( fmembe5403795861166450045fset_a @ C @ A )
        & ~ ( fmembe5403795861166450045fset_a @ C @ B ) ) ) ).

% fminus_iff
thf(fact_373_fminus__iff,axiom,
    ! [C: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ C @ ( minus_8021415692904097795od_c_a @ A @ B ) )
      = ( ( fmembe4375279799106287869od_c_a @ C @ A )
        & ~ ( fmembe4375279799106287869od_c_a @ C @ B ) ) ) ).

% fminus_iff
thf(fact_374_fminus__iff,axiom,
    ! [C: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fmember_fset_a @ C @ ( minus_5856361552470604977fset_a @ A @ B ) )
      = ( ( fmember_fset_a @ C @ A )
        & ~ ( fmember_fset_a @ C @ B ) ) ) ).

% fminus_iff
thf(fact_375_fminusI,axiom,
    ! [C: a,A: fset_a,B: fset_a] :
      ( ( fmember_a @ C @ A )
     => ( ~ ( fmember_a @ C @ B )
       => ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) ) ) ) ).

% fminusI
thf(fact_376_fminusI,axiom,
    ! [C: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ C @ A )
     => ( ~ ( fmembe5403795861166450045fset_a @ C @ B )
       => ( fmembe5403795861166450045fset_a @ C @ ( minus_7225097421573603715fset_a @ A @ B ) ) ) ) ).

% fminusI
thf(fact_377_fminusI,axiom,
    ! [C: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ C @ A )
     => ( ~ ( fmembe4375279799106287869od_c_a @ C @ B )
       => ( fmembe4375279799106287869od_c_a @ C @ ( minus_8021415692904097795od_c_a @ A @ B ) ) ) ) ).

% fminusI
thf(fact_378_fminusI,axiom,
    ! [C: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fmember_fset_a @ C @ A )
     => ( ~ ( fmember_fset_a @ C @ B )
       => ( fmember_fset_a @ C @ ( minus_5856361552470604977fset_a @ A @ B ) ) ) ) ).

% fminusI
thf(fact_379_fimage__fempty,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( fimage7236410608496998447od_b_c @ F @ bot_bo1116342619594069896od_c_a )
      = bot_bo192962769195302586od_b_c ) ).

% fimage_fempty
thf(fact_380_fimage__fempty,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a] :
      ( ( fimage3947511181509736139fset_a @ F @ bot_bo192962769195302586od_b_c )
      = bot_bo2316247384970961928fset_a ) ).

% fimage_fempty
thf(fact_381_fimage__fempty,axiom,
    ! [F: produc2669932201312201782od_c_a > a] :
      ( ( fimage9200358715597384645_c_a_a @ F @ bot_bo1116342619594069896od_c_a )
      = bot_bot_fset_a ) ).

% fimage_fempty
thf(fact_382_fimage__fempty,axiom,
    ! [F: produc21584292878202806fset_a > fset_a] :
      ( ( fimage5315711910825859755fset_a @ F @ bot_bo2316247384970961928fset_a )
      = bot_bot_fset_fset_a ) ).

% fimage_fempty
thf(fact_383_fimage__fempty,axiom,
    ! [F: a > a] :
      ( ( fimage_a_a @ F @ bot_bot_fset_a )
      = bot_bot_fset_a ) ).

% fimage_fempty
thf(fact_384_fimage__fempty,axiom,
    ! [F: a > fset_a] :
      ( ( fimage_a_fset_a @ F @ bot_bot_fset_a )
      = bot_bot_fset_fset_a ) ).

% fimage_fempty
thf(fact_385_fimage__fempty,axiom,
    ! [F: fset_a > fset_P2596122609397385116fset_a] :
      ( ( fimage8078386234238232501fset_a @ F @ bot_bot_fset_fset_a )
      = bot_bo6991086108236063086fset_a ) ).

% fimage_fempty
thf(fact_386_fimage__fempty,axiom,
    ! [F: fset_a > a] :
      ( ( fimage_fset_a_a @ F @ bot_bot_fset_fset_a )
      = bot_bot_fset_a ) ).

% fimage_fempty
thf(fact_387_fimage__fempty,axiom,
    ! [F: fset_a > fset_a] :
      ( ( fimage_fset_a_fset_a @ F @ bot_bot_fset_fset_a )
      = bot_bot_fset_fset_a ) ).

% fimage_fempty
thf(fact_388_fempty__is__fimage,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,A: fset_P8043821951902805788od_c_a] :
      ( ( bot_bo192962769195302586od_b_c
        = ( fimage7236410608496998447od_b_c @ F @ A ) )
      = ( A = bot_bo1116342619594069896od_c_a ) ) ).

% fempty_is_fimage
thf(fact_389_fempty__is__fimage,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a,A: fset_P8649754740804009550od_b_c] :
      ( ( bot_bo2316247384970961928fset_a
        = ( fimage3947511181509736139fset_a @ F @ A ) )
      = ( A = bot_bo192962769195302586od_b_c ) ) ).

% fempty_is_fimage
thf(fact_390_fempty__is__fimage,axiom,
    ! [F: fset_a > fset_P2596122609397385116fset_a,A: fset_fset_a] :
      ( ( bot_bo6991086108236063086fset_a
        = ( fimage8078386234238232501fset_a @ F @ A ) )
      = ( A = bot_bot_fset_fset_a ) ) ).

% fempty_is_fimage
thf(fact_391_fempty__is__fimage,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: fset_P8043821951902805788od_c_a] :
      ( ( bot_bot_fset_a
        = ( fimage9200358715597384645_c_a_a @ F @ A ) )
      = ( A = bot_bo1116342619594069896od_c_a ) ) ).

% fempty_is_fimage
thf(fact_392_fempty__is__fimage,axiom,
    ! [F: a > a,A: fset_a] :
      ( ( bot_bot_fset_a
        = ( fimage_a_a @ F @ A ) )
      = ( A = bot_bot_fset_a ) ) ).

% fempty_is_fimage
thf(fact_393_fempty__is__fimage,axiom,
    ! [F: fset_a > a,A: fset_fset_a] :
      ( ( bot_bot_fset_a
        = ( fimage_fset_a_a @ F @ A ) )
      = ( A = bot_bot_fset_fset_a ) ) ).

% fempty_is_fimage
thf(fact_394_fempty__is__fimage,axiom,
    ! [F: produc21584292878202806fset_a > fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( bot_bot_fset_fset_a
        = ( fimage5315711910825859755fset_a @ F @ A ) )
      = ( A = bot_bo2316247384970961928fset_a ) ) ).

% fempty_is_fimage
thf(fact_395_fempty__is__fimage,axiom,
    ! [F: a > fset_a,A: fset_a] :
      ( ( bot_bot_fset_fset_a
        = ( fimage_a_fset_a @ F @ A ) )
      = ( A = bot_bot_fset_a ) ) ).

% fempty_is_fimage
thf(fact_396_fempty__is__fimage,axiom,
    ! [F: fset_a > fset_a,A: fset_fset_a] :
      ( ( bot_bot_fset_fset_a
        = ( fimage_fset_a_fset_a @ F @ A ) )
      = ( A = bot_bot_fset_fset_a ) ) ).

% fempty_is_fimage
thf(fact_397_fimage__is__fempty,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,A: fset_P8043821951902805788od_c_a] :
      ( ( ( fimage7236410608496998447od_b_c @ F @ A )
        = bot_bo192962769195302586od_b_c )
      = ( A = bot_bo1116342619594069896od_c_a ) ) ).

% fimage_is_fempty
thf(fact_398_fimage__is__fempty,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a,A: fset_P8649754740804009550od_b_c] :
      ( ( ( fimage3947511181509736139fset_a @ F @ A )
        = bot_bo2316247384970961928fset_a )
      = ( A = bot_bo192962769195302586od_b_c ) ) ).

% fimage_is_fempty
thf(fact_399_fimage__is__fempty,axiom,
    ! [F: fset_a > fset_P2596122609397385116fset_a,A: fset_fset_a] :
      ( ( ( fimage8078386234238232501fset_a @ F @ A )
        = bot_bo6991086108236063086fset_a )
      = ( A = bot_bot_fset_fset_a ) ) ).

% fimage_is_fempty
thf(fact_400_fimage__is__fempty,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: fset_P8043821951902805788od_c_a] :
      ( ( ( fimage9200358715597384645_c_a_a @ F @ A )
        = bot_bot_fset_a )
      = ( A = bot_bo1116342619594069896od_c_a ) ) ).

% fimage_is_fempty
thf(fact_401_fimage__is__fempty,axiom,
    ! [F: a > a,A: fset_a] :
      ( ( ( fimage_a_a @ F @ A )
        = bot_bot_fset_a )
      = ( A = bot_bot_fset_a ) ) ).

% fimage_is_fempty
thf(fact_402_fimage__is__fempty,axiom,
    ! [F: fset_a > a,A: fset_fset_a] :
      ( ( ( fimage_fset_a_a @ F @ A )
        = bot_bot_fset_a )
      = ( A = bot_bot_fset_fset_a ) ) ).

% fimage_is_fempty
thf(fact_403_fimage__is__fempty,axiom,
    ! [F: produc21584292878202806fset_a > fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( ( fimage5315711910825859755fset_a @ F @ A )
        = bot_bot_fset_fset_a )
      = ( A = bot_bo2316247384970961928fset_a ) ) ).

% fimage_is_fempty
thf(fact_404_fimage__is__fempty,axiom,
    ! [F: a > fset_a,A: fset_a] :
      ( ( ( fimage_a_fset_a @ F @ A )
        = bot_bot_fset_fset_a )
      = ( A = bot_bot_fset_a ) ) ).

% fimage_is_fempty
thf(fact_405_fimage__is__fempty,axiom,
    ! [F: fset_a > fset_a,A: fset_fset_a] :
      ( ( ( fimage_fset_a_fset_a @ F @ A )
        = bot_bot_fset_fset_a )
      = ( A = bot_bot_fset_fset_a ) ) ).

% fimage_is_fempty
thf(fact_406_funion__fempty,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( ( sup_su6818703939759823728fset_a @ A @ B )
        = bot_bo2316247384970961928fset_a )
      = ( ( A = bot_bo2316247384970961928fset_a )
        & ( B = bot_bo2316247384970961928fset_a ) ) ) ).

% funion_fempty
thf(fact_407_funion__fempty,axiom,
    ! [A: fset_a,B: fset_a] :
      ( ( ( sup_sup_fset_a @ A @ B )
        = bot_bot_fset_a )
      = ( ( A = bot_bot_fset_a )
        & ( B = bot_bot_fset_a ) ) ) ).

% funion_fempty
thf(fact_408_funion__fempty,axiom,
    ! [A: fset_fset_a,B: fset_fset_a] :
      ( ( ( sup_sup_fset_fset_a @ A @ B )
        = bot_bot_fset_fset_a )
      = ( ( A = bot_bot_fset_fset_a )
        & ( B = bot_bot_fset_fset_a ) ) ) ).

% funion_fempty
thf(fact_409_fempty__fminus,axiom,
    ! [A: fset_a] :
      ( ( minus_minus_fset_a @ bot_bot_fset_a @ A )
      = bot_bot_fset_a ) ).

% fempty_fminus
thf(fact_410_fempty__fminus,axiom,
    ! [A: fset_fset_a] :
      ( ( minus_5856361552470604977fset_a @ bot_bot_fset_fset_a @ A )
      = bot_bot_fset_fset_a ) ).

% fempty_fminus
thf(fact_411_fminus__cancel,axiom,
    ! [A: fset_a] :
      ( ( minus_minus_fset_a @ A @ A )
      = bot_bot_fset_a ) ).

% fminus_cancel
thf(fact_412_fminus__cancel,axiom,
    ! [A: fset_fset_a] :
      ( ( minus_5856361552470604977fset_a @ A @ A )
      = bot_bot_fset_fset_a ) ).

% fminus_cancel
thf(fact_413_fminus__fempty,axiom,
    ! [A: fset_a] :
      ( ( minus_minus_fset_a @ A @ bot_bot_fset_a )
      = A ) ).

% fminus_fempty
thf(fact_414_fminus__fempty,axiom,
    ! [A: fset_fset_a] :
      ( ( minus_5856361552470604977fset_a @ A @ bot_bot_fset_fset_a )
      = A ) ).

% fminus_fempty
thf(fact_415_funion__fminus__cancel2,axiom,
    ! [B: fset_P2596122609397385116fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ ( minus_7225097421573603715fset_a @ B @ A ) @ A )
      = ( sup_su6818703939759823728fset_a @ B @ A ) ) ).

% funion_fminus_cancel2
thf(fact_416_funion__fminus__cancel2,axiom,
    ! [B: fset_a,A: fset_a] :
      ( ( sup_sup_fset_a @ ( minus_minus_fset_a @ B @ A ) @ A )
      = ( sup_sup_fset_a @ B @ A ) ) ).

% funion_fminus_cancel2
thf(fact_417_funion__fminus__cancel2,axiom,
    ! [B: fset_fset_a,A: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ ( minus_5856361552470604977fset_a @ B @ A ) @ A )
      = ( sup_sup_fset_fset_a @ B @ A ) ) ).

% funion_fminus_cancel2
thf(fact_418_funion__fminus__cancel,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ A @ ( minus_7225097421573603715fset_a @ B @ A ) )
      = ( sup_su6818703939759823728fset_a @ A @ B ) ) ).

% funion_fminus_cancel
thf(fact_419_funion__fminus__cancel,axiom,
    ! [A: fset_a,B: fset_a] :
      ( ( sup_sup_fset_a @ A @ ( minus_minus_fset_a @ B @ A ) )
      = ( sup_sup_fset_a @ A @ B ) ) ).

% funion_fminus_cancel
thf(fact_420_funion__fminus__cancel,axiom,
    ! [A: fset_fset_a,B: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ A @ ( minus_5856361552470604977fset_a @ B @ A ) )
      = ( sup_sup_fset_fset_a @ A @ B ) ) ).

% funion_fminus_cancel
thf(fact_421_prod_Ocollapse,axiom,
    ! [Prod: product_prod_b_c] :
      ( ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_422_prod_Ocollapse,axiom,
    ! [Prod: produc4838349196543436939fset_a] :
      ( ( produc1535188672328598523fset_a @ ( product_fst_c_fset_a @ Prod ) @ ( product_snd_c_fset_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_423_prod_Ocollapse,axiom,
    ! [Prod: product_prod_c_a] :
      ( ( product_Pair_c_a @ ( product_fst_c_a @ Prod ) @ ( product_snd_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_424_prod_Ocollapse,axiom,
    ! [Prod: produc1137077444975408883fset_a] :
      ( ( produc4496741132823077603fset_a @ ( produc5616988955220391631fset_a @ Prod ) @ ( produc8704244169652099857fset_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_425_prod_Ocollapse,axiom,
    ! [Prod: produc2133359284458116493od_c_a] :
      ( ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ Prod ) @ ( produc5221446324160585515od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_426_prod_Ocollapse,axiom,
    ! [Prod: produc2669932201312201782od_c_a] :
      ( ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ Prod ) @ ( produc8088130389382527188od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_427_prod_Ocollapse,axiom,
    ! [Prod: produc3205353004677322104od_c_a] :
      ( ( produc7911353075612250984od_c_a @ ( produc8171714036265584980od_c_a @ Prod ) @ ( produc4643634399596806294od_c_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_428_prod_Ocollapse,axiom,
    ! [Prod: produc21584292878202806fset_a] :
      ( ( produc3008349866153638182fset_a @ ( produc1739398754328533778fset_a @ Prod ) @ ( produc8188425890649629780fset_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_429_fset_Oset__map,axiom,
    ! [F: produc21584292878202806fset_a > fset_a,V: fset_P2596122609397385116fset_a] :
      ( ( fset_fset_a2 @ ( fimage5315711910825859755fset_a @ F @ V ) )
      = ( image_523240206663320883fset_a @ F @ ( fset_P7830090094348214821fset_a @ V ) ) ) ).

% fset.set_map
thf(fact_430_fset_Oset__map,axiom,
    ! [F: fset_a > fset_P2596122609397385116fset_a,V: fset_fset_a] :
      ( ( fset_f1459617199513185931fset_a @ ( fimage8078386234238232501fset_a @ F @ V ) )
      = ( image_6386003088472927805fset_a @ F @ ( fset_fset_a2 @ V ) ) ) ).

% fset.set_map
thf(fact_431_fset_Oset__map,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,V: fset_P8043821951902805788od_c_a] :
      ( ( fset_P8805262608305245655od_b_c @ ( fimage7236410608496998447od_b_c @ F @ V ) )
      = ( image_4981958329637652151od_b_c @ F @ ( fset_P7369495421554515365od_c_a @ V ) ) ) ).

% fset.set_map
thf(fact_432_fset_Oset__map,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a,V: fset_P8649754740804009550od_b_c] :
      ( ( fset_P7830090094348214821fset_a @ ( fimage3947511181509736139fset_a @ F @ V ) )
      = ( image_2255128035744431443fset_a @ F @ ( fset_P8805262608305245655od_b_c @ V ) ) ) ).

% fset.set_map
thf(fact_433_fset_Oset__map,axiom,
    ! [F: produc2669932201312201782od_c_a > a,V: fset_P8043821951902805788od_c_a] :
      ( ( fset_a2 @ ( fimage9200358715597384645_c_a_a @ F @ V ) )
      = ( image_9171994905208062541_c_a_a @ F @ ( fset_P7369495421554515365od_c_a @ V ) ) ) ).

% fset.set_map
thf(fact_434_fset_Oset__map,axiom,
    ! [F: a > a,V: fset_a] :
      ( ( fset_a2 @ ( fimage_a_a @ F @ V ) )
      = ( image_a_a @ F @ ( fset_a2 @ V ) ) ) ).

% fset.set_map
thf(fact_435_fimage_Orep__eq,axiom,
    ! [X2: produc21584292878202806fset_a > fset_a,Xa4: fset_P2596122609397385116fset_a] :
      ( ( fset_fset_a2 @ ( fimage5315711910825859755fset_a @ X2 @ Xa4 ) )
      = ( image_523240206663320883fset_a @ X2 @ ( fset_P7830090094348214821fset_a @ Xa4 ) ) ) ).

% fimage.rep_eq
thf(fact_436_fimage_Orep__eq,axiom,
    ! [X2: fset_a > fset_P2596122609397385116fset_a,Xa4: fset_fset_a] :
      ( ( fset_f1459617199513185931fset_a @ ( fimage8078386234238232501fset_a @ X2 @ Xa4 ) )
      = ( image_6386003088472927805fset_a @ X2 @ ( fset_fset_a2 @ Xa4 ) ) ) ).

% fimage.rep_eq
thf(fact_437_fimage_Orep__eq,axiom,
    ! [X2: produc2669932201312201782od_c_a > product_prod_b_c,Xa4: fset_P8043821951902805788od_c_a] :
      ( ( fset_P8805262608305245655od_b_c @ ( fimage7236410608496998447od_b_c @ X2 @ Xa4 ) )
      = ( image_4981958329637652151od_b_c @ X2 @ ( fset_P7369495421554515365od_c_a @ Xa4 ) ) ) ).

% fimage.rep_eq
thf(fact_438_fimage_Orep__eq,axiom,
    ! [X2: product_prod_b_c > produc21584292878202806fset_a,Xa4: fset_P8649754740804009550od_b_c] :
      ( ( fset_P7830090094348214821fset_a @ ( fimage3947511181509736139fset_a @ X2 @ Xa4 ) )
      = ( image_2255128035744431443fset_a @ X2 @ ( fset_P8805262608305245655od_b_c @ Xa4 ) ) ) ).

% fimage.rep_eq
thf(fact_439_fimage_Orep__eq,axiom,
    ! [X2: produc2669932201312201782od_c_a > a,Xa4: fset_P8043821951902805788od_c_a] :
      ( ( fset_a2 @ ( fimage9200358715597384645_c_a_a @ X2 @ Xa4 ) )
      = ( image_9171994905208062541_c_a_a @ X2 @ ( fset_P7369495421554515365od_c_a @ Xa4 ) ) ) ).

% fimage.rep_eq
thf(fact_440_fimage_Orep__eq,axiom,
    ! [X2: a > a,Xa4: fset_a] :
      ( ( fset_a2 @ ( fimage_a_a @ X2 @ Xa4 ) )
      = ( image_a_a @ X2 @ ( fset_a2 @ Xa4 ) ) ) ).

% fimage.rep_eq
thf(fact_441__092_060open_062make__observable__transitions_A_Iftransitions_AM_J_A_It__target_A_124_096_124_Ats_A_124_N_124_Adones_J_Adones_Ats_A_061_Ats_A_124_092_060union_062_124_Aqtrans_092_060close_062,axiom,
    ( ( make_o2341063825228466502_a_b_c @ ( ftransitions_a_b_c @ m )
      @ ( minus_5856361552470604977fset_a
        @ ( fimage5315711910825859755fset_a
          @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
          @ tsa )
        @ donesa )
      @ donesa
      @ tsa )
    = ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) ) ).

% \<open>make_observable_transitions (ftransitions M) (t_target |`| ts |-| dones) dones ts = ts |\<union>| qtrans\<close>
thf(fact_442_False,axiom,
    ( ( minus_5856361552470604977fset_a
      @ ( fimage5315711910825859755fset_a
        @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
        @ qtrans )
      @ ( sup_sup_fset_fset_a @ donesa
        @ ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ tsa )
          @ donesa ) ) )
    = bot_bot_fset_fset_a ) ).

% False
thf(fact_443_Cons_Oprems_I1_J,axiom,
    ( fmember_fset_a @ qc
    @ ( sup_sup_fset_fset_a @ donesa
      @ ( minus_5856361552470604977fset_a
        @ ( fimage5315711910825859755fset_a
          @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
          @ tsa )
        @ donesa ) ) ) ).

% Cons.prems(1)
thf(fact_444__092_060open_062qb_____A_124_092_060in_062_124_Adones_A_124_092_060union_062_124_A_It__target_A_124_096_124_Ats_A_124_N_124_Adones_J_092_060close_062,axiom,
    ( fmember_fset_a @ qb
    @ ( sup_sup_fset_fset_a @ donesa
      @ ( minus_5856361552470604977fset_a
        @ ( fimage5315711910825859755fset_a
          @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
          @ tsa )
        @ donesa ) ) ) ).

% \<open>qb__ |\<in>| dones |\<union>| (t_target |`| ts |-| dones)\<close>
thf(fact_445_transpose_Ocases,axiom,
    ! [X2: list_l7625926270567617602fset_a] :
      ( ( X2 != nil_li7246149390463043906fset_a )
     => ( ! [Xss: list_l7625926270567617602fset_a] :
            ( X2
           != ( cons_l1218140764940152050fset_a @ nil_Pr4260649278601056188fset_a @ Xss ) )
       => ~ ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a,Xss: list_l7625926270567617602fset_a] :
              ( X2
             != ( cons_l1218140764940152050fset_a @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_446_transpose_Ocases,axiom,
    ! [X2: list_l8907847357763382004od_b_c] :
      ( ( X2 != nil_li8071162985807626740od_b_c )
     => ( ! [Xss: list_l8907847357763382004od_b_c] :
            ( X2
           != ( cons_l342771267123639716od_b_c @ nil_Product_prod_b_c @ Xss ) )
       => ~ ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Xss: list_l8907847357763382004od_b_c] :
              ( X2
             != ( cons_l342771267123639716od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_447_funion__left__commute,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a,C2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ A @ ( sup_su6818703939759823728fset_a @ B @ C2 ) )
      = ( sup_su6818703939759823728fset_a @ B @ ( sup_su6818703939759823728fset_a @ A @ C2 ) ) ) ).

% funion_left_commute
thf(fact_448_funion__left__commute,axiom,
    ! [A: fset_fset_a,B: fset_fset_a,C2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ A @ ( sup_sup_fset_fset_a @ B @ C2 ) )
      = ( sup_sup_fset_fset_a @ B @ ( sup_sup_fset_fset_a @ A @ C2 ) ) ) ).

% funion_left_commute
thf(fact_449_funion__left__commute,axiom,
    ! [A: fset_a,B: fset_a,C2: fset_a] :
      ( ( sup_sup_fset_a @ A @ ( sup_sup_fset_a @ B @ C2 ) )
      = ( sup_sup_fset_a @ B @ ( sup_sup_fset_a @ A @ C2 ) ) ) ).

% funion_left_commute
thf(fact_450_funion__left__absorb,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ A @ ( sup_su6818703939759823728fset_a @ A @ B ) )
      = ( sup_su6818703939759823728fset_a @ A @ B ) ) ).

% funion_left_absorb
thf(fact_451_funion__left__absorb,axiom,
    ! [A: fset_fset_a,B: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ A @ ( sup_sup_fset_fset_a @ A @ B ) )
      = ( sup_sup_fset_fset_a @ A @ B ) ) ).

% funion_left_absorb
thf(fact_452_funion__left__absorb,axiom,
    ! [A: fset_a,B: fset_a] :
      ( ( sup_sup_fset_a @ A @ ( sup_sup_fset_a @ A @ B ) )
      = ( sup_sup_fset_a @ A @ B ) ) ).

% funion_left_absorb
thf(fact_453_funion__commute,axiom,
    ( sup_su6818703939759823728fset_a
    = ( ^ [A3: fset_P2596122609397385116fset_a,B4: fset_P2596122609397385116fset_a] : ( sup_su6818703939759823728fset_a @ B4 @ A3 ) ) ) ).

% funion_commute
thf(fact_454_funion__commute,axiom,
    ( sup_sup_fset_fset_a
    = ( ^ [A3: fset_fset_a,B4: fset_fset_a] : ( sup_sup_fset_fset_a @ B4 @ A3 ) ) ) ).

% funion_commute
thf(fact_455_funion__commute,axiom,
    ( sup_sup_fset_a
    = ( ^ [A3: fset_a,B4: fset_a] : ( sup_sup_fset_a @ B4 @ A3 ) ) ) ).

% funion_commute
thf(fact_456_funion__fminus,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a,C2: fset_P2596122609397385116fset_a] :
      ( ( minus_7225097421573603715fset_a @ ( sup_su6818703939759823728fset_a @ A @ B ) @ C2 )
      = ( sup_su6818703939759823728fset_a @ ( minus_7225097421573603715fset_a @ A @ C2 ) @ ( minus_7225097421573603715fset_a @ B @ C2 ) ) ) ).

% funion_fminus
thf(fact_457_funion__fminus,axiom,
    ! [A: fset_a,B: fset_a,C2: fset_a] :
      ( ( minus_minus_fset_a @ ( sup_sup_fset_a @ A @ B ) @ C2 )
      = ( sup_sup_fset_a @ ( minus_minus_fset_a @ A @ C2 ) @ ( minus_minus_fset_a @ B @ C2 ) ) ) ).

% funion_fminus
thf(fact_458_funion__fminus,axiom,
    ! [A: fset_fset_a,B: fset_fset_a,C2: fset_fset_a] :
      ( ( minus_5856361552470604977fset_a @ ( sup_sup_fset_fset_a @ A @ B ) @ C2 )
      = ( sup_sup_fset_fset_a @ ( minus_5856361552470604977fset_a @ A @ C2 ) @ ( minus_5856361552470604977fset_a @ B @ C2 ) ) ) ).

% funion_fminus
thf(fact_459_funion__absorb,axiom,
    ! [A: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ A @ A )
      = A ) ).

% funion_absorb
thf(fact_460_funion__absorb,axiom,
    ! [A: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ A @ A )
      = A ) ).

% funion_absorb
thf(fact_461_funion__absorb,axiom,
    ! [A: fset_a] :
      ( ( sup_sup_fset_a @ A @ A )
      = A ) ).

% funion_absorb
thf(fact_462_fimage__funion,axiom,
    ! [F: a > a,A: fset_a,B: fset_a] :
      ( ( fimage_a_a @ F @ ( sup_sup_fset_a @ A @ B ) )
      = ( sup_sup_fset_a @ ( fimage_a_a @ F @ A ) @ ( fimage_a_a @ F @ B ) ) ) ).

% fimage_funion
thf(fact_463_fimage__funion,axiom,
    ! [F: fset_a > a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fimage_fset_a_a @ F @ ( sup_sup_fset_fset_a @ A @ B ) )
      = ( sup_sup_fset_a @ ( fimage_fset_a_a @ F @ A ) @ ( fimage_fset_a_a @ F @ B ) ) ) ).

% fimage_funion
thf(fact_464_fimage__funion,axiom,
    ! [F: a > fset_a,A: fset_a,B: fset_a] :
      ( ( fimage_a_fset_a @ F @ ( sup_sup_fset_a @ A @ B ) )
      = ( sup_sup_fset_fset_a @ ( fimage_a_fset_a @ F @ A ) @ ( fimage_a_fset_a @ F @ B ) ) ) ).

% fimage_funion
thf(fact_465_fimage__funion,axiom,
    ! [F: fset_a > fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fimage_fset_a_fset_a @ F @ ( sup_sup_fset_fset_a @ A @ B ) )
      = ( sup_sup_fset_fset_a @ ( fimage_fset_a_fset_a @ F @ A ) @ ( fimage_fset_a_fset_a @ F @ B ) ) ) ).

% fimage_funion
thf(fact_466_fimage__funion,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fimage9200358715597384645_c_a_a @ F @ ( sup_su7793179031141480176od_c_a @ A @ B ) )
      = ( sup_sup_fset_a @ ( fimage9200358715597384645_c_a_a @ F @ A ) @ ( fimage9200358715597384645_c_a_a @ F @ B ) ) ) ).

% fimage_funion
thf(fact_467_fimage__funion,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fimage7236410608496998447od_b_c @ F @ ( sup_su7793179031141480176od_c_a @ A @ B ) )
      = ( sup_su3559077578824199714od_b_c @ ( fimage7236410608496998447od_b_c @ F @ A ) @ ( fimage7236410608496998447od_b_c @ F @ B ) ) ) ).

% fimage_funion
thf(fact_468_fimage__funion,axiom,
    ! [F: produc21584292878202806fset_a > a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fimage5721101993989878341et_a_a @ F @ ( sup_su6818703939759823728fset_a @ A @ B ) )
      = ( sup_sup_fset_a @ ( fimage5721101993989878341et_a_a @ F @ A ) @ ( fimage5721101993989878341et_a_a @ F @ B ) ) ) ).

% fimage_funion
thf(fact_469_fimage__funion,axiom,
    ! [F: a > produc21584292878202806fset_a,A: fset_a,B: fset_a] :
      ( ( fimage5428608421149153717fset_a @ F @ ( sup_sup_fset_a @ A @ B ) )
      = ( sup_su6818703939759823728fset_a @ ( fimage5428608421149153717fset_a @ F @ A ) @ ( fimage5428608421149153717fset_a @ F @ B ) ) ) ).

% fimage_funion
thf(fact_470_fimage__funion,axiom,
    ! [F: produc21584292878202806fset_a > fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fimage5315711910825859755fset_a @ F @ ( sup_su6818703939759823728fset_a @ A @ B ) )
      = ( sup_sup_fset_fset_a @ ( fimage5315711910825859755fset_a @ F @ A ) @ ( fimage5315711910825859755fset_a @ F @ B ) ) ) ).

% fimage_funion
thf(fact_471_fimage__funion,axiom,
    ! [F: fset_a > produc21584292878202806fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fimage3827890395217950799fset_a @ F @ ( sup_sup_fset_fset_a @ A @ B ) )
      = ( sup_su6818703939759823728fset_a @ ( fimage3827890395217950799fset_a @ F @ A ) @ ( fimage3827890395217950799fset_a @ F @ B ) ) ) ).

% fimage_funion
thf(fact_472_funion__assoc,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a,C2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ ( sup_su6818703939759823728fset_a @ A @ B ) @ C2 )
      = ( sup_su6818703939759823728fset_a @ A @ ( sup_su6818703939759823728fset_a @ B @ C2 ) ) ) ).

% funion_assoc
thf(fact_473_funion__assoc,axiom,
    ! [A: fset_fset_a,B: fset_fset_a,C2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ ( sup_sup_fset_fset_a @ A @ B ) @ C2 )
      = ( sup_sup_fset_fset_a @ A @ ( sup_sup_fset_fset_a @ B @ C2 ) ) ) ).

% funion_assoc
thf(fact_474_funion__assoc,axiom,
    ! [A: fset_a,B: fset_a,C2: fset_a] :
      ( ( sup_sup_fset_a @ ( sup_sup_fset_a @ A @ B ) @ C2 )
      = ( sup_sup_fset_a @ A @ ( sup_sup_fset_a @ B @ C2 ) ) ) ).

% funion_assoc
thf(fact_475_list_Osimps_I8_J,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( map_Pr776894537227705078od_b_c @ F @ nil_Pr1342775757158464060od_c_a )
      = nil_Product_prod_b_c ) ).

% list.simps(8)
thf(fact_476_list_Osimps_I8_J,axiom,
    ! [F: produc21584292878202806fset_a > product_prod_b_c] :
      ( ( map_Pr7697975548294988406od_b_c @ F @ nil_Pr4260649278601056188fset_a )
      = nil_Product_prod_b_c ) ).

% list.simps(8)
thf(fact_477_prod__induct4,axiom,
    ! [P2: produc21584292878202806fset_a > $o,X2: produc21584292878202806fset_a] :
      ( ! [A6: fset_a,B5: b,C3: c,D: fset_a] : ( P2 @ ( produc3008349866153638182fset_a @ A6 @ ( produc4496741132823077603fset_a @ B5 @ ( produc1535188672328598523fset_a @ C3 @ D ) ) ) )
     => ( P2 @ X2 ) ) ).

% prod_induct4
thf(fact_478_prod__induct3,axiom,
    ! [P2: produc21584292878202806fset_a > $o,X2: produc21584292878202806fset_a] :
      ( ! [A6: fset_a,B5: b,C3: produc4838349196543436939fset_a] : ( P2 @ ( produc3008349866153638182fset_a @ A6 @ ( produc4496741132823077603fset_a @ B5 @ C3 ) ) )
     => ( P2 @ X2 ) ) ).

% prod_induct3
thf(fact_479_prod__induct3,axiom,
    ! [P2: produc1137077444975408883fset_a > $o,X2: produc1137077444975408883fset_a] :
      ( ! [A6: b,B5: c,C3: fset_a] : ( P2 @ ( produc4496741132823077603fset_a @ A6 @ ( produc1535188672328598523fset_a @ B5 @ C3 ) ) )
     => ( P2 @ X2 ) ) ).

% prod_induct3
thf(fact_480_prod__cases4,axiom,
    ! [Y2: produc21584292878202806fset_a] :
      ~ ! [A6: fset_a,B5: b,C3: c,D: fset_a] :
          ( Y2
         != ( produc3008349866153638182fset_a @ A6 @ ( produc4496741132823077603fset_a @ B5 @ ( produc1535188672328598523fset_a @ C3 @ D ) ) ) ) ).

% prod_cases4
thf(fact_481_prod__cases3,axiom,
    ! [Y2: produc21584292878202806fset_a] :
      ~ ! [A6: fset_a,B5: b,C3: produc4838349196543436939fset_a] :
          ( Y2
         != ( produc3008349866153638182fset_a @ A6 @ ( produc4496741132823077603fset_a @ B5 @ C3 ) ) ) ).

% prod_cases3
thf(fact_482_prod__cases3,axiom,
    ! [Y2: produc1137077444975408883fset_a] :
      ~ ! [A6: b,B5: c,C3: fset_a] :
          ( Y2
         != ( produc4496741132823077603fset_a @ A6 @ ( produc1535188672328598523fset_a @ B5 @ C3 ) ) ) ).

% prod_cases3
thf(fact_483_Pair__inject,axiom,
    ! [A2: b,B2: c,A5: b,B3: c] :
      ( ( ( product_Pair_b_c @ A2 @ B2 )
        = ( product_Pair_b_c @ A5 @ B3 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_484_Pair__inject,axiom,
    ! [A2: fset_a,B2: produc1137077444975408883fset_a,A5: fset_a,B3: produc1137077444975408883fset_a] :
      ( ( ( produc3008349866153638182fset_a @ A2 @ B2 )
        = ( produc3008349866153638182fset_a @ A5 @ B3 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_485_Pair__inject,axiom,
    ! [A2: b,B2: produc4838349196543436939fset_a,A5: b,B3: produc4838349196543436939fset_a] :
      ( ( ( produc4496741132823077603fset_a @ A2 @ B2 )
        = ( produc4496741132823077603fset_a @ A5 @ B3 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_486_Pair__inject,axiom,
    ! [A2: c,B2: fset_a,A5: c,B3: fset_a] :
      ( ( ( produc1535188672328598523fset_a @ A2 @ B2 )
        = ( produc1535188672328598523fset_a @ A5 @ B3 ) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_487_prod__cases,axiom,
    ! [P2: product_prod_b_c > $o,P: product_prod_b_c] :
      ( ! [A6: b,B5: c] : ( P2 @ ( product_Pair_b_c @ A6 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_488_prod__cases,axiom,
    ! [P2: produc21584292878202806fset_a > $o,P: produc21584292878202806fset_a] :
      ( ! [A6: fset_a,B5: produc1137077444975408883fset_a] : ( P2 @ ( produc3008349866153638182fset_a @ A6 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_489_prod__cases,axiom,
    ! [P2: produc1137077444975408883fset_a > $o,P: produc1137077444975408883fset_a] :
      ( ! [A6: b,B5: produc4838349196543436939fset_a] : ( P2 @ ( produc4496741132823077603fset_a @ A6 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_490_prod__cases,axiom,
    ! [P2: produc4838349196543436939fset_a > $o,P: produc4838349196543436939fset_a] :
      ( ! [A6: c,B5: fset_a] : ( P2 @ ( produc1535188672328598523fset_a @ A6 @ B5 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_491_surj__pair,axiom,
    ! [P: product_prod_b_c] :
    ? [X3: b,Y3: c] :
      ( P
      = ( product_Pair_b_c @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_492_surj__pair,axiom,
    ! [P: produc21584292878202806fset_a] :
    ? [X3: fset_a,Y3: produc1137077444975408883fset_a] :
      ( P
      = ( produc3008349866153638182fset_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_493_surj__pair,axiom,
    ! [P: produc1137077444975408883fset_a] :
    ? [X3: b,Y3: produc4838349196543436939fset_a] :
      ( P
      = ( produc4496741132823077603fset_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_494_surj__pair,axiom,
    ! [P: produc4838349196543436939fset_a] :
    ? [X3: c,Y3: fset_a] :
      ( P
      = ( produc1535188672328598523fset_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_495_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_b_c] :
      ~ ! [A6: b,B5: c] :
          ( Y2
         != ( product_Pair_b_c @ A6 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_496_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc21584292878202806fset_a] :
      ~ ! [A6: fset_a,B5: produc1137077444975408883fset_a] :
          ( Y2
         != ( produc3008349866153638182fset_a @ A6 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_497_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc1137077444975408883fset_a] :
      ~ ! [A6: b,B5: produc4838349196543436939fset_a] :
          ( Y2
         != ( produc4496741132823077603fset_a @ A6 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_498_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc4838349196543436939fset_a] :
      ~ ! [A6: c,B5: fset_a] :
          ( Y2
         != ( produc1535188672328598523fset_a @ A6 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_499_fimage__fimage,axiom,
    ! [F: a > a,G: produc2669932201312201782od_c_a > a,A: fset_P8043821951902805788od_c_a] :
      ( ( fimage_a_a @ F @ ( fimage9200358715597384645_c_a_a @ G @ A ) )
      = ( fimage9200358715597384645_c_a_a
        @ ^ [X: produc2669932201312201782od_c_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_500_fimage__fimage,axiom,
    ! [F: a > product_prod_b_c,G: produc2669932201312201782od_c_a > a,A: fset_P8043821951902805788od_c_a] :
      ( ( fimage3663987841053390183od_b_c @ F @ ( fimage9200358715597384645_c_a_a @ G @ A ) )
      = ( fimage7236410608496998447od_b_c
        @ ^ [X: produc2669932201312201782od_c_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_501_fimage__fimage,axiom,
    ! [F: product_prod_b_c > a,G: produc2669932201312201782od_c_a > product_prod_b_c,A: fset_P8043821951902805788od_c_a] :
      ( ( fimage6813683498707228179_b_c_a @ F @ ( fimage7236410608496998447od_b_c @ G @ A ) )
      = ( fimage9200358715597384645_c_a_a
        @ ^ [X: produc2669932201312201782od_c_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_502_fimage__fimage,axiom,
    ! [F: fset_a > fset_a,G: produc21584292878202806fset_a > fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( fimage_fset_a_fset_a @ F @ ( fimage5315711910825859755fset_a @ G @ A ) )
      = ( fimage5315711910825859755fset_a
        @ ^ [X: produc21584292878202806fset_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_503_fimage__fimage,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,G: produc2669932201312201782od_c_a > product_prod_b_c,A: fset_P8043821951902805788od_c_a] :
      ( ( fimage2995889515442742397od_b_c @ F @ ( fimage7236410608496998447od_b_c @ G @ A ) )
      = ( fimage7236410608496998447od_b_c
        @ ^ [X: produc2669932201312201782od_c_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_504_fimage__fimage,axiom,
    ! [F: produc21584292878202806fset_a > fset_a,G: product_prod_b_c > produc21584292878202806fset_a,A: fset_P8649754740804009550od_b_c] :
      ( ( fimage5315711910825859755fset_a @ F @ ( fimage3947511181509736139fset_a @ G @ A ) )
      = ( fimage4261884857909657465fset_a
        @ ^ [X: product_prod_b_c] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_505_fimage__fimage,axiom,
    ! [F: fset_a > fset_P2596122609397385116fset_a,G: fset_a > fset_a,A: fset_fset_a] :
      ( ( fimage8078386234238232501fset_a @ F @ ( fimage_fset_a_fset_a @ G @ A ) )
      = ( fimage8078386234238232501fset_a
        @ ^ [X: fset_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_506_fimage__fimage,axiom,
    ! [F: produc2669932201312201782od_c_a > a,G: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fimage9200358715597384645_c_a_a @ F @ ( fimage2423425181458651133od_c_a @ G @ A ) )
      = ( fimage9200358715597384645_c_a_a
        @ ^ [X: produc2669932201312201782od_c_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_507_fimage__fimage,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a,G: product_prod_b_c > product_prod_b_c,A: fset_P8649754740804009550od_b_c] :
      ( ( fimage3947511181509736139fset_a @ F @ ( fimage2995889515442742397od_b_c @ G @ A ) )
      = ( fimage3947511181509736139fset_a
        @ ^ [X: product_prod_b_c] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_508_fimage__fimage,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,G: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fimage7236410608496998447od_b_c @ F @ ( fimage2423425181458651133od_c_a @ G @ A ) )
      = ( fimage7236410608496998447od_b_c
        @ ^ [X: produc2669932201312201782od_c_a] : ( F @ ( G @ X ) )
        @ A ) ) ).

% fimage_fimage
thf(fact_509_list_Osimps_I9_J,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,X21: produc2669932201312201782od_c_a,X22: list_P6327159017948738492od_c_a] :
      ( ( map_Pr776894537227705078od_b_c @ F @ ( cons_P8659537020926305772od_c_a @ X21 @ X22 ) )
      = ( cons_P4529483553340347422od_b_c @ ( F @ X21 ) @ ( map_Pr776894537227705078od_b_c @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_510_list_Osimps_I9_J,axiom,
    ! [F: produc21584292878202806fset_a > produc21584292878202806fset_a,X21: produc21584292878202806fset_a,X22: list_P1058411382900698172fset_a] :
      ( ( map_Pr4563436947057144260fset_a @ F @ ( cons_P3377334492504755052fset_a @ X21 @ X22 ) )
      = ( cons_P3377334492504755052fset_a @ ( F @ X21 ) @ ( map_Pr4563436947057144260fset_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_511_list_Osimps_I9_J,axiom,
    ! [F: produc21584292878202806fset_a > product_prod_b_c,X21: produc21584292878202806fset_a,X22: list_P1058411382900698172fset_a] :
      ( ( map_Pr7697975548294988406od_b_c @ F @ ( cons_P3377334492504755052fset_a @ X21 @ X22 ) )
      = ( cons_P4529483553340347422od_b_c @ ( F @ X21 ) @ ( map_Pr7697975548294988406od_b_c @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_512_list_Osimps_I9_J,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a,X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( ( map_Pr5295542084739857042fset_a @ F @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) )
      = ( cons_P3377334492504755052fset_a @ ( F @ X21 ) @ ( map_Pr5295542084739857042fset_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_513_list_Osimps_I9_J,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( ( map_Pr6384634858812015940od_b_c @ F @ ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) )
      = ( cons_P4529483553340347422od_b_c @ ( F @ X21 ) @ ( map_Pr6384634858812015940od_b_c @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_514_Cons__eq__map__D,axiom,
    ! [X2: produc21584292878202806fset_a,Xs: list_P1058411382900698172fset_a,F: produc21584292878202806fset_a > produc21584292878202806fset_a,Ys: list_P1058411382900698172fset_a] :
      ( ( ( cons_P3377334492504755052fset_a @ X2 @ Xs )
        = ( map_Pr4563436947057144260fset_a @ F @ Ys ) )
     => ? [Z3: produc21584292878202806fset_a,Zs: list_P1058411382900698172fset_a] :
          ( ( Ys
            = ( cons_P3377334492504755052fset_a @ Z3 @ Zs ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr4563436947057144260fset_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_515_Cons__eq__map__D,axiom,
    ! [X2: produc21584292878202806fset_a,Xs: list_P1058411382900698172fset_a,F: product_prod_b_c > produc21584292878202806fset_a,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_P3377334492504755052fset_a @ X2 @ Xs )
        = ( map_Pr5295542084739857042fset_a @ F @ Ys ) )
     => ? [Z3: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Ys
            = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr5295542084739857042fset_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_516_Cons__eq__map__D,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: produc2669932201312201782od_c_a > product_prod_b_c,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_Pr776894537227705078od_b_c @ F @ Ys ) )
     => ? [Z3: produc2669932201312201782od_c_a,Zs: list_P6327159017948738492od_c_a] :
          ( ( Ys
            = ( cons_P8659537020926305772od_c_a @ Z3 @ Zs ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr776894537227705078od_b_c @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_517_Cons__eq__map__D,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: produc21584292878202806fset_a > product_prod_b_c,Ys: list_P1058411382900698172fset_a] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_Pr7697975548294988406od_b_c @ F @ Ys ) )
     => ? [Z3: produc21584292878202806fset_a,Zs: list_P1058411382900698172fset_a] :
          ( ( Ys
            = ( cons_P3377334492504755052fset_a @ Z3 @ Zs ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr7697975548294988406od_b_c @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_518_Cons__eq__map__D,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_Pr6384634858812015940od_b_c @ F @ Ys ) )
     => ? [Z3: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Ys
            = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr6384634858812015940od_b_c @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_519_map__eq__Cons__D,axiom,
    ! [F: produc21584292878202806fset_a > produc21584292878202806fset_a,Xs: list_P1058411382900698172fset_a,Y2: produc21584292878202806fset_a,Ys: list_P1058411382900698172fset_a] :
      ( ( ( map_Pr4563436947057144260fset_a @ F @ Xs )
        = ( cons_P3377334492504755052fset_a @ Y2 @ Ys ) )
     => ? [Z3: produc21584292878202806fset_a,Zs: list_P1058411382900698172fset_a] :
          ( ( Xs
            = ( cons_P3377334492504755052fset_a @ Z3 @ Zs ) )
          & ( ( F @ Z3 )
            = Y2 )
          & ( ( map_Pr4563436947057144260fset_a @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_520_map__eq__Cons__D,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a,Xs: list_P903359562653991662od_b_c,Y2: produc21584292878202806fset_a,Ys: list_P1058411382900698172fset_a] :
      ( ( ( map_Pr5295542084739857042fset_a @ F @ Xs )
        = ( cons_P3377334492504755052fset_a @ Y2 @ Ys ) )
     => ? [Z3: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Xs
            = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs ) )
          & ( ( F @ Z3 )
            = Y2 )
          & ( ( map_Pr5295542084739857042fset_a @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_521_map__eq__Cons__D,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a,Y2: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr776894537227705078od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y2 @ Ys ) )
     => ? [Z3: produc2669932201312201782od_c_a,Zs: list_P6327159017948738492od_c_a] :
          ( ( Xs
            = ( cons_P8659537020926305772od_c_a @ Z3 @ Zs ) )
          & ( ( F @ Z3 )
            = Y2 )
          & ( ( map_Pr776894537227705078od_b_c @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_522_map__eq__Cons__D,axiom,
    ! [F: produc21584292878202806fset_a > product_prod_b_c,Xs: list_P1058411382900698172fset_a,Y2: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr7697975548294988406od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y2 @ Ys ) )
     => ? [Z3: produc21584292878202806fset_a,Zs: list_P1058411382900698172fset_a] :
          ( ( Xs
            = ( cons_P3377334492504755052fset_a @ Z3 @ Zs ) )
          & ( ( F @ Z3 )
            = Y2 )
          & ( ( map_Pr7697975548294988406od_b_c @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_523_map__eq__Cons__D,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c,Y2: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr6384634858812015940od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y2 @ Ys ) )
     => ? [Z3: product_prod_b_c,Zs: list_P903359562653991662od_b_c] :
          ( ( Xs
            = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs ) )
          & ( ( F @ Z3 )
            = Y2 )
          & ( ( map_Pr6384634858812015940od_b_c @ F @ Zs )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_524_Cons__eq__map__conv,axiom,
    ! [X2: produc21584292878202806fset_a,Xs: list_P1058411382900698172fset_a,F: produc21584292878202806fset_a > produc21584292878202806fset_a,Ys: list_P1058411382900698172fset_a] :
      ( ( ( cons_P3377334492504755052fset_a @ X2 @ Xs )
        = ( map_Pr4563436947057144260fset_a @ F @ Ys ) )
      = ( ? [Z4: produc21584292878202806fset_a,Zs2: list_P1058411382900698172fset_a] :
            ( ( Ys
              = ( cons_P3377334492504755052fset_a @ Z4 @ Zs2 ) )
            & ( X2
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr4563436947057144260fset_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_525_Cons__eq__map__conv,axiom,
    ! [X2: produc21584292878202806fset_a,Xs: list_P1058411382900698172fset_a,F: product_prod_b_c > produc21584292878202806fset_a,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_P3377334492504755052fset_a @ X2 @ Xs )
        = ( map_Pr5295542084739857042fset_a @ F @ Ys ) )
      = ( ? [Z4: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Ys
              = ( cons_P4529483553340347422od_b_c @ Z4 @ Zs2 ) )
            & ( X2
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr5295542084739857042fset_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_526_Cons__eq__map__conv,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: produc2669932201312201782od_c_a > product_prod_b_c,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_Pr776894537227705078od_b_c @ F @ Ys ) )
      = ( ? [Z4: produc2669932201312201782od_c_a,Zs2: list_P6327159017948738492od_c_a] :
            ( ( Ys
              = ( cons_P8659537020926305772od_c_a @ Z4 @ Zs2 ) )
            & ( X2
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr776894537227705078od_b_c @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_527_Cons__eq__map__conv,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: produc21584292878202806fset_a > product_prod_b_c,Ys: list_P1058411382900698172fset_a] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_Pr7697975548294988406od_b_c @ F @ Ys ) )
      = ( ? [Z4: produc21584292878202806fset_a,Zs2: list_P1058411382900698172fset_a] :
            ( ( Ys
              = ( cons_P3377334492504755052fset_a @ Z4 @ Zs2 ) )
            & ( X2
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr7697975548294988406od_b_c @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_528_Cons__eq__map__conv,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
        = ( map_Pr6384634858812015940od_b_c @ F @ Ys ) )
      = ( ? [Z4: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Ys
              = ( cons_P4529483553340347422od_b_c @ Z4 @ Zs2 ) )
            & ( X2
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr6384634858812015940od_b_c @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_529_map__eq__Cons__conv,axiom,
    ! [F: produc21584292878202806fset_a > produc21584292878202806fset_a,Xs: list_P1058411382900698172fset_a,Y2: produc21584292878202806fset_a,Ys: list_P1058411382900698172fset_a] :
      ( ( ( map_Pr4563436947057144260fset_a @ F @ Xs )
        = ( cons_P3377334492504755052fset_a @ Y2 @ Ys ) )
      = ( ? [Z4: produc21584292878202806fset_a,Zs2: list_P1058411382900698172fset_a] :
            ( ( Xs
              = ( cons_P3377334492504755052fset_a @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y2 )
            & ( ( map_Pr4563436947057144260fset_a @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_530_map__eq__Cons__conv,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a,Xs: list_P903359562653991662od_b_c,Y2: produc21584292878202806fset_a,Ys: list_P1058411382900698172fset_a] :
      ( ( ( map_Pr5295542084739857042fset_a @ F @ Xs )
        = ( cons_P3377334492504755052fset_a @ Y2 @ Ys ) )
      = ( ? [Z4: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( cons_P4529483553340347422od_b_c @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y2 )
            & ( ( map_Pr5295542084739857042fset_a @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_531_map__eq__Cons__conv,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,Xs: list_P6327159017948738492od_c_a,Y2: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr776894537227705078od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y2 @ Ys ) )
      = ( ? [Z4: produc2669932201312201782od_c_a,Zs2: list_P6327159017948738492od_c_a] :
            ( ( Xs
              = ( cons_P8659537020926305772od_c_a @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y2 )
            & ( ( map_Pr776894537227705078od_b_c @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_532_map__eq__Cons__conv,axiom,
    ! [F: produc21584292878202806fset_a > product_prod_b_c,Xs: list_P1058411382900698172fset_a,Y2: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr7697975548294988406od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y2 @ Ys ) )
      = ( ? [Z4: produc21584292878202806fset_a,Zs2: list_P1058411382900698172fset_a] :
            ( ( Xs
              = ( cons_P3377334492504755052fset_a @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y2 )
            & ( ( map_Pr7697975548294988406od_b_c @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_533_map__eq__Cons__conv,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c,Y2: product_prod_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr6384634858812015940od_b_c @ F @ Xs )
        = ( cons_P4529483553340347422od_b_c @ Y2 @ Ys ) )
      = ( ? [Z4: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( cons_P4529483553340347422od_b_c @ Z4 @ Zs2 ) )
            & ( ( F @ Z4 )
              = Y2 )
            & ( ( map_Pr6384634858812015940od_b_c @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_534_list_Odistinct_I1_J,axiom,
    ! [X21: produc21584292878202806fset_a,X22: list_P1058411382900698172fset_a] :
      ( nil_Pr4260649278601056188fset_a
     != ( cons_P3377334492504755052fset_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_535_list_Odistinct_I1_J,axiom,
    ! [X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( nil_Product_prod_b_c
     != ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_536_list_OdiscI,axiom,
    ! [List: list_P1058411382900698172fset_a,X21: produc21584292878202806fset_a,X22: list_P1058411382900698172fset_a] :
      ( ( List
        = ( cons_P3377334492504755052fset_a @ X21 @ X22 ) )
     => ( List != nil_Pr4260649278601056188fset_a ) ) ).

% list.discI
thf(fact_537_list_OdiscI,axiom,
    ! [List: list_P903359562653991662od_b_c,X21: product_prod_b_c,X22: list_P903359562653991662od_b_c] :
      ( ( List
        = ( cons_P4529483553340347422od_b_c @ X21 @ X22 ) )
     => ( List != nil_Product_prod_b_c ) ) ).

% list.discI
thf(fact_538_list_Oexhaust,axiom,
    ! [Y2: list_P1058411382900698172fset_a] :
      ( ( Y2 != nil_Pr4260649278601056188fset_a )
     => ~ ! [X212: produc21584292878202806fset_a,X222: list_P1058411382900698172fset_a] :
            ( Y2
           != ( cons_P3377334492504755052fset_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_539_list_Oexhaust,axiom,
    ! [Y2: list_P903359562653991662od_b_c] :
      ( ( Y2 != nil_Product_prod_b_c )
     => ~ ! [X212: product_prod_b_c,X222: list_P903359562653991662od_b_c] :
            ( Y2
           != ( cons_P4529483553340347422od_b_c @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_540_min__list_Ocases,axiom,
    ! [X2: list_P1058411382900698172fset_a] :
      ( ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a] :
          ( X2
         != ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) )
     => ( X2 = nil_Pr4260649278601056188fset_a ) ) ).

% min_list.cases
thf(fact_541_min__list_Ocases,axiom,
    ! [X2: list_P903359562653991662od_b_c] :
      ( ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
          ( X2
         != ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) )
     => ( X2 = nil_Product_prod_b_c ) ) ).

% min_list.cases
thf(fact_542_remdups__adj_Ocases,axiom,
    ! [X2: list_P1058411382900698172fset_a] :
      ( ( X2 != nil_Pr4260649278601056188fset_a )
     => ( ! [X3: produc21584292878202806fset_a] :
            ( X2
           != ( cons_P3377334492504755052fset_a @ X3 @ nil_Pr4260649278601056188fset_a ) )
       => ~ ! [X3: produc21584292878202806fset_a,Y3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a] :
              ( X2
             != ( cons_P3377334492504755052fset_a @ X3 @ ( cons_P3377334492504755052fset_a @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_543_remdups__adj_Ocases,axiom,
    ! [X2: list_P903359562653991662od_b_c] :
      ( ( X2 != nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c] :
            ( X2
           != ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
       => ~ ! [X3: product_prod_b_c,Y3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( X2
             != ( cons_P4529483553340347422od_b_c @ X3 @ ( cons_P4529483553340347422od_b_c @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_544_neq__Nil__conv,axiom,
    ! [Xs: list_P1058411382900698172fset_a] :
      ( ( Xs != nil_Pr4260649278601056188fset_a )
      = ( ? [Y: produc21584292878202806fset_a,Ys2: list_P1058411382900698172fset_a] :
            ( Xs
            = ( cons_P3377334492504755052fset_a @ Y @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_545_neq__Nil__conv,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( Xs != nil_Product_prod_b_c )
      = ( ? [Y: product_prod_b_c,Ys2: list_P903359562653991662od_b_c] :
            ( Xs
            = ( cons_P4529483553340347422od_b_c @ Y @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_546_list__induct2_H,axiom,
    ! [P2: list_P1058411382900698172fset_a > list_P1058411382900698172fset_a > $o,Xs: list_P1058411382900698172fset_a,Ys: list_P1058411382900698172fset_a] :
      ( ( P2 @ nil_Pr4260649278601056188fset_a @ nil_Pr4260649278601056188fset_a )
     => ( ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a] : ( P2 @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ nil_Pr4260649278601056188fset_a )
       => ( ! [Y3: produc21584292878202806fset_a,Ys3: list_P1058411382900698172fset_a] : ( P2 @ nil_Pr4260649278601056188fset_a @ ( cons_P3377334492504755052fset_a @ Y3 @ Ys3 ) )
         => ( ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a,Y3: produc21584292878202806fset_a,Ys3: list_P1058411382900698172fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ ( cons_P3377334492504755052fset_a @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_547_list__induct2_H,axiom,
    ! [P2: list_P1058411382900698172fset_a > list_P903359562653991662od_b_c > $o,Xs: list_P1058411382900698172fset_a,Ys: list_P903359562653991662od_b_c] :
      ( ( P2 @ nil_Pr4260649278601056188fset_a @ nil_Product_prod_b_c )
     => ( ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a] : ( P2 @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ nil_Product_prod_b_c )
       => ( ! [Y3: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] : ( P2 @ nil_Pr4260649278601056188fset_a @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys3 ) )
         => ( ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a,Y3: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_548_list__induct2_H,axiom,
    ! [P2: list_P903359562653991662od_b_c > list_P1058411382900698172fset_a > $o,Xs: list_P903359562653991662od_b_c,Ys: list_P1058411382900698172fset_a] :
      ( ( P2 @ nil_Product_prod_b_c @ nil_Pr4260649278601056188fset_a )
     => ( ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] : ( P2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ nil_Pr4260649278601056188fset_a )
       => ( ! [Y3: produc21584292878202806fset_a,Ys3: list_P1058411382900698172fset_a] : ( P2 @ nil_Product_prod_b_c @ ( cons_P3377334492504755052fset_a @ Y3 @ Ys3 ) )
         => ( ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y3: produc21584292878202806fset_a,Ys3: list_P1058411382900698172fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ ( cons_P3377334492504755052fset_a @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_549_list__induct2_H,axiom,
    ! [P2: list_P903359562653991662od_b_c > list_P903359562653991662od_b_c > $o,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( P2 @ nil_Product_prod_b_c @ nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] : ( P2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ nil_Product_prod_b_c )
       => ( ! [Y3: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] : ( P2 @ nil_Product_prod_b_c @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys3 ) )
         => ( ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_550_list__nonempty__induct,axiom,
    ! [Xs: list_P1058411382900698172fset_a,P2: list_P1058411382900698172fset_a > $o] :
      ( ( Xs != nil_Pr4260649278601056188fset_a )
     => ( ! [X3: produc21584292878202806fset_a] : ( P2 @ ( cons_P3377334492504755052fset_a @ X3 @ nil_Pr4260649278601056188fset_a ) )
       => ( ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a] :
              ( ( Xs2 != nil_Pr4260649278601056188fset_a )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_551_list__nonempty__induct,axiom,
    ! [Xs: list_P903359562653991662od_b_c,P2: list_P903359562653991662od_b_c > $o] :
      ( ( Xs != nil_Product_prod_b_c )
     => ( ! [X3: product_prod_b_c] : ( P2 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) )
       => ( ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( ( Xs2 != nil_Product_prod_b_c )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_552_observable__alt__def,axiom,
    ( observable_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [Q1: a,X: b,Y: c,Q12: a,Q13: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ Q12 ) ) ) @ ( transitions_a_b_c @ M ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ Q13 ) ) ) @ ( transitions_a_b_c @ M ) ) )
         => ( Q12 = Q13 ) ) ) ) ).

% observable_alt_def
thf(fact_553_observable__alt__def,axiom,
    ( observ2964502264774598958_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q1: product_prod_c_a,X: a,Y: b,Q12: product_prod_c_a,Q13: product_prod_c_a] :
          ( ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ Q12 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) )
            & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ Q13 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) )
         => ( Q12 = Q13 ) ) ) ) ).

% observable_alt_def
thf(fact_554_observable__alt__def,axiom,
    ( observ1610022369204726958et_a_b
    = ( ^ [M: fsm_Pr2398083450103406218et_a_b] :
        ! [Q1: produc4838349196543436939fset_a,X: fset_a,Y: b,Q12: produc4838349196543436939fset_a,Q13: produc4838349196543436939fset_a] :
          ( ( ( member808877655204049083fset_a @ ( produc1662326766993966978fset_a @ Q1 @ ( produc3008349866153638182fset_a @ X @ ( produc4496741132823077603fset_a @ Y @ Q12 ) ) ) @ ( transi3322184091305251195et_a_b @ M ) )
            & ( member808877655204049083fset_a @ ( produc1662326766993966978fset_a @ Q1 @ ( produc3008349866153638182fset_a @ X @ ( produc4496741132823077603fset_a @ Y @ Q13 ) ) ) @ ( transi3322184091305251195et_a_b @ M ) ) )
         => ( Q12 = Q13 ) ) ) ) ).

% observable_alt_def
thf(fact_555_observable__alt__def,axiom,
    ( observ4396094731789362477_a_b_c
    = ( ^ [M: fsm_fset_a_b_c] :
        ! [Q1: fset_a,X: b,Y: c,Q12: fset_a,Q13: fset_a] :
          ( ( ( member4899939873214261983fset_a @ ( produc3008349866153638182fset_a @ Q1 @ ( produc4496741132823077603fset_a @ X @ ( produc1535188672328598523fset_a @ Y @ Q12 ) ) ) @ ( transi6427852111903025530_a_b_c @ M ) )
            & ( member4899939873214261983fset_a @ ( produc3008349866153638182fset_a @ Q1 @ ( produc4496741132823077603fset_a @ X @ ( produc1535188672328598523fset_a @ Y @ Q13 ) ) ) @ ( transi6427852111903025530_a_b_c @ M ) ) )
         => ( Q12 = Q13 ) ) ) ) ).

% observable_alt_def
thf(fact_556_single__input__alt__def,axiom,
    ( single_input_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [Q1: a,X: b,X5: b,Y: c,Y6: c,Q12: a,Q13: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ Q12 ) ) ) @ ( transitions_a_b_c @ M ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X5 @ ( product_Pair_c_a @ Y6 @ Q13 ) ) ) @ ( transitions_a_b_c @ M ) ) )
         => ( X = X5 ) ) ) ) ).

% single_input_alt_def
thf(fact_557_single__input__alt__def,axiom,
    ( single999786218259502647_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q1: product_prod_c_a,X: a,X5: a,Y: b,Y6: b,Q12: product_prod_c_a,Q13: product_prod_c_a] :
          ( ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y @ Q12 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) )
            & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X5 @ ( produc3440865914000881661od_c_a @ Y6 @ Q13 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) )
         => ( X = X5 ) ) ) ) ).

% single_input_alt_def
thf(fact_558_single__input__alt__def,axiom,
    ( single5089827247979212983et_a_b
    = ( ^ [M: fsm_Pr2398083450103406218et_a_b] :
        ! [Q1: produc4838349196543436939fset_a,X: fset_a,X5: fset_a,Y: b,Y6: b,Q12: produc4838349196543436939fset_a,Q13: produc4838349196543436939fset_a] :
          ( ( ( member808877655204049083fset_a @ ( produc1662326766993966978fset_a @ Q1 @ ( produc3008349866153638182fset_a @ X @ ( produc4496741132823077603fset_a @ Y @ Q12 ) ) ) @ ( transi3322184091305251195et_a_b @ M ) )
            & ( member808877655204049083fset_a @ ( produc1662326766993966978fset_a @ Q1 @ ( produc3008349866153638182fset_a @ X5 @ ( produc4496741132823077603fset_a @ Y6 @ Q13 ) ) ) @ ( transi3322184091305251195et_a_b @ M ) ) )
         => ( X = X5 ) ) ) ) ).

% single_input_alt_def
thf(fact_559_single__input__alt__def,axiom,
    ( single3763647048569287350_a_b_c
    = ( ^ [M: fsm_fset_a_b_c] :
        ! [Q1: fset_a,X: b,X5: b,Y: c,Y6: c,Q12: fset_a,Q13: fset_a] :
          ( ( ( member4899939873214261983fset_a @ ( produc3008349866153638182fset_a @ Q1 @ ( produc4496741132823077603fset_a @ X @ ( produc1535188672328598523fset_a @ Y @ Q12 ) ) ) @ ( transi6427852111903025530_a_b_c @ M ) )
            & ( member4899939873214261983fset_a @ ( produc3008349866153638182fset_a @ Q1 @ ( produc4496741132823077603fset_a @ X5 @ ( produc1535188672328598523fset_a @ Y6 @ Q13 ) ) ) @ ( transi6427852111903025530_a_b_c @ M ) ) )
         => ( X = X5 ) ) ) ) ).

% single_input_alt_def
thf(fact_560_deterministic__alt__def,axiom,
    ( deterministic_a_b_c
    = ( ^ [M: fsm_a_b_c] :
        ! [Q1: a,X: b,Y6: c,Y7: c,Q12: a,Q13: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y6 @ Q12 ) ) ) @ ( transitions_a_b_c @ M ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q1 @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y7 @ Q13 ) ) ) @ ( transitions_a_b_c @ M ) ) )
         => ( ( Y6 = Y7 )
            & ( Q12 = Q13 ) ) ) ) ) ).

% deterministic_alt_def
thf(fact_561_deterministic__alt__def,axiom,
    ( determ4061018006534944583_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b] :
        ! [Q1: product_prod_c_a,X: a,Y6: b,Y7: b,Q12: product_prod_c_a,Q13: product_prod_c_a] :
          ( ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y6 @ Q12 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) )
            & ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q1 @ ( produc5762801557034676134od_c_a @ X @ ( produc3440865914000881661od_c_a @ Y7 @ Q13 ) ) ) @ ( transi2941759273464634107_a_a_b @ M ) ) )
         => ( ( Y6 = Y7 )
            & ( Q12 = Q13 ) ) ) ) ) ).

% deterministic_alt_def
thf(fact_562_deterministic__alt__def,axiom,
    ( determ7969653133781418439et_a_b
    = ( ^ [M: fsm_Pr2398083450103406218et_a_b] :
        ! [Q1: produc4838349196543436939fset_a,X: fset_a,Y6: b,Y7: b,Q12: produc4838349196543436939fset_a,Q13: produc4838349196543436939fset_a] :
          ( ( ( member808877655204049083fset_a @ ( produc1662326766993966978fset_a @ Q1 @ ( produc3008349866153638182fset_a @ X @ ( produc4496741132823077603fset_a @ Y6 @ Q12 ) ) ) @ ( transi3322184091305251195et_a_b @ M ) )
            & ( member808877655204049083fset_a @ ( produc1662326766993966978fset_a @ Q1 @ ( produc3008349866153638182fset_a @ X @ ( produc4496741132823077603fset_a @ Y7 @ Q13 ) ) ) @ ( transi3322184091305251195et_a_b @ M ) ) )
         => ( ( Y6 = Y7 )
            & ( Q12 = Q13 ) ) ) ) ) ).

% deterministic_alt_def
thf(fact_563_deterministic__alt__def,axiom,
    ( determ4918356626253405126_a_b_c
    = ( ^ [M: fsm_fset_a_b_c] :
        ! [Q1: fset_a,X: b,Y6: c,Y7: c,Q12: fset_a,Q13: fset_a] :
          ( ( ( member4899939873214261983fset_a @ ( produc3008349866153638182fset_a @ Q1 @ ( produc4496741132823077603fset_a @ X @ ( produc1535188672328598523fset_a @ Y6 @ Q12 ) ) ) @ ( transi6427852111903025530_a_b_c @ M ) )
            & ( member4899939873214261983fset_a @ ( produc3008349866153638182fset_a @ Q1 @ ( produc4496741132823077603fset_a @ X @ ( produc1535188672328598523fset_a @ Y7 @ Q13 ) ) ) @ ( transi6427852111903025530_a_b_c @ M ) ) )
         => ( ( Y6 = Y7 )
            & ( Q12 = Q13 ) ) ) ) ) ).

% deterministic_alt_def
thf(fact_564_path__prepend__t,axiom,
    ! [M2: fsm_a_b_c,Q5: a,P: list_P6327159017948738492od_c_a,Q: a,X2: b,Y2: c] :
      ( ( path_a_b_c @ M2 @ Q5 @ P )
     => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X2 @ ( product_Pair_c_a @ Y2 @ Q5 ) ) ) @ ( transitions_a_b_c @ M2 ) )
       => ( path_a_b_c @ M2 @ Q @ ( cons_P8659537020926305772od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X2 @ ( product_Pair_c_a @ Y2 @ Q5 ) ) ) @ P ) ) ) ) ).

% path_prepend_t
thf(fact_565_path__prepend__t,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,Q5: product_prod_c_a,P: list_P1562850763667305982od_c_a,Q: product_prod_c_a,X2: a,Y2: b] :
      ( ( path_P1107610407223809260_a_a_b @ M2 @ Q5 @ P )
     => ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ Q @ ( produc5762801557034676134od_c_a @ X2 @ ( produc3440865914000881661od_c_a @ Y2 @ Q5 ) ) ) @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ( path_P1107610407223809260_a_a_b @ M2 @ Q @ ( cons_P5525787621418763054od_c_a @ ( produc7911353075612250984od_c_a @ Q @ ( produc5762801557034676134od_c_a @ X2 @ ( produc3440865914000881661od_c_a @ Y2 @ Q5 ) ) ) @ P ) ) ) ) ).

% path_prepend_t
thf(fact_566_path__prepend__t,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,Q5: produc4838349196543436939fset_a,P: list_P5281568020286996248fset_a,Q: produc4838349196543436939fset_a,X2: fset_a,Y2: b] :
      ( ( path_P379120284129538668et_a_b @ M2 @ Q5 @ P )
     => ( ( member808877655204049083fset_a @ ( produc1662326766993966978fset_a @ Q @ ( produc3008349866153638182fset_a @ X2 @ ( produc4496741132823077603fset_a @ Y2 @ Q5 ) ) ) @ ( transi3322184091305251195et_a_b @ M2 ) )
       => ( path_P379120284129538668et_a_b @ M2 @ Q @ ( cons_P8607030675745668296fset_a @ ( produc1662326766993966978fset_a @ Q @ ( produc3008349866153638182fset_a @ X2 @ ( produc4496741132823077603fset_a @ Y2 @ Q5 ) ) ) @ P ) ) ) ) ).

% path_prepend_t
thf(fact_567_path__prepend__t,axiom,
    ! [M2: fsm_fset_a_b_c,Q5: fset_a,P: list_P1058411382900698172fset_a,Q: fset_a,X2: b,Y2: c] :
      ( ( path_fset_a_b_c @ M2 @ Q5 @ P )
     => ( ( member4899939873214261983fset_a @ ( produc3008349866153638182fset_a @ Q @ ( produc4496741132823077603fset_a @ X2 @ ( produc1535188672328598523fset_a @ Y2 @ Q5 ) ) ) @ ( transi6427852111903025530_a_b_c @ M2 ) )
       => ( path_fset_a_b_c @ M2 @ Q @ ( cons_P3377334492504755052fset_a @ ( produc3008349866153638182fset_a @ Q @ ( produc4496741132823077603fset_a @ X2 @ ( produc1535188672328598523fset_a @ Y2 @ Q5 ) ) ) @ P ) ) ) ) ).

% path_prepend_t
thf(fact_568_rev__fimage__eqI,axiom,
    ! [X2: a,A: fset_a,B2: a,F: a > a] :
      ( ( fmember_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmember_a @ B2 @ ( fimage_a_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_569_rev__fimage__eqI,axiom,
    ! [X2: a,A: fset_a,B2: fset_a,F: a > fset_a] :
      ( ( fmember_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmember_fset_a @ B2 @ ( fimage_a_fset_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_570_rev__fimage__eqI,axiom,
    ! [X2: fset_a,A: fset_fset_a,B2: a,F: fset_a > a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmember_a @ B2 @ ( fimage_fset_a_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_571_rev__fimage__eqI,axiom,
    ! [X2: fset_a,A: fset_fset_a,B2: fset_a,F: fset_a > fset_a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmember_fset_a @ B2 @ ( fimage_fset_a_fset_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_572_rev__fimage__eqI,axiom,
    ! [X2: a,A: fset_a,B2: produc2669932201312201782od_c_a,F: a > produc2669932201312201782od_c_a] :
      ( ( fmember_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmembe4375279799106287869od_c_a @ B2 @ ( fimage4212853655448708405od_c_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_573_rev__fimage__eqI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B2: a,F: produc2669932201312201782od_c_a > a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmember_a @ B2 @ ( fimage9200358715597384645_c_a_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_574_rev__fimage__eqI,axiom,
    ! [X2: fset_a,A: fset_fset_a,B2: produc2669932201312201782od_c_a,F: fset_a > produc2669932201312201782od_c_a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmembe4375279799106287869od_c_a @ B2 @ ( fimage3543014039228378575od_c_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_575_rev__fimage__eqI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B2: fset_a,F: produc2669932201312201782od_c_a > fset_a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmember_fset_a @ B2 @ ( fimage2540242761017857067fset_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_576_rev__fimage__eqI,axiom,
    ! [X2: a,A: fset_a,B2: produc21584292878202806fset_a,F: a > produc21584292878202806fset_a] :
      ( ( fmember_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmembe5403795861166450045fset_a @ B2 @ ( fimage5428608421149153717fset_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_577_rev__fimage__eqI,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B2: a,F: produc21584292878202806fset_a > a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ A )
     => ( ( B2
          = ( F @ X2 ) )
       => ( fmember_a @ B2 @ ( fimage5721101993989878341et_a_a @ F @ A ) ) ) ) ).

% rev_fimage_eqI
thf(fact_578_fimage__cong,axiom,
    ! [M2: fset_P8649754740804009550od_b_c,N: fset_P8649754740804009550od_b_c,F: product_prod_b_c > produc21584292878202806fset_a,G: product_prod_b_c > produc21584292878202806fset_a] :
      ( ( M2 = N )
     => ( ! [X3: product_prod_b_c] :
            ( ( fmembe4950632942993856815od_b_c @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( fimage3947511181509736139fset_a @ F @ M2 )
          = ( fimage3947511181509736139fset_a @ G @ N ) ) ) ) ).

% fimage_cong
thf(fact_579_fimage__cong,axiom,
    ! [M2: fset_fset_a,N: fset_fset_a,F: fset_a > fset_P2596122609397385116fset_a,G: fset_a > fset_P2596122609397385116fset_a] :
      ( ( M2 = N )
     => ( ! [X3: fset_a] :
            ( ( fmember_fset_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( fimage8078386234238232501fset_a @ F @ M2 )
          = ( fimage8078386234238232501fset_a @ G @ N ) ) ) ) ).

% fimage_cong
thf(fact_580_fimage__cong,axiom,
    ! [M2: fset_P2596122609397385116fset_a,N: fset_P2596122609397385116fset_a,F: produc21584292878202806fset_a > fset_a,G: produc21584292878202806fset_a > fset_a] :
      ( ( M2 = N )
     => ( ! [X3: produc21584292878202806fset_a] :
            ( ( fmembe5403795861166450045fset_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( fimage5315711910825859755fset_a @ F @ M2 )
          = ( fimage5315711910825859755fset_a @ G @ N ) ) ) ) ).

% fimage_cong
thf(fact_581_fimage__cong,axiom,
    ! [M2: fset_P8043821951902805788od_c_a,N: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > a,G: produc2669932201312201782od_c_a > a] :
      ( ( M2 = N )
     => ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( fmembe4375279799106287869od_c_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( fimage9200358715597384645_c_a_a @ F @ M2 )
          = ( fimage9200358715597384645_c_a_a @ G @ N ) ) ) ) ).

% fimage_cong
thf(fact_582_fimage__cong,axiom,
    ! [M2: fset_P8043821951902805788od_c_a,N: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c,G: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( M2 = N )
     => ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( fmembe4375279799106287869od_c_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( fimage7236410608496998447od_b_c @ F @ M2 )
          = ( fimage7236410608496998447od_b_c @ G @ N ) ) ) ) ).

% fimage_cong
thf(fact_583_fimageI,axiom,
    ! [X2: a,A: fset_a,F: a > a] :
      ( ( fmember_a @ X2 @ A )
     => ( fmember_a @ ( F @ X2 ) @ ( fimage_a_a @ F @ A ) ) ) ).

% fimageI
thf(fact_584_fimageI,axiom,
    ! [X2: a,A: fset_a,F: a > fset_a] :
      ( ( fmember_a @ X2 @ A )
     => ( fmember_fset_a @ ( F @ X2 ) @ ( fimage_a_fset_a @ F @ A ) ) ) ).

% fimageI
thf(fact_585_fimageI,axiom,
    ! [X2: fset_a,A: fset_fset_a,F: fset_a > a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( fmember_a @ ( F @ X2 ) @ ( fimage_fset_a_a @ F @ A ) ) ) ).

% fimageI
thf(fact_586_fimageI,axiom,
    ! [X2: fset_a,A: fset_fset_a,F: fset_a > fset_a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( fmember_fset_a @ ( F @ X2 ) @ ( fimage_fset_a_fset_a @ F @ A ) ) ) ).

% fimageI
thf(fact_587_fimageI,axiom,
    ! [X2: a,A: fset_a,F: a > produc2669932201312201782od_c_a] :
      ( ( fmember_a @ X2 @ A )
     => ( fmembe4375279799106287869od_c_a @ ( F @ X2 ) @ ( fimage4212853655448708405od_c_a @ F @ A ) ) ) ).

% fimageI
thf(fact_588_fimageI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( fmember_a @ ( F @ X2 ) @ ( fimage9200358715597384645_c_a_a @ F @ A ) ) ) ).

% fimageI
thf(fact_589_fimageI,axiom,
    ! [X2: fset_a,A: fset_fset_a,F: fset_a > produc2669932201312201782od_c_a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( fmembe4375279799106287869od_c_a @ ( F @ X2 ) @ ( fimage3543014039228378575od_c_a @ F @ A ) ) ) ).

% fimageI
thf(fact_590_fimageI,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > fset_a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( fmember_fset_a @ ( F @ X2 ) @ ( fimage2540242761017857067fset_a @ F @ A ) ) ) ).

% fimageI
thf(fact_591_fimageI,axiom,
    ! [X2: a,A: fset_a,F: a > produc21584292878202806fset_a] :
      ( ( fmember_a @ X2 @ A )
     => ( fmembe5403795861166450045fset_a @ ( F @ X2 ) @ ( fimage5428608421149153717fset_a @ F @ A ) ) ) ).

% fimageI
thf(fact_592_fimageI,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,F: produc21584292878202806fset_a > a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ A )
     => ( fmember_a @ ( F @ X2 ) @ ( fimage5721101993989878341et_a_a @ F @ A ) ) ) ).

% fimageI
thf(fact_593_fimageE,axiom,
    ! [B2: a,F: a > a,A: fset_a] :
      ( ( fmember_a @ B2 @ ( fimage_a_a @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmember_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_594_fimageE,axiom,
    ! [B2: a,F: fset_a > a,A: fset_fset_a] :
      ( ( fmember_a @ B2 @ ( fimage_fset_a_a @ F @ A ) )
     => ~ ! [X3: fset_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmember_fset_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_595_fimageE,axiom,
    ! [B2: fset_a,F: a > fset_a,A: fset_a] :
      ( ( fmember_fset_a @ B2 @ ( fimage_a_fset_a @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmember_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_596_fimageE,axiom,
    ! [B2: fset_a,F: fset_a > fset_a,A: fset_fset_a] :
      ( ( fmember_fset_a @ B2 @ ( fimage_fset_a_fset_a @ F @ A ) )
     => ~ ! [X3: fset_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmember_fset_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_597_fimageE,axiom,
    ! [B2: a,F: produc2669932201312201782od_c_a > a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmember_a @ B2 @ ( fimage9200358715597384645_c_a_a @ F @ A ) )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmembe4375279799106287869od_c_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_598_fimageE,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: a > produc2669932201312201782od_c_a,A: fset_a] :
      ( ( fmembe4375279799106287869od_c_a @ B2 @ ( fimage4212853655448708405od_c_a @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmember_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_599_fimageE,axiom,
    ! [B2: fset_a,F: produc2669932201312201782od_c_a > fset_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmember_fset_a @ B2 @ ( fimage2540242761017857067fset_a @ F @ A ) )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmembe4375279799106287869od_c_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_600_fimageE,axiom,
    ! [B2: produc2669932201312201782od_c_a,F: fset_a > produc2669932201312201782od_c_a,A: fset_fset_a] :
      ( ( fmembe4375279799106287869od_c_a @ B2 @ ( fimage3543014039228378575od_c_a @ F @ A ) )
     => ~ ! [X3: fset_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmember_fset_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_601_fimageE,axiom,
    ! [B2: product_prod_b_c,F: produc2669932201312201782od_c_a > product_prod_b_c,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4950632942993856815od_b_c @ B2 @ ( fimage7236410608496998447od_b_c @ F @ A ) )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmembe4375279799106287869od_c_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_602_fimageE,axiom,
    ! [B2: a,F: produc21584292878202806fset_a > a,A: fset_P2596122609397385116fset_a] :
      ( ( fmember_a @ B2 @ ( fimage5721101993989878341et_a_a @ F @ A ) )
     => ~ ! [X3: produc21584292878202806fset_a] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( fmembe5403795861166450045fset_a @ X3 @ A ) ) ) ).

% fimageE
thf(fact_603_fset_Omap__cong,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Ya: fset_P2596122609397385116fset_a,F: produc21584292878202806fset_a > fset_a,G: produc21584292878202806fset_a > fset_a] :
      ( ( X2 = Ya )
     => ( ! [Z3: produc21584292878202806fset_a] :
            ( ( member4899939873214261983fset_a @ Z3 @ ( fset_P7830090094348214821fset_a @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( fimage5315711910825859755fset_a @ F @ X2 )
          = ( fimage5315711910825859755fset_a @ G @ Ya ) ) ) ) ).

% fset.map_cong
thf(fact_604_fset_Omap__cong,axiom,
    ! [X2: fset_P8043821951902805788od_c_a,Ya: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > a,G: produc2669932201312201782od_c_a > a] :
      ( ( X2 = Ya )
     => ( ! [Z3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ Z3 @ ( fset_P7369495421554515365od_c_a @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( fimage9200358715597384645_c_a_a @ F @ X2 )
          = ( fimage9200358715597384645_c_a_a @ G @ Ya ) ) ) ) ).

% fset.map_cong
thf(fact_605_fset_Omap__cong,axiom,
    ! [X2: fset_fset_a,Ya: fset_fset_a,F: fset_a > fset_P2596122609397385116fset_a,G: fset_a > fset_P2596122609397385116fset_a] :
      ( ( X2 = Ya )
     => ( ! [Z3: fset_a] :
            ( ( member_fset_a @ Z3 @ ( fset_fset_a2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( fimage8078386234238232501fset_a @ F @ X2 )
          = ( fimage8078386234238232501fset_a @ G @ Ya ) ) ) ) ).

% fset.map_cong
thf(fact_606_fset_Omap__cong,axiom,
    ! [X2: fset_P8043821951902805788od_c_a,Ya: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c,G: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( X2 = Ya )
     => ( ! [Z3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ Z3 @ ( fset_P7369495421554515365od_c_a @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( fimage7236410608496998447od_b_c @ F @ X2 )
          = ( fimage7236410608496998447od_b_c @ G @ Ya ) ) ) ) ).

% fset.map_cong
thf(fact_607_fset_Omap__cong,axiom,
    ! [X2: fset_P8649754740804009550od_b_c,Ya: fset_P8649754740804009550od_b_c,F: product_prod_b_c > produc21584292878202806fset_a,G: product_prod_b_c > produc21584292878202806fset_a] :
      ( ( X2 = Ya )
     => ( ! [Z3: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ Z3 @ ( fset_P8805262608305245655od_b_c @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( fimage3947511181509736139fset_a @ F @ X2 )
          = ( fimage3947511181509736139fset_a @ G @ Ya ) ) ) ) ).

% fset.map_cong
thf(fact_608_fset_Omap__cong0,axiom,
    ! [X2: fset_P2596122609397385116fset_a,F: produc21584292878202806fset_a > fset_a,G: produc21584292878202806fset_a > fset_a] :
      ( ! [Z3: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ Z3 @ ( fset_P7830090094348214821fset_a @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( fimage5315711910825859755fset_a @ F @ X2 )
        = ( fimage5315711910825859755fset_a @ G @ X2 ) ) ) ).

% fset.map_cong0
thf(fact_609_fset_Omap__cong0,axiom,
    ! [X2: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > a,G: produc2669932201312201782od_c_a > a] :
      ( ! [Z3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ Z3 @ ( fset_P7369495421554515365od_c_a @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( fimage9200358715597384645_c_a_a @ F @ X2 )
        = ( fimage9200358715597384645_c_a_a @ G @ X2 ) ) ) ).

% fset.map_cong0
thf(fact_610_fset_Omap__cong0,axiom,
    ! [X2: fset_fset_a,F: fset_a > fset_P2596122609397385116fset_a,G: fset_a > fset_P2596122609397385116fset_a] :
      ( ! [Z3: fset_a] :
          ( ( member_fset_a @ Z3 @ ( fset_fset_a2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( fimage8078386234238232501fset_a @ F @ X2 )
        = ( fimage8078386234238232501fset_a @ G @ X2 ) ) ) ).

% fset.map_cong0
thf(fact_611_fset_Omap__cong0,axiom,
    ! [X2: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c,G: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ! [Z3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ Z3 @ ( fset_P7369495421554515365od_c_a @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( fimage7236410608496998447od_b_c @ F @ X2 )
        = ( fimage7236410608496998447od_b_c @ G @ X2 ) ) ) ).

% fset.map_cong0
thf(fact_612_fset_Omap__cong0,axiom,
    ! [X2: fset_P8649754740804009550od_b_c,F: product_prod_b_c > produc21584292878202806fset_a,G: product_prod_b_c > produc21584292878202806fset_a] :
      ( ! [Z3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ Z3 @ ( fset_P8805262608305245655od_b_c @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( fimage3947511181509736139fset_a @ F @ X2 )
        = ( fimage3947511181509736139fset_a @ G @ X2 ) ) ) ).

% fset.map_cong0
thf(fact_613_fset_Oinj__map__strong,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Xa4: fset_P2596122609397385116fset_a,F: produc21584292878202806fset_a > fset_a,Fa: produc21584292878202806fset_a > fset_a] :
      ( ! [Z3: produc21584292878202806fset_a,Za: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ Z3 @ ( fset_P7830090094348214821fset_a @ X2 ) )
         => ( ( member4899939873214261983fset_a @ Za @ ( fset_P7830090094348214821fset_a @ Xa4 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( fimage5315711910825859755fset_a @ F @ X2 )
          = ( fimage5315711910825859755fset_a @ Fa @ Xa4 ) )
       => ( X2 = Xa4 ) ) ) ).

% fset.inj_map_strong
thf(fact_614_fset_Oinj__map__strong,axiom,
    ! [X2: fset_P8043821951902805788od_c_a,Xa4: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > a,Fa: produc2669932201312201782od_c_a > a] :
      ( ! [Z3: produc2669932201312201782od_c_a,Za: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ Z3 @ ( fset_P7369495421554515365od_c_a @ X2 ) )
         => ( ( member5502040268208861279od_c_a @ Za @ ( fset_P7369495421554515365od_c_a @ Xa4 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( fimage9200358715597384645_c_a_a @ F @ X2 )
          = ( fimage9200358715597384645_c_a_a @ Fa @ Xa4 ) )
       => ( X2 = Xa4 ) ) ) ).

% fset.inj_map_strong
thf(fact_615_fset_Oinj__map__strong,axiom,
    ! [X2: fset_fset_a,Xa4: fset_fset_a,F: fset_a > fset_P2596122609397385116fset_a,Fa: fset_a > fset_P2596122609397385116fset_a] :
      ( ! [Z3: fset_a,Za: fset_a] :
          ( ( member_fset_a @ Z3 @ ( fset_fset_a2 @ X2 ) )
         => ( ( member_fset_a @ Za @ ( fset_fset_a2 @ Xa4 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( fimage8078386234238232501fset_a @ F @ X2 )
          = ( fimage8078386234238232501fset_a @ Fa @ Xa4 ) )
       => ( X2 = Xa4 ) ) ) ).

% fset.inj_map_strong
thf(fact_616_fset_Oinj__map__strong,axiom,
    ! [X2: fset_P8043821951902805788od_c_a,Xa4: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c,Fa: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ! [Z3: produc2669932201312201782od_c_a,Za: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ Z3 @ ( fset_P7369495421554515365od_c_a @ X2 ) )
         => ( ( member5502040268208861279od_c_a @ Za @ ( fset_P7369495421554515365od_c_a @ Xa4 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( fimage7236410608496998447od_b_c @ F @ X2 )
          = ( fimage7236410608496998447od_b_c @ Fa @ Xa4 ) )
       => ( X2 = Xa4 ) ) ) ).

% fset.inj_map_strong
thf(fact_617_fset_Oinj__map__strong,axiom,
    ! [X2: fset_P8649754740804009550od_b_c,Xa4: fset_P8649754740804009550od_b_c,F: product_prod_b_c > produc21584292878202806fset_a,Fa: product_prod_b_c > produc21584292878202806fset_a] :
      ( ! [Z3: product_prod_b_c,Za: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ Z3 @ ( fset_P8805262608305245655od_b_c @ X2 ) )
         => ( ( member7862447941013992593od_b_c @ Za @ ( fset_P8805262608305245655od_b_c @ Xa4 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( fimage3947511181509736139fset_a @ F @ X2 )
          = ( fimage3947511181509736139fset_a @ Fa @ Xa4 ) )
       => ( X2 = Xa4 ) ) ) ).

% fset.inj_map_strong
thf(fact_618_fset_Omap__ident__strong,axiom,
    ! [T4: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ! [Z3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ Z3 @ ( fset_P7369495421554515365od_c_a @ T4 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( fimage2423425181458651133od_c_a @ F @ T4 )
        = T4 ) ) ).

% fset.map_ident_strong
thf(fact_619_fset_Omap__ident__strong,axiom,
    ! [T4: fset_P9065897409413591390od_c_a,F: produc3205353004677322104od_c_a > produc3205353004677322104od_c_a] :
      ( ! [Z3: produc3205353004677322104od_c_a] :
          ( ( member6508725278956418977od_c_a @ Z3 @ ( fset_P4425921658469021927od_c_a @ T4 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( fimage5758177350403239037od_c_a @ F @ T4 )
        = T4 ) ) ).

% fset.map_ident_strong
thf(fact_620_fset_Omap__ident__strong,axiom,
    ! [T4: fset_P2596122609397385116fset_a,F: produc21584292878202806fset_a > produc21584292878202806fset_a] :
      ( ! [Z3: produc21584292878202806fset_a] :
          ( ( member4899939873214261983fset_a @ Z3 @ ( fset_P7830090094348214821fset_a @ T4 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( fimage1420713159605766909fset_a @ F @ T4 )
        = T4 ) ) ).

% fset.map_ident_strong
thf(fact_621_fset_Omap__ident__strong,axiom,
    ! [T4: fset_a,F: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( fset_a2 @ T4 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( fimage_a_a @ F @ T4 )
        = T4 ) ) ).

% fset.map_ident_strong
thf(fact_622_funionI2,axiom,
    ! [C: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ C @ B )
     => ( fmembe4375279799106287869od_c_a @ C @ ( sup_su7793179031141480176od_c_a @ A @ B ) ) ) ).

% funionI2
thf(fact_623_funionI2,axiom,
    ! [C: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ C @ B )
     => ( fmembe5403795861166450045fset_a @ C @ ( sup_su6818703939759823728fset_a @ A @ B ) ) ) ).

% funionI2
thf(fact_624_funionI2,axiom,
    ! [C: fset_a,B: fset_fset_a,A: fset_fset_a] :
      ( ( fmember_fset_a @ C @ B )
     => ( fmember_fset_a @ C @ ( sup_sup_fset_fset_a @ A @ B ) ) ) ).

% funionI2
thf(fact_625_funionI2,axiom,
    ! [C: a,B: fset_a,A: fset_a] :
      ( ( fmember_a @ C @ B )
     => ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) ) ) ).

% funionI2
thf(fact_626_funionI1,axiom,
    ! [C: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ C @ A )
     => ( fmembe4375279799106287869od_c_a @ C @ ( sup_su7793179031141480176od_c_a @ A @ B ) ) ) ).

% funionI1
thf(fact_627_funionI1,axiom,
    ! [C: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ C @ A )
     => ( fmembe5403795861166450045fset_a @ C @ ( sup_su6818703939759823728fset_a @ A @ B ) ) ) ).

% funionI1
thf(fact_628_funionI1,axiom,
    ! [C: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fmember_fset_a @ C @ A )
     => ( fmember_fset_a @ C @ ( sup_sup_fset_fset_a @ A @ B ) ) ) ).

% funionI1
thf(fact_629_funionI1,axiom,
    ! [C: a,A: fset_a,B: fset_a] :
      ( ( fmember_a @ C @ A )
     => ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) ) ) ).

% funionI1
thf(fact_630_funionE,axiom,
    ! [C: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ C @ ( sup_su7793179031141480176od_c_a @ A @ B ) )
     => ( ~ ( fmembe4375279799106287869od_c_a @ C @ A )
       => ( fmembe4375279799106287869od_c_a @ C @ B ) ) ) ).

% funionE
thf(fact_631_funionE,axiom,
    ! [C: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ C @ ( sup_su6818703939759823728fset_a @ A @ B ) )
     => ( ~ ( fmembe5403795861166450045fset_a @ C @ A )
       => ( fmembe5403795861166450045fset_a @ C @ B ) ) ) ).

% funionE
thf(fact_632_funionE,axiom,
    ! [C: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fmember_fset_a @ C @ ( sup_sup_fset_fset_a @ A @ B ) )
     => ( ~ ( fmember_fset_a @ C @ A )
       => ( fmember_fset_a @ C @ B ) ) ) ).

% funionE
thf(fact_633_funionE,axiom,
    ! [C: a,A: fset_a,B: fset_a] :
      ( ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) )
     => ( ~ ( fmember_a @ C @ A )
       => ( fmember_a @ C @ B ) ) ) ).

% funionE
thf(fact_634_snd__conv,axiom,
    ! [X1: b,X23: c] :
      ( ( product_snd_b_c @ ( product_Pair_b_c @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_635_snd__conv,axiom,
    ! [X1: c,X23: fset_a] :
      ( ( product_snd_c_fset_a @ ( produc1535188672328598523fset_a @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_636_snd__conv,axiom,
    ! [X1: c,X23: a] :
      ( ( product_snd_c_a @ ( product_Pair_c_a @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_637_snd__conv,axiom,
    ! [X1: b,X23: produc4838349196543436939fset_a] :
      ( ( produc8704244169652099857fset_a @ ( produc4496741132823077603fset_a @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_638_snd__conv,axiom,
    ! [X1: b,X23: product_prod_c_a] :
      ( ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_639_snd__conv,axiom,
    ! [X1: a,X23: produc2133359284458116493od_c_a] :
      ( ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_640_snd__conv,axiom,
    ! [X1: product_prod_c_a,X23: produc2669932201312201782od_c_a] :
      ( ( produc4643634399596806294od_c_a @ ( produc7911353075612250984od_c_a @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_641_snd__conv,axiom,
    ! [X1: fset_a,X23: produc1137077444975408883fset_a] :
      ( ( produc8188425890649629780fset_a @ ( produc3008349866153638182fset_a @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_642_snd__eqD,axiom,
    ! [X2: b,Y2: c,A2: c] :
      ( ( ( product_snd_b_c @ ( product_Pair_b_c @ X2 @ Y2 ) )
        = A2 )
     => ( Y2 = A2 ) ) ).

% snd_eqD
thf(fact_643_snd__eqD,axiom,
    ! [X2: c,Y2: fset_a,A2: fset_a] :
      ( ( ( product_snd_c_fset_a @ ( produc1535188672328598523fset_a @ X2 @ Y2 ) )
        = A2 )
     => ( Y2 = A2 ) ) ).

% snd_eqD
thf(fact_644_snd__eqD,axiom,
    ! [X2: c,Y2: a,A2: a] :
      ( ( ( product_snd_c_a @ ( product_Pair_c_a @ X2 @ Y2 ) )
        = A2 )
     => ( Y2 = A2 ) ) ).

% snd_eqD
thf(fact_645_snd__eqD,axiom,
    ! [X2: b,Y2: produc4838349196543436939fset_a,A2: produc4838349196543436939fset_a] :
      ( ( ( produc8704244169652099857fset_a @ ( produc4496741132823077603fset_a @ X2 @ Y2 ) )
        = A2 )
     => ( Y2 = A2 ) ) ).

% snd_eqD
thf(fact_646_snd__eqD,axiom,
    ! [X2: b,Y2: product_prod_c_a,A2: product_prod_c_a] :
      ( ( ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X2 @ Y2 ) )
        = A2 )
     => ( Y2 = A2 ) ) ).

% snd_eqD
thf(fact_647_snd__eqD,axiom,
    ! [X2: a,Y2: produc2133359284458116493od_c_a,A2: produc2133359284458116493od_c_a] :
      ( ( ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X2 @ Y2 ) )
        = A2 )
     => ( Y2 = A2 ) ) ).

% snd_eqD
thf(fact_648_snd__eqD,axiom,
    ! [X2: product_prod_c_a,Y2: produc2669932201312201782od_c_a,A2: produc2669932201312201782od_c_a] :
      ( ( ( produc4643634399596806294od_c_a @ ( produc7911353075612250984od_c_a @ X2 @ Y2 ) )
        = A2 )
     => ( Y2 = A2 ) ) ).

% snd_eqD
thf(fact_649_snd__eqD,axiom,
    ! [X2: fset_a,Y2: produc1137077444975408883fset_a,A2: produc1137077444975408883fset_a] :
      ( ( ( produc8188425890649629780fset_a @ ( produc3008349866153638182fset_a @ X2 @ Y2 ) )
        = A2 )
     => ( Y2 = A2 ) ) ).

% snd_eqD
thf(fact_650_fst__conv,axiom,
    ! [X1: b,X23: c] :
      ( ( product_fst_b_c @ ( product_Pair_b_c @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_651_fst__conv,axiom,
    ! [X1: c,X23: fset_a] :
      ( ( product_fst_c_fset_a @ ( produc1535188672328598523fset_a @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_652_fst__conv,axiom,
    ! [X1: c,X23: a] :
      ( ( product_fst_c_a @ ( product_Pair_c_a @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_653_fst__conv,axiom,
    ! [X1: b,X23: produc4838349196543436939fset_a] :
      ( ( produc5616988955220391631fset_a @ ( produc4496741132823077603fset_a @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_654_fst__conv,axiom,
    ! [X1: b,X23: product_prod_c_a] :
      ( ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_655_fst__conv,axiom,
    ! [X1: a,X23: produc2133359284458116493od_c_a] :
      ( ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_656_fst__conv,axiom,
    ! [X1: product_prod_c_a,X23: produc2669932201312201782od_c_a] :
      ( ( produc8171714036265584980od_c_a @ ( produc7911353075612250984od_c_a @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_657_fst__conv,axiom,
    ! [X1: fset_a,X23: produc1137077444975408883fset_a] :
      ( ( produc1739398754328533778fset_a @ ( produc3008349866153638182fset_a @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_658_fst__eqD,axiom,
    ! [X2: b,Y2: c,A2: b] :
      ( ( ( product_fst_b_c @ ( product_Pair_b_c @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_659_fst__eqD,axiom,
    ! [X2: c,Y2: fset_a,A2: c] :
      ( ( ( product_fst_c_fset_a @ ( produc1535188672328598523fset_a @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_660_fst__eqD,axiom,
    ! [X2: c,Y2: a,A2: c] :
      ( ( ( product_fst_c_a @ ( product_Pair_c_a @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_661_fst__eqD,axiom,
    ! [X2: b,Y2: produc4838349196543436939fset_a,A2: b] :
      ( ( ( produc5616988955220391631fset_a @ ( produc4496741132823077603fset_a @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_662_fst__eqD,axiom,
    ! [X2: b,Y2: product_prod_c_a,A2: b] :
      ( ( ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_663_fst__eqD,axiom,
    ! [X2: a,Y2: produc2133359284458116493od_c_a,A2: a] :
      ( ( ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_664_fst__eqD,axiom,
    ! [X2: product_prod_c_a,Y2: produc2669932201312201782od_c_a,A2: product_prod_c_a] :
      ( ( ( produc8171714036265584980od_c_a @ ( produc7911353075612250984od_c_a @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_665_fst__eqD,axiom,
    ! [X2: fset_a,Y2: produc1137077444975408883fset_a,A2: fset_a] :
      ( ( ( produc1739398754328533778fset_a @ ( produc3008349866153638182fset_a @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_666_fminusD2,axiom,
    ! [C: a,A: fset_a,B: fset_a] :
      ( ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) )
     => ~ ( fmember_a @ C @ B ) ) ).

% fminusD2
thf(fact_667_fminusD2,axiom,
    ! [C: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ C @ ( minus_7225097421573603715fset_a @ A @ B ) )
     => ~ ( fmembe5403795861166450045fset_a @ C @ B ) ) ).

% fminusD2
thf(fact_668_fminusD2,axiom,
    ! [C: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ C @ ( minus_8021415692904097795od_c_a @ A @ B ) )
     => ~ ( fmembe4375279799106287869od_c_a @ C @ B ) ) ).

% fminusD2
thf(fact_669_fminusD2,axiom,
    ! [C: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fmember_fset_a @ C @ ( minus_5856361552470604977fset_a @ A @ B ) )
     => ~ ( fmember_fset_a @ C @ B ) ) ).

% fminusD2
thf(fact_670_fminusD1,axiom,
    ! [C: a,A: fset_a,B: fset_a] :
      ( ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) )
     => ( fmember_a @ C @ A ) ) ).

% fminusD1
thf(fact_671_fminusD1,axiom,
    ! [C: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ C @ ( minus_7225097421573603715fset_a @ A @ B ) )
     => ( fmembe5403795861166450045fset_a @ C @ A ) ) ).

% fminusD1
thf(fact_672_fminusD1,axiom,
    ! [C: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ C @ ( minus_8021415692904097795od_c_a @ A @ B ) )
     => ( fmembe4375279799106287869od_c_a @ C @ A ) ) ).

% fminusD1
thf(fact_673_fminusD1,axiom,
    ! [C: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fmember_fset_a @ C @ ( minus_5856361552470604977fset_a @ A @ B ) )
     => ( fmember_fset_a @ C @ A ) ) ).

% fminusD1
thf(fact_674_fminusE,axiom,
    ! [C: a,A: fset_a,B: fset_a] :
      ( ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) )
     => ~ ( ( fmember_a @ C @ A )
         => ( fmember_a @ C @ B ) ) ) ).

% fminusE
thf(fact_675_fminusE,axiom,
    ! [C: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ C @ ( minus_7225097421573603715fset_a @ A @ B ) )
     => ~ ( ( fmembe5403795861166450045fset_a @ C @ A )
         => ( fmembe5403795861166450045fset_a @ C @ B ) ) ) ).

% fminusE
thf(fact_676_fminusE,axiom,
    ! [C: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ C @ ( minus_8021415692904097795od_c_a @ A @ B ) )
     => ~ ( ( fmembe4375279799106287869od_c_a @ C @ A )
         => ( fmembe4375279799106287869od_c_a @ C @ B ) ) ) ).

% fminusE
thf(fact_677_fminusE,axiom,
    ! [C: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( fmember_fset_a @ C @ ( minus_5856361552470604977fset_a @ A @ B ) )
     => ~ ( ( fmember_fset_a @ C @ A )
         => ( fmember_fset_a @ C @ B ) ) ) ).

% fminusE
thf(fact_678_funion__fempty__left,axiom,
    ! [B: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ bot_bo2316247384970961928fset_a @ B )
      = B ) ).

% funion_fempty_left
thf(fact_679_funion__fempty__left,axiom,
    ! [B: fset_a] :
      ( ( sup_sup_fset_a @ bot_bot_fset_a @ B )
      = B ) ).

% funion_fempty_left
thf(fact_680_funion__fempty__left,axiom,
    ! [B: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ bot_bot_fset_fset_a @ B )
      = B ) ).

% funion_fempty_left
thf(fact_681_funion__fempty__right,axiom,
    ! [A: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ A @ bot_bo2316247384970961928fset_a )
      = A ) ).

% funion_fempty_right
thf(fact_682_funion__fempty__right,axiom,
    ! [A: fset_a] :
      ( ( sup_sup_fset_a @ A @ bot_bot_fset_a )
      = A ) ).

% funion_fempty_right
thf(fact_683_funion__fempty__right,axiom,
    ! [A: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ A @ bot_bot_fset_fset_a )
      = A ) ).

% funion_fempty_right
thf(fact_684_pair__list__eqI,axiom,
    ! [Xs: list_P3569802872000582161fset_a,Ys: list_P3569802872000582161fset_a] :
      ( ( ( map_Pr3277271672726795257et_a_c @ product_fst_c_fset_a @ Xs )
        = ( map_Pr3277271672726795257et_a_c @ product_fst_c_fset_a @ Ys ) )
     => ( ( ( map_Pr5736923103159262045fset_a @ product_snd_c_fset_a @ Xs )
          = ( map_Pr5736923103159262045fset_a @ product_snd_c_fset_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_685_pair__list__eqI,axiom,
    ! [Xs: list_P125642481956313003od_c_a,Ys: list_P125642481956313003od_c_a] :
      ( ( ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Xs )
        = ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a @ Ys ) )
     => ( ( ( map_Pr5866436826731527133_c_a_a @ product_snd_c_a @ Xs )
          = ( map_Pr5866436826731527133_c_a_a @ product_snd_c_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_686_pair__list__eqI,axiom,
    ! [Xs: list_P5940095693748001145fset_a,Ys: list_P5940095693748001145fset_a] :
      ( ( ( map_Pr3857243663455625872et_a_b @ produc5616988955220391631fset_a @ Xs )
        = ( map_Pr3857243663455625872et_a_b @ produc5616988955220391631fset_a @ Ys ) )
     => ( ( ( map_Pr2484813270135829148fset_a @ produc8704244169652099857fset_a @ Xs )
          = ( map_Pr2484813270135829148fset_a @ produc8704244169652099857fset_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_687_pair__list__eqI,axiom,
    ! [Xs: list_P5767727266746070163od_c_a,Ys: list_P5767727266746070163od_c_a] :
      ( ( ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Xs )
        = ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a @ Ys ) )
     => ( ( ( map_Pr6684165712643126428od_c_a @ produc5221446324160585515od_c_a @ Xs )
          = ( map_Pr6684165712643126428od_c_a @ produc5221446324160585515od_c_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_688_pair__list__eqI,axiom,
    ! [Xs: list_P6327159017948738492od_c_a,Ys: list_P6327159017948738492od_c_a] :
      ( ( ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Xs )
        = ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a @ Ys ) )
     => ( ( ( map_Pr3684547939035685915od_c_a @ produc8088130389382527188od_c_a @ Xs )
          = ( map_Pr3684547939035685915od_c_a @ produc8088130389382527188od_c_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_689_pair__list__eqI,axiom,
    ! [Xs: list_P1562850763667305982od_c_a,Ys: list_P1562850763667305982od_c_a] :
      ( ( ( map_Pr7059746129634632049od_c_a @ produc8171714036265584980od_c_a @ Xs )
        = ( map_Pr7059746129634632049od_c_a @ produc8171714036265584980od_c_a @ Ys ) )
     => ( ( ( map_Pr2148725087721669634od_c_a @ produc4643634399596806294od_c_a @ Xs )
          = ( map_Pr2148725087721669634od_c_a @ produc4643634399596806294od_c_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_690_pair__list__eqI,axiom,
    ! [Xs: list_P1058411382900698172fset_a,Ys: list_P1058411382900698172fset_a] :
      ( ( ( map_Pr4352688505896054258fset_a @ produc1739398754328533778fset_a @ Xs )
        = ( map_Pr4352688505896054258fset_a @ produc1739398754328533778fset_a @ Ys ) )
     => ( ( ( map_Pr1712903356274806273fset_a @ produc8188425890649629780fset_a @ Xs )
          = ( map_Pr1712903356274806273fset_a @ produc8188425890649629780fset_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_691_pathlike_Onil,axiom,
    ! [Ts: fset_P2596122609397385116fset_a,Q: fset_a] : ( pathlike_fset_a_b_c @ Ts @ Q @ nil_Pr4260649278601056188fset_a ) ).

% pathlike.nil
thf(fact_692_language__state__containment,axiom,
    ! [M2: fsm_Pr1175075309171164859fset_a,Q: produc1137077444975408883fset_a,P: list_P8742742886961787479fset_a,Io: list_P3569802872000582161fset_a] :
      ( ( path_P1724798301931140637fset_a @ M2 @ Q @ P )
     => ( ( ( map_Pr2785363058057876286fset_a
            @ ^ [T: produc7303452399287242961fset_a] : ( produc1535188672328598523fset_a @ ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T ) ) @ ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T ) ) ) )
            @ P )
          = Io )
       => ( member2596827339850278714fset_a @ Io @ ( lS_Pro8627259047355266299fset_a @ M2 @ Q ) ) ) ) ).

% language_state_containment
thf(fact_693_language__state__containment,axiom,
    ! [M2: fsm_Pr7813689363856283786_c_a_a,Q: produc2133359284458116493od_c_a,P: list_P1143992737879525208od_c_a,Io: list_P8814837271076367046_c_a_a] :
      ( ( path_P5541858805596636140_c_a_a @ M2 @ Q @ P )
     => ( ( ( map_Pr132765484133579058_c_a_a
            @ ^ [T: produc1135898663509404882od_c_a] : ( produc7661602042108997680_c_a_a @ ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) ) )
            @ P )
          = Io )
       => ( member5697445122164201327_c_a_a @ Io @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ Q ) ) ) ) ).

% language_state_containment
thf(fact_694_language__state__containment,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,Q: produc4838349196543436939fset_a,P: list_P5281568020286996248fset_a,Io: list_P7633400287818446792et_a_b] :
      ( ( path_P379120284129538668et_a_b @ M2 @ Q @ P )
     => ( ( ( map_Pr6485817111475779316et_a_b
            @ ^ [T: produc7245291065743820818fset_a] : ( produc4282089192724377778et_a_b @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T ) ) @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) )
            @ P )
          = Io )
       => ( member6660424755668143345et_a_b @ Io @ ( lS_Pro8532755013169807178et_a_b @ M2 @ Q ) ) ) ) ).

% language_state_containment
thf(fact_695_language__state__containment,axiom,
    ! [M2: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a,Io: list_P903359562653991662od_b_c] :
      ( ( path_a_b_c @ M2 @ Q @ P )
     => ( ( ( map_Pr776894537227705078od_b_c
            @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
            @ P )
          = Io )
       => ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ Q ) ) ) ) ).

% language_state_containment
thf(fact_696_language__state__containment,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P: list_P1562850763667305982od_c_a,Io: list_P1467974523212632366od_a_b] :
      ( ( path_P1107610407223809260_a_a_b @ M2 @ Q @ P )
     => ( ( ( map_Pr3411285261028437108od_a_b
            @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
            @ P )
          = Io )
       => ( member6895035109809442519od_a_b @ Io @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ Q ) ) ) ) ).

% language_state_containment
thf(fact_697_language__state__containment,axiom,
    ! [M2: fsm_fset_a_b_c,Q: fset_a,P: list_P1058411382900698172fset_a,Io: list_P903359562653991662od_b_c] :
      ( ( path_fset_a_b_c @ M2 @ Q @ P )
     => ( ( ( map_Pr7697975548294988406od_b_c
            @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
            @ P )
          = Io )
       => ( member6330420149250801815od_b_c @ Io @ ( lS_fset_a_b_c @ M2 @ Q ) ) ) ) ).

% language_state_containment
thf(fact_698_language__intro,axiom,
    ! [M2: fsm_Pr1175075309171164859fset_a,Q: produc1137077444975408883fset_a,P: list_P8742742886961787479fset_a] :
      ( ( path_P1724798301931140637fset_a @ M2 @ Q @ P )
     => ( member2596827339850278714fset_a
        @ ( map_Pr2785363058057876286fset_a
          @ ^ [T: produc7303452399287242961fset_a] : ( produc1535188672328598523fset_a @ ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T ) ) @ ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T ) ) ) )
          @ P )
        @ ( lS_Pro8627259047355266299fset_a @ M2 @ Q ) ) ) ).

% language_intro
thf(fact_699_language__intro,axiom,
    ! [M2: fsm_Pr7813689363856283786_c_a_a,Q: produc2133359284458116493od_c_a,P: list_P1143992737879525208od_c_a] :
      ( ( path_P5541858805596636140_c_a_a @ M2 @ Q @ P )
     => ( member5697445122164201327_c_a_a
        @ ( map_Pr132765484133579058_c_a_a
          @ ^ [T: produc1135898663509404882od_c_a] : ( produc7661602042108997680_c_a_a @ ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) ) )
          @ P )
        @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ Q ) ) ) ).

% language_intro
thf(fact_700_language__intro,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,Q: produc4838349196543436939fset_a,P: list_P5281568020286996248fset_a] :
      ( ( path_P379120284129538668et_a_b @ M2 @ Q @ P )
     => ( member6660424755668143345et_a_b
        @ ( map_Pr6485817111475779316et_a_b
          @ ^ [T: produc7245291065743820818fset_a] : ( produc4282089192724377778et_a_b @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T ) ) @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) )
          @ P )
        @ ( lS_Pro8532755013169807178et_a_b @ M2 @ Q ) ) ) ).

% language_intro
thf(fact_701_language__intro,axiom,
    ! [M2: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M2 @ Q @ P )
     => ( member6330420149250801815od_b_c
        @ ( map_Pr776894537227705078od_b_c
          @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
          @ P )
        @ ( lS_a_b_c @ M2 @ Q ) ) ) ).

% language_intro
thf(fact_702_language__intro,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P: list_P1562850763667305982od_c_a] :
      ( ( path_P1107610407223809260_a_a_b @ M2 @ Q @ P )
     => ( member6895035109809442519od_a_b
        @ ( map_Pr3411285261028437108od_a_b
          @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
          @ P )
        @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ Q ) ) ) ).

% language_intro
thf(fact_703_language__intro,axiom,
    ! [M2: fsm_fset_a_b_c,Q: fset_a,P: list_P1058411382900698172fset_a] :
      ( ( path_fset_a_b_c @ M2 @ Q @ P )
     => ( member6330420149250801815od_b_c
        @ ( map_Pr7697975548294988406od_b_c
          @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
          @ P )
        @ ( lS_fset_a_b_c @ M2 @ Q ) ) ) ).

% language_intro
thf(fact_704_surjective__pairing,axiom,
    ! [T4: product_prod_b_c] :
      ( T4
      = ( product_Pair_b_c @ ( product_fst_b_c @ T4 ) @ ( product_snd_b_c @ T4 ) ) ) ).

% surjective_pairing
thf(fact_705_surjective__pairing,axiom,
    ! [T4: produc4838349196543436939fset_a] :
      ( T4
      = ( produc1535188672328598523fset_a @ ( product_fst_c_fset_a @ T4 ) @ ( product_snd_c_fset_a @ T4 ) ) ) ).

% surjective_pairing
thf(fact_706_surjective__pairing,axiom,
    ! [T4: product_prod_c_a] :
      ( T4
      = ( product_Pair_c_a @ ( product_fst_c_a @ T4 ) @ ( product_snd_c_a @ T4 ) ) ) ).

% surjective_pairing
thf(fact_707_surjective__pairing,axiom,
    ! [T4: produc1137077444975408883fset_a] :
      ( T4
      = ( produc4496741132823077603fset_a @ ( produc5616988955220391631fset_a @ T4 ) @ ( produc8704244169652099857fset_a @ T4 ) ) ) ).

% surjective_pairing
thf(fact_708_surjective__pairing,axiom,
    ! [T4: produc2133359284458116493od_c_a] :
      ( T4
      = ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ T4 ) @ ( produc5221446324160585515od_c_a @ T4 ) ) ) ).

% surjective_pairing
thf(fact_709_surjective__pairing,axiom,
    ! [T4: produc2669932201312201782od_c_a] :
      ( T4
      = ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ T4 ) @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ).

% surjective_pairing
thf(fact_710_surjective__pairing,axiom,
    ! [T4: produc3205353004677322104od_c_a] :
      ( T4
      = ( produc7911353075612250984od_c_a @ ( produc8171714036265584980od_c_a @ T4 ) @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ).

% surjective_pairing
thf(fact_711_surjective__pairing,axiom,
    ! [T4: produc21584292878202806fset_a] :
      ( T4
      = ( produc3008349866153638182fset_a @ ( produc1739398754328533778fset_a @ T4 ) @ ( produc8188425890649629780fset_a @ T4 ) ) ) ).

% surjective_pairing
thf(fact_712_conjI__realizer,axiom,
    ! [P2: b > $o,P: b,Q2: c > $o,Q: c] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_b_c @ ( product_Pair_b_c @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_b_c @ ( product_Pair_b_c @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_713_conjI__realizer,axiom,
    ! [P2: c > $o,P: c,Q2: fset_a > $o,Q: fset_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_c_fset_a @ ( produc1535188672328598523fset_a @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_c_fset_a @ ( produc1535188672328598523fset_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_714_conjI__realizer,axiom,
    ! [P2: c > $o,P: c,Q2: a > $o,Q: a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_c_a @ ( product_Pair_c_a @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_c_a @ ( product_Pair_c_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_715_conjI__realizer,axiom,
    ! [P2: b > $o,P: b,Q2: produc4838349196543436939fset_a > $o,Q: produc4838349196543436939fset_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc5616988955220391631fset_a @ ( produc4496741132823077603fset_a @ P @ Q ) ) )
          & ( Q2 @ ( produc8704244169652099857fset_a @ ( produc4496741132823077603fset_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_716_conjI__realizer,axiom,
    ! [P2: b > $o,P: b,Q2: product_prod_c_a > $o,Q: product_prod_c_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ P @ Q ) ) )
          & ( Q2 @ ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_717_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q2: produc2133359284458116493od_c_a > $o,Q: produc2133359284458116493od_c_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ P @ Q ) ) )
          & ( Q2 @ ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_718_conjI__realizer,axiom,
    ! [P2: product_prod_c_a > $o,P: product_prod_c_a,Q2: produc2669932201312201782od_c_a > $o,Q: produc2669932201312201782od_c_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc8171714036265584980od_c_a @ ( produc7911353075612250984od_c_a @ P @ Q ) ) )
          & ( Q2 @ ( produc4643634399596806294od_c_a @ ( produc7911353075612250984od_c_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_719_conjI__realizer,axiom,
    ! [P2: fset_a > $o,P: fset_a,Q2: produc1137077444975408883fset_a > $o,Q: produc1137077444975408883fset_a] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc1739398754328533778fset_a @ ( produc3008349866153638182fset_a @ P @ Q ) ) )
          & ( Q2 @ ( produc8188425890649629780fset_a @ ( produc3008349866153638182fset_a @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_720_exI__realizer,axiom,
    ! [P2: c > b > $o,Y2: c,X2: b] :
      ( ( P2 @ Y2 @ X2 )
     => ( P2 @ ( product_snd_b_c @ ( product_Pair_b_c @ X2 @ Y2 ) ) @ ( product_fst_b_c @ ( product_Pair_b_c @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_721_exI__realizer,axiom,
    ! [P2: fset_a > c > $o,Y2: fset_a,X2: c] :
      ( ( P2 @ Y2 @ X2 )
     => ( P2 @ ( product_snd_c_fset_a @ ( produc1535188672328598523fset_a @ X2 @ Y2 ) ) @ ( product_fst_c_fset_a @ ( produc1535188672328598523fset_a @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_722_exI__realizer,axiom,
    ! [P2: a > c > $o,Y2: a,X2: c] :
      ( ( P2 @ Y2 @ X2 )
     => ( P2 @ ( product_snd_c_a @ ( product_Pair_c_a @ X2 @ Y2 ) ) @ ( product_fst_c_a @ ( product_Pair_c_a @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_723_exI__realizer,axiom,
    ! [P2: produc4838349196543436939fset_a > b > $o,Y2: produc4838349196543436939fset_a,X2: b] :
      ( ( P2 @ Y2 @ X2 )
     => ( P2 @ ( produc8704244169652099857fset_a @ ( produc4496741132823077603fset_a @ X2 @ Y2 ) ) @ ( produc5616988955220391631fset_a @ ( produc4496741132823077603fset_a @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_724_exI__realizer,axiom,
    ! [P2: product_prod_c_a > b > $o,Y2: product_prod_c_a,X2: b] :
      ( ( P2 @ Y2 @ X2 )
     => ( P2 @ ( produc5221446324160585515od_c_a @ ( produc3440865914000881661od_c_a @ X2 @ Y2 ) ) @ ( produc301949074746585577od_c_a @ ( produc3440865914000881661od_c_a @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_725_exI__realizer,axiom,
    ! [P2: produc2133359284458116493od_c_a > a > $o,Y2: produc2133359284458116493od_c_a,X2: a] :
      ( ( P2 @ Y2 @ X2 )
     => ( P2 @ ( produc8088130389382527188od_c_a @ ( produc5762801557034676134od_c_a @ X2 @ Y2 ) ) @ ( produc4225943634317630866od_c_a @ ( produc5762801557034676134od_c_a @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_726_exI__realizer,axiom,
    ! [P2: produc2669932201312201782od_c_a > product_prod_c_a > $o,Y2: produc2669932201312201782od_c_a,X2: product_prod_c_a] :
      ( ( P2 @ Y2 @ X2 )
     => ( P2 @ ( produc4643634399596806294od_c_a @ ( produc7911353075612250984od_c_a @ X2 @ Y2 ) ) @ ( produc8171714036265584980od_c_a @ ( produc7911353075612250984od_c_a @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_727_exI__realizer,axiom,
    ! [P2: produc1137077444975408883fset_a > fset_a > $o,Y2: produc1137077444975408883fset_a,X2: fset_a] :
      ( ( P2 @ Y2 @ X2 )
     => ( P2 @ ( produc8188425890649629780fset_a @ ( produc3008349866153638182fset_a @ X2 @ Y2 ) ) @ ( produc1739398754328533778fset_a @ ( produc3008349866153638182fset_a @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_728_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_b_c] :
      ( Prod
      = ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_729_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4838349196543436939fset_a] :
      ( Prod
      = ( produc1535188672328598523fset_a @ ( product_fst_c_fset_a @ Prod ) @ ( product_snd_c_fset_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_730_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_c_a] :
      ( Prod
      = ( product_Pair_c_a @ ( product_fst_c_a @ Prod ) @ ( product_snd_c_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_731_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1137077444975408883fset_a] :
      ( Prod
      = ( produc4496741132823077603fset_a @ ( produc5616988955220391631fset_a @ Prod ) @ ( produc8704244169652099857fset_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_732_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2133359284458116493od_c_a] :
      ( Prod
      = ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ Prod ) @ ( produc5221446324160585515od_c_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_733_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2669932201312201782od_c_a] :
      ( Prod
      = ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ Prod ) @ ( produc8088130389382527188od_c_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_734_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3205353004677322104od_c_a] :
      ( Prod
      = ( produc7911353075612250984od_c_a @ ( produc8171714036265584980od_c_a @ Prod ) @ ( produc4643634399596806294od_c_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_735_prod_Oexhaust__sel,axiom,
    ! [Prod: produc21584292878202806fset_a] :
      ( Prod
      = ( produc3008349866153638182fset_a @ ( produc1739398754328533778fset_a @ Prod ) @ ( produc8188425890649629780fset_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_736_LS__single__transition,axiom,
    ! [X2: c,Y2: fset_a,M2: fsm_Pr1175075309171164859fset_a,Q: produc1137077444975408883fset_a] :
      ( ( member2596827339850278714fset_a @ ( cons_P4277760901013084993fset_a @ ( produc1535188672328598523fset_a @ X2 @ Y2 ) @ nil_Pr2782780582842296209fset_a ) @ ( lS_Pro8627259047355266299fset_a @ M2 @ Q ) )
      = ( ? [X: produc7303452399287242961fset_a] :
            ( ( member9109638507678586746fset_a @ X @ ( transi1231084882770062892fset_a @ M2 ) )
            & ( ( produc2105839935146153389fset_a @ X )
              = Q )
            & ( ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ X ) )
              = X2 )
            & ( ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ X ) ) )
              = Y2 ) ) ) ) ).

% LS_single_transition
thf(fact_737_LS__single__transition,axiom,
    ! [X2: product_prod_c_a,Y2: a,M2: fsm_Pr7813689363856283786_c_a_a,Q: produc2133359284458116493od_c_a] :
      ( ( member5697445122164201327_c_a_a @ ( cons_P1791870815957980406_c_a_a @ ( produc7661602042108997680_c_a_a @ X2 @ Y2 ) @ nil_Pr9019007095756880198_c_a_a ) @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ Q ) )
      = ( ? [X: produc1135898663509404882od_c_a] :
            ( ( member6699836568822684667od_c_a @ X @ ( transi5048145386435558395_c_a_a @ M2 ) )
            & ( ( produc5475348670448042158od_c_a @ X )
              = Q )
            & ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ X ) )
              = X2 )
            & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ X ) ) )
              = Y2 ) ) ) ) ).

% LS_single_transition
thf(fact_738_LS__single__transition,axiom,
    ! [X2: fset_a,Y2: b,M2: fsm_Pr2398083450103406218et_a_b,Q: produc4838349196543436939fset_a] :
      ( ( member6660424755668143345et_a_b @ ( cons_P1519664230097792120et_a_b @ ( produc4282089192724377778et_a_b @ X2 @ Y2 ) @ nil_Pr24683911927003336et_a_b ) @ ( lS_Pro8532755013169807178et_a_b @ M2 @ Q ) )
      = ( ? [X: produc7245291065743820818fset_a] :
            ( ( member808877655204049083fset_a @ X @ ( transi3322184091305251195et_a_b @ M2 ) )
            & ( ( produc8095675384246233966fset_a @ X )
              = Q )
            & ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ X ) )
              = X2 )
            & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ X ) ) )
              = Y2 ) ) ) ) ).

% LS_single_transition
thf(fact_739_LS__single__transition,axiom,
    ! [X2: b,Y2: c,M2: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y2 ) @ nil_Product_prod_b_c ) @ ( lS_a_b_c @ M2 @ Q ) )
      = ( ? [X: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X @ ( transitions_a_b_c @ M2 ) )
            & ( ( produc4225943634317630866od_c_a @ X )
              = Q )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X ) )
              = X2 )
            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X ) ) )
              = Y2 ) ) ) ) ).

% LS_single_transition
thf(fact_740_LS__single__transition,axiom,
    ! [X2: a,Y2: b,M2: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a] :
      ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) @ nil_Product_prod_a_b ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ Q ) )
      = ( ? [X: produc3205353004677322104od_c_a] :
            ( ( member6508725278956418977od_c_a @ X @ ( transi2941759273464634107_a_a_b @ M2 ) )
            & ( ( produc8171714036265584980od_c_a @ X )
              = Q )
            & ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ X ) )
              = X2 )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ X ) ) )
              = Y2 ) ) ) ) ).

% LS_single_transition
thf(fact_741_LS__single__transition,axiom,
    ! [X2: b,Y2: c,M2: fsm_fset_a_b_c,Q: fset_a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y2 ) @ nil_Product_prod_b_c ) @ ( lS_fset_a_b_c @ M2 @ Q ) )
      = ( ? [X: produc21584292878202806fset_a] :
            ( ( member4899939873214261983fset_a @ X @ ( transi6427852111903025530_a_b_c @ M2 ) )
            & ( ( produc1739398754328533778fset_a @ X )
              = Q )
            & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ X ) )
              = X2 )
            & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ X ) ) )
              = Y2 ) ) ) ) ).

% LS_single_transition
thf(fact_742_Compr__fimage__eq,axiom,
    ! [F: a > a,A: fset_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( fmember_a @ X @ ( fimage_a_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_a_a @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( fmember_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_743_Compr__fimage__eq,axiom,
    ! [F: fset_a > a,A: fset_fset_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( fmember_a @ X @ ( fimage_fset_a_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_fset_a_a @ F
        @ ( collect_fset_a
          @ ^ [X: fset_a] :
              ( ( fmember_fset_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_744_Compr__fimage__eq,axiom,
    ! [F: a > fset_a,A: fset_a,P2: fset_a > $o] :
      ( ( collect_fset_a
        @ ^ [X: fset_a] :
            ( ( fmember_fset_a @ X @ ( fimage_a_fset_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_a_fset_a @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( fmember_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_745_Compr__fimage__eq,axiom,
    ! [F: fset_a > fset_a,A: fset_fset_a,P2: fset_a > $o] :
      ( ( collect_fset_a
        @ ^ [X: fset_a] :
            ( ( fmember_fset_a @ X @ ( fimage_fset_a_fset_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_fset_a_fset_a @ F
        @ ( collect_fset_a
          @ ^ [X: fset_a] :
              ( ( fmember_fset_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_746_Compr__fimage__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: fset_P8043821951902805788od_c_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( fmember_a @ X @ ( fimage9200358715597384645_c_a_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_9171994905208062541_c_a_a @ F
        @ ( collec8405073525749625505od_c_a
          @ ^ [X: produc2669932201312201782od_c_a] :
              ( ( fmembe4375279799106287869od_c_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_747_Compr__fimage__eq,axiom,
    ! [F: a > produc2669932201312201782od_c_a,A: fset_a,P2: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X: produc2669932201312201782od_c_a] :
            ( ( fmembe4375279799106287869od_c_a @ X @ ( fimage4212853655448708405od_c_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_4184489845059386301od_c_a @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( fmember_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_748_Compr__fimage__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > fset_a,A: fset_P8043821951902805788od_c_a,P2: fset_a > $o] :
      ( ( collect_fset_a
        @ ^ [X: fset_a] :
            ( ( fmember_fset_a @ X @ ( fimage2540242761017857067fset_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_1164951314715494067fset_a @ F
        @ ( collec8405073525749625505od_c_a
          @ ^ [X: produc2669932201312201782od_c_a] :
              ( ( fmembe4375279799106287869od_c_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_749_Compr__fimage__eq,axiom,
    ! [F: fset_a > produc2669932201312201782od_c_a,A: fset_fset_a,P2: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X: produc2669932201312201782od_c_a] :
            ( ( fmembe4375279799106287869od_c_a @ X @ ( fimage3543014039228378575od_c_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_2167722592926015575od_c_a @ F
        @ ( collect_fset_a
          @ ^ [X: fset_a] :
              ( ( fmember_fset_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_750_Compr__fimage__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,A: fset_P8043821951902805788od_c_a,P2: product_prod_b_c > $o] :
      ( ( collec548942224018234067od_b_c
        @ ^ [X: product_prod_b_c] :
            ( ( fmembe4950632942993856815od_b_c @ X @ ( fimage7236410608496998447od_b_c @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_4981958329637652151od_b_c @ F
        @ ( collec8405073525749625505od_c_a
          @ ^ [X: produc2669932201312201782od_c_a] :
              ( ( fmembe4375279799106287869od_c_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_751_Compr__fimage__eq,axiom,
    ! [F: produc21584292878202806fset_a > a,A: fset_P2596122609397385116fset_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( fmember_a @ X @ ( fimage5721101993989878341et_a_a @ F @ A ) )
            & ( P2 @ X ) ) )
      = ( image_3466649715130532045et_a_a @ F
        @ ( collec8021744632519760673fset_a
          @ ^ [X: produc21584292878202806fset_a] :
              ( ( fmembe5403795861166450045fset_a @ X @ A )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_fimage_eq
thf(fact_752_ftransitions__set,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( fset_P4425921658469021927od_c_a @ ( ftrans4915350037625726329_a_a_b @ M2 ) )
      = ( transi2941759273464634107_a_a_b @ M2 ) ) ).

% ftransitions_set
thf(fact_753_ftransitions__set,axiom,
    ! [M2: fsm_a_b_c] :
      ( ( fset_P7369495421554515365od_c_a @ ( ftransitions_a_b_c @ M2 ) )
      = ( transitions_a_b_c @ M2 ) ) ).

% ftransitions_set
thf(fact_754_observable__path__unique,axiom,
    ! [M2: fsm_Pr1175075309171164859fset_a,Q: produc1137077444975408883fset_a,P: list_P8742742886961787479fset_a,P5: list_P8742742886961787479fset_a] :
      ( ( observ1975363445242043999fset_a @ M2 )
     => ( ( path_P1724798301931140637fset_a @ M2 @ Q @ P )
       => ( ( path_P1724798301931140637fset_a @ M2 @ Q @ P5 )
         => ( ( ( map_Pr2785363058057876286fset_a
                @ ^ [T: produc7303452399287242961fset_a] : ( produc1535188672328598523fset_a @ ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T ) ) @ ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T ) ) ) )
                @ P )
              = ( map_Pr2785363058057876286fset_a
                @ ^ [T: produc7303452399287242961fset_a] : ( produc1535188672328598523fset_a @ ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T ) ) @ ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T ) ) ) )
                @ P5 ) )
           => ( P = P5 ) ) ) ) ) ).

% observable_path_unique
thf(fact_755_observable__path__unique,axiom,
    ! [M2: fsm_Pr7813689363856283786_c_a_a,Q: produc2133359284458116493od_c_a,P: list_P1143992737879525208od_c_a,P5: list_P1143992737879525208od_c_a] :
      ( ( observ5792423948907539502_c_a_a @ M2 )
     => ( ( path_P5541858805596636140_c_a_a @ M2 @ Q @ P )
       => ( ( path_P5541858805596636140_c_a_a @ M2 @ Q @ P5 )
         => ( ( ( map_Pr132765484133579058_c_a_a
                @ ^ [T: produc1135898663509404882od_c_a] : ( produc7661602042108997680_c_a_a @ ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) ) )
                @ P )
              = ( map_Pr132765484133579058_c_a_a
                @ ^ [T: produc1135898663509404882od_c_a] : ( produc7661602042108997680_c_a_a @ ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) ) )
                @ P5 ) )
           => ( P = P5 ) ) ) ) ) ).

% observable_path_unique
thf(fact_756_observable__path__unique,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,Q: produc4838349196543436939fset_a,P: list_P5281568020286996248fset_a,P5: list_P5281568020286996248fset_a] :
      ( ( observ1610022369204726958et_a_b @ M2 )
     => ( ( path_P379120284129538668et_a_b @ M2 @ Q @ P )
       => ( ( path_P379120284129538668et_a_b @ M2 @ Q @ P5 )
         => ( ( ( map_Pr6485817111475779316et_a_b
                @ ^ [T: produc7245291065743820818fset_a] : ( produc4282089192724377778et_a_b @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T ) ) @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) )
                @ P )
              = ( map_Pr6485817111475779316et_a_b
                @ ^ [T: produc7245291065743820818fset_a] : ( produc4282089192724377778et_a_b @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T ) ) @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) )
                @ P5 ) )
           => ( P = P5 ) ) ) ) ) ).

% observable_path_unique
thf(fact_757_observable__path__unique,axiom,
    ! [M2: fsm_a_b_c,Q: a,P: list_P6327159017948738492od_c_a,P5: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( path_a_b_c @ M2 @ Q @ P )
       => ( ( path_a_b_c @ M2 @ Q @ P5 )
         => ( ( ( map_Pr776894537227705078od_b_c
                @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                @ P )
              = ( map_Pr776894537227705078od_b_c
                @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                @ P5 ) )
           => ( P = P5 ) ) ) ) ) ).

% observable_path_unique
thf(fact_758_observable__path__unique,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a,P: list_P1562850763667305982od_c_a,P5: list_P1562850763667305982od_c_a] :
      ( ( observ2964502264774598958_a_a_b @ M2 )
     => ( ( path_P1107610407223809260_a_a_b @ M2 @ Q @ P )
       => ( ( path_P1107610407223809260_a_a_b @ M2 @ Q @ P5 )
         => ( ( ( map_Pr3411285261028437108od_a_b
                @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                @ P )
              = ( map_Pr3411285261028437108od_a_b
                @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                @ P5 ) )
           => ( P = P5 ) ) ) ) ) ).

% observable_path_unique
thf(fact_759_observable__path__unique,axiom,
    ! [M2: fsm_fset_a_b_c,Q: fset_a,P: list_P1058411382900698172fset_a,P5: list_P1058411382900698172fset_a] :
      ( ( observ4396094731789362477_a_b_c @ M2 )
     => ( ( path_fset_a_b_c @ M2 @ Q @ P )
       => ( ( path_fset_a_b_c @ M2 @ Q @ P5 )
         => ( ( ( map_Pr7697975548294988406od_b_c
                @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
                @ P )
              = ( map_Pr7697975548294988406od_b_c
                @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
                @ P5 ) )
           => ( P = P5 ) ) ) ) ) ).

% observable_path_unique
thf(fact_760_not__Cons__self2,axiom,
    ! [X2: produc21584292878202806fset_a,Xs: list_P1058411382900698172fset_a] :
      ( ( cons_P3377334492504755052fset_a @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_761_not__Cons__self2,axiom,
    ! [X2: product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( cons_P4529483553340347422od_b_c @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_762_language__next__transition__ob,axiom,
    ! [X2: c,Y2: fset_a,Ios: list_P3569802872000582161fset_a,M2: fsm_Pr1175075309171164859fset_a,Q: produc1137077444975408883fset_a] :
      ( ( member2596827339850278714fset_a @ ( cons_P4277760901013084993fset_a @ ( produc1535188672328598523fset_a @ X2 @ Y2 ) @ Ios ) @ ( lS_Pro8627259047355266299fset_a @ M2 @ Q ) )
     => ~ ! [T3: produc7303452399287242961fset_a] :
            ( ( ( produc2105839935146153389fset_a @ T3 )
              = Q )
           => ( ( member9109638507678586746fset_a @ T3 @ ( transi1231084882770062892fset_a @ M2 ) )
             => ( ( ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T3 ) )
                  = X2 )
               => ( ( ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T3 ) ) )
                    = Y2 )
                 => ~ ( member2596827339850278714fset_a @ Ios @ ( lS_Pro8627259047355266299fset_a @ M2 @ ( produc8188425890649629780fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T3 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_763_language__next__transition__ob,axiom,
    ! [X2: product_prod_c_a,Y2: a,Ios: list_P8814837271076367046_c_a_a,M2: fsm_Pr7813689363856283786_c_a_a,Q: produc2133359284458116493od_c_a] :
      ( ( member5697445122164201327_c_a_a @ ( cons_P1791870815957980406_c_a_a @ ( produc7661602042108997680_c_a_a @ X2 @ Y2 ) @ Ios ) @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ Q ) )
     => ~ ! [T3: produc1135898663509404882od_c_a] :
            ( ( ( produc5475348670448042158od_c_a @ T3 )
              = Q )
           => ( ( member6699836568822684667od_c_a @ T3 @ ( transi5048145386435558395_c_a_a @ M2 ) )
             => ( ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T3 ) )
                  = X2 )
               => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T3 ) ) )
                    = Y2 )
                 => ~ ( member5697445122164201327_c_a_a @ Ios @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T3 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_764_language__next__transition__ob,axiom,
    ! [X2: fset_a,Y2: b,Ios: list_P7633400287818446792et_a_b,M2: fsm_Pr2398083450103406218et_a_b,Q: produc4838349196543436939fset_a] :
      ( ( member6660424755668143345et_a_b @ ( cons_P1519664230097792120et_a_b @ ( produc4282089192724377778et_a_b @ X2 @ Y2 ) @ Ios ) @ ( lS_Pro8532755013169807178et_a_b @ M2 @ Q ) )
     => ~ ! [T3: produc7245291065743820818fset_a] :
            ( ( ( produc8095675384246233966fset_a @ T3 )
              = Q )
           => ( ( member808877655204049083fset_a @ T3 @ ( transi3322184091305251195et_a_b @ M2 ) )
             => ( ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T3 ) )
                  = X2 )
               => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T3 ) ) )
                    = Y2 )
                 => ~ ( member6660424755668143345et_a_b @ Ios @ ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T3 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_765_language__next__transition__ob,axiom,
    ! [X2: b,Y2: c,Ios: list_P903359562653991662od_b_c,M2: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y2 ) @ Ios ) @ ( lS_a_b_c @ M2 @ Q ) )
     => ~ ! [T3: produc2669932201312201782od_c_a] :
            ( ( ( produc4225943634317630866od_c_a @ T3 )
              = Q )
           => ( ( member5502040268208861279od_c_a @ T3 @ ( transitions_a_b_c @ M2 ) )
             => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) )
                  = X2 )
               => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) )
                    = Y2 )
                 => ~ ( member6330420149250801815od_b_c @ Ios @ ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T3 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_766_language__next__transition__ob,axiom,
    ! [X2: a,Y2: b,Ios: list_P1467974523212632366od_a_b,M2: fsm_Pr1886043760914421514_a_a_b,Q: product_prod_c_a] :
      ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ X2 @ Y2 ) @ Ios ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ Q ) )
     => ~ ! [T3: produc3205353004677322104od_c_a] :
            ( ( ( produc8171714036265584980od_c_a @ T3 )
              = Q )
           => ( ( member6508725278956418977od_c_a @ T3 @ ( transi2941759273464634107_a_a_b @ M2 ) )
             => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T3 ) )
                  = X2 )
               => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T3 ) ) )
                    = Y2 )
                 => ~ ( member6895035109809442519od_a_b @ Ios @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T3 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_767_language__next__transition__ob,axiom,
    ! [X2: b,Y2: c,Ios: list_P903359562653991662od_b_c,M2: fsm_fset_a_b_c,Q: fset_a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X2 @ Y2 ) @ Ios ) @ ( lS_fset_a_b_c @ M2 @ Q ) )
     => ~ ! [T3: produc21584292878202806fset_a] :
            ( ( ( produc1739398754328533778fset_a @ T3 )
              = Q )
           => ( ( member4899939873214261983fset_a @ T3 @ ( transi6427852111903025530_a_b_c @ M2 ) )
             => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T3 ) )
                  = X2 )
               => ( ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) )
                    = Y2 )
                 => ~ ( member6330420149250801815od_b_c @ Ios @ ( lS_fset_a_b_c @ M2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) ) ) ) ) ) ) ) ) ).

% language_next_transition_ob
thf(fact_768_LS__prepend__transition,axiom,
    ! [T4: produc21584292878202806fset_a,M2: fsm_fset_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( member4899939873214261983fset_a @ T4 @ ( transi6427852111903025530_a_b_c @ M2 ) )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_fset_a_b_c @ M2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) )
       => ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) @ Io ) @ ( lS_fset_a_b_c @ M2 @ ( produc1739398754328533778fset_a @ T4 ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_769_LS__prepend__transition,axiom,
    ! [T4: produc2669932201312201782od_c_a,M2: fsm_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) )
       => ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) @ Io ) @ ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T4 ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_770_LS__prepend__transition,axiom,
    ! [T4: produc7245291065743820818fset_a,M2: fsm_Pr2398083450103406218et_a_b,Io: list_P7633400287818446792et_a_b] :
      ( ( member808877655204049083fset_a @ T4 @ ( transi3322184091305251195et_a_b @ M2 ) )
     => ( ( member6660424755668143345et_a_b @ Io @ ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) ) )
       => ( member6660424755668143345et_a_b @ ( cons_P1519664230097792120et_a_b @ ( produc4282089192724377778et_a_b @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) ) @ Io ) @ ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8095675384246233966fset_a @ T4 ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_771_LS__prepend__transition,axiom,
    ! [T4: produc3205353004677322104od_c_a,M2: fsm_Pr1886043760914421514_a_a_b,Io: list_P1467974523212632366od_a_b] :
      ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
     => ( ( member6895035109809442519od_a_b @ Io @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) )
       => ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) @ Io ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T4 ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_772_LS__prepend__transition,axiom,
    ! [T4: produc1135898663509404882od_c_a,M2: fsm_Pr7813689363856283786_c_a_a,Io: list_P8814837271076367046_c_a_a] :
      ( ( member6699836568822684667od_c_a @ T4 @ ( transi5048145386435558395_c_a_a @ M2 ) )
     => ( ( member5697445122164201327_c_a_a @ Io @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) ) )
       => ( member5697445122164201327_c_a_a @ ( cons_P1791870815957980406_c_a_a @ ( produc7661602042108997680_c_a_a @ ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) ) @ Io ) @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc5475348670448042158od_c_a @ T4 ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_773_LS__prepend__transition,axiom,
    ! [T4: produc7303452399287242961fset_a,M2: fsm_Pr1175075309171164859fset_a,Io: list_P3569802872000582161fset_a] :
      ( ( member9109638507678586746fset_a @ T4 @ ( transi1231084882770062892fset_a @ M2 ) )
     => ( ( member2596827339850278714fset_a @ Io @ ( lS_Pro8627259047355266299fset_a @ M2 @ ( produc8188425890649629780fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) ) ) )
       => ( member2596827339850278714fset_a @ ( cons_P4277760901013084993fset_a @ ( produc1535188672328598523fset_a @ ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) @ ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) ) ) @ Io ) @ ( lS_Pro8627259047355266299fset_a @ M2 @ ( produc2105839935146153389fset_a @ T4 ) ) ) ) ) ).

% LS_prepend_transition
thf(fact_774_observable__language__transition__target,axiom,
    ! [M2: fsm_Pr1175075309171164859fset_a,T4: produc7303452399287242961fset_a,Io: list_P3569802872000582161fset_a] :
      ( ( observ1975363445242043999fset_a @ M2 )
     => ( ( member9109638507678586746fset_a @ T4 @ ( transi1231084882770062892fset_a @ M2 ) )
       => ( ( member2596827339850278714fset_a @ ( cons_P4277760901013084993fset_a @ ( produc1535188672328598523fset_a @ ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) @ ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) ) ) @ Io ) @ ( lS_Pro8627259047355266299fset_a @ M2 @ ( produc2105839935146153389fset_a @ T4 ) ) )
         => ( member2596827339850278714fset_a @ Io @ ( lS_Pro8627259047355266299fset_a @ M2 @ ( produc8188425890649629780fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_775_observable__language__transition__target,axiom,
    ! [M2: fsm_Pr7813689363856283786_c_a_a,T4: produc1135898663509404882od_c_a,Io: list_P8814837271076367046_c_a_a] :
      ( ( observ5792423948907539502_c_a_a @ M2 )
     => ( ( member6699836568822684667od_c_a @ T4 @ ( transi5048145386435558395_c_a_a @ M2 ) )
       => ( ( member5697445122164201327_c_a_a @ ( cons_P1791870815957980406_c_a_a @ ( produc7661602042108997680_c_a_a @ ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) ) @ Io ) @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc5475348670448042158od_c_a @ T4 ) ) )
         => ( member5697445122164201327_c_a_a @ Io @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_776_observable__language__transition__target,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,T4: produc7245291065743820818fset_a,Io: list_P7633400287818446792et_a_b] :
      ( ( observ1610022369204726958et_a_b @ M2 )
     => ( ( member808877655204049083fset_a @ T4 @ ( transi3322184091305251195et_a_b @ M2 ) )
       => ( ( member6660424755668143345et_a_b @ ( cons_P1519664230097792120et_a_b @ ( produc4282089192724377778et_a_b @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) ) @ Io ) @ ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8095675384246233966fset_a @ T4 ) ) )
         => ( member6660424755668143345et_a_b @ Io @ ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_777_observable__language__transition__target,axiom,
    ! [M2: fsm_a_b_c,T4: produc2669932201312201782od_c_a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
       => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) @ Io ) @ ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T4 ) ) )
         => ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_778_observable__language__transition__target,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T4: produc3205353004677322104od_c_a,Io: list_P1467974523212632366od_a_b] :
      ( ( observ2964502264774598958_a_a_b @ M2 )
     => ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) @ Io ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T4 ) ) )
         => ( member6895035109809442519od_a_b @ Io @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_779_observable__language__transition__target,axiom,
    ! [M2: fsm_fset_a_b_c,T4: produc21584292878202806fset_a,Io: list_P903359562653991662od_b_c] :
      ( ( observ4396094731789362477_a_b_c @ M2 )
     => ( ( member4899939873214261983fset_a @ T4 @ ( transi6427852111903025530_a_b_c @ M2 ) )
       => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) @ Io ) @ ( lS_fset_a_b_c @ M2 @ ( produc1739398754328533778fset_a @ T4 ) ) )
         => ( member6330420149250801815od_b_c @ Io @ ( lS_fset_a_b_c @ M2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_780_single__transition__path,axiom,
    ! [T4: produc2669932201312201782od_c_a,M2: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
     => ( path_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T4 ) @ ( cons_P8659537020926305772od_c_a @ T4 @ nil_Pr1342775757158464060od_c_a ) ) ) ).

% single_transition_path
thf(fact_781_single__transition__path,axiom,
    ! [T4: produc3205353004677322104od_c_a,M2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
     => ( path_P1107610407223809260_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T4 ) @ ( cons_P5525787621418763054od_c_a @ T4 @ nil_Pr7913020553331505022od_c_a ) ) ) ).

% single_transition_path
thf(fact_782_single__transition__path,axiom,
    ! [T4: produc21584292878202806fset_a,M2: fsm_fset_a_b_c] :
      ( ( member4899939873214261983fset_a @ T4 @ ( transi6427852111903025530_a_b_c @ M2 ) )
     => ( path_fset_a_b_c @ M2 @ ( produc1739398754328533778fset_a @ T4 ) @ ( cons_P3377334492504755052fset_a @ T4 @ nil_Pr4260649278601056188fset_a ) ) ) ).

% single_transition_path
thf(fact_783_i1,axiom,
    ! [T4: produc21584292878202806fset_a] :
      ( ( fmembe5403795861166450045fset_a @ T4 @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) )
     => ( ( fmember_fset_a @ ( produc1739398754328533778fset_a @ T4 )
          @ ( sup_sup_fset_fset_a @ donesa
            @ ( minus_5856361552470604977fset_a
              @ ( fimage5315711910825859755fset_a
                @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                @ tsa )
              @ donesa ) ) )
        & ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
         != bot_bot_fset_a )
        & ( ( fset_a2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) )
          = ( image_9171994905208062541_c_a_a
            @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
            @ ( collec8405073525749625505od_c_a
              @ ^ [T5: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ T5 @ ( transitions_a_b_c @ m ) )
                  & ( fmember_a @ ( produc4225943634317630866od_c_a @ T5 ) @ ( produc1739398754328533778fset_a @ T4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) )
                    = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) )
                    = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) ) ) ) ) ) ) ).

% i1
thf(fact_784_i2,axiom,
    ! [Q: fset_a,T2: produc2669932201312201782od_c_a] :
      ( ( fmember_fset_a @ Q
        @ ( sup_sup_fset_fset_a @ donesa
          @ ( minus_5856361552470604977fset_a
            @ ( fimage5315711910825859755fset_a
              @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
              @ tsa )
            @ donesa ) ) )
     => ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ m ) )
       => ( ( fmember_a @ ( produc4225943634317630866od_c_a @ T2 ) @ Q )
         => ? [T3: produc21584292878202806fset_a] :
              ( ( fmembe5403795861166450045fset_a @ T3 @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) )
              & ( ( produc1739398754328533778fset_a @ T3 )
                = Q )
              & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T3 ) )
                = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) )
              & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) )
                = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) ) ) ) ) ) ).

% i2
thf(fact_785__092_060open_062_092_060exists_062p_H_O_Apathlike_A_Imake__observable__transitions_A_Iftransitions_AM_J_A_It__target_A_124_096_124_Ats_A_124_N_124_Adones_J_Adones_Ats_J_Aq_Ap_H_A_092_060and_062_Ap__io_Ap_H_A_061_AioT_A_D_AioP_092_060close_062,axiom,
    ? [P6: list_P1058411382900698172fset_a] :
      ( ( pathlike_fset_a_b_c
        @ ( make_o2341063825228466502_a_b_c @ ( ftransitions_a_b_c @ m )
          @ ( minus_5856361552470604977fset_a
            @ ( fimage5315711910825859755fset_a
              @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
              @ tsa )
            @ donesa )
          @ donesa
          @ tsa )
        @ qc
        @ P6 )
      & ( ( map_Pr7697975548294988406od_b_c
          @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
          @ P6 )
        = ( cons_P4529483553340347422od_b_c @ ioTa @ ioPa ) ) ) ).

% \<open>\<exists>p'. pathlike (make_observable_transitions (ftransitions M) (t_target |`| ts |-| dones) dones ts) q p' \<and> p_io p' = ioT # ioP\<close>
thf(fact_786__092_060open_062_092_060exists_062p_H_O_Apathlike_A_Its_A_124_092_060union_062_124_Aqtrans_J_A_It__target_AtP_J_Ap_H_A_092_060and_062_Ap__io_Ap_H_A_061_AioP_092_060close_062,axiom,
    ? [P6: list_P1058411382900698172fset_a] :
      ( ( pathlike_fset_a_b_c @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ tP ) ) ) @ P6 )
      & ( ( map_Pr7697975548294988406od_b_c
          @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
          @ P6 )
        = ioPa ) ) ).

% \<open>\<exists>p'. pathlike (ts |\<union>| qtrans) (t_target tP) p' \<and> p_io p' = ioP\<close>
thf(fact_787_res__cases,axiom,
    ( ( ( ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ qtrans )
          @ ( sup_sup_fset_fset_a @ donesa
            @ ( minus_5856361552470604977fset_a
              @ ( fimage5315711910825859755fset_a
                @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                @ tsa )
              @ donesa ) ) )
        = bot_bot_fset_fset_a )
     => ( ( make_o2341063825228466502_a_b_c @ ( ftransitions_a_b_c @ m )
          @ ( minus_5856361552470604977fset_a
            @ ( fimage5315711910825859755fset_a
              @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
              @ tsa )
            @ donesa )
          @ donesa
          @ tsa )
        = ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) ) )
    & ( ( ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ qtrans )
          @ ( sup_sup_fset_fset_a @ donesa
            @ ( minus_5856361552470604977fset_a
              @ ( fimage5315711910825859755fset_a
                @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                @ tsa )
              @ donesa ) ) )
       != bot_bot_fset_fset_a )
     => ( ( make_o2341063825228466502_a_b_c @ ( ftransitions_a_b_c @ m )
          @ ( minus_5856361552470604977fset_a
            @ ( fimage5315711910825859755fset_a
              @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
              @ tsa )
            @ donesa )
          @ donesa
          @ tsa )
        = ( make_o2341063825228466502_a_b_c @ ( ftransitions_a_b_c @ m )
          @ ( minus_5856361552470604977fset_a
            @ ( fimage5315711910825859755fset_a
              @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
              @ qtrans )
            @ ( sup_sup_fset_fset_a @ donesa
              @ ( minus_5856361552470604977fset_a
                @ ( fimage5315711910825859755fset_a
                  @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                  @ tsa )
                @ donesa ) ) )
          @ ( sup_sup_fset_fset_a @ donesa
            @ ( minus_5856361552470604977fset_a
              @ ( fimage5315711910825859755fset_a
                @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                @ tsa )
              @ donesa ) )
          @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) ) ) ) ) ).

% res_cases
thf(fact_788_paths__for__input__def,axiom,
    ( paths_1738280729674400731fset_a
    = ( ^ [M: fsm_Pr1175075309171164859fset_a,Q3: produc1137077444975408883fset_a,Xs3: list_c] :
          ( collec5240617237521426114fset_a
          @ ^ [P4: list_P8742742886961787479fset_a] :
              ( ( path_P1724798301931140637fset_a @ M @ Q3 @ P4 )
              & ( ( map_Pr3277271672726795257et_a_c @ product_fst_c_fset_a
                  @ ( map_Pr2785363058057876286fset_a
                    @ ^ [T: produc7303452399287242961fset_a] : ( produc1535188672328598523fset_a @ ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T ) ) @ ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T ) ) ) )
                    @ P4 ) )
                = Xs3 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_789_paths__for__input__def,axiom,
    ( paths_8814928053986176783_a_c_a
    = ( ^ [M: fsm_Pr7461828476839194351_a_c_a,Q3: produc2133359284458116493od_c_a,Xs3: list_c] :
          ( collec6149110611429497512od_c_a
          @ ^ [P4: list_P1786258881060786621od_c_a] :
              ( ( path_P2505919752976167761_a_c_a @ M @ Q3 @ P4 )
              & ( ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a
                  @ ( map_Pr5719429167398746866od_c_a
                    @ ^ [T: produc3729687299602777783od_c_a] : ( product_Pair_c_a @ ( produc7173261134821384633od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xs3 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_790_paths__for__input__def,axiom,
    ( paths_8440692088494772896od_c_a
    = ( ^ [M: fsm_Pr7194325094602570112od_c_a,Q3: produc2669932201312201782od_c_a,Xs3: list_b] :
          ( collec3319302300263626690od_c_a
          @ ^ [P4: list_P8560821412064581591od_c_a] :
              ( ( path_P1747656828206969570od_c_a @ M @ Q3 @ P4 )
              & ( ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a
                  @ ( map_Pr6962381543282861376od_c_a
                    @ ^ [T: produc4887275193591018193od_c_a] : ( produc3440865914000881661od_c_a @ ( produc2967918992726476220od_c_a @ ( produc5236404386356303343od_c_a @ T ) ) @ ( produc8171714036265584980od_c_a @ ( produc7936356256688089854od_c_a @ ( produc5236404386356303343od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xs3 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_791_paths__for__input__def,axiom,
    ( paths_5555341233339896234_c_a_a
    = ( ^ [M: fsm_Pr7813689363856283786_c_a_a,Q3: produc2133359284458116493od_c_a,Xs3: list_P125642481956313003od_c_a] :
          ( collec5243987561044225347od_c_a
          @ ^ [P4: list_P1143992737879525208od_c_a] :
              ( ( path_P5541858805596636140_c_a_a @ M @ Q3 @ P4 )
              & ( ( map_Pr5000116531144708777od_c_a @ produc4522685202854701596_c_a_a
                  @ ( map_Pr132765484133579058_c_a_a
                    @ ^ [T: produc1135898663509404882od_c_a] : ( produc7661602042108997680_c_a_a @ ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xs3 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_792_paths__for__input__def,axiom,
    ( paths_6688128585139547690et_a_b
    = ( ^ [M: fsm_Pr2398083450103406218et_a_b,Q3: produc4838349196543436939fset_a,Xs3: list_fset_a] :
          ( collec5334678913176435075fset_a
          @ ^ [P4: list_P5281568020286996248fset_a] :
              ( ( path_P379120284129538668et_a_b @ M @ Q3 @ P4 )
              & ( ( map_Pr6967926024347102566fset_a @ product_fst_fset_a_b
                  @ ( map_Pr6485817111475779316et_a_b
                    @ ^ [T: produc7245291065743820818fset_a] : ( produc4282089192724377778et_a_b @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T ) ) @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) )
                    @ P4 ) )
                = Xs3 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_793_paths__for__input__def,axiom,
    ( paths_6942934977415288259_a_b_c
    = ( ^ [M: fsm_a_b_c,Q3: a,Xs3: list_b] :
          ( collec6273869032445462695od_c_a
          @ ^ [P4: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M @ Q3 @ P4 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr776894537227705078od_b_c
                    @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xs3 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_794_paths__for__input__def,axiom,
    ( paths_607830118076874410_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b,Q3: product_prod_c_a,Xs3: list_a] :
          ( collec453890259647268073od_c_a
          @ ^ [P4: list_P1562850763667305982od_c_a] :
              ( ( path_P1107610407223809260_a_a_b @ M @ Q3 @ P4 )
              & ( ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b
                  @ ( map_Pr3411285261028437108od_a_b
                    @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                    @ P4 ) )
                = Xs3 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_795_paths__for__input__def,axiom,
    ( paths_6964290788893503657_a_b_c
    = ( ^ [M: fsm_fset_a_b_c,Q3: fset_a,Xs3: list_b] :
          ( collec5941573760022830631fset_a
          @ ^ [P4: list_P1058411382900698172fset_a] :
              ( ( path_fset_a_b_c @ M @ Q3 @ P4 )
              & ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
                  @ ( map_Pr7697975548294988406od_b_c
                    @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
                    @ P4 ) )
                = Xs3 ) ) ) ) ) ).

% paths_for_input_def
thf(fact_796__092_060open_062t__target_AtP_A_124_092_060in_062_124_Adones_A_124_092_060union_062_124_A_It__target_A_124_096_124_Ats_A_124_N_124_Adones_J_092_060close_062,axiom,
    ( fmember_fset_a @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ tP ) ) )
    @ ( sup_sup_fset_fset_a @ donesa
      @ ( minus_5856361552470604977fset_a
        @ ( fimage5315711910825859755fset_a
          @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
          @ tsa )
        @ donesa ) ) ) ).

% \<open>t_target tP |\<in>| dones |\<union>| (t_target |`| ts |-| dones)\<close>
thf(fact_797_i3,axiom,
    ( ( minus_5856361552470604977fset_a
      @ ( fimage5315711910825859755fset_a
        @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
        @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) )
      @ ( sup_sup_fset_fset_a @ donesa
        @ ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ tsa )
          @ donesa ) ) )
    = ( minus_5856361552470604977fset_a
      @ ( fimage5315711910825859755fset_a
        @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
        @ qtrans )
      @ ( sup_sup_fset_fset_a @ donesa
        @ ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ tsa )
          @ donesa ) ) ) ) ).

% i3
thf(fact_798__092_060open_062_092_060exists_062q_H_Ap_O_Aq_H_A_124_092_060in_062_124_Aq_A_092_060and_062_Apath_AM_Aq_H_Ap_A_092_060and_062_Ap__io_Ap_A_061_AioT_A_D_AioP_A_092_060Longrightarrow_062_A_092_060exists_062p_H_O_Apathlike_A_Imake__observable__transitions_A_Iftransitions_AM_J_A_It__target_A_124_096_124_Ats_A_124_N_124_Adones_J_Adones_Ats_J_Aq_Ap_H_A_092_060and_062_Ap__io_Ap_H_A_061_AioT_A_D_AioP_092_060close_062,axiom,
    ( ? [Q6: a,P7: list_P6327159017948738492od_c_a] :
        ( ( fmember_a @ Q6 @ qc )
        & ( path_a_b_c @ m @ Q6 @ P7 )
        & ( ( map_Pr776894537227705078od_b_c
            @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
            @ P7 )
          = ( cons_P4529483553340347422od_b_c @ ioTa @ ioPa ) ) )
   => ? [P6: list_P1058411382900698172fset_a] :
        ( ( pathlike_fset_a_b_c
          @ ( make_o2341063825228466502_a_b_c @ ( ftransitions_a_b_c @ m )
            @ ( minus_5856361552470604977fset_a
              @ ( fimage5315711910825859755fset_a
                @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                @ tsa )
              @ donesa )
            @ donesa
            @ tsa )
          @ qc
          @ P6 )
        & ( ( map_Pr7697975548294988406od_b_c
            @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
            @ P6 )
          = ( cons_P4529483553340347422od_b_c @ ioTa @ ioPa ) ) ) ) ).

% \<open>\<exists>q' p. q' |\<in>| q \<and> path M q' p \<and> p_io p = ioT # ioP \<Longrightarrow> \<exists>p'. pathlike (make_observable_transitions (ftransitions M) (t_target |`| ts |-| dones) dones ts) q p' \<and> p_io p' = ioT # ioP\<close>
thf(fact_799_i7,axiom,
    ~ ( fmember_fset_a @ bot_bot_fset_a
      @ ( sup_sup_fset_fset_a @ donesa
        @ ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ tsa )
          @ donesa ) ) ) ).

% i7
thf(fact_800__092_060open_062p__io_Ap_H_A_061_AioT_A_D_AioP_092_060close_062,axiom,
    ( ( map_Pr7697975548294988406od_b_c
      @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
      @ p )
    = ( cons_P4529483553340347422od_b_c @ ioTa @ ioPa ) ) ).

% \<open>p_io p' = ioT # ioP\<close>
thf(fact_801__092_060open_062p__io_ApP_A_061_AioP_092_060close_062,axiom,
    ( ( map_Pr7697975548294988406od_b_c
      @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
      @ pP )
    = ioPa ) ).

% \<open>p_io pP = ioP\<close>
thf(fact_802__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_H_O_A_092_060lbrakk_062pathlike_A_Its_A_124_092_060union_062_124_Aqtrans_J_Aq_Ap_H_059_Ap__io_Ap_H_A_061_AioT_A_D_AioP_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [P6: list_P1058411382900698172fset_a] :
        ( ( pathlike_fset_a_b_c @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) @ qc @ P6 )
       => ( ( map_Pr7697975548294988406od_b_c
            @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
            @ P6 )
         != ( cons_P4529483553340347422od_b_c @ ioTa @ ioPa ) ) ) ).

% \<open>\<And>thesis. (\<And>p'. \<lbrakk>pathlike (ts |\<union>| qtrans) q p'; p_io p' = ioT # ioP\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_803_Cons_OIH,axiom,
    ! [Q: fset_a] :
      ( ( fmember_fset_a @ Q
        @ ( sup_sup_fset_fset_a @ donesa
          @ ( minus_5856361552470604977fset_a
            @ ( fimage5315711910825859755fset_a
              @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
              @ tsa )
            @ donesa ) ) )
     => ( ( Q != bot_bot_fset_a )
       => ( ( ? [Q7: a,P4: list_P6327159017948738492od_c_a] :
                ( ( fmember_a @ Q7 @ Q )
                & ( path_a_b_c @ m @ Q7 @ P4 )
                & ( ( map_Pr776894537227705078od_b_c
                    @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                    @ P4 )
                  = ioPa ) ) )
          = ( ? [P8: list_P1058411382900698172fset_a] :
                ( ( pathlike_fset_a_b_c
                  @ ( make_o2341063825228466502_a_b_c @ ( ftransitions_a_b_c @ m )
                    @ ( minus_5856361552470604977fset_a
                      @ ( fimage5315711910825859755fset_a
                        @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                        @ tsa )
                      @ donesa )
                    @ donesa
                    @ tsa )
                  @ Q
                  @ P8 )
                & ( ( map_Pr7697975548294988406od_b_c
                    @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
                    @ P8 )
                  = ioPa ) ) ) ) ) ) ).

% Cons.IH
thf(fact_804_successively_Ocases,axiom,
    ! [X2: produc2958018031394921973fset_a] :
      ( ! [P9: produc21584292878202806fset_a > produc21584292878202806fset_a > $o] :
          ( X2
         != ( produc5126147958370928741fset_a @ P9 @ nil_Pr4260649278601056188fset_a ) )
     => ( ! [P9: produc21584292878202806fset_a > produc21584292878202806fset_a > $o,X3: produc21584292878202806fset_a] :
            ( X2
           != ( produc5126147958370928741fset_a @ P9 @ ( cons_P3377334492504755052fset_a @ X3 @ nil_Pr4260649278601056188fset_a ) ) )
       => ~ ! [P9: produc21584292878202806fset_a > produc21584292878202806fset_a > $o,X3: produc21584292878202806fset_a,Y3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a] :
              ( X2
             != ( produc5126147958370928741fset_a @ P9 @ ( cons_P3377334492504755052fset_a @ X3 @ ( cons_P3377334492504755052fset_a @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_805_successively_Ocases,axiom,
    ! [X2: produc2416979464461413415od_b_c] :
      ( ! [P9: product_prod_b_c > product_prod_b_c > $o] :
          ( X2
         != ( produc6092892548594537111od_b_c @ P9 @ nil_Product_prod_b_c ) )
     => ( ! [P9: product_prod_b_c > product_prod_b_c > $o,X3: product_prod_b_c] :
            ( X2
           != ( produc6092892548594537111od_b_c @ P9 @ ( cons_P4529483553340347422od_b_c @ X3 @ nil_Product_prod_b_c ) ) )
       => ~ ! [P9: product_prod_b_c > product_prod_b_c > $o,X3: product_prod_b_c,Y3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
              ( X2
             != ( produc6092892548594537111od_b_c @ P9 @ ( cons_P4529483553340347422od_b_c @ X3 @ ( cons_P4529483553340347422od_b_c @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_806_sorted__wrt_Ocases,axiom,
    ! [X2: produc2958018031394921973fset_a] :
      ( ! [P9: produc21584292878202806fset_a > produc21584292878202806fset_a > $o] :
          ( X2
         != ( produc5126147958370928741fset_a @ P9 @ nil_Pr4260649278601056188fset_a ) )
     => ~ ! [P9: produc21584292878202806fset_a > produc21584292878202806fset_a > $o,X3: produc21584292878202806fset_a,Ys3: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc5126147958370928741fset_a @ P9 @ ( cons_P3377334492504755052fset_a @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_807_sorted__wrt_Ocases,axiom,
    ! [X2: produc2416979464461413415od_b_c] :
      ( ! [P9: product_prod_b_c > product_prod_b_c > $o] :
          ( X2
         != ( produc6092892548594537111od_b_c @ P9 @ nil_Product_prod_b_c ) )
     => ~ ! [P9: product_prod_b_c > product_prod_b_c > $o,X3: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc6092892548594537111od_b_c @ P9 @ ( cons_P4529483553340347422od_b_c @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_808_shuffles_Ocases,axiom,
    ! [X2: produc5258545326496974119fset_a] :
      ( ! [Ys3: list_P1058411382900698172fset_a] :
          ( X2
         != ( produc316883261966145815fset_a @ nil_Pr4260649278601056188fset_a @ Ys3 ) )
     => ( ! [Xs2: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc316883261966145815fset_a @ Xs2 @ nil_Pr4260649278601056188fset_a ) )
       => ~ ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a,Y3: produc21584292878202806fset_a,Ys3: list_P1058411382900698172fset_a] :
              ( X2
             != ( produc316883261966145815fset_a @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ ( cons_P3377334492504755052fset_a @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_809_shuffles_Ocases,axiom,
    ! [X2: produc5878855887281848999od_b_c] :
      ( ! [Ys3: list_P903359562653991662od_b_c] :
          ( X2
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys3 ) )
     => ( ! [Xs2: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc3076307724246486423od_b_c @ Xs2 @ nil_Product_prod_b_c ) )
       => ~ ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] :
              ( X2
             != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_810_splice_Ocases,axiom,
    ! [X2: produc5258545326496974119fset_a] :
      ( ! [Ys3: list_P1058411382900698172fset_a] :
          ( X2
         != ( produc316883261966145815fset_a @ nil_Pr4260649278601056188fset_a @ Ys3 ) )
     => ~ ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a,Ys3: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc316883261966145815fset_a @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ Ys3 ) ) ) ).

% splice.cases
thf(fact_811_splice_Ocases,axiom,
    ! [X2: produc5878855887281848999od_b_c] :
      ( ! [Ys3: list_P903359562653991662od_b_c] :
          ( X2
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Ys3 ) )
     => ~ ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Ys3: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ Ys3 ) ) ) ).

% splice.cases
thf(fact_812_i4,axiom,
    ! [Q: fset_a] :
      ( ( fmember_fset_a @ Q
        @ ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ ( sup_su6818703939759823728fset_a @ tsa @ qtrans ) )
          @ ( sup_sup_fset_fset_a @ donesa
            @ ( minus_5856361552470604977fset_a
              @ ( fimage5315711910825859755fset_a
                @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                @ tsa )
              @ donesa ) ) ) )
     => ( fmember_fset_a @ Q
        @ ( fPow_a
          @ ( sup_sup_fset_a @ ( fimage9200358715597384645_c_a_a @ produc4225943634317630866od_c_a @ ( ftransitions_a_b_c @ m ) )
            @ ( fimage9200358715597384645_c_a_a
              @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
              @ ( ftransitions_a_b_c @ m ) ) ) ) ) ) ).

% i4
thf(fact_813_assms_I3_J,axiom,
    ! [Q: fset_a] :
      ( ( fmember_fset_a @ Q
        @ ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ ts )
          @ dones ) )
     => ( fmember_fset_a @ Q
        @ ( fPow_a
          @ ( sup_sup_fset_a @ ( fimage9200358715597384645_c_a_a @ produc4225943634317630866od_c_a @ ( ftransitions_a_b_c @ m ) )
            @ ( fimage9200358715597384645_c_a_a
              @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
              @ ( ftransitions_a_b_c @ m ) ) ) ) ) ) ).

% assms(3)
thf(fact_814__C1_Oprems_C_I3_J,axiom,
    ! [Q: fset_a] :
      ( ( fmember_fset_a @ Q
        @ ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ tsa )
          @ donesa ) )
     => ( fmember_fset_a @ Q
        @ ( fPow_a
          @ ( sup_sup_fset_a @ ( fimage9200358715597384645_c_a_a @ produc4225943634317630866od_c_a @ ( ftransitions_a_b_c @ m ) )
            @ ( fimage9200358715597384645_c_a_a
              @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
              @ ( ftransitions_a_b_c @ m ) ) ) ) ) ) ).

% "1.prems"(3)
thf(fact_815_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_a_b_c,F: a > a] :
      ( ( transitions_a_b_c @ ( rename4556187690751568563_b_c_a @ M2 @ F ) )
      = ( image_2169818278710580869od_c_a
        @ ^ [T: produc2669932201312201782od_c_a] : ( produc5762801557034676134od_c_a @ ( F @ ( produc4225943634317630866od_c_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_Pair_c_a @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( F @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) )
        @ ( transitions_a_b_c @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_816_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_a_b_c,F: a > fset_a] :
      ( ( transi6427852111903025530_a_b_c @ ( rename7194843435349443609fset_a @ M2 @ F ) )
      = ( image_3290014519244618501fset_a
        @ ^ [T: produc2669932201312201782od_c_a] : ( produc3008349866153638182fset_a @ ( F @ ( produc4225943634317630866od_c_a @ T ) ) @ ( produc4496741132823077603fset_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( produc1535188672328598523fset_a @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( F @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) )
        @ ( transitions_a_b_c @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_817_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_fset_a_b_c,F: fset_a > a] :
      ( ( transitions_a_b_c @ ( rename358903315601536205_b_c_a @ M2 @ F ) )
      = ( image_9201318158128649477od_c_a
        @ ^ [T: produc21584292878202806fset_a] : ( produc5762801557034676134od_c_a @ ( F @ ( produc1739398754328533778fset_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_Pair_c_a @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) @ ( F @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) ) ) ) )
        @ ( transi6427852111903025530_a_b_c @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_818_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,F: product_prod_c_a > c] :
      ( ( transitions_c_a_b @ ( rename6043294581747517582_a_b_c @ M2 @ F ) )
      = ( image_3434781565917999685od_b_c
        @ ^ [T: produc3205353004677322104od_c_a] : ( produc8300696160155326504od_b_c @ ( F @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc6867761804911831425od_b_c @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( F @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
        @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_819_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_fset_a_b_c,F: fset_a > fset_a] :
      ( ( transi6427852111903025530_a_b_c @ ( rename5030638990823766323fset_a @ M2 @ F ) )
      = ( image_1862373013431515525fset_a
        @ ^ [T: produc21584292878202806fset_a] : ( produc3008349866153638182fset_a @ ( F @ ( produc1739398754328533778fset_a @ T ) ) @ ( produc4496741132823077603fset_a @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( produc1535188672328598523fset_a @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) @ ( F @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) ) ) ) )
        @ ( transi6427852111903025530_a_b_c @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_820_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,F: produc4838349196543436939fset_a > c] :
      ( ( transi5791655338373036656et_a_b @ ( rename5129839117423441934_a_b_c @ M2 @ F ) )
      = ( image_3917889189756418501od_b_c
        @ ^ [T: produc7245291065743820818fset_a] : ( produc356619284632400834od_b_c @ ( F @ ( produc8095675384246233966fset_a @ T ) ) @ ( produc2872419864197951771od_b_c @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T ) ) @ ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) @ ( F @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) ) ) ) )
        @ ( transi3322184091305251195et_a_b @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_821_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,F: product_prod_c_a > product_prod_c_a] :
      ( ( transi2941759273464634107_a_a_b @ ( rename3079062586811799859od_c_a @ M2 @ F ) )
      = ( image_5098445917802613509od_c_a
        @ ^ [T: produc3205353004677322104od_c_a] : ( produc7911353075612250984od_c_a @ ( F @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc5762801557034676134od_c_a @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( F @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
        @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_822_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_Pr6915214480684328304_a_a_b,F: produc4838349196543436939fset_a > product_prod_c_a] :
      ( ( transi2941759273464634107_a_a_b @ ( rename1277623853359151693od_c_a @ M2 @ F ) )
      = ( image_2285320794969873669od_c_a
        @ ^ [T: produc3581717587131018616fset_a] : ( produc7911353075612250984od_c_a @ ( F @ ( produc3818667683119453012fset_a @ T ) ) @ ( produc5762801557034676134od_c_a @ ( produc3103901700553635960fset_a @ ( produc8787104947081066646fset_a @ T ) ) @ ( produc3440865914000881661od_c_a @ ( produc5616988955220391631fset_a @ ( produc1797847783904905914fset_a @ ( produc8787104947081066646fset_a @ T ) ) ) @ ( F @ ( produc8704244169652099857fset_a @ ( produc1797847783904905914fset_a @ ( produc8787104947081066646fset_a @ T ) ) ) ) ) ) )
        @ ( transi9000197413582903393_a_a_b @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_823_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,F: product_prod_c_a > produc4838349196543436939fset_a] :
      ( ( transi9000197413582903393_a_a_b @ ( rename6619924244263907609fset_a @ M2 @ F ) )
      = ( image_3464665086703242757fset_a
        @ ^ [T: produc3205353004677322104od_c_a] : ( produc516798377661499752fset_a @ ( F @ ( produc8171714036265584980od_c_a @ T ) ) @ ( produc4472983735029739660fset_a @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc4496741132823077603fset_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) @ ( F @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) )
        @ ( transi2941759273464634107_a_a_b @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_824_rename__states__simps_I5_J,axiom,
    ! [M2: fsm_Pr5428598191854926628et_a_b,F: product_prod_c_a > produc4838349196543436939fset_a] :
      ( ( transi3322184091305251195et_a_b @ ( rename7783063812205685119fset_a @ M2 @ F ) )
      = ( image_4876467461384845829fset_a
        @ ^ [T: produc3786058203479759378od_c_a] : ( produc1662326766993966978fset_a @ ( F @ ( produc3166900594103290734od_c_a @ T ) ) @ ( produc3008349866153638182fset_a @ ( produc6630392285145617708od_c_a @ ( produc7046871763585333680od_c_a @ T ) ) @ ( produc4496741132823077603fset_a @ ( produc301949074746585577od_c_a @ ( produc5324338368496887662od_c_a @ ( produc7046871763585333680od_c_a @ T ) ) ) @ ( F @ ( produc5221446324160585515od_c_a @ ( produc5324338368496887662od_c_a @ ( produc7046871763585333680od_c_a @ T ) ) ) ) ) ) )
        @ ( transi1223780177809997589et_a_b @ M2 ) ) ) ).

% rename_states_simps(5)
thf(fact_825_paths__for__io__def,axiom,
    ( paths_2388022959984353505fset_a
    = ( ^ [M: fsm_Pr1175075309171164859fset_a,Q3: produc1137077444975408883fset_a,Io2: list_P3569802872000582161fset_a] :
          ( collec5240617237521426114fset_a
          @ ^ [P4: list_P8742742886961787479fset_a] :
              ( ( path_P1724798301931140637fset_a @ M @ Q3 @ P4 )
              & ( ( map_Pr2785363058057876286fset_a
                  @ ^ [T: produc7303452399287242961fset_a] : ( produc1535188672328598523fset_a @ ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T ) ) @ ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T ) ) ) )
                  @ P4 )
                = Io2 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_826_paths__for__io__def,axiom,
    ( paths_6205083463649849008_c_a_a
    = ( ^ [M: fsm_Pr7813689363856283786_c_a_a,Q3: produc2133359284458116493od_c_a,Io2: list_P8814837271076367046_c_a_a] :
          ( collec5243987561044225347od_c_a
          @ ^ [P4: list_P1143992737879525208od_c_a] :
              ( ( path_P5541858805596636140_c_a_a @ M @ Q3 @ P4 )
              & ( ( map_Pr132765484133579058_c_a_a
                  @ ^ [T: produc1135898663509404882od_c_a] : ( produc7661602042108997680_c_a_a @ ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) ) )
                  @ P4 )
                = Io2 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_827_paths__for__io__def,axiom,
    ( paths_5009151303363445040et_a_b
    = ( ^ [M: fsm_Pr2398083450103406218et_a_b,Q3: produc4838349196543436939fset_a,Io2: list_P7633400287818446792et_a_b] :
          ( collec5334678913176435075fset_a
          @ ^ [P4: list_P5281568020286996248fset_a] :
              ( ( path_P379120284129538668et_a_b @ M @ Q3 @ P4 )
              & ( ( map_Pr6485817111475779316et_a_b
                  @ ^ [T: produc7245291065743820818fset_a] : ( produc4282089192724377778et_a_b @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T ) ) @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) )
                  @ P4 )
                = Io2 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_828_paths__for__io__def,axiom,
    ( paths_for_io_a_b_c
    = ( ^ [M: fsm_a_b_c,Q3: a,Io2: list_P903359562653991662od_b_c] :
          ( collec6273869032445462695od_c_a
          @ ^ [P4: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M @ Q3 @ P4 )
              & ( ( map_Pr776894537227705078od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ P4 )
                = Io2 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_829_paths__for__io__def,axiom,
    ( paths_593845848187029424_a_a_b
    = ( ^ [M: fsm_Pr1886043760914421514_a_a_b,Q3: product_prod_c_a,Io2: list_P1467974523212632366od_a_b] :
          ( collec453890259647268073od_c_a
          @ ^ [P4: list_P1562850763667305982od_c_a] :
              ( ( path_P1107610407223809260_a_a_b @ M @ Q3 @ P4 )
              & ( ( map_Pr3411285261028437108od_a_b
                  @ ^ [T: produc3205353004677322104od_c_a] : ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) )
                  @ P4 )
                = Io2 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_830_paths__for__io__def,axiom,
    ( paths_5762723032515753135_a_b_c
    = ( ^ [M: fsm_fset_a_b_c,Q3: fset_a,Io2: list_P903359562653991662od_b_c] :
          ( collec5941573760022830631fset_a
          @ ^ [P4: list_P1058411382900698172fset_a] :
              ( ( path_fset_a_b_c @ M @ Q3 @ P4 )
              & ( ( map_Pr7697975548294988406od_b_c
                  @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
                  @ P4 )
                = Io2 ) ) ) ) ) ).

% paths_for_io_def
thf(fact_831_sup__bot_Oright__neutral,axiom,
    ! [A2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ A2 @ bot_bo2316247384970961928fset_a )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_832_sup__bot_Oright__neutral,axiom,
    ! [A2: fset_a] :
      ( ( sup_sup_fset_a @ A2 @ bot_bot_fset_a )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_833_sup__bot_Oright__neutral,axiom,
    ! [A2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ A2 @ bot_bot_fset_fset_a )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_834_sup__bot_Oright__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_835_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: fset_P2596122609397385116fset_a,B2: fset_P2596122609397385116fset_a] :
      ( ( bot_bo2316247384970961928fset_a
        = ( sup_su6818703939759823728fset_a @ A2 @ B2 ) )
      = ( ( A2 = bot_bo2316247384970961928fset_a )
        & ( B2 = bot_bo2316247384970961928fset_a ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_836_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: fset_a,B2: fset_a] :
      ( ( bot_bot_fset_a
        = ( sup_sup_fset_a @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_fset_a )
        & ( B2 = bot_bot_fset_a ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_837_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: fset_fset_a,B2: fset_fset_a] :
      ( ( bot_bot_fset_fset_a
        = ( sup_sup_fset_fset_a @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_fset_fset_a )
        & ( B2 = bot_bot_fset_fset_a ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_838_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_839_Un__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_840_sup_Oidem,axiom,
    ! [A2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_841_sup_Oidem,axiom,
    ! [A2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_842_sup_Oidem,axiom,
    ! [A2: fset_a] :
      ( ( sup_sup_fset_a @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_843_sup__idem,axiom,
    ! [X2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ X2 @ X2 )
      = X2 ) ).

% sup_idem
thf(fact_844_sup__idem,axiom,
    ! [X2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ X2 @ X2 )
      = X2 ) ).

% sup_idem
thf(fact_845_sup__idem,axiom,
    ! [X2: fset_a] :
      ( ( sup_sup_fset_a @ X2 @ X2 )
      = X2 ) ).

% sup_idem
thf(fact_846_sup_Oleft__idem,axiom,
    ! [A2: fset_P2596122609397385116fset_a,B2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ A2 @ ( sup_su6818703939759823728fset_a @ A2 @ B2 ) )
      = ( sup_su6818703939759823728fset_a @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_847_sup_Oleft__idem,axiom,
    ! [A2: fset_fset_a,B2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ A2 @ ( sup_sup_fset_fset_a @ A2 @ B2 ) )
      = ( sup_sup_fset_fset_a @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_848_sup_Oleft__idem,axiom,
    ! [A2: fset_a,B2: fset_a] :
      ( ( sup_sup_fset_a @ A2 @ ( sup_sup_fset_a @ A2 @ B2 ) )
      = ( sup_sup_fset_a @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_849_sup__left__idem,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ X2 @ ( sup_su6818703939759823728fset_a @ X2 @ Y2 ) )
      = ( sup_su6818703939759823728fset_a @ X2 @ Y2 ) ) ).

% sup_left_idem
thf(fact_850_sup__left__idem,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ X2 @ ( sup_sup_fset_fset_a @ X2 @ Y2 ) )
      = ( sup_sup_fset_fset_a @ X2 @ Y2 ) ) ).

% sup_left_idem
thf(fact_851_sup__left__idem,axiom,
    ! [X2: fset_a,Y2: fset_a] :
      ( ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ X2 @ Y2 ) )
      = ( sup_sup_fset_a @ X2 @ Y2 ) ) ).

% sup_left_idem
thf(fact_852_sup_Oright__idem,axiom,
    ! [A2: fset_P2596122609397385116fset_a,B2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ ( sup_su6818703939759823728fset_a @ A2 @ B2 ) @ B2 )
      = ( sup_su6818703939759823728fset_a @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_853_sup_Oright__idem,axiom,
    ! [A2: fset_fset_a,B2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ ( sup_sup_fset_fset_a @ A2 @ B2 ) @ B2 )
      = ( sup_sup_fset_fset_a @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_854_sup_Oright__idem,axiom,
    ! [A2: fset_a,B2: fset_a] :
      ( ( sup_sup_fset_a @ ( sup_sup_fset_a @ A2 @ B2 ) @ B2 )
      = ( sup_sup_fset_a @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_855_sup__bot__left,axiom,
    ! [X2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ bot_bo2316247384970961928fset_a @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_856_sup__bot__left,axiom,
    ! [X2: fset_a] :
      ( ( sup_sup_fset_a @ bot_bot_fset_a @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_857_sup__bot__left,axiom,
    ! [X2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ bot_bot_fset_fset_a @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_858_sup__bot__left,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_859_sup__bot__right,axiom,
    ! [X2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ X2 @ bot_bo2316247384970961928fset_a )
      = X2 ) ).

% sup_bot_right
thf(fact_860_sup__bot__right,axiom,
    ! [X2: fset_a] :
      ( ( sup_sup_fset_a @ X2 @ bot_bot_fset_a )
      = X2 ) ).

% sup_bot_right
thf(fact_861_sup__bot__right,axiom,
    ! [X2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ X2 @ bot_bot_fset_fset_a )
      = X2 ) ).

% sup_bot_right
thf(fact_862_sup__bot__right,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
      = X2 ) ).

% sup_bot_right
thf(fact_863_bot__eq__sup__iff,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a] :
      ( ( bot_bo2316247384970961928fset_a
        = ( sup_su6818703939759823728fset_a @ X2 @ Y2 ) )
      = ( ( X2 = bot_bo2316247384970961928fset_a )
        & ( Y2 = bot_bo2316247384970961928fset_a ) ) ) ).

% bot_eq_sup_iff
thf(fact_864_bot__eq__sup__iff,axiom,
    ! [X2: fset_a,Y2: fset_a] :
      ( ( bot_bot_fset_a
        = ( sup_sup_fset_a @ X2 @ Y2 ) )
      = ( ( X2 = bot_bot_fset_a )
        & ( Y2 = bot_bot_fset_a ) ) ) ).

% bot_eq_sup_iff
thf(fact_865_bot__eq__sup__iff,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a] :
      ( ( bot_bot_fset_fset_a
        = ( sup_sup_fset_fset_a @ X2 @ Y2 ) )
      = ( ( X2 = bot_bot_fset_fset_a )
        & ( Y2 = bot_bot_fset_fset_a ) ) ) ).

% bot_eq_sup_iff
thf(fact_866_bot__eq__sup__iff,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X2 @ Y2 ) )
      = ( ( X2 = bot_bot_set_nat )
        & ( Y2 = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_867_sup__eq__bot__iff,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a] :
      ( ( ( sup_su6818703939759823728fset_a @ X2 @ Y2 )
        = bot_bo2316247384970961928fset_a )
      = ( ( X2 = bot_bo2316247384970961928fset_a )
        & ( Y2 = bot_bo2316247384970961928fset_a ) ) ) ).

% sup_eq_bot_iff
thf(fact_868_sup__eq__bot__iff,axiom,
    ! [X2: fset_a,Y2: fset_a] :
      ( ( ( sup_sup_fset_a @ X2 @ Y2 )
        = bot_bot_fset_a )
      = ( ( X2 = bot_bot_fset_a )
        & ( Y2 = bot_bot_fset_a ) ) ) ).

% sup_eq_bot_iff
thf(fact_869_sup__eq__bot__iff,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a] :
      ( ( ( sup_sup_fset_fset_a @ X2 @ Y2 )
        = bot_bot_fset_fset_a )
      = ( ( X2 = bot_bot_fset_fset_a )
        & ( Y2 = bot_bot_fset_fset_a ) ) ) ).

% sup_eq_bot_iff
thf(fact_870_sup__eq__bot__iff,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ( sup_sup_set_nat @ X2 @ Y2 )
        = bot_bot_set_nat )
      = ( ( X2 = bot_bot_set_nat )
        & ( Y2 = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_871_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: fset_P2596122609397385116fset_a,B2: fset_P2596122609397385116fset_a] :
      ( ( ( sup_su6818703939759823728fset_a @ A2 @ B2 )
        = bot_bo2316247384970961928fset_a )
      = ( ( A2 = bot_bo2316247384970961928fset_a )
        & ( B2 = bot_bo2316247384970961928fset_a ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_872_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: fset_a,B2: fset_a] :
      ( ( ( sup_sup_fset_a @ A2 @ B2 )
        = bot_bot_fset_a )
      = ( ( A2 = bot_bot_fset_a )
        & ( B2 = bot_bot_fset_a ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_873_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: fset_fset_a,B2: fset_fset_a] :
      ( ( ( sup_sup_fset_fset_a @ A2 @ B2 )
        = bot_bot_fset_fset_a )
      = ( ( A2 = bot_bot_fset_fset_a )
        & ( B2 = bot_bot_fset_fset_a ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_874_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_875_sup__bot_Oleft__neutral,axiom,
    ! [A2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ bot_bo2316247384970961928fset_a @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_876_sup__bot_Oleft__neutral,axiom,
    ! [A2: fset_a] :
      ( ( sup_sup_fset_a @ bot_bot_fset_a @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_877_sup__bot_Oleft__neutral,axiom,
    ! [A2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ bot_bot_fset_fset_a @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_878_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_879_union__fset,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Xa4: fset_P2596122609397385116fset_a] :
      ( ( fset_P7830090094348214821fset_a @ ( sup_su6818703939759823728fset_a @ X2 @ Xa4 ) )
      = ( sup_su4665271905203605226fset_a @ ( fset_P7830090094348214821fset_a @ X2 ) @ ( fset_P7830090094348214821fset_a @ Xa4 ) ) ) ).

% union_fset
thf(fact_880_union__fset,axiom,
    ! [X2: fset_fset_a,Xa4: fset_fset_a] :
      ( ( fset_fset_a2 @ ( sup_sup_fset_fset_a @ X2 @ Xa4 ) )
      = ( sup_sup_set_fset_a @ ( fset_fset_a2 @ X2 ) @ ( fset_fset_a2 @ Xa4 ) ) ) ).

% union_fset
thf(fact_881_union__fset,axiom,
    ! [X2: fset_a,Xa4: fset_a] :
      ( ( fset_a2 @ ( sup_sup_fset_a @ X2 @ Xa4 ) )
      = ( sup_sup_set_a @ ( fset_a2 @ X2 ) @ ( fset_a2 @ Xa4 ) ) ) ).

% union_fset
thf(fact_882_image__Un,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: set_Pr8357624050790843030od_c_a,B: set_Pr8357624050790843030od_c_a] :
      ( ( image_9171994905208062541_c_a_a @ F @ ( sup_su7950883165052597354od_c_a @ A @ B ) )
      = ( sup_sup_set_a @ ( image_9171994905208062541_c_a_a @ F @ A ) @ ( image_9171994905208062541_c_a_a @ F @ B ) ) ) ).

% image_Un
thf(fact_883_Un__empty__left,axiom,
    ! [B: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_884_Un__empty__right,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Un_empty_right
thf(fact_885_fimage__fPow__surj,axiom,
    ! [F: a > a,A: fset_a,B: fset_a] :
      ( ( ( fimage_a_a @ F @ A )
        = B )
     => ( ( fimage_fset_a_fset_a @ ( fimage_a_a @ F ) @ ( fPow_a @ A ) )
        = ( fPow_a @ B ) ) ) ).

% fimage_fPow_surj
thf(fact_886_fimage__fPow__surj,axiom,
    ! [F: a > produc21584292878202806fset_a,A: fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( ( fimage5428608421149153717fset_a @ F @ A )
        = B )
     => ( ( fimage8078386234238232501fset_a @ ( fimage5428608421149153717fset_a @ F ) @ ( fPow_a @ A ) )
        = ( fPow_P2879564663448066779fset_a @ B ) ) ) ).

% fimage_fPow_surj
thf(fact_887_fimage__fPow__surj,axiom,
    ! [F: produc21584292878202806fset_a > fset_a,A: fset_P2596122609397385116fset_a,B: fset_fset_a] :
      ( ( ( fimage5315711910825859755fset_a @ F @ A )
        = B )
     => ( ( fimage2893293597485128491fset_a @ ( fimage5315711910825859755fset_a @ F ) @ ( fPow_P2879564663448066779fset_a @ A ) )
        = ( fPow_fset_a @ B ) ) ) ).

% fimage_fPow_surj
thf(fact_888_fimage__fPow__surj,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A: fset_P8043821951902805788od_c_a,B: fset_a] :
      ( ( ( fimage9200358715597384645_c_a_a @ F @ A )
        = B )
     => ( ( fimage5191618493868882629fset_a @ ( fimage9200358715597384645_c_a_a @ F ) @ ( fPow_P8991780681278733403od_c_a @ A ) )
        = ( fPow_a @ B ) ) ) ).

% fimage_fPow_surj
thf(fact_889_fimage__fPow__surj,axiom,
    ! [F: fset_a > fset_P2596122609397385116fset_a,A: fset_fset_a,B: fset_f3206295485159697154fset_a] :
      ( ( ( fimage8078386234238232501fset_a @ F @ A )
        = B )
     => ( ( fimage6332195442235878325fset_a @ ( fimage8078386234238232501fset_a @ F ) @ ( fPow_fset_a @ A ) )
        = ( fPow_f151930114231407681fset_a @ B ) ) ) ).

% fimage_fPow_surj
thf(fact_890_fimage__fPow__surj,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,A: fset_P8043821951902805788od_c_a,B: fset_P8649754740804009550od_b_c] :
      ( ( ( fimage7236410608496998447od_b_c @ F @ A )
        = B )
     => ( ( fimage4876735402760017199od_b_c @ ( fimage7236410608496998447od_b_c @ F ) @ ( fPow_P8991780681278733403od_c_a @ A ) )
        = ( fPow_P7706984239998531725od_b_c @ B ) ) ) ).

% fimage_fPow_surj
thf(fact_891_fimage__fPow__surj,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a,A: fset_P8649754740804009550od_b_c,B: fset_P2596122609397385116fset_a] :
      ( ( ( fimage3947511181509736139fset_a @ F @ A )
        = B )
     => ( ( fimage7357916213740669387fset_a @ ( fimage3947511181509736139fset_a @ F ) @ ( fPow_P7706984239998531725od_b_c @ A ) )
        = ( fPow_P2879564663448066779fset_a @ B ) ) ) ).

% fimage_fPow_surj
thf(fact_892_fPow__top,axiom,
    ! [A: fset_a] : ( fmember_fset_a @ A @ ( fPow_a @ A ) ) ).

% fPow_top
thf(fact_893_fPow__not__fempty,axiom,
    ! [A: fset_a] :
      ( ( fPow_a @ A )
     != bot_bot_fset_fset_a ) ).

% fPow_not_fempty
thf(fact_894_fPow__bottom,axiom,
    ! [B: fset_a] : ( fmember_fset_a @ bot_bot_fset_a @ ( fPow_a @ B ) ) ).

% fPow_bottom
thf(fact_895_fPow__bottom,axiom,
    ! [B: fset_fset_a] : ( fmember_fset_fset_a @ bot_bot_fset_fset_a @ ( fPow_fset_a @ B ) ) ).

% fPow_bottom
thf(fact_896_inf__sup__aci_I8_J,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ X2 @ ( sup_su6818703939759823728fset_a @ X2 @ Y2 ) )
      = ( sup_su6818703939759823728fset_a @ X2 @ Y2 ) ) ).

% inf_sup_aci(8)
thf(fact_897_inf__sup__aci_I8_J,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ X2 @ ( sup_sup_fset_fset_a @ X2 @ Y2 ) )
      = ( sup_sup_fset_fset_a @ X2 @ Y2 ) ) ).

% inf_sup_aci(8)
thf(fact_898_inf__sup__aci_I8_J,axiom,
    ! [X2: fset_a,Y2: fset_a] :
      ( ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ X2 @ Y2 ) )
      = ( sup_sup_fset_a @ X2 @ Y2 ) ) ).

% inf_sup_aci(8)
thf(fact_899_inf__sup__aci_I7_J,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a,Z2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ X2 @ ( sup_su6818703939759823728fset_a @ Y2 @ Z2 ) )
      = ( sup_su6818703939759823728fset_a @ Y2 @ ( sup_su6818703939759823728fset_a @ X2 @ Z2 ) ) ) ).

% inf_sup_aci(7)
thf(fact_900_inf__sup__aci_I7_J,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a,Z2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ X2 @ ( sup_sup_fset_fset_a @ Y2 @ Z2 ) )
      = ( sup_sup_fset_fset_a @ Y2 @ ( sup_sup_fset_fset_a @ X2 @ Z2 ) ) ) ).

% inf_sup_aci(7)
thf(fact_901_inf__sup__aci_I7_J,axiom,
    ! [X2: fset_a,Y2: fset_a,Z2: fset_a] :
      ( ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ Y2 @ Z2 ) )
      = ( sup_sup_fset_a @ Y2 @ ( sup_sup_fset_a @ X2 @ Z2 ) ) ) ).

% inf_sup_aci(7)
thf(fact_902_inf__sup__aci_I6_J,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a,Z2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ ( sup_su6818703939759823728fset_a @ X2 @ Y2 ) @ Z2 )
      = ( sup_su6818703939759823728fset_a @ X2 @ ( sup_su6818703939759823728fset_a @ Y2 @ Z2 ) ) ) ).

% inf_sup_aci(6)
thf(fact_903_inf__sup__aci_I6_J,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a,Z2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ ( sup_sup_fset_fset_a @ X2 @ Y2 ) @ Z2 )
      = ( sup_sup_fset_fset_a @ X2 @ ( sup_sup_fset_fset_a @ Y2 @ Z2 ) ) ) ).

% inf_sup_aci(6)
thf(fact_904_inf__sup__aci_I6_J,axiom,
    ! [X2: fset_a,Y2: fset_a,Z2: fset_a] :
      ( ( sup_sup_fset_a @ ( sup_sup_fset_a @ X2 @ Y2 ) @ Z2 )
      = ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ Y2 @ Z2 ) ) ) ).

% inf_sup_aci(6)
thf(fact_905_inf__sup__aci_I5_J,axiom,
    ( sup_su6818703939759823728fset_a
    = ( ^ [X: fset_P2596122609397385116fset_a,Y: fset_P2596122609397385116fset_a] : ( sup_su6818703939759823728fset_a @ Y @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_906_inf__sup__aci_I5_J,axiom,
    ( sup_sup_fset_fset_a
    = ( ^ [X: fset_fset_a,Y: fset_fset_a] : ( sup_sup_fset_fset_a @ Y @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_907_inf__sup__aci_I5_J,axiom,
    ( sup_sup_fset_a
    = ( ^ [X: fset_a,Y: fset_a] : ( sup_sup_fset_a @ Y @ X ) ) ) ).

% inf_sup_aci(5)
thf(fact_908_sup_Oassoc,axiom,
    ! [A2: fset_P2596122609397385116fset_a,B2: fset_P2596122609397385116fset_a,C: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ ( sup_su6818703939759823728fset_a @ A2 @ B2 ) @ C )
      = ( sup_su6818703939759823728fset_a @ A2 @ ( sup_su6818703939759823728fset_a @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_909_sup_Oassoc,axiom,
    ! [A2: fset_fset_a,B2: fset_fset_a,C: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ ( sup_sup_fset_fset_a @ A2 @ B2 ) @ C )
      = ( sup_sup_fset_fset_a @ A2 @ ( sup_sup_fset_fset_a @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_910_sup_Oassoc,axiom,
    ! [A2: fset_a,B2: fset_a,C: fset_a] :
      ( ( sup_sup_fset_a @ ( sup_sup_fset_a @ A2 @ B2 ) @ C )
      = ( sup_sup_fset_a @ A2 @ ( sup_sup_fset_a @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_911_sup__assoc,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a,Z2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ ( sup_su6818703939759823728fset_a @ X2 @ Y2 ) @ Z2 )
      = ( sup_su6818703939759823728fset_a @ X2 @ ( sup_su6818703939759823728fset_a @ Y2 @ Z2 ) ) ) ).

% sup_assoc
thf(fact_912_sup__assoc,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a,Z2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ ( sup_sup_fset_fset_a @ X2 @ Y2 ) @ Z2 )
      = ( sup_sup_fset_fset_a @ X2 @ ( sup_sup_fset_fset_a @ Y2 @ Z2 ) ) ) ).

% sup_assoc
thf(fact_913_sup__assoc,axiom,
    ! [X2: fset_a,Y2: fset_a,Z2: fset_a] :
      ( ( sup_sup_fset_a @ ( sup_sup_fset_a @ X2 @ Y2 ) @ Z2 )
      = ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ Y2 @ Z2 ) ) ) ).

% sup_assoc
thf(fact_914_sup_Ocommute,axiom,
    ( sup_su6818703939759823728fset_a
    = ( ^ [A4: fset_P2596122609397385116fset_a,B6: fset_P2596122609397385116fset_a] : ( sup_su6818703939759823728fset_a @ B6 @ A4 ) ) ) ).

% sup.commute
thf(fact_915_sup_Ocommute,axiom,
    ( sup_sup_fset_fset_a
    = ( ^ [A4: fset_fset_a,B6: fset_fset_a] : ( sup_sup_fset_fset_a @ B6 @ A4 ) ) ) ).

% sup.commute
thf(fact_916_sup_Ocommute,axiom,
    ( sup_sup_fset_a
    = ( ^ [A4: fset_a,B6: fset_a] : ( sup_sup_fset_a @ B6 @ A4 ) ) ) ).

% sup.commute
thf(fact_917_sup__commute,axiom,
    ( sup_su6818703939759823728fset_a
    = ( ^ [X: fset_P2596122609397385116fset_a,Y: fset_P2596122609397385116fset_a] : ( sup_su6818703939759823728fset_a @ Y @ X ) ) ) ).

% sup_commute
thf(fact_918_sup__commute,axiom,
    ( sup_sup_fset_fset_a
    = ( ^ [X: fset_fset_a,Y: fset_fset_a] : ( sup_sup_fset_fset_a @ Y @ X ) ) ) ).

% sup_commute
thf(fact_919_sup__commute,axiom,
    ( sup_sup_fset_a
    = ( ^ [X: fset_a,Y: fset_a] : ( sup_sup_fset_a @ Y @ X ) ) ) ).

% sup_commute
thf(fact_920_sup_Oleft__commute,axiom,
    ! [B2: fset_P2596122609397385116fset_a,A2: fset_P2596122609397385116fset_a,C: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ B2 @ ( sup_su6818703939759823728fset_a @ A2 @ C ) )
      = ( sup_su6818703939759823728fset_a @ A2 @ ( sup_su6818703939759823728fset_a @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_921_sup_Oleft__commute,axiom,
    ! [B2: fset_fset_a,A2: fset_fset_a,C: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ B2 @ ( sup_sup_fset_fset_a @ A2 @ C ) )
      = ( sup_sup_fset_fset_a @ A2 @ ( sup_sup_fset_fset_a @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_922_sup_Oleft__commute,axiom,
    ! [B2: fset_a,A2: fset_a,C: fset_a] :
      ( ( sup_sup_fset_a @ B2 @ ( sup_sup_fset_a @ A2 @ C ) )
      = ( sup_sup_fset_a @ A2 @ ( sup_sup_fset_a @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_923_sup__left__commute,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a,Z2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ X2 @ ( sup_su6818703939759823728fset_a @ Y2 @ Z2 ) )
      = ( sup_su6818703939759823728fset_a @ Y2 @ ( sup_su6818703939759823728fset_a @ X2 @ Z2 ) ) ) ).

% sup_left_commute
thf(fact_924_sup__left__commute,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a,Z2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ X2 @ ( sup_sup_fset_fset_a @ Y2 @ Z2 ) )
      = ( sup_sup_fset_fset_a @ Y2 @ ( sup_sup_fset_fset_a @ X2 @ Z2 ) ) ) ).

% sup_left_commute
thf(fact_925_sup__left__commute,axiom,
    ! [X2: fset_a,Y2: fset_a,Z2: fset_a] :
      ( ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ Y2 @ Z2 ) )
      = ( sup_sup_fset_a @ Y2 @ ( sup_sup_fset_a @ X2 @ Z2 ) ) ) ).

% sup_left_commute
thf(fact_926_i5,axiom,
    ! [Q: fset_a] :
      ( ( fmember_fset_a @ Q
        @ ( sup_sup_fset_fset_a @ donesa
          @ ( minus_5856361552470604977fset_a
            @ ( fimage5315711910825859755fset_a
              @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
              @ tsa )
            @ donesa ) ) )
     => ( fmember_fset_a @ Q
        @ ( fPow_a
          @ ( sup_sup_fset_a
            @ ( sup_sup_fset_a @ ( fimage9200358715597384645_c_a_a @ produc4225943634317630866od_c_a @ ( ftransitions_a_b_c @ m ) )
              @ ( fimage9200358715597384645_c_a_a
                @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
                @ ( ftransitions_a_b_c @ m ) ) )
            @ ( finsert_a @ ( initial_a_b_c @ m ) @ bot_bot_fset_a ) ) ) ) ) ).

% i5
thf(fact_927_map__pair__fst,axiom,
    ! [F: b > c,Xs: list_b] :
      ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c
        @ ( map_b_4433156289202544749od_b_c
          @ ^ [X: b] : ( product_Pair_b_c @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_928_map__pair__fst,axiom,
    ! [F: c > fset_a,Xs: list_c] :
      ( ( map_Pr3277271672726795257et_a_c @ product_fst_c_fset_a
        @ ( map_c_8249927862686135247fset_a
          @ ^ [X: c] : ( produc1535188672328598523fset_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_929_map__pair__fst,axiom,
    ! [F: c > a,Xs: list_c] :
      ( ( map_Pr5866436826731527135_c_a_c @ product_fst_c_a
        @ ( map_c_1006260398291594985od_c_a
          @ ^ [X: c] : ( product_Pair_c_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_930_map__pair__fst,axiom,
    ! [F: b > produc4838349196543436939fset_a,Xs: list_b] :
      ( ( map_Pr3857243663455625872et_a_b @ produc5616988955220391631fset_a
        @ ( map_b_5289178268354934136fset_a
          @ ^ [X: b] : ( produc4496741132823077603fset_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_931_map__pair__fst,axiom,
    ! [F: b > product_prod_c_a,Xs: list_b] :
      ( ( map_Pr3016883280456414454_c_a_b @ produc301949074746585577od_c_a
        @ ( map_b_1729217027404323474od_c_a
          @ ^ [X: b] : ( produc3440865914000881661od_c_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_932_map__pair__fst,axiom,
    ! [F: a > produc2133359284458116493od_c_a,Xs: list_a] :
      ( ( map_Pr5179659585931931020_c_a_a @ produc4225943634317630866od_c_a
        @ ( map_a_192154525783254780od_c_a
          @ ^ [X: a] : ( produc5762801557034676134od_c_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_933_map__pair__fst,axiom,
    ! [F: product_prod_c_a > produc2669932201312201782od_c_a,Xs: list_P125642481956313003od_c_a] :
      ( ( map_Pr7059746129634632049od_c_a @ produc8171714036265584980od_c_a
        @ ( map_Pr1543565444835072855od_c_a
          @ ^ [X: product_prod_c_a] : ( produc7911353075612250984od_c_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_934_map__pair__fst,axiom,
    ! [F: fset_a > produc1137077444975408883fset_a,Xs: list_fset_a] :
      ( ( map_Pr4352688505896054258fset_a @ produc1739398754328533778fset_a
        @ ( map_fs2864866990288145302fset_a
          @ ^ [X: fset_a] : ( produc3008349866153638182fset_a @ X @ ( F @ X ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_935__092_060open_062qtrans_A_092_060equiv_062_AffUnion_A_I_I_092_060lambda_062q_O_Alet_Aqts_A_061_Affilter_A_I_092_060lambda_062t_O_At__source_At_A_124_092_060in_062_124_Aq_J_A_Iftransitions_AM_J_Ain_ALet_A_I_I_092_060lambda_062t_O_A_It__input_At_M_At__output_At_J_J_A_124_096_124_Aqts_J_A_I_I_124_096_124_J_A_I_092_060lambda_062_Ix_M_Ay_J_O_A_Iq_M_Ax_M_Ay_M_At__target_A_124_096_124_Affilter_A_I_092_060lambda_062t_O_A_It__input_At_M_At__output_At_J_A_061_A_Ix_M_Ay_J_J_Aqts_J_J_J_J_A_124_096_124_A_It__target_A_124_096_124_Ats_A_124_N_124_Adones_J_J_092_060close_062,axiom,
    ( qtrans
    = ( ffUnio2651180436950201422fset_a
      @ ( fimage8078386234238232501fset_a
        @ ^ [Q3: fset_a] :
            ( fimage3947511181509736139fset_a
            @ ( produc4415396419650312073fset_a
              @ ^ [X: b,Y: c] :
                  ( produc3008349866153638182fset_a @ Q3
                  @ ( produc4496741132823077603fset_a @ X
                    @ ( produc1535188672328598523fset_a @ Y
                      @ ( fimage9200358715597384645_c_a_a
                        @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
                        @ ( ffilte6209691498356321503od_c_a
                          @ ^ [T: produc2669932201312201782od_c_a] :
                              ( ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                              = ( product_Pair_b_c @ X @ Y ) )
                          @ ( ffilte6209691498356321503od_c_a
                            @ ^ [T: produc2669932201312201782od_c_a] : ( fmember_a @ ( produc4225943634317630866od_c_a @ T ) @ Q3 )
                            @ ( ftransitions_a_b_c @ m ) ) ) ) ) ) ) )
            @ ( fimage7236410608496998447od_b_c
              @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
              @ ( ffilte6209691498356321503od_c_a
                @ ^ [T: produc2669932201312201782od_c_a] : ( fmember_a @ ( produc4225943634317630866od_c_a @ T ) @ Q3 )
                @ ( ftransitions_a_b_c @ m ) ) ) )
        @ ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ tsa )
          @ donesa ) ) ) ) ).

% \<open>qtrans \<equiv> ffUnion ((\<lambda>q. let qts = ffilter (\<lambda>t. t_source t |\<in>| q) (ftransitions M) in Let ((\<lambda>t. (t_input t, t_output t)) |`| qts) ((|`|) (\<lambda>(x, y). (q, x, y, t_target |`| ffilter (\<lambda>t. (t_input t, t_output t) = (x, y)) qts)))) |`| (t_target |`| ts |-| dones))\<close>
thf(fact_936_delete__aux_Ocases,axiom,
    ! [X2: produc6475121930738208726od_b_c] :
      ( ! [K: b] :
          ( X2
         != ( produc561751498933632326od_b_c @ K @ nil_Product_prod_b_c ) )
     => ~ ! [K: b,K2: b,V2: c,Xs2: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc561751498933632326od_b_c @ K @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K2 @ V2 ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_937_delete__aux_Ocases,axiom,
    ! [X2: produc6053358620936464511fset_a] :
      ( ! [K: fset_a] :
          ( X2
         != ( produc458405453601871983fset_a @ K @ nil_Pr4260649278601056188fset_a ) )
     => ~ ! [K: fset_a,K2: fset_a,V2: produc1137077444975408883fset_a,Xs2: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc458405453601871983fset_a @ K @ ( cons_P3377334492504755052fset_a @ ( produc3008349866153638182fset_a @ K2 @ V2 ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_938_delete__aux_Ocases,axiom,
    ! [X2: produc6232872190587292001fset_a] :
      ( ! [K: b] :
          ( X2
         != ( produc473098471316008913fset_a @ K @ nil_Pr5741349403539668985fset_a ) )
     => ~ ! [K: b,K2: b,V2: produc4838349196543436939fset_a,Xs2: list_P5940095693748001145fset_a] :
            ( X2
           != ( produc473098471316008913fset_a @ K @ ( cons_P6981728131976004521fset_a @ ( produc4496741132823077603fset_a @ K2 @ V2 ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_939_delete__aux_Ocases,axiom,
    ! [X2: produc3710299173004873528fset_a] :
      ( ! [K: c] :
          ( X2
         != ( produc8319982366668297384fset_a @ K @ nil_Pr2782780582842296209fset_a ) )
     => ~ ! [K: c,K2: c,V2: fset_a,Xs2: list_P3569802872000582161fset_a] :
            ( X2
           != ( produc8319982366668297384fset_a @ K @ ( cons_P4277760901013084993fset_a @ ( produc1535188672328598523fset_a @ K2 @ V2 ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_940_fsm__impl__from__list_H_Ocases,axiom,
    ! [X2: produc6053358620936464511fset_a] :
      ( ! [Q8: fset_a] :
          ( X2
         != ( produc458405453601871983fset_a @ Q8 @ nil_Pr4260649278601056188fset_a ) )
     => ~ ! [Q8: fset_a,T3: produc21584292878202806fset_a,Ts3: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc458405453601871983fset_a @ Q8 @ ( cons_P3377334492504755052fset_a @ T3 @ Ts3 ) ) ) ) ).

% fsm_impl_from_list'.cases
thf(fact_941__C1_Ohyps_C,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Xa4: fset_fset_a,Xb: fset_P2596122609397385116fset_a,Q: fset_a,Io: list_P903359562653991662od_b_c] :
      ( ( X2
        = ( ffUnio2651180436950201422fset_a
          @ ( fimage8078386234238232501fset_a
            @ ^ [Q3: fset_a] :
                ( fimage3947511181509736139fset_a
                @ ( produc4415396419650312073fset_a
                  @ ^ [X: b,Y: c] :
                      ( produc3008349866153638182fset_a @ Q3
                      @ ( produc4496741132823077603fset_a @ X
                        @ ( produc1535188672328598523fset_a @ Y
                          @ ( fimage9200358715597384645_c_a_a
                            @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
                            @ ( ffilte6209691498356321503od_c_a
                              @ ^ [T: produc2669932201312201782od_c_a] :
                                  ( ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                                  = ( product_Pair_b_c @ X @ Y ) )
                              @ ( ffilte6209691498356321503od_c_a
                                @ ^ [T: produc2669932201312201782od_c_a] : ( fmember_a @ ( produc4225943634317630866od_c_a @ T ) @ Q3 )
                                @ ( ftransitions_a_b_c @ m ) ) ) ) ) ) ) )
                @ ( fimage7236410608496998447od_b_c
                  @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                  @ ( ffilte6209691498356321503od_c_a
                    @ ^ [T: produc2669932201312201782od_c_a] : ( fmember_a @ ( produc4225943634317630866od_c_a @ T ) @ Q3 )
                    @ ( ftransitions_a_b_c @ m ) ) ) )
            @ ( minus_5856361552470604977fset_a
              @ ( fimage5315711910825859755fset_a
                @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                @ tsa )
              @ donesa ) ) ) )
     => ( ( Xa4
          = ( sup_sup_fset_fset_a @ donesa
            @ ( minus_5856361552470604977fset_a
              @ ( fimage5315711910825859755fset_a
                @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                @ tsa )
              @ donesa ) ) )
       => ( ( Xb
            = ( sup_su6818703939759823728fset_a @ tsa @ X2 ) )
         => ( ( ( minus_5856361552470604977fset_a
                @ ( fimage5315711910825859755fset_a
                  @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                  @ Xb )
                @ Xa4 )
              = ( minus_5856361552470604977fset_a
                @ ( fimage5315711910825859755fset_a
                  @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                  @ X2 )
                @ Xa4 ) )
           => ( ( ( minus_5856361552470604977fset_a
                  @ ( fimage5315711910825859755fset_a
                    @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                    @ Xb )
                  @ Xa4 )
               != bot_bot_fset_fset_a )
             => ( ! [T3: produc21584292878202806fset_a] :
                    ( ( fmembe5403795861166450045fset_a @ T3 @ Xb )
                   => ( ( fmember_fset_a @ ( produc1739398754328533778fset_a @ T3 ) @ Xa4 )
                      & ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) )
                       != bot_bot_fset_a )
                      & ( ( fset_a2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) ) )
                        = ( image_9171994905208062541_c_a_a
                          @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
                          @ ( collec8405073525749625505od_c_a
                            @ ^ [T5: produc2669932201312201782od_c_a] :
                                ( ( member5502040268208861279od_c_a @ T5 @ ( transitions_a_b_c @ m ) )
                                & ( fmember_a @ ( produc4225943634317630866od_c_a @ T5 ) @ ( produc1739398754328533778fset_a @ T3 ) )
                                & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) )
                                  = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) )
                                & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) )
                                  = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T3 ) ) ) ) ) ) ) ) ) )
               => ( ! [Q8: fset_a] :
                      ( ( fmember_fset_a @ Q8 @ Xa4 )
                     => ! [T6: produc2669932201312201782od_c_a] :
                          ( ( member5502040268208861279od_c_a @ T6 @ ( transitions_a_b_c @ m ) )
                         => ( ( fmember_a @ ( produc4225943634317630866od_c_a @ T6 ) @ Q8 )
                           => ? [T7: produc21584292878202806fset_a] :
                                ( ( fmembe5403795861166450045fset_a @ T7 @ Xb )
                                & ( ( produc1739398754328533778fset_a @ T7 )
                                  = Q8 )
                                & ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T7 ) )
                                  = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T6 ) ) )
                                & ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T7 ) ) )
                                  = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T6 ) ) ) ) ) ) ) )
                 => ( ! [Q8: fset_a] :
                        ( ( fmember_fset_a @ Q8
                          @ ( minus_5856361552470604977fset_a
                            @ ( fimage5315711910825859755fset_a
                              @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                              @ Xb )
                            @ Xa4 ) )
                       => ( fmember_fset_a @ Q8
                          @ ( fPow_a
                            @ ( sup_sup_fset_a @ ( fimage9200358715597384645_c_a_a @ produc4225943634317630866od_c_a @ ( ftransitions_a_b_c @ m ) )
                              @ ( fimage9200358715597384645_c_a_a
                                @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
                                @ ( ftransitions_a_b_c @ m ) ) ) ) ) )
                   => ( ! [Q8: fset_a] :
                          ( ( fmember_fset_a @ Q8 @ Xa4 )
                         => ( fmember_fset_a @ Q8
                            @ ( fPow_a
                              @ ( sup_sup_fset_a
                                @ ( sup_sup_fset_a @ ( fimage9200358715597384645_c_a_a @ produc4225943634317630866od_c_a @ ( ftransitions_a_b_c @ m ) )
                                  @ ( fimage9200358715597384645_c_a_a
                                    @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
                                    @ ( ftransitions_a_b_c @ m ) ) )
                                @ ( finsert_a @ ( initial_a_b_c @ m ) @ bot_bot_fset_a ) ) ) ) )
                     => ( ~ ( fmember_fset_a @ bot_bot_fset_a @ Xa4 )
                       => ( ( fmember_fset_a @ Q @ Xa4 )
                         => ( ( ? [Q7: a] :
                                  ( ( fmember_a @ Q7 @ Q )
                                  & ? [P4: list_P6327159017948738492od_c_a] :
                                      ( ( path_a_b_c @ m @ Q7 @ P4 )
                                      & ( ( map_Pr776894537227705078od_b_c
                                          @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                                          @ P4 )
                                        = Io ) ) ) )
                            = ( ? [P8: list_P1058411382900698172fset_a] :
                                  ( ( pathlike_fset_a_b_c
                                    @ ( make_o2341063825228466502_a_b_c @ ( ftransitions_a_b_c @ m )
                                      @ ( minus_5856361552470604977fset_a
                                        @ ( fimage5315711910825859755fset_a
                                          @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
                                          @ Xb )
                                        @ Xa4 )
                                      @ Xa4
                                      @ Xb )
                                    @ Q
                                    @ P8 )
                                  & ( ( map_Pr7697975548294988406od_b_c
                                      @ ^ [T: produc21584292878202806fset_a] : ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) )
                                      @ P8 )
                                    = Io ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% "1.hyps"
thf(fact_942_Diff__cancel,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ A )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_943_empty__Diff,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_944_Diff__empty,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Diff_empty
thf(fact_945_Un__iff,axiom,
    ! [C: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,B: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C @ ( sup_su7950883165052597354od_c_a @ A @ B ) )
      = ( ( member5502040268208861279od_c_a @ C @ A )
        | ( member5502040268208861279od_c_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_946_Un__iff,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
      = ( ( member_a @ C @ A )
        | ( member_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_947_Un__iff,axiom,
    ! [C: produc3205353004677322104od_c_a,A: set_Pr2640428967771533528od_c_a,B: set_Pr2640428967771533528od_c_a] :
      ( ( member6508725278956418977od_c_a @ C @ ( sup_su7764855436033407660od_c_a @ A @ B ) )
      = ( ( member6508725278956418977od_c_a @ C @ A )
        | ( member6508725278956418977od_c_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_948_Un__iff,axiom,
    ! [C: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a,B: set_Pr4695589342295920918fset_a] :
      ( ( member4899939873214261983fset_a @ C @ ( sup_su4665271905203605226fset_a @ A @ B ) )
      = ( ( member4899939873214261983fset_a @ C @ A )
        | ( member4899939873214261983fset_a @ C @ B ) ) ) ).

% Un_iff
thf(fact_949_UnCI,axiom,
    ! [C: produc2669932201312201782od_c_a,B: set_Pr8357624050790843030od_c_a,A: set_Pr8357624050790843030od_c_a] :
      ( ( ~ ( member5502040268208861279od_c_a @ C @ B )
       => ( member5502040268208861279od_c_a @ C @ A ) )
     => ( member5502040268208861279od_c_a @ C @ ( sup_su7950883165052597354od_c_a @ A @ B ) ) ) ).

% UnCI
thf(fact_950_UnCI,axiom,
    ! [C: a,B: set_a,A: set_a] :
      ( ( ~ ( member_a @ C @ B )
       => ( member_a @ C @ A ) )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnCI
thf(fact_951_UnCI,axiom,
    ! [C: produc3205353004677322104od_c_a,B: set_Pr2640428967771533528od_c_a,A: set_Pr2640428967771533528od_c_a] :
      ( ( ~ ( member6508725278956418977od_c_a @ C @ B )
       => ( member6508725278956418977od_c_a @ C @ A ) )
     => ( member6508725278956418977od_c_a @ C @ ( sup_su7764855436033407660od_c_a @ A @ B ) ) ) ).

% UnCI
thf(fact_952_UnCI,axiom,
    ! [C: produc21584292878202806fset_a,B: set_Pr4695589342295920918fset_a,A: set_Pr4695589342295920918fset_a] :
      ( ( ~ ( member4899939873214261983fset_a @ C @ B )
       => ( member4899939873214261983fset_a @ C @ A ) )
     => ( member4899939873214261983fset_a @ C @ ( sup_su4665271905203605226fset_a @ A @ B ) ) ) ).

% UnCI
thf(fact_953_finsert__absorb2,axiom,
    ! [X2: a,A: fset_a] :
      ( ( finsert_a @ X2 @ ( finsert_a @ X2 @ A ) )
      = ( finsert_a @ X2 @ A ) ) ).

% finsert_absorb2
thf(fact_954_finsertCI,axiom,
    ! [A2: a,B: fset_a,B2: a] :
      ( ( ~ ( fmember_a @ A2 @ B )
       => ( A2 = B2 ) )
     => ( fmember_a @ A2 @ ( finsert_a @ B2 @ B ) ) ) ).

% finsertCI
thf(fact_955_finsertCI,axiom,
    ! [A2: fset_a,B: fset_fset_a,B2: fset_a] :
      ( ( ~ ( fmember_fset_a @ A2 @ B )
       => ( A2 = B2 ) )
     => ( fmember_fset_a @ A2 @ ( finsert_fset_a @ B2 @ B ) ) ) ).

% finsertCI
thf(fact_956_finsertCI,axiom,
    ! [A2: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a,B2: produc21584292878202806fset_a] :
      ( ( ~ ( fmembe5403795861166450045fset_a @ A2 @ B )
       => ( A2 = B2 ) )
     => ( fmembe5403795861166450045fset_a @ A2 @ ( finser2277912206736664318fset_a @ B2 @ B ) ) ) ).

% finsertCI
thf(fact_957_finsertCI,axiom,
    ! [A2: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a,B2: produc2669932201312201782od_c_a] :
      ( ( ~ ( fmembe4375279799106287869od_c_a @ A2 @ B )
       => ( A2 = B2 ) )
     => ( fmembe4375279799106287869od_c_a @ A2 @ ( finser4474152491806270334od_c_a @ B2 @ B ) ) ) ).

% finsertCI
thf(fact_958_finsert__iff,axiom,
    ! [A2: a,B2: a,A: fset_a] :
      ( ( fmember_a @ A2 @ ( finsert_a @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( fmember_a @ A2 @ A ) ) ) ).

% finsert_iff
thf(fact_959_finsert__iff,axiom,
    ! [A2: fset_a,B2: fset_a,A: fset_fset_a] :
      ( ( fmember_fset_a @ A2 @ ( finsert_fset_a @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( fmember_fset_a @ A2 @ A ) ) ) ).

% finsert_iff
thf(fact_960_finsert__iff,axiom,
    ! [A2: produc21584292878202806fset_a,B2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ A2 @ ( finser2277912206736664318fset_a @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( fmembe5403795861166450045fset_a @ A2 @ A ) ) ) ).

% finsert_iff
thf(fact_961_finsert__iff,axiom,
    ! [A2: produc2669932201312201782od_c_a,B2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ A2 @ ( finser4474152491806270334od_c_a @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( fmembe4375279799106287869od_c_a @ A2 @ A ) ) ) ).

% finsert_iff
thf(fact_962_fimage__finsert,axiom,
    ! [F: produc21584292878202806fset_a > fset_a,A2: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( fimage5315711910825859755fset_a @ F @ ( finser2277912206736664318fset_a @ A2 @ B ) )
      = ( finsert_fset_a @ ( F @ A2 ) @ ( fimage5315711910825859755fset_a @ F @ B ) ) ) ).

% fimage_finsert
thf(fact_963_fimage__finsert,axiom,
    ! [F: fset_a > fset_P2596122609397385116fset_a,A2: fset_a,B: fset_fset_a] :
      ( ( fimage8078386234238232501fset_a @ F @ ( finsert_fset_a @ A2 @ B ) )
      = ( finser1277071787618628324fset_a @ ( F @ A2 ) @ ( fimage8078386234238232501fset_a @ F @ B ) ) ) ).

% fimage_finsert
thf(fact_964_fimage__finsert,axiom,
    ! [F: produc2669932201312201782od_c_a > product_prod_b_c,A2: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fimage7236410608496998447od_b_c @ F @ ( finser4474152491806270334od_c_a @ A2 @ B ) )
      = ( finser2611709836832183728od_b_c @ ( F @ A2 ) @ ( fimage7236410608496998447od_b_c @ F @ B ) ) ) ).

% fimage_finsert
thf(fact_965_fimage__finsert,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a,A2: product_prod_b_c,B: fset_P8649754740804009550od_b_c] :
      ( ( fimage3947511181509736139fset_a @ F @ ( finser2611709836832183728od_b_c @ A2 @ B ) )
      = ( finser2277912206736664318fset_a @ ( F @ A2 ) @ ( fimage3947511181509736139fset_a @ F @ B ) ) ) ).

% fimage_finsert
thf(fact_966_fimage__finsert,axiom,
    ! [F: produc2669932201312201782od_c_a > a,A2: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ( fimage9200358715597384645_c_a_a @ F @ ( finser4474152491806270334od_c_a @ A2 @ B ) )
      = ( finsert_a @ ( F @ A2 ) @ ( fimage9200358715597384645_c_a_a @ F @ B ) ) ) ).

% fimage_finsert
thf(fact_967_fimage__finsert,axiom,
    ! [F: a > a,A2: a,B: fset_a] :
      ( ( fimage_a_a @ F @ ( finsert_a @ A2 @ B ) )
      = ( finsert_a @ ( F @ A2 ) @ ( fimage_a_a @ F @ B ) ) ) ).

% fimage_finsert
thf(fact_968_funion__finsert__left,axiom,
    ! [A2: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a,C2: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ ( finser2277912206736664318fset_a @ A2 @ B ) @ C2 )
      = ( finser2277912206736664318fset_a @ A2 @ ( sup_su6818703939759823728fset_a @ B @ C2 ) ) ) ).

% funion_finsert_left
thf(fact_969_funion__finsert__left,axiom,
    ! [A2: fset_a,B: fset_fset_a,C2: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ ( finsert_fset_a @ A2 @ B ) @ C2 )
      = ( finsert_fset_a @ A2 @ ( sup_sup_fset_fset_a @ B @ C2 ) ) ) ).

% funion_finsert_left
thf(fact_970_funion__finsert__left,axiom,
    ! [A2: a,B: fset_a,C2: fset_a] :
      ( ( sup_sup_fset_a @ ( finsert_a @ A2 @ B ) @ C2 )
      = ( finsert_a @ A2 @ ( sup_sup_fset_a @ B @ C2 ) ) ) ).

% funion_finsert_left
thf(fact_971_funion__finsert__right,axiom,
    ! [A: fset_P2596122609397385116fset_a,A2: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( sup_su6818703939759823728fset_a @ A @ ( finser2277912206736664318fset_a @ A2 @ B ) )
      = ( finser2277912206736664318fset_a @ A2 @ ( sup_su6818703939759823728fset_a @ A @ B ) ) ) ).

% funion_finsert_right
thf(fact_972_funion__finsert__right,axiom,
    ! [A: fset_fset_a,A2: fset_a,B: fset_fset_a] :
      ( ( sup_sup_fset_fset_a @ A @ ( finsert_fset_a @ A2 @ B ) )
      = ( finsert_fset_a @ A2 @ ( sup_sup_fset_fset_a @ A @ B ) ) ) ).

% funion_finsert_right
thf(fact_973_funion__finsert__right,axiom,
    ! [A: fset_a,A2: a,B: fset_a] :
      ( ( sup_sup_fset_a @ A @ ( finsert_a @ A2 @ B ) )
      = ( finsert_a @ A2 @ ( sup_sup_fset_a @ A @ B ) ) ) ).

% funion_finsert_right
thf(fact_974_minus__fset,axiom,
    ! [X2: fset_a,Xa4: fset_a] :
      ( ( fset_a2 @ ( minus_minus_fset_a @ X2 @ Xa4 ) )
      = ( minus_minus_set_a @ ( fset_a2 @ X2 ) @ ( fset_a2 @ Xa4 ) ) ) ).

% minus_fset
thf(fact_975_minus__fset,axiom,
    ! [X2: fset_fset_a,Xa4: fset_fset_a] :
      ( ( fset_fset_a2 @ ( minus_5856361552470604977fset_a @ X2 @ Xa4 ) )
      = ( minus_3046475851064503467fset_a @ ( fset_fset_a2 @ X2 ) @ ( fset_fset_a2 @ Xa4 ) ) ) ).

% minus_fset
thf(fact_976_ffmember__filter,axiom,
    ! [X2: a,P2: a > $o,A: fset_a] :
      ( ( fmember_a @ X2 @ ( ffilter_a @ P2 @ A ) )
      = ( ( fmember_a @ X2 @ A )
        & ( P2 @ X2 ) ) ) ).

% ffmember_filter
thf(fact_977_ffmember__filter,axiom,
    ! [X2: fset_a,P2: fset_a > $o,A: fset_fset_a] :
      ( ( fmember_fset_a @ X2 @ ( ffilter_fset_a @ P2 @ A ) )
      = ( ( fmember_fset_a @ X2 @ A )
        & ( P2 @ X2 ) ) ) ).

% ffmember_filter
thf(fact_978_ffmember__filter,axiom,
    ! [X2: produc21584292878202806fset_a,P2: produc21584292878202806fset_a > $o,A: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ ( ffilte3318296817612072799fset_a @ P2 @ A ) )
      = ( ( fmembe5403795861166450045fset_a @ X2 @ A )
        & ( P2 @ X2 ) ) ) ).

% ffmember_filter
thf(fact_979_ffmember__filter,axiom,
    ! [X2: produc2669932201312201782od_c_a,P2: produc2669932201312201782od_c_a > $o,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ ( ffilte6209691498356321503od_c_a @ P2 @ A ) )
      = ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
        & ( P2 @ X2 ) ) ) ).

% ffmember_filter
thf(fact_980_rename__states__simps_I1_J,axiom,
    ! [M2: fsm_a_b_c,F: a > a] :
      ( ( initial_a_b_c @ ( rename4556187690751568563_b_c_a @ M2 @ F ) )
      = ( F @ ( initial_a_b_c @ M2 ) ) ) ).

% rename_states_simps(1)
thf(fact_981_case__prod__conv,axiom,
    ! [F: b > c > produc21584292878202806fset_a,A2: b,B2: c] :
      ( ( produc4415396419650312073fset_a @ F @ ( product_Pair_b_c @ A2 @ B2 ) )
      = ( F @ A2 @ B2 ) ) ).

% case_prod_conv
thf(fact_982_fempty__ffilter,axiom,
    ! [A: fset_P8043821951902805788od_c_a] :
      ( ( ffilte6209691498356321503od_c_a
        @ ^ [Uu: produc2669932201312201782od_c_a] : $false
        @ A )
      = bot_bo1116342619594069896od_c_a ) ).

% fempty_ffilter
thf(fact_983_fempty__ffilter,axiom,
    ! [A: fset_a] :
      ( ( ffilter_a
        @ ^ [Uu: a] : $false
        @ A )
      = bot_bot_fset_a ) ).

% fempty_ffilter
thf(fact_984_fempty__ffilter,axiom,
    ! [A: fset_fset_a] :
      ( ( ffilter_fset_a
        @ ^ [Uu: fset_a] : $false
        @ A )
      = bot_bot_fset_fset_a ) ).

% fempty_ffilter
thf(fact_985_finsert__fimage,axiom,
    ! [X2: product_prod_b_c,A: fset_P8649754740804009550od_b_c,F: product_prod_b_c > produc21584292878202806fset_a] :
      ( ( fmembe4950632942993856815od_b_c @ X2 @ A )
     => ( ( finser2277912206736664318fset_a @ ( F @ X2 ) @ ( fimage3947511181509736139fset_a @ F @ A ) )
        = ( fimage3947511181509736139fset_a @ F @ A ) ) ) ).

% finsert_fimage
thf(fact_986_finsert__fimage,axiom,
    ! [X2: a,A: fset_a,F: a > a] :
      ( ( fmember_a @ X2 @ A )
     => ( ( finsert_a @ ( F @ X2 ) @ ( fimage_a_a @ F @ A ) )
        = ( fimage_a_a @ F @ A ) ) ) ).

% finsert_fimage
thf(fact_987_finsert__fimage,axiom,
    ! [X2: fset_a,A: fset_fset_a,F: fset_a > fset_P2596122609397385116fset_a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( ( finser1277071787618628324fset_a @ ( F @ X2 ) @ ( fimage8078386234238232501fset_a @ F @ A ) )
        = ( fimage8078386234238232501fset_a @ F @ A ) ) ) ).

% finsert_fimage
thf(fact_988_finsert__fimage,axiom,
    ! [X2: fset_a,A: fset_fset_a,F: fset_a > a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( ( finsert_a @ ( F @ X2 ) @ ( fimage_fset_a_a @ F @ A ) )
        = ( fimage_fset_a_a @ F @ A ) ) ) ).

% finsert_fimage
thf(fact_989_finsert__fimage,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,F: produc21584292878202806fset_a > fset_a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ A )
     => ( ( finsert_fset_a @ ( F @ X2 ) @ ( fimage5315711910825859755fset_a @ F @ A ) )
        = ( fimage5315711910825859755fset_a @ F @ A ) ) ) ).

% finsert_fimage
thf(fact_990_finsert__fimage,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,F: produc21584292878202806fset_a > a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ A )
     => ( ( finsert_a @ ( F @ X2 ) @ ( fimage5721101993989878341et_a_a @ F @ A ) )
        = ( fimage5721101993989878341et_a_a @ F @ A ) ) ) ).

% finsert_fimage
thf(fact_991_finsert__fimage,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > product_prod_b_c] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( ( finser2611709836832183728od_b_c @ ( F @ X2 ) @ ( fimage7236410608496998447od_b_c @ F @ A ) )
        = ( fimage7236410608496998447od_b_c @ F @ A ) ) ) ).

% finsert_fimage
thf(fact_992_finsert__fimage,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,F: produc2669932201312201782od_c_a > a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( ( finsert_a @ ( F @ X2 ) @ ( fimage9200358715597384645_c_a_a @ F @ A ) )
        = ( fimage9200358715597384645_c_a_a @ F @ A ) ) ) ).

% finsert_fimage
thf(fact_993_finsert__fminus1,axiom,
    ! [X2: a,B: fset_a,A: fset_a] :
      ( ( fmember_a @ X2 @ B )
     => ( ( minus_minus_fset_a @ ( finsert_a @ X2 @ A ) @ B )
        = ( minus_minus_fset_a @ A @ B ) ) ) ).

% finsert_fminus1
thf(fact_994_finsert__fminus1,axiom,
    ! [X2: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ B )
     => ( ( minus_7225097421573603715fset_a @ ( finser2277912206736664318fset_a @ X2 @ A ) @ B )
        = ( minus_7225097421573603715fset_a @ A @ B ) ) ) ).

% finsert_fminus1
thf(fact_995_finsert__fminus1,axiom,
    ! [X2: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ B )
     => ( ( minus_8021415692904097795od_c_a @ ( finser4474152491806270334od_c_a @ X2 @ A ) @ B )
        = ( minus_8021415692904097795od_c_a @ A @ B ) ) ) ).

% finsert_fminus1
thf(fact_996_finsert__fminus1,axiom,
    ! [X2: fset_a,B: fset_fset_a,A: fset_fset_a] :
      ( ( fmember_fset_a @ X2 @ B )
     => ( ( minus_5856361552470604977fset_a @ ( finsert_fset_a @ X2 @ A ) @ B )
        = ( minus_5856361552470604977fset_a @ A @ B ) ) ) ).

% finsert_fminus1
thf(fact_997_finsert__fminus__single,axiom,
    ! [A2: a,A: fset_a] :
      ( ( finsert_a @ A2 @ ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ bot_bot_fset_a ) ) )
      = ( finsert_a @ A2 @ A ) ) ).

% finsert_fminus_single
thf(fact_998_finsert__fminus__single,axiom,
    ! [A2: fset_a,A: fset_fset_a] :
      ( ( finsert_fset_a @ A2 @ ( minus_5856361552470604977fset_a @ A @ ( finsert_fset_a @ A2 @ bot_bot_fset_fset_a ) ) )
      = ( finsert_fset_a @ A2 @ A ) ) ).

% finsert_fminus_single
thf(fact_999_pair__imageI,axiom,
    ! [A2: b,B2: c,A: set_Product_prod_b_c,F: b > c > a] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_7762865866877615259_b_c_a @ ( produc8837194378803474129_b_c_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1000_pair__imageI,axiom,
    ! [A2: c,B2: fset_a,A: set_Pr1098063973736579051fset_a,F: c > fset_a > a] :
      ( ( member1042037746285471284fset_a @ ( produc1535188672328598523fset_a @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_5808782396235892280et_a_a @ ( produc3491678822951036142et_a_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1001_pair__imageI,axiom,
    ! [A2: b,B2: produc4838349196543436939fset_a,A: set_Pr7381868494853627219fset_a,F: b > produc4838349196543436939fset_a > a] :
      ( ( member7243057332918018972fset_a @ ( produc4496741132823077603fset_a @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_9009886171310279632et_a_a @ ( produc1878885086199489542et_a_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1002_pair__imageI,axiom,
    ! [A2: a,B2: produc2133359284458116493od_c_a,A: set_Pr8357624050790843030od_c_a,F: a > produc2133359284458116493od_c_a > a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_9171994905208062541_c_a_a @ ( produc4731461908963710979_c_a_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1003_pair__imageI,axiom,
    ! [A2: b,B2: c,A: set_Product_prod_b_c,F: b > c > produc2669932201312201782od_c_a] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ A2 @ B2 ) @ A )
     => ( member5502040268208861279od_c_a @ ( F @ A2 @ B2 ) @ ( image_2296332512377132755od_c_a @ ( produc829960961062246153od_c_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1004_pair__imageI,axiom,
    ! [A2: fset_a,B2: produc1137077444975408883fset_a,A: set_Pr4695589342295920918fset_a,F: fset_a > produc1137077444975408883fset_a > a] :
      ( ( member4899939873214261983fset_a @ ( produc3008349866153638182fset_a @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_3466649715130532045et_a_a @ ( produc2456308241817076355et_a_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1005_pair__imageI,axiom,
    ! [A2: c,B2: fset_a,A: set_Pr1098063973736579051fset_a,F: c > fset_a > produc2669932201312201782od_c_a] :
      ( ( member1042037746285471284fset_a @ ( produc1535188672328598523fset_a @ A2 @ B2 ) @ A )
     => ( member5502040268208861279od_c_a @ ( F @ A2 @ B2 ) @ ( image_1461573630340395376od_c_a @ ( produc7147840336095436326od_c_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1006_pair__imageI,axiom,
    ! [A2: product_prod_c_a,B2: produc2669932201312201782od_c_a,A: set_Pr2640428967771533528od_c_a,F: product_prod_c_a > produc2669932201312201782od_c_a > a] :
      ( ( member6508725278956418977od_c_a @ ( produc7911353075612250984od_c_a @ A2 @ B2 ) @ A )
     => ( member_a @ ( F @ A2 @ B2 ) @ ( image_7535793878625517963_c_a_a @ ( produc2725652790355472577_c_a_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1007_pair__imageI,axiom,
    ! [A2: b,B2: c,A: set_Product_prod_b_c,F: b > c > produc21584292878202806fset_a] :
      ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ A2 @ B2 ) @ A )
     => ( member4899939873214261983fset_a @ ( F @ A2 @ B2 ) @ ( image_2255128035744431443fset_a @ ( produc4415396419650312073fset_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1008_pair__imageI,axiom,
    ! [A2: b,B2: produc4838349196543436939fset_a,A: set_Pr7381868494853627219fset_a,F: b > produc4838349196543436939fset_a > produc2669932201312201782od_c_a] :
      ( ( member7243057332918018972fset_a @ ( produc4496741132823077603fset_a @ A2 @ B2 ) @ A )
     => ( member5502040268208861279od_c_a @ ( F @ A2 @ B2 ) @ ( image_1655043632036638472od_c_a @ ( produc8285974001681159486od_c_a @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_1009_ffUnion__funion__distrib,axiom,
    ! [A: fset_f3206295485159697154fset_a,B: fset_f3206295485159697154fset_a] :
      ( ( ffUnio2651180436950201422fset_a @ ( sup_su2038437349791040726fset_a @ A @ B ) )
      = ( sup_su6818703939759823728fset_a @ ( ffUnio2651180436950201422fset_a @ A ) @ ( ffUnio2651180436950201422fset_a @ B ) ) ) ).

% ffUnion_funion_distrib
thf(fact_1010_ffUnion__funion__distrib,axiom,
    ! [A: fset_fset_fset_a,B: fset_fset_fset_a] :
      ( ( ffUnion_fset_a @ ( sup_su6542719571438662916fset_a @ A @ B ) )
      = ( sup_sup_fset_fset_a @ ( ffUnion_fset_a @ A ) @ ( ffUnion_fset_a @ B ) ) ) ).

% ffUnion_funion_distrib
thf(fact_1011_ffUnion__funion__distrib,axiom,
    ! [A: fset_fset_a,B: fset_fset_a] :
      ( ( ffUnion_a @ ( sup_sup_fset_fset_a @ A @ B ) )
      = ( sup_sup_fset_a @ ( ffUnion_a @ A ) @ ( ffUnion_a @ B ) ) ) ).

% ffUnion_funion_distrib
thf(fact_1012_ffunion__empty,axiom,
    ( ( ffUnio2651180436950201422fset_a @ bot_bo6991086108236063086fset_a )
    = bot_bo2316247384970961928fset_a ) ).

% ffunion_empty
thf(fact_1013_ffunion__empty,axiom,
    ( ( ffUnion_fset_a @ bot_bo3176604761809765788fset_a )
    = bot_bot_fset_fset_a ) ).

% ffunion_empty
thf(fact_1014_ffunion__empty,axiom,
    ( ( ffUnion_a @ bot_bot_fset_fset_a )
    = bot_bot_fset_a ) ).

% ffunion_empty
thf(fact_1015__C1_Oprems_C_I4_J,axiom,
    ! [Q: fset_a] :
      ( ( fmember_fset_a @ Q @ donesa )
     => ( fmember_fset_a @ Q
        @ ( fPow_a
          @ ( sup_sup_fset_a
            @ ( sup_sup_fset_a @ ( fimage9200358715597384645_c_a_a @ produc4225943634317630866od_c_a @ ( ftransitions_a_b_c @ m ) )
              @ ( fimage9200358715597384645_c_a_a
                @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
                @ ( ftransitions_a_b_c @ m ) ) )
            @ ( finsert_a @ ( initial_a_b_c @ m ) @ bot_bot_fset_a ) ) ) ) ) ).

% "1.prems"(4)
thf(fact_1016_assms_I4_J,axiom,
    ! [Q: fset_a] :
      ( ( fmember_fset_a @ Q @ dones )
     => ( fmember_fset_a @ Q
        @ ( fPow_a
          @ ( sup_sup_fset_a
            @ ( sup_sup_fset_a @ ( fimage9200358715597384645_c_a_a @ produc4225943634317630866od_c_a @ ( ftransitions_a_b_c @ m ) )
              @ ( fimage9200358715597384645_c_a_a
                @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
                @ ( ftransitions_a_b_c @ m ) ) )
            @ ( finsert_a @ ( initial_a_b_c @ m ) @ bot_bot_fset_a ) ) ) ) ) ).

% assms(4)
thf(fact_1017_qtrans__def,axiom,
    ( qtrans
    = ( ffUnio2651180436950201422fset_a
      @ ( fimage8078386234238232501fset_a
        @ ^ [Q3: fset_a] :
            ( fimage3947511181509736139fset_a
            @ ( produc4415396419650312073fset_a
              @ ^ [X: b,Y: c] :
                  ( produc3008349866153638182fset_a @ Q3
                  @ ( produc4496741132823077603fset_a @ X
                    @ ( produc1535188672328598523fset_a @ Y
                      @ ( fimage9200358715597384645_c_a_a
                        @ ^ [A4: produc2669932201312201782od_c_a] : ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
                        @ ( ffilte6209691498356321503od_c_a
                          @ ^ [T: produc2669932201312201782od_c_a] :
                              ( ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
                              = ( product_Pair_b_c @ X @ Y ) )
                          @ ( ffilte6209691498356321503od_c_a
                            @ ^ [T: produc2669932201312201782od_c_a] : ( fmember_a @ ( produc4225943634317630866od_c_a @ T ) @ Q3 )
                            @ ( ftransitions_a_b_c @ m ) ) ) ) ) ) ) )
            @ ( fimage7236410608496998447od_b_c
              @ ^ [T: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
              @ ( ffilte6209691498356321503od_c_a
                @ ^ [T: produc2669932201312201782od_c_a] : ( fmember_a @ ( produc4225943634317630866od_c_a @ T ) @ Q3 )
                @ ( ftransitions_a_b_c @ m ) ) ) )
        @ ( minus_5856361552470604977fset_a
          @ ( fimage5315711910825859755fset_a
            @ ^ [A4: produc21584292878202806fset_a] : ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ A4 ) ) )
            @ tsa )
          @ donesa ) ) ) ) ).

% qtrans_def
thf(fact_1018_sup__set__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A3: set_a,B4: set_a] :
          ( collect_a
          @ ( sup_sup_a_o
            @ ^ [X: a] : ( member_a @ X @ A3 )
            @ ^ [X: a] : ( member_a @ X @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_1019_sup__set__def,axiom,
    ( sup_su7764855436033407660od_c_a
    = ( ^ [A3: set_Pr2640428967771533528od_c_a,B4: set_Pr2640428967771533528od_c_a] :
          ( collec7519840507084740067od_c_a
          @ ( sup_su1241247566940024625_c_a_o
            @ ^ [X: produc3205353004677322104od_c_a] : ( member6508725278956418977od_c_a @ X @ A3 )
            @ ^ [X: produc3205353004677322104od_c_a] : ( member6508725278956418977od_c_a @ X @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_1020_sup__set__def,axiom,
    ( sup_su4665271905203605226fset_a
    = ( ^ [A3: set_Pr4695589342295920918fset_a,B4: set_Pr4695589342295920918fset_a] :
          ( collec8021744632519760673fset_a
          @ ( sup_su1707976133927056499et_a_o
            @ ^ [X: produc21584292878202806fset_a] : ( member4899939873214261983fset_a @ X @ A3 )
            @ ^ [X: produc21584292878202806fset_a] : ( member4899939873214261983fset_a @ X @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_1021_sup__set__def,axiom,
    ( sup_su7950883165052597354od_c_a
    = ( ^ [A3: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ( sup_su3030496677316786931_c_a_o
            @ ^ [X: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X @ A3 )
            @ ^ [X: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X @ B4 ) ) ) ) ) ).

% sup_set_def
thf(fact_1022_UnI2,axiom,
    ! [C: produc2669932201312201782od_c_a,B: set_Pr8357624050790843030od_c_a,A: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C @ B )
     => ( member5502040268208861279od_c_a @ C @ ( sup_su7950883165052597354od_c_a @ A @ B ) ) ) ).

% UnI2
thf(fact_1023_UnI2,axiom,
    ! [C: a,B: set_a,A: set_a] :
      ( ( member_a @ C @ B )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI2
thf(fact_1024_UnI2,axiom,
    ! [C: produc3205353004677322104od_c_a,B: set_Pr2640428967771533528od_c_a,A: set_Pr2640428967771533528od_c_a] :
      ( ( member6508725278956418977od_c_a @ C @ B )
     => ( member6508725278956418977od_c_a @ C @ ( sup_su7764855436033407660od_c_a @ A @ B ) ) ) ).

% UnI2
thf(fact_1025_UnI2,axiom,
    ! [C: produc21584292878202806fset_a,B: set_Pr4695589342295920918fset_a,A: set_Pr4695589342295920918fset_a] :
      ( ( member4899939873214261983fset_a @ C @ B )
     => ( member4899939873214261983fset_a @ C @ ( sup_su4665271905203605226fset_a @ A @ B ) ) ) ).

% UnI2
thf(fact_1026_UnI1,axiom,
    ! [C: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,B: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C @ A )
     => ( member5502040268208861279od_c_a @ C @ ( sup_su7950883165052597354od_c_a @ A @ B ) ) ) ).

% UnI1
thf(fact_1027_UnI1,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ A )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).

% UnI1
thf(fact_1028_UnI1,axiom,
    ! [C: produc3205353004677322104od_c_a,A: set_Pr2640428967771533528od_c_a,B: set_Pr2640428967771533528od_c_a] :
      ( ( member6508725278956418977od_c_a @ C @ A )
     => ( member6508725278956418977od_c_a @ C @ ( sup_su7764855436033407660od_c_a @ A @ B ) ) ) ).

% UnI1
thf(fact_1029_UnI1,axiom,
    ! [C: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a,B: set_Pr4695589342295920918fset_a] :
      ( ( member4899939873214261983fset_a @ C @ A )
     => ( member4899939873214261983fset_a @ C @ ( sup_su4665271905203605226fset_a @ A @ B ) ) ) ).

% UnI1
thf(fact_1030_UnE,axiom,
    ! [C: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,B: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C @ ( sup_su7950883165052597354od_c_a @ A @ B ) )
     => ( ~ ( member5502040268208861279od_c_a @ C @ A )
       => ( member5502040268208861279od_c_a @ C @ B ) ) ) ).

% UnE
thf(fact_1031_UnE,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
     => ( ~ ( member_a @ C @ A )
       => ( member_a @ C @ B ) ) ) ).

% UnE
thf(fact_1032_UnE,axiom,
    ! [C: produc3205353004677322104od_c_a,A: set_Pr2640428967771533528od_c_a,B: set_Pr2640428967771533528od_c_a] :
      ( ( member6508725278956418977od_c_a @ C @ ( sup_su7764855436033407660od_c_a @ A @ B ) )
     => ( ~ ( member6508725278956418977od_c_a @ C @ A )
       => ( member6508725278956418977od_c_a @ C @ B ) ) ) ).

% UnE
thf(fact_1033_UnE,axiom,
    ! [C: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a,B: set_Pr4695589342295920918fset_a] :
      ( ( member4899939873214261983fset_a @ C @ ( sup_su4665271905203605226fset_a @ A @ B ) )
     => ( ~ ( member4899939873214261983fset_a @ C @ A )
       => ( member4899939873214261983fset_a @ C @ B ) ) ) ).

% UnE
thf(fact_1034_Un__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A3: set_a,B4: set_a] :
          ( collect_a
          @ ^ [X: a] :
              ( ( member_a @ X @ A3 )
              | ( member_a @ X @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_1035_Un__def,axiom,
    ( sup_su7764855436033407660od_c_a
    = ( ^ [A3: set_Pr2640428967771533528od_c_a,B4: set_Pr2640428967771533528od_c_a] :
          ( collec7519840507084740067od_c_a
          @ ^ [X: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X @ A3 )
              | ( member6508725278956418977od_c_a @ X @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_1036_Un__def,axiom,
    ( sup_su4665271905203605226fset_a
    = ( ^ [A3: set_Pr4695589342295920918fset_a,B4: set_Pr4695589342295920918fset_a] :
          ( collec8021744632519760673fset_a
          @ ^ [X: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ X @ A3 )
              | ( member4899939873214261983fset_a @ X @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_1037_Un__def,axiom,
    ( sup_su7950883165052597354od_c_a
    = ( ^ [A3: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [X: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X @ A3 )
              | ( member5502040268208861279od_c_a @ X @ B4 ) ) ) ) ) ).

% Un_def
thf(fact_1038_Collect__disj__eq,axiom,
    ! [P2: produc2669932201312201782od_c_a > $o,Q2: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X: produc2669932201312201782od_c_a] :
            ( ( P2 @ X )
            | ( Q2 @ X ) ) )
      = ( sup_su7950883165052597354od_c_a @ ( collec8405073525749625505od_c_a @ P2 ) @ ( collec8405073525749625505od_c_a @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_1039_finsert__commute,axiom,
    ! [X2: a,Y2: a,A: fset_a] :
      ( ( finsert_a @ X2 @ ( finsert_a @ Y2 @ A ) )
      = ( finsert_a @ Y2 @ ( finsert_a @ X2 @ A ) ) ) ).

% finsert_commute
thf(fact_1040_prod_Ocase__distrib,axiom,
    ! [H: produc21584292878202806fset_a > produc21584292878202806fset_a,F: b > c > produc21584292878202806fset_a,Prod: product_prod_b_c] :
      ( ( H @ ( produc4415396419650312073fset_a @ F @ Prod ) )
      = ( produc4415396419650312073fset_a
        @ ^ [X12: b,X24: c] : ( H @ ( F @ X12 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_1041_finsertE,axiom,
    ! [A2: a,B2: a,A: fset_a] :
      ( ( fmember_a @ A2 @ ( finsert_a @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( fmember_a @ A2 @ A ) ) ) ).

% finsertE
thf(fact_1042_finsertE,axiom,
    ! [A2: fset_a,B2: fset_a,A: fset_fset_a] :
      ( ( fmember_fset_a @ A2 @ ( finsert_fset_a @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( fmember_fset_a @ A2 @ A ) ) ) ).

% finsertE
thf(fact_1043_finsertE,axiom,
    ! [A2: produc21584292878202806fset_a,B2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ A2 @ ( finser2277912206736664318fset_a @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( fmembe5403795861166450045fset_a @ A2 @ A ) ) ) ).

% finsertE
thf(fact_1044_finsertE,axiom,
    ! [A2: produc2669932201312201782od_c_a,B2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ A2 @ ( finser4474152491806270334od_c_a @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( fmembe4375279799106287869od_c_a @ A2 @ A ) ) ) ).

% finsertE
thf(fact_1045_finsertI1,axiom,
    ! [A2: a,B: fset_a] : ( fmember_a @ A2 @ ( finsert_a @ A2 @ B ) ) ).

% finsertI1
thf(fact_1046_finsertI1,axiom,
    ! [A2: fset_a,B: fset_fset_a] : ( fmember_fset_a @ A2 @ ( finsert_fset_a @ A2 @ B ) ) ).

% finsertI1
thf(fact_1047_finsertI1,axiom,
    ! [A2: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a] : ( fmembe5403795861166450045fset_a @ A2 @ ( finser2277912206736664318fset_a @ A2 @ B ) ) ).

% finsertI1
thf(fact_1048_finsertI1,axiom,
    ! [A2: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a] : ( fmembe4375279799106287869od_c_a @ A2 @ ( finser4474152491806270334od_c_a @ A2 @ B ) ) ).

% finsertI1
thf(fact_1049_finsertI2,axiom,
    ! [A2: a,B: fset_a,B2: a] :
      ( ( fmember_a @ A2 @ B )
     => ( fmember_a @ A2 @ ( finsert_a @ B2 @ B ) ) ) ).

% finsertI2
thf(fact_1050_finsertI2,axiom,
    ! [A2: fset_a,B: fset_fset_a,B2: fset_a] :
      ( ( fmember_fset_a @ A2 @ B )
     => ( fmember_fset_a @ A2 @ ( finsert_fset_a @ B2 @ B ) ) ) ).

% finsertI2
thf(fact_1051_finsertI2,axiom,
    ! [A2: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a,B2: produc21584292878202806fset_a] :
      ( ( fmembe5403795861166450045fset_a @ A2 @ B )
     => ( fmembe5403795861166450045fset_a @ A2 @ ( finser2277912206736664318fset_a @ B2 @ B ) ) ) ).

% finsertI2
thf(fact_1052_finsertI2,axiom,
    ! [A2: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a,B2: produc2669932201312201782od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ A2 @ B )
     => ( fmembe4375279799106287869od_c_a @ A2 @ ( finser4474152491806270334od_c_a @ B2 @ B ) ) ) ).

% finsertI2
thf(fact_1053_set__finsert,axiom,
    ! [X2: a,A: fset_a] :
      ( ( fmember_a @ X2 @ A )
     => ~ ! [B7: fset_a] :
            ( ( A
              = ( finsert_a @ X2 @ B7 ) )
           => ( fmember_a @ X2 @ B7 ) ) ) ).

% set_finsert
thf(fact_1054_set__finsert,axiom,
    ! [X2: fset_a,A: fset_fset_a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ~ ! [B7: fset_fset_a] :
            ( ( A
              = ( finsert_fset_a @ X2 @ B7 ) )
           => ( fmember_fset_a @ X2 @ B7 ) ) ) ).

% set_finsert
thf(fact_1055_set__finsert,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ A )
     => ~ ! [B7: fset_P2596122609397385116fset_a] :
            ( ( A
              = ( finser2277912206736664318fset_a @ X2 @ B7 ) )
           => ( fmembe5403795861166450045fset_a @ X2 @ B7 ) ) ) ).

% set_finsert
thf(fact_1056_set__finsert,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ~ ! [B7: fset_P8043821951902805788od_c_a] :
            ( ( A
              = ( finser4474152491806270334od_c_a @ X2 @ B7 ) )
           => ( fmembe4375279799106287869od_c_a @ X2 @ B7 ) ) ) ).

% set_finsert
thf(fact_1057_finsert__ident,axiom,
    ! [X2: a,A: fset_a,B: fset_a] :
      ( ~ ( fmember_a @ X2 @ A )
     => ( ~ ( fmember_a @ X2 @ B )
       => ( ( ( finsert_a @ X2 @ A )
            = ( finsert_a @ X2 @ B ) )
          = ( A = B ) ) ) ) ).

% finsert_ident
thf(fact_1058_finsert__ident,axiom,
    ! [X2: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ~ ( fmember_fset_a @ X2 @ A )
     => ( ~ ( fmember_fset_a @ X2 @ B )
       => ( ( ( finsert_fset_a @ X2 @ A )
            = ( finsert_fset_a @ X2 @ B ) )
          = ( A = B ) ) ) ) ).

% finsert_ident
thf(fact_1059_finsert__ident,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ~ ( fmembe5403795861166450045fset_a @ X2 @ A )
     => ( ~ ( fmembe5403795861166450045fset_a @ X2 @ B )
       => ( ( ( finser2277912206736664318fset_a @ X2 @ A )
            = ( finser2277912206736664318fset_a @ X2 @ B ) )
          = ( A = B ) ) ) ) ).

% finsert_ident
thf(fact_1060_finsert__ident,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ~ ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( ~ ( fmembe4375279799106287869od_c_a @ X2 @ B )
       => ( ( ( finser4474152491806270334od_c_a @ X2 @ A )
            = ( finser4474152491806270334od_c_a @ X2 @ B ) )
          = ( A = B ) ) ) ) ).

% finsert_ident
thf(fact_1061_finsert__absorb,axiom,
    ! [A2: a,A: fset_a] :
      ( ( fmember_a @ A2 @ A )
     => ( ( finsert_a @ A2 @ A )
        = A ) ) ).

% finsert_absorb
thf(fact_1062_finsert__absorb,axiom,
    ! [A2: fset_a,A: fset_fset_a] :
      ( ( fmember_fset_a @ A2 @ A )
     => ( ( finsert_fset_a @ A2 @ A )
        = A ) ) ).

% finsert_absorb
thf(fact_1063_finsert__absorb,axiom,
    ! [A2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ A2 @ A )
     => ( ( finser2277912206736664318fset_a @ A2 @ A )
        = A ) ) ).

% finsert_absorb
thf(fact_1064_finsert__absorb,axiom,
    ! [A2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ A2 @ A )
     => ( ( finser4474152491806270334od_c_a @ A2 @ A )
        = A ) ) ).

% finsert_absorb
thf(fact_1065_finsert__eq__iff,axiom,
    ! [A2: a,A: fset_a,B2: a,B: fset_a] :
      ( ~ ( fmember_a @ A2 @ A )
     => ( ~ ( fmember_a @ B2 @ B )
       => ( ( ( finsert_a @ A2 @ A )
            = ( finsert_a @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C4: fset_a] :
                  ( ( A
                    = ( finsert_a @ B2 @ C4 ) )
                  & ~ ( fmember_a @ B2 @ C4 )
                  & ( B
                    = ( finsert_a @ A2 @ C4 ) )
                  & ~ ( fmember_a @ A2 @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_1066_finsert__eq__iff,axiom,
    ! [A2: fset_a,A: fset_fset_a,B2: fset_a,B: fset_fset_a] :
      ( ~ ( fmember_fset_a @ A2 @ A )
     => ( ~ ( fmember_fset_a @ B2 @ B )
       => ( ( ( finsert_fset_a @ A2 @ A )
            = ( finsert_fset_a @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C4: fset_fset_a] :
                  ( ( A
                    = ( finsert_fset_a @ B2 @ C4 ) )
                  & ~ ( fmember_fset_a @ B2 @ C4 )
                  & ( B
                    = ( finsert_fset_a @ A2 @ C4 ) )
                  & ~ ( fmember_fset_a @ A2 @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_1067_finsert__eq__iff,axiom,
    ! [A2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B2: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a] :
      ( ~ ( fmembe5403795861166450045fset_a @ A2 @ A )
     => ( ~ ( fmembe5403795861166450045fset_a @ B2 @ B )
       => ( ( ( finser2277912206736664318fset_a @ A2 @ A )
            = ( finser2277912206736664318fset_a @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C4: fset_P2596122609397385116fset_a] :
                  ( ( A
                    = ( finser2277912206736664318fset_a @ B2 @ C4 ) )
                  & ~ ( fmembe5403795861166450045fset_a @ B2 @ C4 )
                  & ( B
                    = ( finser2277912206736664318fset_a @ A2 @ C4 ) )
                  & ~ ( fmembe5403795861166450045fset_a @ A2 @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_1068_finsert__eq__iff,axiom,
    ! [A2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,B2: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a] :
      ( ~ ( fmembe4375279799106287869od_c_a @ A2 @ A )
     => ( ~ ( fmembe4375279799106287869od_c_a @ B2 @ B )
       => ( ( ( finser4474152491806270334od_c_a @ A2 @ A )
            = ( finser4474152491806270334od_c_a @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C4: fset_P8043821951902805788od_c_a] :
                  ( ( A
                    = ( finser4474152491806270334od_c_a @ B2 @ C4 ) )
                  & ~ ( fmembe4375279799106287869od_c_a @ B2 @ C4 )
                  & ( B
                    = ( finser4474152491806270334od_c_a @ A2 @ C4 ) )
                  & ~ ( fmembe4375279799106287869od_c_a @ A2 @ C4 ) ) ) ) ) ) ) ).

% finsert_eq_iff
thf(fact_1069_mk__disjoint__finsert,axiom,
    ! [A2: a,A: fset_a] :
      ( ( fmember_a @ A2 @ A )
     => ? [B7: fset_a] :
          ( ( A
            = ( finsert_a @ A2 @ B7 ) )
          & ~ ( fmember_a @ A2 @ B7 ) ) ) ).

% mk_disjoint_finsert
thf(fact_1070_mk__disjoint__finsert,axiom,
    ! [A2: fset_a,A: fset_fset_a] :
      ( ( fmember_fset_a @ A2 @ A )
     => ? [B7: fset_fset_a] :
          ( ( A
            = ( finsert_fset_a @ A2 @ B7 ) )
          & ~ ( fmember_fset_a @ A2 @ B7 ) ) ) ).

% mk_disjoint_finsert
thf(fact_1071_mk__disjoint__finsert,axiom,
    ! [A2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ A2 @ A )
     => ? [B7: fset_P2596122609397385116fset_a] :
          ( ( A
            = ( finser2277912206736664318fset_a @ A2 @ B7 ) )
          & ~ ( fmembe5403795861166450045fset_a @ A2 @ B7 ) ) ) ).

% mk_disjoint_finsert
thf(fact_1072_mk__disjoint__finsert,axiom,
    ! [A2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ A2 @ A )
     => ? [B7: fset_P8043821951902805788od_c_a] :
          ( ( A
            = ( finser4474152491806270334od_c_a @ A2 @ B7 ) )
          & ~ ( fmembe4375279799106287869od_c_a @ A2 @ B7 ) ) ) ).

% mk_disjoint_finsert
thf(fact_1073_finsert__not__fempty,axiom,
    ! [A2: a,A: fset_a] :
      ( ( finsert_a @ A2 @ A )
     != bot_bot_fset_a ) ).

% finsert_not_fempty
thf(fact_1074_finsert__not__fempty,axiom,
    ! [A2: fset_a,A: fset_fset_a] :
      ( ( finsert_fset_a @ A2 @ A )
     != bot_bot_fset_fset_a ) ).

% finsert_not_fempty
thf(fact_1075_fsingleton__inject,axiom,
    ! [A2: a,B2: a] :
      ( ( ( finsert_a @ A2 @ bot_bot_fset_a )
        = ( finsert_a @ B2 @ bot_bot_fset_a ) )
     => ( A2 = B2 ) ) ).

% fsingleton_inject
thf(fact_1076_fsingleton__inject,axiom,
    ! [A2: fset_a,B2: fset_a] :
      ( ( ( finsert_fset_a @ A2 @ bot_bot_fset_fset_a )
        = ( finsert_fset_a @ B2 @ bot_bot_fset_fset_a ) )
     => ( A2 = B2 ) ) ).

% fsingleton_inject
thf(fact_1077_fdoubleton__eq__iff,axiom,
    ! [A2: a,B2: a,C: a,D2: a] :
      ( ( ( finsert_a @ A2 @ ( finsert_a @ B2 @ bot_bot_fset_a ) )
        = ( finsert_a @ C @ ( finsert_a @ D2 @ bot_bot_fset_a ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% fdoubleton_eq_iff
thf(fact_1078_fdoubleton__eq__iff,axiom,
    ! [A2: fset_a,B2: fset_a,C: fset_a,D2: fset_a] :
      ( ( ( finsert_fset_a @ A2 @ ( finsert_fset_a @ B2 @ bot_bot_fset_fset_a ) )
        = ( finsert_fset_a @ C @ ( finsert_fset_a @ D2 @ bot_bot_fset_fset_a ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% fdoubleton_eq_iff
thf(fact_1079_fset__exhaust,axiom,
    ! [S3: fset_a] :
      ( ( S3 != bot_bot_fset_a )
     => ~ ! [X3: a,S4: fset_a] :
            ( S3
           != ( finsert_a @ X3 @ S4 ) ) ) ).

% fset_exhaust
thf(fact_1080_fset__exhaust,axiom,
    ! [S3: fset_fset_a] :
      ( ( S3 != bot_bot_fset_fset_a )
     => ~ ! [X3: fset_a,S4: fset_fset_a] :
            ( S3
           != ( finsert_fset_a @ X3 @ S4 ) ) ) ).

% fset_exhaust
thf(fact_1081_FSet_Ofset__induct,axiom,
    ! [P2: fset_a > $o,S3: fset_a] :
      ( ( P2 @ bot_bot_fset_a )
     => ( ! [X3: a,S5: fset_a] :
            ( ( P2 @ S5 )
           => ( P2 @ ( finsert_a @ X3 @ S5 ) ) )
       => ( P2 @ S3 ) ) ) ).

% FSet.fset_induct
thf(fact_1082_FSet_Ofset__induct,axiom,
    ! [P2: fset_fset_a > $o,S3: fset_fset_a] :
      ( ( P2 @ bot_bot_fset_fset_a )
     => ( ! [X3: fset_a,S5: fset_fset_a] :
            ( ( P2 @ S5 )
           => ( P2 @ ( finsert_fset_a @ X3 @ S5 ) ) )
       => ( P2 @ S3 ) ) ) ).

% FSet.fset_induct
thf(fact_1083_old_Oprod_Ocase,axiom,
    ! [F: b > c > produc21584292878202806fset_a,X1: b,X23: c] :
      ( ( produc4415396419650312073fset_a @ F @ ( product_Pair_b_c @ X1 @ X23 ) )
      = ( F @ X1 @ X23 ) ) ).

% old.prod.case
thf(fact_1084_cond__case__prod__eta,axiom,
    ! [F: b > c > produc21584292878202806fset_a,G: product_prod_b_c > produc21584292878202806fset_a] :
      ( ! [X3: b,Y3: c] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_b_c @ X3 @ Y3 ) ) )
     => ( ( produc4415396419650312073fset_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_1085_case__prod__eta,axiom,
    ! [F: product_prod_b_c > produc21584292878202806fset_a] :
      ( ( produc4415396419650312073fset_a
        @ ^ [X: b,Y: c] : ( F @ ( product_Pair_b_c @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_1086_case__prodE2,axiom,
    ! [Q2: produc21584292878202806fset_a > $o,P2: b > c > produc21584292878202806fset_a,Z2: product_prod_b_c] :
      ( ( Q2 @ ( produc4415396419650312073fset_a @ P2 @ Z2 ) )
     => ~ ! [X3: b,Y3: c] :
            ( ( Z2
              = ( product_Pair_b_c @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_1087_snd__def,axiom,
    ( product_snd_c_fset_a
    = ( produc1647539370155386068fset_a
      @ ^ [X12: c,X24: fset_a] : X24 ) ) ).

% snd_def
thf(fact_1088_snd__def,axiom,
    ( product_snd_c_a
    = ( produc1561580084227405268_c_a_a
      @ ^ [X12: c,X24: a] : X24 ) ) ).

% snd_def
thf(fact_1089_snd__def,axiom,
    ( produc8704244169652099857fset_a
    = ( produc3489749502604197267fset_a
      @ ^ [X12: b,X24: produc4838349196543436939fset_a] : X24 ) ) ).

% snd_def
thf(fact_1090_snd__def,axiom,
    ( produc5221446324160585515od_c_a
    = ( produc5632948794012708883od_c_a
      @ ^ [X12: b,X24: product_prod_c_a] : X24 ) ) ).

% snd_def
thf(fact_1091_snd__def,axiom,
    ( produc8088130389382527188od_c_a
    = ( produc3172194304981725330od_c_a
      @ ^ [X12: a,X24: produc2133359284458116493od_c_a] : X24 ) ) ).

% snd_def
thf(fact_1092_snd__def,axiom,
    ( produc4643634399596806294od_c_a
    = ( produc3245024683913906425od_c_a
      @ ^ [X12: product_prod_c_a,X24: produc2669932201312201782od_c_a] : X24 ) ) ).

% snd_def
thf(fact_1093_snd__def,axiom,
    ( produc8188425890649629780fset_a
    = ( produc7500112268220036600fset_a
      @ ^ [X12: fset_a,X24: produc1137077444975408883fset_a] : X24 ) ) ).

% snd_def
thf(fact_1094_fst__def,axiom,
    ( product_fst_c_fset_a
    = ( produc3491678822951036144et_a_c
      @ ^ [X12: c,X24: fset_a] : X12 ) ) ).

% fst_def
thf(fact_1095_fst__def,axiom,
    ( product_fst_c_a
    = ( produc1561580084227405270_c_a_c
      @ ^ [X12: c,X24: a] : X12 ) ) ).

% fst_def
thf(fact_1096_fst__def,axiom,
    ( produc5616988955220391631fset_a
    = ( produc1878885086199489543et_a_b
      @ ^ [X12: b,X24: produc4838349196543436939fset_a] : X12 ) ) ).

% fst_def
thf(fact_1097_fst__def,axiom,
    ( produc301949074746585577od_c_a
    = ( produc2354600891577745261_c_a_b
      @ ^ [X12: b,X24: product_prod_c_a] : X12 ) ) ).

% fst_def
thf(fact_1098_fst__def,axiom,
    ( produc4225943634317630866od_c_a
    = ( produc4731461908963710979_c_a_a
      @ ^ [X12: a,X24: produc2133359284458116493od_c_a] : X12 ) ) ).

% fst_def
thf(fact_1099_fst__def,axiom,
    ( produc8171714036265584980od_c_a
    = ( produc5138706694719407208od_c_a
      @ ^ [X12: product_prod_c_a,X24: produc2669932201312201782od_c_a] : X12 ) ) ).

% fst_def
thf(fact_1100_fst__def,axiom,
    ( produc1739398754328533778fset_a
    = ( produc2027362624671691753fset_a
      @ ^ [X12: fset_a,X24: produc1137077444975408883fset_a] : X12 ) ) ).

% fst_def
thf(fact_1101_eq__ffilter,axiom,
    ! [P2: a > $o,A: fset_a,Q2: a > $o] :
      ( ( ( ffilter_a @ P2 @ A )
        = ( ffilter_a @ Q2 @ A ) )
      = ( ! [X: a] :
            ( ( fmember_a @ X @ A )
           => ( ( P2 @ X )
              = ( Q2 @ X ) ) ) ) ) ).

% eq_ffilter
thf(fact_1102_eq__ffilter,axiom,
    ! [P2: fset_a > $o,A: fset_fset_a,Q2: fset_a > $o] :
      ( ( ( ffilter_fset_a @ P2 @ A )
        = ( ffilter_fset_a @ Q2 @ A ) )
      = ( ! [X: fset_a] :
            ( ( fmember_fset_a @ X @ A )
           => ( ( P2 @ X )
              = ( Q2 @ X ) ) ) ) ) ).

% eq_ffilter
thf(fact_1103_eq__ffilter,axiom,
    ! [P2: produc21584292878202806fset_a > $o,A: fset_P2596122609397385116fset_a,Q2: produc21584292878202806fset_a > $o] :
      ( ( ( ffilte3318296817612072799fset_a @ P2 @ A )
        = ( ffilte3318296817612072799fset_a @ Q2 @ A ) )
      = ( ! [X: produc21584292878202806fset_a] :
            ( ( fmembe5403795861166450045fset_a @ X @ A )
           => ( ( P2 @ X )
              = ( Q2 @ X ) ) ) ) ) ).

% eq_ffilter
thf(fact_1104_eq__ffilter,axiom,
    ! [P2: produc2669932201312201782od_c_a > $o,A: fset_P8043821951902805788od_c_a,Q2: produc2669932201312201782od_c_a > $o] :
      ( ( ( ffilte6209691498356321503od_c_a @ P2 @ A )
        = ( ffilte6209691498356321503od_c_a @ Q2 @ A ) )
      = ( ! [X: produc2669932201312201782od_c_a] :
            ( ( fmembe4375279799106287869od_c_a @ X @ A )
           => ( ( P2 @ X )
              = ( Q2 @ X ) ) ) ) ) ).

% eq_ffilter
thf(fact_1105_ffUnion__fmember__ob,axiom,
    ! [X2: a,XS: fset_fset_a] :
      ( ( fmember_a @ X2 @ ( ffUnion_a @ XS ) )
     => ? [X6: fset_a] :
          ( ( fmember_fset_a @ X6 @ XS )
          & ( fmember_a @ X2 @ X6 ) ) ) ).

% ffUnion_fmember_ob
thf(fact_1106_ffUnion__fmember__ob,axiom,
    ! [X2: fset_a,XS: fset_fset_fset_a] :
      ( ( fmember_fset_a @ X2 @ ( ffUnion_fset_a @ XS ) )
     => ? [X6: fset_fset_a] :
          ( ( fmember_fset_fset_a @ X6 @ XS )
          & ( fmember_fset_a @ X2 @ X6 ) ) ) ).

% ffUnion_fmember_ob
thf(fact_1107_ffUnion__fmember__ob,axiom,
    ! [X2: produc21584292878202806fset_a,XS: fset_f3206295485159697154fset_a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ ( ffUnio2651180436950201422fset_a @ XS ) )
     => ? [X6: fset_P2596122609397385116fset_a] :
          ( ( fmembe865372479932229091fset_a @ X6 @ XS )
          & ( fmembe5403795861166450045fset_a @ X2 @ X6 ) ) ) ).

% ffUnion_fmember_ob
thf(fact_1108_ffUnion__fmember__ob,axiom,
    ! [X2: produc2669932201312201782od_c_a,XS: fset_f8091011237881300354od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ ( ffUnio1999024989708583118od_c_a @ XS ) )
     => ? [X6: fset_P8043821951902805788od_c_a] :
          ( ( fmembe2338955566798629987od_c_a @ X6 @ XS )
          & ( fmembe4375279799106287869od_c_a @ X2 @ X6 ) ) ) ).

% ffUnion_fmember_ob
thf(fact_1109_fset__induct__stronger,axiom,
    ! [P2: fset_P2596122609397385116fset_a > $o,S3: fset_P2596122609397385116fset_a] :
      ( ( P2 @ bot_bo2316247384970961928fset_a )
     => ( ! [X3: produc21584292878202806fset_a,S5: fset_P2596122609397385116fset_a] :
            ( ~ ( fmembe5403795861166450045fset_a @ X3 @ S5 )
           => ( ( P2 @ S5 )
             => ( P2 @ ( finser2277912206736664318fset_a @ X3 @ S5 ) ) ) )
       => ( P2 @ S3 ) ) ) ).

% fset_induct_stronger
thf(fact_1110_fset__induct__stronger,axiom,
    ! [P2: fset_P8043821951902805788od_c_a > $o,S3: fset_P8043821951902805788od_c_a] :
      ( ( P2 @ bot_bo1116342619594069896od_c_a )
     => ( ! [X3: produc2669932201312201782od_c_a,S5: fset_P8043821951902805788od_c_a] :
            ( ~ ( fmembe4375279799106287869od_c_a @ X3 @ S5 )
           => ( ( P2 @ S5 )
             => ( P2 @ ( finser4474152491806270334od_c_a @ X3 @ S5 ) ) ) )
       => ( P2 @ S3 ) ) ) ).

% fset_induct_stronger
thf(fact_1111_fset__induct__stronger,axiom,
    ! [P2: fset_a > $o,S3: fset_a] :
      ( ( P2 @ bot_bot_fset_a )
     => ( ! [X3: a,S5: fset_a] :
            ( ~ ( fmember_a @ X3 @ S5 )
           => ( ( P2 @ S5 )
             => ( P2 @ ( finsert_a @ X3 @ S5 ) ) ) )
       => ( P2 @ S3 ) ) ) ).

% fset_induct_stronger
thf(fact_1112_fset__induct__stronger,axiom,
    ! [P2: fset_fset_a > $o,S3: fset_fset_a] :
      ( ( P2 @ bot_bot_fset_fset_a )
     => ( ! [X3: fset_a,S5: fset_fset_a] :
            ( ~ ( fmember_fset_a @ X3 @ S5 )
           => ( ( P2 @ S5 )
             => ( P2 @ ( finsert_fset_a @ X3 @ S5 ) ) ) )
       => ( P2 @ S3 ) ) ) ).

% fset_induct_stronger
thf(fact_1113_fset__strong__cases,axiom,
    ! [Xs: fset_P2596122609397385116fset_a] :
      ( ( Xs != bot_bo2316247384970961928fset_a )
     => ~ ! [Ys3: fset_P2596122609397385116fset_a,X3: produc21584292878202806fset_a] :
            ( ~ ( fmembe5403795861166450045fset_a @ X3 @ Ys3 )
           => ( Xs
             != ( finser2277912206736664318fset_a @ X3 @ Ys3 ) ) ) ) ).

% fset_strong_cases
thf(fact_1114_fset__strong__cases,axiom,
    ! [Xs: fset_P8043821951902805788od_c_a] :
      ( ( Xs != bot_bo1116342619594069896od_c_a )
     => ~ ! [Ys3: fset_P8043821951902805788od_c_a,X3: produc2669932201312201782od_c_a] :
            ( ~ ( fmembe4375279799106287869od_c_a @ X3 @ Ys3 )
           => ( Xs
             != ( finser4474152491806270334od_c_a @ X3 @ Ys3 ) ) ) ) ).

% fset_strong_cases
thf(fact_1115_fset__strong__cases,axiom,
    ! [Xs: fset_a] :
      ( ( Xs != bot_bot_fset_a )
     => ~ ! [Ys3: fset_a,X3: a] :
            ( ~ ( fmember_a @ X3 @ Ys3 )
           => ( Xs
             != ( finsert_a @ X3 @ Ys3 ) ) ) ) ).

% fset_strong_cases
thf(fact_1116_fset__strong__cases,axiom,
    ! [Xs: fset_fset_a] :
      ( ( Xs != bot_bot_fset_fset_a )
     => ~ ! [Ys3: fset_fset_a,X3: fset_a] :
            ( ~ ( fmember_fset_a @ X3 @ Ys3 )
           => ( Xs
             != ( finsert_fset_a @ X3 @ Ys3 ) ) ) ) ).

% fset_strong_cases
thf(fact_1117_fsingleton__iff,axiom,
    ! [B2: produc21584292878202806fset_a,A2: produc21584292878202806fset_a] :
      ( ( fmembe5403795861166450045fset_a @ B2 @ ( finser2277912206736664318fset_a @ A2 @ bot_bo2316247384970961928fset_a ) )
      = ( B2 = A2 ) ) ).

% fsingleton_iff
thf(fact_1118_fsingleton__iff,axiom,
    ! [B2: produc2669932201312201782od_c_a,A2: produc2669932201312201782od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ B2 @ ( finser4474152491806270334od_c_a @ A2 @ bot_bo1116342619594069896od_c_a ) )
      = ( B2 = A2 ) ) ).

% fsingleton_iff
thf(fact_1119_fsingleton__iff,axiom,
    ! [B2: a,A2: a] :
      ( ( fmember_a @ B2 @ ( finsert_a @ A2 @ bot_bot_fset_a ) )
      = ( B2 = A2 ) ) ).

% fsingleton_iff
thf(fact_1120_fsingleton__iff,axiom,
    ! [B2: fset_a,A2: fset_a] :
      ( ( fmember_fset_a @ B2 @ ( finsert_fset_a @ A2 @ bot_bot_fset_fset_a ) )
      = ( B2 = A2 ) ) ).

% fsingleton_iff
thf(fact_1121_fset__induct2,axiom,
    ! [P2: fset_a > fset_a > $o,Xsa: fset_a,Ysa: fset_a] :
      ( ( P2 @ bot_bot_fset_a @ bot_bot_fset_a )
     => ( ! [X3: a,Xs2: fset_a] :
            ( ~ ( fmember_a @ X3 @ Xs2 )
           => ( P2 @ ( finsert_a @ X3 @ Xs2 ) @ bot_bot_fset_a ) )
       => ( ! [Y3: a,Ys3: fset_a] :
              ( ~ ( fmember_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bot_fset_a @ ( finsert_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: a,Xs2: fset_a,Y3: a,Ys3: fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmember_a @ X3 @ Xs2 )
                 => ( ~ ( fmember_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finsert_a @ X3 @ Xs2 ) @ ( finsert_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1122_fset__induct2,axiom,
    ! [P2: fset_a > fset_fset_a > $o,Xsa: fset_a,Ysa: fset_fset_a] :
      ( ( P2 @ bot_bot_fset_a @ bot_bot_fset_fset_a )
     => ( ! [X3: a,Xs2: fset_a] :
            ( ~ ( fmember_a @ X3 @ Xs2 )
           => ( P2 @ ( finsert_a @ X3 @ Xs2 ) @ bot_bot_fset_fset_a ) )
       => ( ! [Y3: fset_a,Ys3: fset_fset_a] :
              ( ~ ( fmember_fset_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bot_fset_a @ ( finsert_fset_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: a,Xs2: fset_a,Y3: fset_a,Ys3: fset_fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmember_a @ X3 @ Xs2 )
                 => ( ~ ( fmember_fset_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finsert_a @ X3 @ Xs2 ) @ ( finsert_fset_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1123_fset__induct2,axiom,
    ! [P2: fset_fset_a > fset_a > $o,Xsa: fset_fset_a,Ysa: fset_a] :
      ( ( P2 @ bot_bot_fset_fset_a @ bot_bot_fset_a )
     => ( ! [X3: fset_a,Xs2: fset_fset_a] :
            ( ~ ( fmember_fset_a @ X3 @ Xs2 )
           => ( P2 @ ( finsert_fset_a @ X3 @ Xs2 ) @ bot_bot_fset_a ) )
       => ( ! [Y3: a,Ys3: fset_a] :
              ( ~ ( fmember_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bot_fset_fset_a @ ( finsert_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: fset_a,Xs2: fset_fset_a,Y3: a,Ys3: fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmember_fset_a @ X3 @ Xs2 )
                 => ( ~ ( fmember_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finsert_fset_a @ X3 @ Xs2 ) @ ( finsert_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1124_fset__induct2,axiom,
    ! [P2: fset_fset_a > fset_fset_a > $o,Xsa: fset_fset_a,Ysa: fset_fset_a] :
      ( ( P2 @ bot_bot_fset_fset_a @ bot_bot_fset_fset_a )
     => ( ! [X3: fset_a,Xs2: fset_fset_a] :
            ( ~ ( fmember_fset_a @ X3 @ Xs2 )
           => ( P2 @ ( finsert_fset_a @ X3 @ Xs2 ) @ bot_bot_fset_fset_a ) )
       => ( ! [Y3: fset_a,Ys3: fset_fset_a] :
              ( ~ ( fmember_fset_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bot_fset_fset_a @ ( finsert_fset_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: fset_a,Xs2: fset_fset_a,Y3: fset_a,Ys3: fset_fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmember_fset_a @ X3 @ Xs2 )
                 => ( ~ ( fmember_fset_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finsert_fset_a @ X3 @ Xs2 ) @ ( finsert_fset_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1125_fset__induct2,axiom,
    ! [P2: fset_P8043821951902805788od_c_a > fset_a > $o,Xsa: fset_P8043821951902805788od_c_a,Ysa: fset_a] :
      ( ( P2 @ bot_bo1116342619594069896od_c_a @ bot_bot_fset_a )
     => ( ! [X3: produc2669932201312201782od_c_a,Xs2: fset_P8043821951902805788od_c_a] :
            ( ~ ( fmembe4375279799106287869od_c_a @ X3 @ Xs2 )
           => ( P2 @ ( finser4474152491806270334od_c_a @ X3 @ Xs2 ) @ bot_bot_fset_a ) )
       => ( ! [Y3: a,Ys3: fset_a] :
              ( ~ ( fmember_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bo1116342619594069896od_c_a @ ( finsert_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: produc2669932201312201782od_c_a,Xs2: fset_P8043821951902805788od_c_a,Y3: a,Ys3: fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmembe4375279799106287869od_c_a @ X3 @ Xs2 )
                 => ( ~ ( fmember_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finser4474152491806270334od_c_a @ X3 @ Xs2 ) @ ( finsert_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1126_fset__induct2,axiom,
    ! [P2: fset_a > fset_P8043821951902805788od_c_a > $o,Xsa: fset_a,Ysa: fset_P8043821951902805788od_c_a] :
      ( ( P2 @ bot_bot_fset_a @ bot_bo1116342619594069896od_c_a )
     => ( ! [X3: a,Xs2: fset_a] :
            ( ~ ( fmember_a @ X3 @ Xs2 )
           => ( P2 @ ( finsert_a @ X3 @ Xs2 ) @ bot_bo1116342619594069896od_c_a ) )
       => ( ! [Y3: produc2669932201312201782od_c_a,Ys3: fset_P8043821951902805788od_c_a] :
              ( ~ ( fmembe4375279799106287869od_c_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bot_fset_a @ ( finser4474152491806270334od_c_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: a,Xs2: fset_a,Y3: produc2669932201312201782od_c_a,Ys3: fset_P8043821951902805788od_c_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmember_a @ X3 @ Xs2 )
                 => ( ~ ( fmembe4375279799106287869od_c_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finsert_a @ X3 @ Xs2 ) @ ( finser4474152491806270334od_c_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1127_fset__induct2,axiom,
    ! [P2: fset_P8043821951902805788od_c_a > fset_fset_a > $o,Xsa: fset_P8043821951902805788od_c_a,Ysa: fset_fset_a] :
      ( ( P2 @ bot_bo1116342619594069896od_c_a @ bot_bot_fset_fset_a )
     => ( ! [X3: produc2669932201312201782od_c_a,Xs2: fset_P8043821951902805788od_c_a] :
            ( ~ ( fmembe4375279799106287869od_c_a @ X3 @ Xs2 )
           => ( P2 @ ( finser4474152491806270334od_c_a @ X3 @ Xs2 ) @ bot_bot_fset_fset_a ) )
       => ( ! [Y3: fset_a,Ys3: fset_fset_a] :
              ( ~ ( fmember_fset_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bo1116342619594069896od_c_a @ ( finsert_fset_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: produc2669932201312201782od_c_a,Xs2: fset_P8043821951902805788od_c_a,Y3: fset_a,Ys3: fset_fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmembe4375279799106287869od_c_a @ X3 @ Xs2 )
                 => ( ~ ( fmember_fset_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finser4474152491806270334od_c_a @ X3 @ Xs2 ) @ ( finsert_fset_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1128_fset__induct2,axiom,
    ! [P2: fset_fset_a > fset_P8043821951902805788od_c_a > $o,Xsa: fset_fset_a,Ysa: fset_P8043821951902805788od_c_a] :
      ( ( P2 @ bot_bot_fset_fset_a @ bot_bo1116342619594069896od_c_a )
     => ( ! [X3: fset_a,Xs2: fset_fset_a] :
            ( ~ ( fmember_fset_a @ X3 @ Xs2 )
           => ( P2 @ ( finsert_fset_a @ X3 @ Xs2 ) @ bot_bo1116342619594069896od_c_a ) )
       => ( ! [Y3: produc2669932201312201782od_c_a,Ys3: fset_P8043821951902805788od_c_a] :
              ( ~ ( fmembe4375279799106287869od_c_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bot_fset_fset_a @ ( finser4474152491806270334od_c_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: fset_a,Xs2: fset_fset_a,Y3: produc2669932201312201782od_c_a,Ys3: fset_P8043821951902805788od_c_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmember_fset_a @ X3 @ Xs2 )
                 => ( ~ ( fmembe4375279799106287869od_c_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finsert_fset_a @ X3 @ Xs2 ) @ ( finser4474152491806270334od_c_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1129_fset__induct2,axiom,
    ! [P2: fset_P2596122609397385116fset_a > fset_a > $o,Xsa: fset_P2596122609397385116fset_a,Ysa: fset_a] :
      ( ( P2 @ bot_bo2316247384970961928fset_a @ bot_bot_fset_a )
     => ( ! [X3: produc21584292878202806fset_a,Xs2: fset_P2596122609397385116fset_a] :
            ( ~ ( fmembe5403795861166450045fset_a @ X3 @ Xs2 )
           => ( P2 @ ( finser2277912206736664318fset_a @ X3 @ Xs2 ) @ bot_bot_fset_a ) )
       => ( ! [Y3: a,Ys3: fset_a] :
              ( ~ ( fmember_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bo2316247384970961928fset_a @ ( finsert_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: produc21584292878202806fset_a,Xs2: fset_P2596122609397385116fset_a,Y3: a,Ys3: fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmembe5403795861166450045fset_a @ X3 @ Xs2 )
                 => ( ~ ( fmember_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finser2277912206736664318fset_a @ X3 @ Xs2 ) @ ( finsert_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1130_fset__induct2,axiom,
    ! [P2: fset_a > fset_P2596122609397385116fset_a > $o,Xsa: fset_a,Ysa: fset_P2596122609397385116fset_a] :
      ( ( P2 @ bot_bot_fset_a @ bot_bo2316247384970961928fset_a )
     => ( ! [X3: a,Xs2: fset_a] :
            ( ~ ( fmember_a @ X3 @ Xs2 )
           => ( P2 @ ( finsert_a @ X3 @ Xs2 ) @ bot_bo2316247384970961928fset_a ) )
       => ( ! [Y3: produc21584292878202806fset_a,Ys3: fset_P2596122609397385116fset_a] :
              ( ~ ( fmembe5403795861166450045fset_a @ Y3 @ Ys3 )
             => ( P2 @ bot_bot_fset_a @ ( finser2277912206736664318fset_a @ Y3 @ Ys3 ) ) )
         => ( ! [X3: a,Xs2: fset_a,Y3: produc21584292878202806fset_a,Ys3: fset_P2596122609397385116fset_a] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( ~ ( fmember_a @ X3 @ Xs2 )
                 => ( ~ ( fmembe5403795861166450045fset_a @ Y3 @ Ys3 )
                   => ( P2 @ ( finsert_a @ X3 @ Xs2 ) @ ( finser2277912206736664318fset_a @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ Xsa @ Ysa ) ) ) ) ) ).

% fset_induct2
thf(fact_1131_finsert__fminus__if,axiom,
    ! [X2: a,B: fset_a,A: fset_a] :
      ( ( ( fmember_a @ X2 @ B )
       => ( ( minus_minus_fset_a @ ( finsert_a @ X2 @ A ) @ B )
          = ( minus_minus_fset_a @ A @ B ) ) )
      & ( ~ ( fmember_a @ X2 @ B )
       => ( ( minus_minus_fset_a @ ( finsert_a @ X2 @ A ) @ B )
          = ( finsert_a @ X2 @ ( minus_minus_fset_a @ A @ B ) ) ) ) ) ).

% finsert_fminus_if
thf(fact_1132_finsert__fminus__if,axiom,
    ! [X2: produc21584292878202806fset_a,B: fset_P2596122609397385116fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( ( fmembe5403795861166450045fset_a @ X2 @ B )
       => ( ( minus_7225097421573603715fset_a @ ( finser2277912206736664318fset_a @ X2 @ A ) @ B )
          = ( minus_7225097421573603715fset_a @ A @ B ) ) )
      & ( ~ ( fmembe5403795861166450045fset_a @ X2 @ B )
       => ( ( minus_7225097421573603715fset_a @ ( finser2277912206736664318fset_a @ X2 @ A ) @ B )
          = ( finser2277912206736664318fset_a @ X2 @ ( minus_7225097421573603715fset_a @ A @ B ) ) ) ) ) ).

% finsert_fminus_if
thf(fact_1133_finsert__fminus__if,axiom,
    ! [X2: produc2669932201312201782od_c_a,B: fset_P8043821951902805788od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( ( fmembe4375279799106287869od_c_a @ X2 @ B )
       => ( ( minus_8021415692904097795od_c_a @ ( finser4474152491806270334od_c_a @ X2 @ A ) @ B )
          = ( minus_8021415692904097795od_c_a @ A @ B ) ) )
      & ( ~ ( fmembe4375279799106287869od_c_a @ X2 @ B )
       => ( ( minus_8021415692904097795od_c_a @ ( finser4474152491806270334od_c_a @ X2 @ A ) @ B )
          = ( finser4474152491806270334od_c_a @ X2 @ ( minus_8021415692904097795od_c_a @ A @ B ) ) ) ) ) ).

% finsert_fminus_if
thf(fact_1134_finsert__fminus__if,axiom,
    ! [X2: fset_a,B: fset_fset_a,A: fset_fset_a] :
      ( ( ( fmember_fset_a @ X2 @ B )
       => ( ( minus_5856361552470604977fset_a @ ( finsert_fset_a @ X2 @ A ) @ B )
          = ( minus_5856361552470604977fset_a @ A @ B ) ) )
      & ( ~ ( fmember_fset_a @ X2 @ B )
       => ( ( minus_5856361552470604977fset_a @ ( finsert_fset_a @ X2 @ A ) @ B )
          = ( finsert_fset_a @ X2 @ ( minus_5856361552470604977fset_a @ A @ B ) ) ) ) ) ).

% finsert_fminus_if
thf(fact_1135_finsert__is__funion,axiom,
    ( finser2277912206736664318fset_a
    = ( ^ [A4: produc21584292878202806fset_a] : ( sup_su6818703939759823728fset_a @ ( finser2277912206736664318fset_a @ A4 @ bot_bo2316247384970961928fset_a ) ) ) ) ).

% finsert_is_funion
thf(fact_1136_finsert__is__funion,axiom,
    ( finsert_a
    = ( ^ [A4: a] : ( sup_sup_fset_a @ ( finsert_a @ A4 @ bot_bot_fset_a ) ) ) ) ).

% finsert_is_funion
thf(fact_1137_finsert__is__funion,axiom,
    ( finsert_fset_a
    = ( ^ [A4: fset_a] : ( sup_sup_fset_fset_a @ ( finsert_fset_a @ A4 @ bot_bot_fset_fset_a ) ) ) ) ).

% finsert_is_funion
thf(fact_1138_fsingleton__funion__iff,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a] :
      ( ( ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a )
        = ( sup_su6818703939759823728fset_a @ A @ B ) )
      = ( ( ( A = bot_bo2316247384970961928fset_a )
          & ( B
            = ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) ) )
        | ( ( A
            = ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) )
          & ( B = bot_bo2316247384970961928fset_a ) )
        | ( ( A
            = ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) )
          & ( B
            = ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) ) ) ) ) ).

% fsingleton_funion_iff
thf(fact_1139_fsingleton__funion__iff,axiom,
    ! [X2: a,A: fset_a,B: fset_a] :
      ( ( ( finsert_a @ X2 @ bot_bot_fset_a )
        = ( sup_sup_fset_a @ A @ B ) )
      = ( ( ( A = bot_bot_fset_a )
          & ( B
            = ( finsert_a @ X2 @ bot_bot_fset_a ) ) )
        | ( ( A
            = ( finsert_a @ X2 @ bot_bot_fset_a ) )
          & ( B = bot_bot_fset_a ) )
        | ( ( A
            = ( finsert_a @ X2 @ bot_bot_fset_a ) )
          & ( B
            = ( finsert_a @ X2 @ bot_bot_fset_a ) ) ) ) ) ).

% fsingleton_funion_iff
thf(fact_1140_fsingleton__funion__iff,axiom,
    ! [X2: fset_a,A: fset_fset_a,B: fset_fset_a] :
      ( ( ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a )
        = ( sup_sup_fset_fset_a @ A @ B ) )
      = ( ( ( A = bot_bot_fset_fset_a )
          & ( B
            = ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) ) )
        | ( ( A
            = ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) )
          & ( B = bot_bot_fset_fset_a ) )
        | ( ( A
            = ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) )
          & ( B
            = ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) ) ) ) ) ).

% fsingleton_funion_iff
thf(fact_1141_funion__fsingleton__iff,axiom,
    ! [A: fset_P2596122609397385116fset_a,B: fset_P2596122609397385116fset_a,X2: produc21584292878202806fset_a] :
      ( ( ( sup_su6818703939759823728fset_a @ A @ B )
        = ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) )
      = ( ( ( A = bot_bo2316247384970961928fset_a )
          & ( B
            = ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) ) )
        | ( ( A
            = ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) )
          & ( B = bot_bo2316247384970961928fset_a ) )
        | ( ( A
            = ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) )
          & ( B
            = ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) ) ) ) ) ).

% funion_fsingleton_iff
thf(fact_1142_funion__fsingleton__iff,axiom,
    ! [A: fset_a,B: fset_a,X2: a] :
      ( ( ( sup_sup_fset_a @ A @ B )
        = ( finsert_a @ X2 @ bot_bot_fset_a ) )
      = ( ( ( A = bot_bot_fset_a )
          & ( B
            = ( finsert_a @ X2 @ bot_bot_fset_a ) ) )
        | ( ( A
            = ( finsert_a @ X2 @ bot_bot_fset_a ) )
          & ( B = bot_bot_fset_a ) )
        | ( ( A
            = ( finsert_a @ X2 @ bot_bot_fset_a ) )
          & ( B
            = ( finsert_a @ X2 @ bot_bot_fset_a ) ) ) ) ) ).

% funion_fsingleton_iff
thf(fact_1143_funion__fsingleton__iff,axiom,
    ! [A: fset_fset_a,B: fset_fset_a,X2: fset_a] :
      ( ( ( sup_sup_fset_fset_a @ A @ B )
        = ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) )
      = ( ( ( A = bot_bot_fset_fset_a )
          & ( B
            = ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) ) )
        | ( ( A
            = ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) )
          & ( B = bot_bot_fset_fset_a ) )
        | ( ( A
            = ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) )
          & ( B
            = ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) ) ) ) ) ).

% funion_fsingleton_iff
thf(fact_1144_fminus__finsert,axiom,
    ! [A: fset_a,A2: a,B: fset_a] :
      ( ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ B ) )
      = ( minus_minus_fset_a @ ( minus_minus_fset_a @ A @ B ) @ ( finsert_a @ A2 @ bot_bot_fset_a ) ) ) ).

% fminus_finsert
thf(fact_1145_fminus__finsert,axiom,
    ! [A: fset_fset_a,A2: fset_a,B: fset_fset_a] :
      ( ( minus_5856361552470604977fset_a @ A @ ( finsert_fset_a @ A2 @ B ) )
      = ( minus_5856361552470604977fset_a @ ( minus_5856361552470604977fset_a @ A @ B ) @ ( finsert_fset_a @ A2 @ bot_bot_fset_fset_a ) ) ) ).

% fminus_finsert
thf(fact_1146_fminus__finsert2,axiom,
    ! [A: fset_a,A2: a,B: fset_a] :
      ( ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ B ) )
      = ( minus_minus_fset_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ bot_bot_fset_a ) ) @ B ) ) ).

% fminus_finsert2
thf(fact_1147_fminus__finsert2,axiom,
    ! [A: fset_fset_a,A2: fset_a,B: fset_fset_a] :
      ( ( minus_5856361552470604977fset_a @ A @ ( finsert_fset_a @ A2 @ B ) )
      = ( minus_5856361552470604977fset_a @ ( minus_5856361552470604977fset_a @ A @ ( finsert_fset_a @ A2 @ bot_bot_fset_fset_a ) ) @ B ) ) ).

% fminus_finsert2
thf(fact_1148_language__contains__empty__sequence,axiom,
    ! [M2: fsm_a_b_c] : ( member6330420149250801815od_b_c @ nil_Product_prod_b_c @ ( lS_a_b_c @ M2 @ ( initial_a_b_c @ M2 ) ) ) ).

% language_contains_empty_sequence
thf(fact_1149_split__beta,axiom,
    ( produc4415396419650312073fset_a
    = ( ^ [F2: b > c > produc21584292878202806fset_a,Prod3: product_prod_b_c] : ( F2 @ ( product_fst_b_c @ Prod3 ) @ ( product_snd_b_c @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_1150_case__prod__beta,axiom,
    ( produc4415396419650312073fset_a
    = ( ^ [F2: b > c > produc21584292878202806fset_a,P4: product_prod_b_c] : ( F2 @ ( product_fst_b_c @ P4 ) @ ( product_snd_b_c @ P4 ) ) ) ) ).

% case_prod_beta
thf(fact_1151_fPow__finsert,axiom,
    ! [A2: a,A: fset_a] :
      ( ( fPow_a @ ( finsert_a @ A2 @ A ) )
      = ( sup_sup_fset_fset_a @ ( fPow_a @ A ) @ ( fimage_fset_a_fset_a @ ( finsert_a @ A2 ) @ ( fPow_a @ A ) ) ) ) ).

% fPow_finsert
thf(fact_1152_fimage__constant__conv,axiom,
    ! [A: fset_P8043821951902805788od_c_a,C: product_prod_b_c] :
      ( ( ( A = bot_bo1116342619594069896od_c_a )
       => ( ( fimage7236410608496998447od_b_c
            @ ^ [X: produc2669932201312201782od_c_a] : C
            @ A )
          = bot_bo192962769195302586od_b_c ) )
      & ( ( A != bot_bo1116342619594069896od_c_a )
       => ( ( fimage7236410608496998447od_b_c
            @ ^ [X: produc2669932201312201782od_c_a] : C
            @ A )
          = ( finser2611709836832183728od_b_c @ C @ bot_bo192962769195302586od_b_c ) ) ) ) ).

% fimage_constant_conv
thf(fact_1153_fimage__constant__conv,axiom,
    ! [A: fset_P8649754740804009550od_b_c,C: produc21584292878202806fset_a] :
      ( ( ( A = bot_bo192962769195302586od_b_c )
       => ( ( fimage3947511181509736139fset_a
            @ ^ [X: product_prod_b_c] : C
            @ A )
          = bot_bo2316247384970961928fset_a ) )
      & ( ( A != bot_bo192962769195302586od_b_c )
       => ( ( fimage3947511181509736139fset_a
            @ ^ [X: product_prod_b_c] : C
            @ A )
          = ( finser2277912206736664318fset_a @ C @ bot_bo2316247384970961928fset_a ) ) ) ) ).

% fimage_constant_conv
thf(fact_1154_fimage__constant__conv,axiom,
    ! [A: fset_P8043821951902805788od_c_a,C: a] :
      ( ( ( A = bot_bo1116342619594069896od_c_a )
       => ( ( fimage9200358715597384645_c_a_a
            @ ^ [X: produc2669932201312201782od_c_a] : C
            @ A )
          = bot_bot_fset_a ) )
      & ( ( A != bot_bo1116342619594069896od_c_a )
       => ( ( fimage9200358715597384645_c_a_a
            @ ^ [X: produc2669932201312201782od_c_a] : C
            @ A )
          = ( finsert_a @ C @ bot_bot_fset_a ) ) ) ) ).

% fimage_constant_conv
thf(fact_1155_fimage__constant__conv,axiom,
    ! [A: fset_P2596122609397385116fset_a,C: fset_a] :
      ( ( ( A = bot_bo2316247384970961928fset_a )
       => ( ( fimage5315711910825859755fset_a
            @ ^ [X: produc21584292878202806fset_a] : C
            @ A )
          = bot_bot_fset_fset_a ) )
      & ( ( A != bot_bo2316247384970961928fset_a )
       => ( ( fimage5315711910825859755fset_a
            @ ^ [X: produc21584292878202806fset_a] : C
            @ A )
          = ( finsert_fset_a @ C @ bot_bot_fset_fset_a ) ) ) ) ).

% fimage_constant_conv
thf(fact_1156_fimage__constant__conv,axiom,
    ! [A: fset_a,C: a] :
      ( ( ( A = bot_bot_fset_a )
       => ( ( fimage_a_a
            @ ^ [X: a] : C
            @ A )
          = bot_bot_fset_a ) )
      & ( ( A != bot_bot_fset_a )
       => ( ( fimage_a_a
            @ ^ [X: a] : C
            @ A )
          = ( finsert_a @ C @ bot_bot_fset_a ) ) ) ) ).

% fimage_constant_conv
thf(fact_1157_fimage__constant__conv,axiom,
    ! [A: fset_a,C: fset_a] :
      ( ( ( A = bot_bot_fset_a )
       => ( ( fimage_a_fset_a
            @ ^ [X: a] : C
            @ A )
          = bot_bot_fset_fset_a ) )
      & ( ( A != bot_bot_fset_a )
       => ( ( fimage_a_fset_a
            @ ^ [X: a] : C
            @ A )
          = ( finsert_fset_a @ C @ bot_bot_fset_fset_a ) ) ) ) ).

% fimage_constant_conv
thf(fact_1158_fimage__constant__conv,axiom,
    ! [A: fset_fset_a,C: fset_P2596122609397385116fset_a] :
      ( ( ( A = bot_bot_fset_fset_a )
       => ( ( fimage8078386234238232501fset_a
            @ ^ [X: fset_a] : C
            @ A )
          = bot_bo6991086108236063086fset_a ) )
      & ( ( A != bot_bot_fset_fset_a )
       => ( ( fimage8078386234238232501fset_a
            @ ^ [X: fset_a] : C
            @ A )
          = ( finser1277071787618628324fset_a @ C @ bot_bo6991086108236063086fset_a ) ) ) ) ).

% fimage_constant_conv
thf(fact_1159_fimage__constant__conv,axiom,
    ! [A: fset_fset_a,C: a] :
      ( ( ( A = bot_bot_fset_fset_a )
       => ( ( fimage_fset_a_a
            @ ^ [X: fset_a] : C
            @ A )
          = bot_bot_fset_a ) )
      & ( ( A != bot_bot_fset_fset_a )
       => ( ( fimage_fset_a_a
            @ ^ [X: fset_a] : C
            @ A )
          = ( finsert_a @ C @ bot_bot_fset_a ) ) ) ) ).

% fimage_constant_conv
thf(fact_1160_fimage__constant__conv,axiom,
    ! [A: fset_fset_a,C: fset_a] :
      ( ( ( A = bot_bot_fset_fset_a )
       => ( ( fimage_fset_a_fset_a
            @ ^ [X: fset_a] : C
            @ A )
          = bot_bot_fset_fset_a ) )
      & ( ( A != bot_bot_fset_fset_a )
       => ( ( fimage_fset_a_fset_a
            @ ^ [X: fset_a] : C
            @ A )
          = ( finsert_fset_a @ C @ bot_bot_fset_fset_a ) ) ) ) ).

% fimage_constant_conv
thf(fact_1161_case__prod__unfold,axiom,
    ( produc4415396419650312073fset_a
    = ( ^ [C5: b > c > produc21584292878202806fset_a,P4: product_prod_b_c] : ( C5 @ ( product_fst_b_c @ P4 ) @ ( product_snd_b_c @ P4 ) ) ) ) ).

% case_prod_unfold
thf(fact_1162_case__prod__beta_H,axiom,
    ( produc4415396419650312073fset_a
    = ( ^ [F2: b > c > produc21584292878202806fset_a,X: product_prod_b_c] : ( F2 @ ( product_fst_b_c @ X ) @ ( product_snd_b_c @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_1163_exE__realizer,axiom,
    ! [P2: c > b > $o,P: product_prod_b_c,Q2: produc21584292878202806fset_a > $o,F: b > c > produc21584292878202806fset_a] :
      ( ( P2 @ ( product_snd_b_c @ P ) @ ( product_fst_b_c @ P ) )
     => ( ! [X3: b,Y3: c] :
            ( ( P2 @ Y3 @ X3 )
           => ( Q2 @ ( F @ X3 @ Y3 ) ) )
       => ( Q2 @ ( produc4415396419650312073fset_a @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_1164_fminus__finsert__absorb,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ~ ( fmembe5403795861166450045fset_a @ X2 @ A )
     => ( ( minus_7225097421573603715fset_a @ ( finser2277912206736664318fset_a @ X2 @ A ) @ ( finser2277912206736664318fset_a @ X2 @ bot_bo2316247384970961928fset_a ) )
        = A ) ) ).

% fminus_finsert_absorb
thf(fact_1165_fminus__finsert__absorb,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ~ ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( ( minus_8021415692904097795od_c_a @ ( finser4474152491806270334od_c_a @ X2 @ A ) @ ( finser4474152491806270334od_c_a @ X2 @ bot_bo1116342619594069896od_c_a ) )
        = A ) ) ).

% fminus_finsert_absorb
thf(fact_1166_fminus__finsert__absorb,axiom,
    ! [X2: a,A: fset_a] :
      ( ~ ( fmember_a @ X2 @ A )
     => ( ( minus_minus_fset_a @ ( finsert_a @ X2 @ A ) @ ( finsert_a @ X2 @ bot_bot_fset_a ) )
        = A ) ) ).

% fminus_finsert_absorb
thf(fact_1167_fminus__finsert__absorb,axiom,
    ! [X2: fset_a,A: fset_fset_a] :
      ( ~ ( fmember_fset_a @ X2 @ A )
     => ( ( minus_5856361552470604977fset_a @ ( finsert_fset_a @ X2 @ A ) @ ( finsert_fset_a @ X2 @ bot_bot_fset_fset_a ) )
        = A ) ) ).

% fminus_finsert_absorb
thf(fact_1168_finsert__fminus,axiom,
    ! [A2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a] :
      ( ( fmembe5403795861166450045fset_a @ A2 @ A )
     => ( ( finser2277912206736664318fset_a @ A2 @ ( minus_7225097421573603715fset_a @ A @ ( finser2277912206736664318fset_a @ A2 @ bot_bo2316247384970961928fset_a ) ) )
        = A ) ) ).

% finsert_fminus
thf(fact_1169_finsert__fminus,axiom,
    ! [A2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a] :
      ( ( fmembe4375279799106287869od_c_a @ A2 @ A )
     => ( ( finser4474152491806270334od_c_a @ A2 @ ( minus_8021415692904097795od_c_a @ A @ ( finser4474152491806270334od_c_a @ A2 @ bot_bo1116342619594069896od_c_a ) ) )
        = A ) ) ).

% finsert_fminus
thf(fact_1170_finsert__fminus,axiom,
    ! [A2: a,A: fset_a] :
      ( ( fmember_a @ A2 @ A )
     => ( ( finsert_a @ A2 @ ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ bot_bot_fset_a ) ) )
        = A ) ) ).

% finsert_fminus
thf(fact_1171_finsert__fminus,axiom,
    ! [A2: fset_a,A: fset_fset_a] :
      ( ( fmember_fset_a @ A2 @ A )
     => ( ( finsert_fset_a @ A2 @ ( minus_5856361552470604977fset_a @ A @ ( finsert_fset_a @ A2 @ bot_bot_fset_fset_a ) ) )
        = A ) ) ).

% finsert_fminus
thf(fact_1172_prod_Osplit__sel,axiom,
    ! [P2: produc21584292878202806fset_a > $o,F: b > c > produc21584292878202806fset_a,Prod: product_prod_b_c] :
      ( ( P2 @ ( produc4415396419650312073fset_a @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) )
       => ( P2 @ ( F @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_1173_prod_Osplit__sel__asm,axiom,
    ! [P2: produc21584292878202806fset_a > $o,F: b > c > produc21584292878202806fset_a,Prod: product_prod_b_c] :
      ( ( P2 @ ( produc4415396419650312073fset_a @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_1174_fimage__constant,axiom,
    ! [X2: a,A: fset_a,C: a] :
      ( ( fmember_a @ X2 @ A )
     => ( ( fimage_a_a
          @ ^ [X: a] : C
          @ A )
        = ( finsert_a @ C @ bot_bot_fset_a ) ) ) ).

% fimage_constant
thf(fact_1175_fimage__constant,axiom,
    ! [X2: fset_a,A: fset_fset_a,C: a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( ( fimage_fset_a_a
          @ ^ [X: fset_a] : C
          @ A )
        = ( finsert_a @ C @ bot_bot_fset_a ) ) ) ).

% fimage_constant
thf(fact_1176_fimage__constant,axiom,
    ! [X2: a,A: fset_a,C: fset_a] :
      ( ( fmember_a @ X2 @ A )
     => ( ( fimage_a_fset_a
          @ ^ [X: a] : C
          @ A )
        = ( finsert_fset_a @ C @ bot_bot_fset_fset_a ) ) ) ).

% fimage_constant
thf(fact_1177_fimage__constant,axiom,
    ! [X2: fset_a,A: fset_fset_a,C: fset_a] :
      ( ( fmember_fset_a @ X2 @ A )
     => ( ( fimage_fset_a_fset_a
          @ ^ [X: fset_a] : C
          @ A )
        = ( finsert_fset_a @ C @ bot_bot_fset_fset_a ) ) ) ).

% fimage_constant
thf(fact_1178_fimage__constant,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,C: a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( ( fimage9200358715597384645_c_a_a
          @ ^ [X: produc2669932201312201782od_c_a] : C
          @ A )
        = ( finsert_a @ C @ bot_bot_fset_a ) ) ) ).

% fimage_constant
thf(fact_1179_fimage__constant,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,C: fset_a] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( ( fimage2540242761017857067fset_a
          @ ^ [X: produc2669932201312201782od_c_a] : C
          @ A )
        = ( finsert_fset_a @ C @ bot_bot_fset_fset_a ) ) ) ).

% fimage_constant
thf(fact_1180_fimage__constant,axiom,
    ! [X2: produc2669932201312201782od_c_a,A: fset_P8043821951902805788od_c_a,C: product_prod_b_c] :
      ( ( fmembe4375279799106287869od_c_a @ X2 @ A )
     => ( ( fimage7236410608496998447od_b_c
          @ ^ [X: produc2669932201312201782od_c_a] : C
          @ A )
        = ( finser2611709836832183728od_b_c @ C @ bot_bo192962769195302586od_b_c ) ) ) ).

% fimage_constant
thf(fact_1181_fimage__constant,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,C: a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ A )
     => ( ( fimage5721101993989878341et_a_a
          @ ^ [X: produc21584292878202806fset_a] : C
          @ A )
        = ( finsert_a @ C @ bot_bot_fset_a ) ) ) ).

% fimage_constant
thf(fact_1182_fimage__constant,axiom,
    ! [X2: produc21584292878202806fset_a,A: fset_P2596122609397385116fset_a,C: fset_a] :
      ( ( fmembe5403795861166450045fset_a @ X2 @ A )
     => ( ( fimage5315711910825859755fset_a
          @ ^ [X: produc21584292878202806fset_a] : C
          @ A )
        = ( finsert_fset_a @ C @ bot_bot_fset_fset_a ) ) ) ).

% fimage_constant
thf(fact_1183_fimage__constant,axiom,
    ! [X2: product_prod_b_c,A: fset_P8649754740804009550od_b_c,C: produc21584292878202806fset_a] :
      ( ( fmembe4950632942993856815od_b_c @ X2 @ A )
     => ( ( fimage3947511181509736139fset_a
          @ ^ [X: product_prod_b_c] : C
          @ A )
        = ( finser2277912206736664318fset_a @ C @ bot_bo2316247384970961928fset_a ) ) ) ).

% fimage_constant
thf(fact_1184_find__index_Ocases,axiom,
    ! [X2: produc3153060970645322794fset_a] :
      ( ! [F3: produc21584292878202806fset_a > $o] :
          ( X2
         != ( produc1715491115440840796fset_a @ F3 @ nil_Pr4260649278601056188fset_a ) )
     => ~ ! [F3: produc21584292878202806fset_a > $o,X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc1715491115440840796fset_a @ F3 @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) ) ) ) ).

% find_index.cases
thf(fact_1185_find__index_Ocases,axiom,
    ! [X2: produc7816210539016439438od_b_c] :
      ( ! [F3: product_prod_b_c > $o] :
          ( X2
         != ( produc7392021306980666816od_b_c @ F3 @ nil_Product_prod_b_c ) )
     => ~ ! [F3: product_prod_b_c > $o,X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc7392021306980666816od_b_c @ F3 @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) ) ) ) ).

% find_index.cases
thf(fact_1186_linear__order__from__list__position_H_Ocases,axiom,
    ! [X2: list_P1058411382900698172fset_a] :
      ( ( X2 != nil_Pr4260649278601056188fset_a )
     => ~ ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a] :
            ( X2
           != ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_1187_linear__order__from__list__position_H_Ocases,axiom,
    ! [X2: list_P903359562653991662od_b_c] :
      ( ( X2 != nil_Product_prod_b_c )
     => ~ ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
            ( X2
           != ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_1188_h__obs_Ocases,axiom,
    ! [X2: produc4954405829418270953fset_a] :
      ~ ! [M3: fSM_fs5209479309116428276fset_a,Q8: fset_a,X3: b,Y3: produc4838349196543436939fset_a] :
          ( X2
         != ( produc5186979826147233881fset_a @ M3 @ ( produc3008349866153638182fset_a @ Q8 @ ( produc4496741132823077603fset_a @ X3 @ Y3 ) ) ) ) ).

% h_obs.cases
thf(fact_1189_h__obs_Ocases,axiom,
    ! [X2: produc6162238119313743795fset_a] :
      ~ ! [M3: fSM_fs1444178598041484711fset_a,Q8: b,X3: c,Y3: fset_a] :
          ( X2
         != ( produc6404092902902889635fset_a @ M3 @ ( produc4496741132823077603fset_a @ Q8 @ ( produc1535188672328598523fset_a @ X3 @ Y3 ) ) ) ) ).

% h_obs.cases
thf(fact_1190_map__default_Ocases,axiom,
    ! [X2: produc3730712799252344065fset_a] :
      ( ! [K: fset_a,V2: produc1137077444975408883fset_a,F3: produc1137077444975408883fset_a > produc1137077444975408883fset_a] :
          ( X2
         != ( produc5811172702134561915fset_a @ K @ ( produc5597073046559312616fset_a @ V2 @ ( produc2855626948256020478fset_a @ F3 @ nil_Pr4260649278601056188fset_a ) ) ) )
     => ~ ! [K: fset_a,V2: produc1137077444975408883fset_a,F3: produc1137077444975408883fset_a > produc1137077444975408883fset_a,P3: produc21584292878202806fset_a,Ps: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc5811172702134561915fset_a @ K @ ( produc5597073046559312616fset_a @ V2 @ ( produc2855626948256020478fset_a @ F3 @ ( cons_P3377334492504755052fset_a @ P3 @ Ps ) ) ) ) ) ) ).

% map_default.cases
thf(fact_1191_map__default_Ocases,axiom,
    ! [X2: produc2009451851958091931od_b_c] :
      ( ! [K: b,V2: c,F3: c > c] :
          ( X2
         != ( produc1384939585547926677od_b_c @ K @ ( produc3590379597927054951od_b_c @ V2 @ ( produc1101678828523717360od_b_c @ F3 @ nil_Product_prod_b_c ) ) ) )
     => ~ ! [K: b,V2: c,F3: c > c,P3: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc1384939585547926677od_b_c @ K @ ( produc3590379597927054951od_b_c @ V2 @ ( produc1101678828523717360od_b_c @ F3 @ ( cons_P4529483553340347422od_b_c @ P3 @ Ps ) ) ) ) ) ) ).

% map_default.cases
thf(fact_1192_map__entry_Ocases,axiom,
    ! [X2: produc4932829996385814111fset_a] :
      ( ! [K: fset_a,F3: produc1137077444975408883fset_a > produc1137077444975408883fset_a] :
          ( X2
         != ( produc1160115607834325593fset_a @ K @ ( produc2855626948256020478fset_a @ F3 @ nil_Pr4260649278601056188fset_a ) ) )
     => ~ ! [K: fset_a,F3: produc1137077444975408883fset_a > produc1137077444975408883fset_a,P3: produc21584292878202806fset_a,Ps: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc1160115607834325593fset_a @ K @ ( produc2855626948256020478fset_a @ F3 @ ( cons_P3377334492504755052fset_a @ P3 @ Ps ) ) ) ) ) ).

% map_entry.cases
thf(fact_1193_map__entry_Ocases,axiom,
    ! [X2: produc6452300523622238572od_b_c] :
      ( ! [K: b,F3: c > c] :
          ( X2
         != ( produc7015427828931889894od_b_c @ K @ ( produc1101678828523717360od_b_c @ F3 @ nil_Product_prod_b_c ) ) )
     => ~ ! [K: b,F3: c > c,P3: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc7015427828931889894od_b_c @ K @ ( produc1101678828523717360od_b_c @ F3 @ ( cons_P4529483553340347422od_b_c @ P3 @ Ps ) ) ) ) ) ).

% map_entry.cases
thf(fact_1194_clearjunk_Ocases,axiom,
    ! [X2: list_P1058411382900698172fset_a] :
      ( ( X2 != nil_Pr4260649278601056188fset_a )
     => ~ ! [P3: produc21584292878202806fset_a,Ps: list_P1058411382900698172fset_a] :
            ( X2
           != ( cons_P3377334492504755052fset_a @ P3 @ Ps ) ) ) ).

% clearjunk.cases
thf(fact_1195_clearjunk_Ocases,axiom,
    ! [X2: list_P903359562653991662od_b_c] :
      ( ( X2 != nil_Product_prod_b_c )
     => ~ ! [P3: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
            ( X2
           != ( cons_P4529483553340347422od_b_c @ P3 @ Ps ) ) ) ).

% clearjunk.cases
thf(fact_1196_find__remove_H_Ocases,axiom,
    ! [X2: produc1871883748873956117fset_a] :
      ( ! [P9: produc21584292878202806fset_a > $o,Uu2: list_P1058411382900698172fset_a] :
          ( X2
         != ( produc3416685118976706887fset_a @ P9 @ ( produc316883261966145815fset_a @ nil_Pr4260649278601056188fset_a @ Uu2 ) ) )
     => ~ ! [P9: produc21584292878202806fset_a > $o,X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a,Prev: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc3416685118976706887fset_a @ P9 @ ( produc316883261966145815fset_a @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_1197_find__remove_H_Ocases,axiom,
    ! [X2: produc5559707038353031495od_b_c] :
      ( ! [P9: product_prod_b_c > $o,Uu2: list_P903359562653991662od_b_c] :
          ( X2
         != ( produc1005531610559690873od_b_c @ P9 @ ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Uu2 ) ) )
     => ~ ! [P9: product_prod_b_c > $o,X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Prev: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc1005531610559690873od_b_c @ P9 @ ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_1198_is__prefix_Ocases,axiom,
    ! [X2: produc5258545326496974119fset_a] :
      ( ! [Uu2: list_P1058411382900698172fset_a] :
          ( X2
         != ( produc316883261966145815fset_a @ nil_Pr4260649278601056188fset_a @ Uu2 ) )
     => ( ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a] :
            ( X2
           != ( produc316883261966145815fset_a @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ nil_Pr4260649278601056188fset_a ) )
       => ~ ! [X3: produc21584292878202806fset_a,Xs2: list_P1058411382900698172fset_a,Y3: produc21584292878202806fset_a,Ys3: list_P1058411382900698172fset_a] :
              ( X2
             != ( produc316883261966145815fset_a @ ( cons_P3377334492504755052fset_a @ X3 @ Xs2 ) @ ( cons_P3377334492504755052fset_a @ Y3 @ Ys3 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_1199_is__prefix_Ocases,axiom,
    ! [X2: produc5878855887281848999od_b_c] :
      ( ! [Uu2: list_P903359562653991662od_b_c] :
          ( X2
         != ( produc3076307724246486423od_b_c @ nil_Product_prod_b_c @ Uu2 ) )
     => ( ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c] :
            ( X2
           != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ nil_Product_prod_b_c ) )
       => ~ ! [X3: product_prod_b_c,Xs2: list_P903359562653991662od_b_c,Y3: product_prod_b_c,Ys3: list_P903359562653991662od_b_c] :
              ( X2
             != ( produc3076307724246486423od_b_c @ ( cons_P4529483553340347422od_b_c @ X3 @ Xs2 ) @ ( cons_P4529483553340347422od_b_c @ Y3 @ Ys3 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_1200_observable__language__next,axiom,
    ! [Io: produc4838349196543436939fset_a,Ios: list_P3569802872000582161fset_a,M2: fsm_Pr1175075309171164859fset_a,T4: produc7303452399287242961fset_a] :
      ( ( member2596827339850278714fset_a @ ( cons_P4277760901013084993fset_a @ Io @ Ios ) @ ( lS_Pro8627259047355266299fset_a @ M2 @ ( produc2105839935146153389fset_a @ T4 ) ) )
     => ( ( observ1975363445242043999fset_a @ M2 )
       => ( ( member9109638507678586746fset_a @ T4 @ ( transi1231084882770062892fset_a @ M2 ) )
         => ( ( ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T4 ) )
              = ( product_fst_c_fset_a @ Io ) )
           => ( ( ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) )
                = ( product_snd_c_fset_a @ Io ) )
             => ( member2596827339850278714fset_a @ Ios @ ( lS_Pro8627259047355266299fset_a @ ( from_F6154415312349873588fset_a @ M2 @ ( produc8188425890649629780fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) ) ) @ ( initia8236402991889688966fset_a @ ( from_F6154415312349873588fset_a @ M2 @ ( produc8188425890649629780fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_1201_observable__language__next,axiom,
    ! [Io: product_prod_c_a,Ios: list_P125642481956313003od_c_a,M2: fsm_Pr7461828476839194351_a_c_a,T4: produc3729687299602777783od_c_a] :
      ( ( member5552703068553123156od_c_a @ ( cons_P1742027962761213787od_c_a @ Io @ Ios ) @ ( lS_Pro1436182445161660463_a_c_a @ M2 @ ( produc1784202036732648467od_c_a @ T4 ) ) )
     => ( ( observ3736821838051356051_a_c_a @ M2 )
       => ( ( member5500486793640445664od_c_a @ T4 @ ( transi5448983560151880288_a_c_a @ M2 ) )
         => ( ( ( produc7173261134821384633od_c_a @ ( produc2799690148840255317od_c_a @ T4 ) )
              = ( product_fst_c_a @ Io ) )
           => ( ( ( produc4225943634317630866od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T4 ) ) )
                = ( product_snd_c_a @ Io ) )
             => ( member5552703068553123156od_c_a @ Ios @ ( lS_Pro1436182445161660463_a_c_a @ ( from_F8501008907474727400_a_c_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T4 ) ) ) ) @ ( initia6168580702928107450_a_c_a @ ( from_F8501008907474727400_a_c_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4398916234287704827od_c_a @ ( produc2799690148840255317od_c_a @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_1202_observable__language__next,axiom,
    ! [Io: produc2133359284458116493od_c_a,Ios: list_P5767727266746070163od_c_a,M2: fsm_Pr7194325094602570112od_c_a,T4: produc4887275193591018193od_c_a] :
      ( ( member2650335117833904444od_c_a @ ( cons_P7376644893796496707od_c_a @ Io @ Ios ) @ ( lS_Pro6466918861714006976od_c_a @ M2 @ ( produc5911674319360489645od_c_a @ T4 ) ) )
     => ( ( observ2000839013862770980od_c_a @ M2 )
       => ( ( member5709309679808970746od_c_a @ T4 @ ( transi6473412698945717233od_c_a @ M2 ) )
         => ( ( ( produc2967918992726476220od_c_a @ ( produc5236404386356303343od_c_a @ T4 ) )
              = ( produc301949074746585577od_c_a @ Io ) )
           => ( ( ( produc8171714036265584980od_c_a @ ( produc7936356256688089854od_c_a @ ( produc5236404386356303343od_c_a @ T4 ) ) )
                = ( produc5221446324160585515od_c_a @ Io ) )
             => ( member2650335117833904444od_c_a @ Ios @ ( lS_Pro6466918861714006976od_c_a @ ( from_F7030966089064027001od_c_a @ M2 @ ( produc4643634399596806294od_c_a @ ( produc7936356256688089854od_c_a @ ( produc5236404386356303343od_c_a @ T4 ) ) ) ) @ ( initia5033892106405598539od_c_a @ ( from_F7030966089064027001od_c_a @ M2 @ ( produc4643634399596806294od_c_a @ ( produc7936356256688089854od_c_a @ ( produc5236404386356303343od_c_a @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_1203_observable__language__next,axiom,
    ! [Io: produc5771957243474376000_c_a_a,Ios: list_P8814837271076367046_c_a_a,M2: fsm_Pr7813689363856283786_c_a_a,T4: produc1135898663509404882od_c_a] :
      ( ( member5697445122164201327_c_a_a @ ( cons_P1791870815957980406_c_a_a @ Io @ Ios ) @ ( lS_Pro3220947514165985994_c_a_a @ M2 @ ( produc5475348670448042158od_c_a @ T4 ) ) )
     => ( ( observ5792423948907539502_c_a_a @ M2 )
       => ( ( member6699836568822684667od_c_a @ T4 @ ( transi5048145386435558395_c_a_a @ M2 ) )
         => ( ( ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) )
              = ( produc4522685202854701596_c_a_a @ Io ) )
           => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) )
                = ( produc218810415413925726_c_a_a @ Io ) )
             => ( member5697445122164201327_c_a_a @ Ios @ ( lS_Pro3220947514165985994_c_a_a @ ( from_F748103779160593283_c_a_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) ) @ ( initia2830091458700408661_c_a_a @ ( from_F748103779160593283_c_a_a @ M2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_1204_observable__language__next,axiom,
    ! [Io: produc2080252525628144066et_a_b,Ios: list_P7633400287818446792et_a_b,M2: fsm_Pr2398083450103406218et_a_b,T4: produc7245291065743820818fset_a] :
      ( ( member6660424755668143345et_a_b @ ( cons_P1519664230097792120et_a_b @ Io @ Ios ) @ ( lS_Pro8532755013169807178et_a_b @ M2 @ ( produc8095675384246233966fset_a @ T4 ) ) )
     => ( ( observ1610022369204726958et_a_b @ M2 )
       => ( ( member808877655204049083fset_a @ T4 @ ( transi3322184091305251195et_a_b @ M2 ) )
         => ( ( ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T4 ) )
              = ( product_fst_fset_a_b @ Io ) )
           => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) )
                = ( product_snd_fset_a_b @ Io ) )
             => ( member6660424755668143345et_a_b @ Ios @ ( lS_Pro8532755013169807178et_a_b @ ( from_F6374209438628098307et_a_b @ M2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) ) @ ( initia4041781234081478357et_a_b @ ( from_F6374209438628098307et_a_b @ M2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_1205_observable__language__next,axiom,
    ! [Io: product_prod_b_c,Ios: list_P903359562653991662od_b_c,M2: fsm_a_b_c,T4: produc2669932201312201782od_c_a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ Io @ Ios ) @ ( lS_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T4 ) ) )
     => ( ( observable_a_b_c @ M2 )
       => ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
         => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) )
              = ( product_fst_b_c @ Io ) )
           => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) )
                = ( product_snd_b_c @ Io ) )
             => ( member6330420149250801815od_b_c @ Ios @ ( lS_a_b_c @ ( from_FSM_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_1206_observable__language__next,axiom,
    ! [Io: product_prod_a_b,Ios: list_P1467974523212632366od_a_b,M2: fsm_Pr1886043760914421514_a_a_b,T4: produc3205353004677322104od_c_a] :
      ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ Io @ Ios ) @ ( lS_Pro5114676609533226954_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T4 ) ) )
     => ( ( observ2964502264774598958_a_a_b @ M2 )
       => ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
         => ( ( ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) )
              = ( product_fst_a_b @ Io ) )
           => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) )
                = ( product_snd_a_b @ Io ) )
             => ( member6895035109809442519od_a_b @ Ios @ ( lS_Pro5114676609533226954_a_a_b @ ( from_F6727869081933152899_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) @ ( initia681974817753967189_a_a_b @ ( from_F6727869081933152899_a_a_b @ M2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_1207_observable__language__next,axiom,
    ! [Io: product_prod_b_c,Ios: list_P903359562653991662od_b_c,M2: fsm_fset_a_b_c,T4: produc21584292878202806fset_a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ Io @ Ios ) @ ( lS_fset_a_b_c @ M2 @ ( produc1739398754328533778fset_a @ T4 ) ) )
     => ( ( observ4396094731789362477_a_b_c @ M2 )
       => ( ( member4899939873214261983fset_a @ T4 @ ( transi6427852111903025530_a_b_c @ M2 ) )
         => ( ( ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) )
              = ( product_fst_b_c @ Io ) )
           => ( ( ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
                = ( product_snd_b_c @ Io ) )
             => ( member6330420149250801815od_b_c @ Ios @ ( lS_fset_a_b_c @ ( from_FSM_fset_a_b_c @ M2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) @ ( initial_fset_a_b_c @ ( from_FSM_fset_a_b_c @ M2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_1208_filter__states__simps_I5_J,axiom,
    ! [P2: fset_a > $o,M2: fsm_fset_a_b_c] :
      ( ( P2 @ ( initial_fset_a_b_c @ M2 ) )
     => ( ( transi6427852111903025530_a_b_c @ ( filter8103885693533907990_a_b_c @ M2 @ P2 ) )
        = ( collec8021744632519760673fset_a
          @ ^ [T: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ T @ ( transi6427852111903025530_a_b_c @ M2 ) )
              & ( P2 @ ( produc1739398754328533778fset_a @ T ) )
              & ( P2 @ ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T ) ) ) ) ) ) ) ) ).

% filter_states_simps(5)
thf(fact_1209_filter__states__simps_I5_J,axiom,
    ! [P2: a > $o,M2: fsm_a_b_c] :
      ( ( P2 @ ( initial_a_b_c @ M2 ) )
     => ( ( transitions_a_b_c @ ( filter_states_a_b_c @ M2 @ P2 ) )
        = ( collec8405073525749625505od_c_a
          @ ^ [T: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M2 ) )
              & ( P2 @ ( produc4225943634317630866od_c_a @ T ) )
              & ( P2 @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ) ).

% filter_states_simps(5)
thf(fact_1210_filter__states__simps_I5_J,axiom,
    ! [P2: produc4838349196543436939fset_a > $o,M2: fsm_Pr2398083450103406218et_a_b] :
      ( ( P2 @ ( initia4041781234081478357et_a_b @ M2 ) )
     => ( ( transi3322184091305251195et_a_b @ ( filter8150164458569013783et_a_b @ M2 @ P2 ) )
        = ( collec123419402997478397fset_a
          @ ^ [T: produc7245291065743820818fset_a] :
              ( ( member808877655204049083fset_a @ T @ ( transi3322184091305251195et_a_b @ M2 ) )
              & ( P2 @ ( produc8095675384246233966fset_a @ T ) )
              & ( P2 @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T ) ) ) ) ) ) ) ) ).

% filter_states_simps(5)
thf(fact_1211_filter__states__simps_I5_J,axiom,
    ! [P2: product_prod_c_a > $o,M2: fsm_Pr1886043760914421514_a_a_b] :
      ( ( P2 @ ( initia681974817753967189_a_a_b @ M2 ) )
     => ( ( transi2941759273464634107_a_a_b @ ( filter1780482953621413783_a_a_b @ M2 @ P2 ) )
        = ( collec7519840507084740067od_c_a
          @ ^ [T: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ T @ ( transi2941759273464634107_a_a_b @ M2 ) )
              & ( P2 @ ( produc8171714036265584980od_c_a @ T ) )
              & ( P2 @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T ) ) ) ) ) ) ) ) ).

% filter_states_simps(5)
thf(fact_1212_filter__states__simps_I5_J,axiom,
    ! [P2: produc2133359284458116493od_c_a > $o,M2: fsm_Pr7813689363856283786_c_a_a] :
      ( ( P2 @ ( initia2830091458700408661_c_a_a @ M2 ) )
     => ( ( transi5048145386435558395_c_a_a @ ( filter8279811424865667223_c_a_a @ M2 @ P2 ) )
        = ( collec8817176272877519933od_c_a
          @ ^ [T: produc1135898663509404882od_c_a] :
              ( ( member6699836568822684667od_c_a @ T @ ( transi5048145386435558395_c_a_a @ M2 ) )
              & ( P2 @ ( produc5475348670448042158od_c_a @ T ) )
              & ( P2 @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T ) ) ) ) ) ) ) ) ).

% filter_states_simps(5)
thf(fact_1213_after__transition__exhaust,axiom,
    ! [M2: fsm_fset_a_b_c,T4: produc21584292878202806fset_a] :
      ( ( observ4396094731789362477_a_b_c @ M2 )
     => ( ( member4899939873214261983fset_a @ T4 @ ( transi6427852111903025530_a_b_c @ M2 ) )
       => ( ( product_snd_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) )
          = ( after_fset_a_b_c @ M2 @ ( produc1739398754328533778fset_a @ T4 ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) @ ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ T4 ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_1214_after__transition__exhaust,axiom,
    ! [M2: fsm_a_b_c,T4: produc2669932201312201782od_c_a] :
      ( ( observable_a_b_c @ M2 )
     => ( ( member5502040268208861279od_c_a @ T4 @ ( transitions_a_b_c @ M2 ) )
       => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) )
          = ( after_a_b_c @ M2 @ ( produc4225943634317630866od_c_a @ T4 ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T4 ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_1215_after__transition__exhaust,axiom,
    ! [M2: fsm_Pr2398083450103406218et_a_b,T4: produc7245291065743820818fset_a] :
      ( ( observ1610022369204726958et_a_b @ M2 )
     => ( ( member808877655204049083fset_a @ T4 @ ( transi3322184091305251195et_a_b @ M2 ) )
       => ( ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) )
          = ( after_6279612785922921917et_a_b @ M2 @ ( produc8095675384246233966fset_a @ T4 ) @ ( cons_P1519664230097792120et_a_b @ ( produc4282089192724377778et_a_b @ ( produc1739398754328533778fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) @ ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ ( produc8178304657718331312fset_a @ T4 ) ) ) ) @ nil_Pr24683911927003336et_a_b ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_1216_after__transition__exhaust,axiom,
    ! [M2: fsm_Pr1886043760914421514_a_a_b,T4: produc3205353004677322104od_c_a] :
      ( ( observ2964502264774598958_a_a_b @ M2 )
     => ( ( member6508725278956418977od_c_a @ T4 @ ( transi2941759273464634107_a_a_b @ M2 ) )
       => ( ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) )
          = ( after_3087770633486430525_a_a_b @ M2 @ ( produc8171714036265584980od_c_a @ T4 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ T4 ) ) ) ) @ nil_Product_prod_a_b ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_1217_after__transition__exhaust,axiom,
    ! [M2: fsm_Pr7813689363856283786_c_a_a,T4: produc1135898663509404882od_c_a] :
      ( ( observ5792423948907539502_c_a_a @ M2 )
     => ( ( member6699836568822684667od_c_a @ T4 @ ( transi5048145386435558395_c_a_a @ M2 ) )
       => ( ( produc8088130389382527188od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) )
          = ( after_1728454483267612221_c_a_a @ M2 @ ( produc5475348670448042158od_c_a @ T4 ) @ ( cons_P1791870815957980406_c_a_a @ ( produc7661602042108997680_c_a_a @ ( produc8171714036265584980od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) @ ( produc4225943634317630866od_c_a @ ( produc4643634399596806294od_c_a @ ( produc9116639341042876912od_c_a @ T4 ) ) ) ) @ nil_Pr9019007095756880198_c_a_a ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_1218_after__transition__exhaust,axiom,
    ! [M2: fsm_Pr1175075309171164859fset_a,T4: produc7303452399287242961fset_a] :
      ( ( observ1975363445242043999fset_a @ M2 )
     => ( ( member9109638507678586746fset_a @ T4 @ ( transi1231084882770062892fset_a @ M2 ) )
       => ( ( produc8188425890649629780fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) )
          = ( after_7134766016456892526fset_a @ M2 @ ( produc2105839935146153389fset_a @ T4 ) @ ( cons_P4277760901013084993fset_a @ ( produc1535188672328598523fset_a @ ( produc2756759926546231865fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) @ ( produc1739398754328533778fset_a @ ( produc8452052326732228987fset_a @ ( produc2537342046628548079fset_a @ T4 ) ) ) ) @ nil_Pr2782780582842296209fset_a ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_1219_DiffI,axiom,
    ! [C: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,B: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C @ A )
     => ( ~ ( member5502040268208861279od_c_a @ C @ B )
       => ( member5502040268208861279od_c_a @ C @ ( minus_2240829453383958269od_c_a @ A @ B ) ) ) ) ).

% DiffI
thf(fact_1220_DiffI,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ A )
     => ( ~ ( member_a @ C @ B )
       => ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).

% DiffI
thf(fact_1221_DiffI,axiom,
    ! [C: produc3205353004677322104od_c_a,A: set_Pr2640428967771533528od_c_a,B: set_Pr2640428967771533528od_c_a] :
      ( ( member6508725278956418977od_c_a @ C @ A )
     => ( ~ ( member6508725278956418977od_c_a @ C @ B )
       => ( member6508725278956418977od_c_a @ C @ ( minus_3506473535942176575od_c_a @ A @ B ) ) ) ) ).

% DiffI
thf(fact_1222_DiffI,axiom,
    ! [C: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a,B: set_Pr4695589342295920918fset_a] :
      ( ( member4899939873214261983fset_a @ C @ A )
     => ( ~ ( member4899939873214261983fset_a @ C @ B )
       => ( member4899939873214261983fset_a @ C @ ( minus_2452200411918847613fset_a @ A @ B ) ) ) ) ).

% DiffI
thf(fact_1223_Diff__iff,axiom,
    ! [C: produc2669932201312201782od_c_a,A: set_Pr8357624050790843030od_c_a,B: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ C @ ( minus_2240829453383958269od_c_a @ A @ B ) )
      = ( ( member5502040268208861279od_c_a @ C @ A )
        & ~ ( member5502040268208861279od_c_a @ C @ B ) ) ) ).

% Diff_iff
thf(fact_1224_Diff__iff,axiom,
    ! [C: a,A: set_a,B: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
      = ( ( member_a @ C @ A )
        & ~ ( member_a @ C @ B ) ) ) ).

% Diff_iff
thf(fact_1225_Diff__iff,axiom,
    ! [C: produc3205353004677322104od_c_a,A: set_Pr2640428967771533528od_c_a,B: set_Pr2640428967771533528od_c_a] :
      ( ( member6508725278956418977od_c_a @ C @ ( minus_3506473535942176575od_c_a @ A @ B ) )
      = ( ( member6508725278956418977od_c_a @ C @ A )
        & ~ ( member6508725278956418977od_c_a @ C @ B ) ) ) ).

% Diff_iff
thf(fact_1226_Diff__iff,axiom,
    ! [C: produc21584292878202806fset_a,A: set_Pr4695589342295920918fset_a,B: set_Pr4695589342295920918fset_a] :
      ( ( member4899939873214261983fset_a @ C @ ( minus_2452200411918847613fset_a @ A @ B ) )
      = ( ( member4899939873214261983fset_a @ C @ A )
        & ~ ( member4899939873214261983fset_a @ C @ B ) ) ) ).

% Diff_iff
thf(fact_1227_case__prodI,axiom,
    ! [F: b > c > $o,A2: b,B2: c] :
      ( ( F @ A2 @ B2 )
     => ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1228_case__prodI,axiom,
    ! [F: fset_a > produc1137077444975408883fset_a > $o,A2: fset_a,B2: produc1137077444975408883fset_a] :
      ( ( F @ A2 @ B2 )
     => ( produc2145210010652847517et_a_o @ F @ ( produc3008349866153638182fset_a @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1229_case__prodI,axiom,
    ! [F: b > produc4838349196543436939fset_a > $o,A2: b,B2: produc4838349196543436939fset_a] :
      ( ( F @ A2 @ B2 )
     => ( produc3143923879407689888et_a_o @ F @ ( produc4496741132823077603fset_a @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1230_case__prodI,axiom,
    ! [F: c > fset_a > $o,A2: c,B2: fset_a] :
      ( ( F @ A2 @ B2 )
     => ( produc8124322634180564360et_a_o @ F @ ( produc1535188672328598523fset_a @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_1231_case__prodI2,axiom,
    ! [P: product_prod_b_c,C: b > c > $o] :
      ( ! [A6: b,B5: c] :
          ( ( P
            = ( product_Pair_b_c @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc1873627588981547243_b_c_o @ C @ P ) ) ).

% case_prodI2
thf(fact_1232_case__prodI2,axiom,
    ! [P: produc21584292878202806fset_a,C: fset_a > produc1137077444975408883fset_a > $o] :
      ( ! [A6: fset_a,B5: produc1137077444975408883fset_a] :
          ( ( P
            = ( produc3008349866153638182fset_a @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc2145210010652847517et_a_o @ C @ P ) ) ).

% case_prodI2
thf(fact_1233_case__prodI2,axiom,
    ! [P: produc1137077444975408883fset_a,C: b > produc4838349196543436939fset_a > $o] :
      ( ! [A6: b,B5: produc4838349196543436939fset_a] :
          ( ( P
            = ( produc4496741132823077603fset_a @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc3143923879407689888et_a_o @ C @ P ) ) ).

% case_prodI2
thf(fact_1234_case__prodI2,axiom,
    ! [P: produc4838349196543436939fset_a,C: c > fset_a > $o] :
      ( ! [A6: c,B5: fset_a] :
          ( ( P
            = ( produc1535188672328598523fset_a @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc8124322634180564360et_a_o @ C @ P ) ) ).

% case_prodI2
thf(fact_1235_mem__case__prodI,axiom,
    ! [Z2: a,C: b > c > set_a,A2: b,B2: c] :
      ( ( member_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z2 @ ( produc3958908706888076337_set_a @ C @ ( product_Pair_b_c @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1236_mem__case__prodI,axiom,
    ! [Z2: a,C: c > fset_a > set_a,A2: c,B2: fset_a] :
      ( ( member_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z2 @ ( produc3853195960805756494_set_a @ C @ ( produc1535188672328598523fset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1237_mem__case__prodI,axiom,
    ! [Z2: a,C: b > produc4838349196543436939fset_a > set_a,A2: b,B2: produc4838349196543436939fset_a] :
      ( ( member_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z2 @ ( produc5502495250099645798_set_a @ C @ ( produc4496741132823077603fset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1238_mem__case__prodI,axiom,
    ! [Z2: produc2669932201312201782od_c_a,C: b > c > set_Pr8357624050790843030od_c_a,A2: b,B2: c] :
      ( ( member5502040268208861279od_c_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member5502040268208861279od_c_a @ Z2 @ ( produc7297509611420049769od_c_a @ C @ ( product_Pair_b_c @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1239_mem__case__prodI,axiom,
    ! [Z2: a,C: fset_a > produc1137077444975408883fset_a > set_a,A2: fset_a,B2: produc1137077444975408883fset_a] :
      ( ( member_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z2 @ ( produc6102014494714763235_set_a @ C @ ( produc3008349866153638182fset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1240_mem__case__prodI,axiom,
    ! [Z2: produc2669932201312201782od_c_a,C: c > fset_a > set_Pr8357624050790843030od_c_a,A2: c,B2: fset_a] :
      ( ( member5502040268208861279od_c_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member5502040268208861279od_c_a @ Z2 @ ( produc1759818263262180486od_c_a @ C @ ( produc1535188672328598523fset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1241_mem__case__prodI,axiom,
    ! [Z2: produc21584292878202806fset_a,C: b > c > set_Pr4695589342295920918fset_a,A2: b,B2: c] :
      ( ( member4899939873214261983fset_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member4899939873214261983fset_a @ Z2 @ ( produc4234317041742767337fset_a @ C @ ( product_Pair_b_c @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1242_mem__case__prodI,axiom,
    ! [Z2: produc2669932201312201782od_c_a,C: b > produc4838349196543436939fset_a > set_Pr8357624050790843030od_c_a,A2: b,B2: produc4838349196543436939fset_a] :
      ( ( member5502040268208861279od_c_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member5502040268208861279od_c_a @ Z2 @ ( produc1132286984903642014od_c_a @ C @ ( produc4496741132823077603fset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1243_mem__case__prodI,axiom,
    ! [Z2: produc21584292878202806fset_a,C: c > fset_a > set_Pr4695589342295920918fset_a,A2: c,B2: fset_a] :
      ( ( member4899939873214261983fset_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member4899939873214261983fset_a @ Z2 @ ( produc8424478845293794054fset_a @ C @ ( produc1535188672328598523fset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1244_mem__case__prodI,axiom,
    ! [Z2: produc3205353004677322104od_c_a,C: b > c > set_Pr2640428967771533528od_c_a,A2: b,B2: c] :
      ( ( member6508725278956418977od_c_a @ Z2 @ ( C @ A2 @ B2 ) )
     => ( member6508725278956418977od_c_a @ Z2 @ ( produc2368348868963170731od_c_a @ C @ ( product_Pair_b_c @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_1245_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z2: a,C: b > c > set_a] :
      ( ! [A6: b,B5: c] :
          ( ( P
            = ( product_Pair_b_c @ A6 @ B5 ) )
         => ( member_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member_a @ Z2 @ ( produc3958908706888076337_set_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1246_mem__case__prodI2,axiom,
    ! [P: produc4838349196543436939fset_a,Z2: a,C: c > fset_a > set_a] :
      ( ! [A6: c,B5: fset_a] :
          ( ( P
            = ( produc1535188672328598523fset_a @ A6 @ B5 ) )
         => ( member_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member_a @ Z2 @ ( produc3853195960805756494_set_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1247_mem__case__prodI2,axiom,
    ! [P: produc1137077444975408883fset_a,Z2: a,C: b > produc4838349196543436939fset_a > set_a] :
      ( ! [A6: b,B5: produc4838349196543436939fset_a] :
          ( ( P
            = ( produc4496741132823077603fset_a @ A6 @ B5 ) )
         => ( member_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member_a @ Z2 @ ( produc5502495250099645798_set_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1248_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z2: produc2669932201312201782od_c_a,C: b > c > set_Pr8357624050790843030od_c_a] :
      ( ! [A6: b,B5: c] :
          ( ( P
            = ( product_Pair_b_c @ A6 @ B5 ) )
         => ( member5502040268208861279od_c_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member5502040268208861279od_c_a @ Z2 @ ( produc7297509611420049769od_c_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1249_mem__case__prodI2,axiom,
    ! [P: produc21584292878202806fset_a,Z2: a,C: fset_a > produc1137077444975408883fset_a > set_a] :
      ( ! [A6: fset_a,B5: produc1137077444975408883fset_a] :
          ( ( P
            = ( produc3008349866153638182fset_a @ A6 @ B5 ) )
         => ( member_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member_a @ Z2 @ ( produc6102014494714763235_set_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1250_mem__case__prodI2,axiom,
    ! [P: produc4838349196543436939fset_a,Z2: produc2669932201312201782od_c_a,C: c > fset_a > set_Pr8357624050790843030od_c_a] :
      ( ! [A6: c,B5: fset_a] :
          ( ( P
            = ( produc1535188672328598523fset_a @ A6 @ B5 ) )
         => ( member5502040268208861279od_c_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member5502040268208861279od_c_a @ Z2 @ ( produc1759818263262180486od_c_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1251_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z2: produc21584292878202806fset_a,C: b > c > set_Pr4695589342295920918fset_a] :
      ( ! [A6: b,B5: c] :
          ( ( P
            = ( product_Pair_b_c @ A6 @ B5 ) )
         => ( member4899939873214261983fset_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member4899939873214261983fset_a @ Z2 @ ( produc4234317041742767337fset_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1252_mem__case__prodI2,axiom,
    ! [P: produc1137077444975408883fset_a,Z2: produc2669932201312201782od_c_a,C: b > produc4838349196543436939fset_a > set_Pr8357624050790843030od_c_a] :
      ( ! [A6: b,B5: produc4838349196543436939fset_a] :
          ( ( P
            = ( produc4496741132823077603fset_a @ A6 @ B5 ) )
         => ( member5502040268208861279od_c_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member5502040268208861279od_c_a @ Z2 @ ( produc1132286984903642014od_c_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1253_mem__case__prodI2,axiom,
    ! [P: produc4838349196543436939fset_a,Z2: produc21584292878202806fset_a,C: c > fset_a > set_Pr4695589342295920918fset_a] :
      ( ! [A6: c,B5: fset_a] :
          ( ( P
            = ( produc1535188672328598523fset_a @ A6 @ B5 ) )
         => ( member4899939873214261983fset_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member4899939873214261983fset_a @ Z2 @ ( produc8424478845293794054fset_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1254_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z2: produc3205353004677322104od_c_a,C: b > c > set_Pr2640428967771533528od_c_a] :
      ( ! [A6: b,B5: c] :
          ( ( P
            = ( product_Pair_b_c @ A6 @ B5 ) )
         => ( member6508725278956418977od_c_a @ Z2 @ ( C @ A6 @ B5 ) ) )
     => ( member6508725278956418977od_c_a @ Z2 @ ( produc2368348868963170731od_c_a @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_1255_ffunion__insert,axiom,
    ! [A2: fset_P2596122609397385116fset_a,B: fset_f3206295485159697154fset_a] :
      ( ( ffUnio2651180436950201422fset_a @ ( finser1277071787618628324fset_a @ A2 @ B ) )
      = ( sup_su6818703939759823728fset_a @ A2 @ ( ffUnio2651180436950201422fset_a @ B ) ) ) ).

% ffunion_insert
thf(fact_1256_ffunion__insert,axiom,
    ! [A2: fset_fset_a,B: fset_fset_fset_a] :
      ( ( ffUnion_fset_a @ ( finsert_fset_fset_a @ A2 @ B ) )
      = ( sup_sup_fset_fset_a @ A2 @ ( ffUnion_fset_a @ B ) ) ) ).

% ffunion_insert
thf(fact_1257_ffunion__insert,axiom,
    ! [A2: fset_a,B: fset_fset_a] :
      ( ( ffUnion_a @ ( finsert_fset_a @ A2 @ B ) )
      = ( sup_sup_fset_a @ A2 @ ( ffUnion_a @ B ) ) ) ).

% ffunion_insert
thf(fact_1258_filter__states__simps_I1_J,axiom,
    ! [P2: a > $o,M2: fsm_a_b_c] :
      ( ( P2 @ ( initial_a_b_c @ M2 ) )
     => ( ( initial_a_b_c @ ( filter_states_a_b_c @ M2 @ P2 ) )
        = ( initial_a_b_c @ M2 ) ) ) ).

% filter_states_simps(1)
thf(fact_1259_fPow__fempty,axiom,
    ( ( fPow_a @ bot_bot_fset_a )
    = ( finsert_fset_a @ bot_bot_fset_a @ bot_bot_fset_fset_a ) ) ).

% fPow_fempty
thf(fact_1260_fPow__fempty,axiom,
    ( ( fPow_fset_a @ bot_bot_fset_fset_a )
    = ( finsert_fset_fset_a @ bot_bot_fset_fset_a @ bot_bo3176604761809765788fset_a ) ) ).

% fPow_fempty
thf(fact_1261_set__diff__eq,axiom,
    ( minus_3506473535942176575od_c_a
    = ( ^ [A3: set_Pr2640428967771533528od_c_a,B4: set_Pr2640428967771533528od_c_a] :
          ( collec7519840507084740067od_c_a
          @ ^ [X: produc3205353004677322104od_c_a] :
              ( ( member6508725278956418977od_c_a @ X @ A3 )
              & ~ ( member6508725278956418977od_c_a @ X @ B4 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1262_set__diff__eq,axiom,
    ( minus_2452200411918847613fset_a
    = ( ^ [A3: set_Pr4695589342295920918fset_a,B4: set_Pr4695589342295920918fset_a] :
          ( collec8021744632519760673fset_a
          @ ^ [X: produc21584292878202806fset_a] :
              ( ( member4899939873214261983fset_a @ X @ A3 )
              & ~ ( member4899939873214261983fset_a @ X @ B4 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1263_set__diff__eq,axiom,
    ( minus_2240829453383958269od_c_a
    = ( ^ [A3: set_Pr8357624050790843030od_c_a,B4: set_Pr8357624050790843030od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [X: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X @ A3 )
              & ~ ( member5502040268208861279od_c_a @ X @ B4 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1264_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty

% Helper facts (15)
thf(help_If_2_1_If_001tf__a_T,axiom,
    ! [X2: a,Y2: a] :
      ( ( if_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001tf__a_T,axiom,
    ! [X2: a,Y2: a] :
      ( ( if_a @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_Itf__a_J_T,axiom,
    ! [X2: fset_a,Y2: fset_a] :
      ( ( if_fset_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_Itf__a_J_T,axiom,
    ! [X2: fset_a,Y2: fset_a] :
      ( ( if_fset_a @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__FSet__Ofset_Itf__a_J_J_T,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a] :
      ( ( if_fset_fset_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__FSet__Ofset_Itf__a_J_J_T,axiom,
    ! [X2: fset_fset_a,Y2: fset_fset_a] :
      ( ( if_fset_fset_a @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_T,axiom,
    ! [X2: produc2669932201312201782od_c_a,Y2: produc2669932201312201782od_c_a] :
      ( ( if_Pro5723405847749414652od_c_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_T,axiom,
    ! [X2: produc2669932201312201782od_c_a,Y2: produc2669932201312201782od_c_a] :
      ( ( if_Pro5723405847749414652od_c_a @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_T,axiom,
    ! [X2: fset_P8043821951902805788od_c_a,Y2: fset_P8043821951902805788od_c_a] :
      ( ( if_fse3511714763770754786od_c_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_T,axiom,
    ! [X2: fset_P8043821951902805788od_c_a,Y2: fset_P8043821951902805788od_c_a] :
      ( ( if_fse3511714763770754786od_c_a @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_J_T,axiom,
    ! [X2: produc21584292878202806fset_a,Y2: produc21584292878202806fset_a] :
      ( ( if_Pro8211175460565361788fset_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_J_T,axiom,
    ! [X2: produc21584292878202806fset_a,Y2: produc21584292878202806fset_a] :
      ( ( if_Pro8211175460565361788fset_a @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_3_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_J_J_T,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a] :
      ( ( if_fse1100752710543153506fset_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__FSet__Ofset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__FSet__Ofset_Itf__a_J_J_J_J_J_T,axiom,
    ! [X2: fset_P2596122609397385116fset_a,Y2: fset_P2596122609397385116fset_a] :
      ( ( if_fse1100752710543153506fset_a @ $true @ X2 @ Y2 )
      = X2 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [TM: produc2669932201312201782od_c_a] :
      ( ( fmember_a @ ( produc4225943634317630866od_c_a @ TM ) @ qc )
     => ( ( member5502040268208861279od_c_a @ TM @ ( transitions_a_b_c @ m ) )
       => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ TM ) )
            = ( produc5616988955220391631fset_a @ ( produc8188425890649629780fset_a @ tP ) ) )
         => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ TM ) ) )
              = ( product_fst_c_fset_a @ ( produc8704244169652099857fset_a @ ( produc8188425890649629780fset_a @ tP ) ) ) )
           => ( ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ TM ) ) )
                = q2 )
             => thesis ) ) ) ) ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------