TPTP Problem File: SLH0477^1.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/0039_Minimisation/prob_01796_081167__19513064_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1880 ( 431 unt; 604 typ;   0 def)
%            Number of atoms       : 4421 (1437 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 16638 ( 351   ~;  35   |; 537   &;13501   @)
%                                         (   0 <=>;2214  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   7 avg)
%            Number of types       :   91 (  90 usr)
%            Number of type conns  : 1301 (1301   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  516 ( 514 usr;  21 con; 0-5 aty)
%            Number of variables   : 3706 ( 302   ^;3292   !; 112   ?;3706   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:28:08.443
%------------------------------------------------------------------------------
% Could-be-implicit typings (90)
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_001tf__b,type,
    image_9171994905208062542_c_a_b: ( produc2669932201312201782od_c_a > b ) > set_Pr8357624050790843030od_c_a > set_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_001tf__c,type,
    image_9171994905208062543_c_a_c: ( produc2669932201312201782od_c_a > c ) > set_Pr8357624050790843030od_c_a > set_c ).

thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
    image_set_a_a: ( set_a > a ) > set_set_a > set_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__Code____Numeral__Ointeger,type,
    image_a_Code_integer: ( a > code_integer ) > set_a > set_Code_integer ).

thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
    image_a_nat: ( a > nat ) > set_a > set_nat ).

thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
    image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).

thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
    image_a_set_a: ( a > set_a ) > set_a > set_set_a ).

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

thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
    image_b_nat: ( b > nat ) > set_b > set_nat ).

thf(sy_c_Set_Oimage_001tf__b_001tf__c,type,
    image_b_c: ( b > c ) > set_b > set_c ).

thf(sy_c_Set_Oimage_001tf__c_001t__Nat__Onat,type,
    image_c_nat: ( c > nat ) > set_c > set_nat ).

thf(sy_c_Set_Oimage_001tf__c_001tf__a,type,
    image_c_a: ( c > a ) > set_c > set_a ).

thf(sy_c_Set_Oimage_001tf__c_001tf__b,type,
    image_c_b: ( c > b ) > set_c > set_b ).

thf(sy_c_Set_Oimage_001tf__c_001tf__c,type,
    image_c_c: ( c > c ) > set_c > set_c ).

thf(sy_c_Set_Oinsert_001t__Code____Numeral__Ointeger,type,
    insert_Code_integer: code_integer > set_Code_integer > set_Code_integer ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Code____Numeral__Ointeger_J_J_J_J,type,
    insert2637641413281142478nteger: list_P7609586222656207102nteger > set_li2014469892457175902nteger > set_li2014469892457175902nteger ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J,type,
    insert8610875376133238836_c_nat: list_P1412026030649758308_c_nat > set_li7359738830946830788_c_nat > set_li7359738830946830788_c_nat ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Set__Oset_Itf__a_J_J_J_J_J,type,
    insert3727591976216729996_set_a: list_P6923848007992943548_set_a > set_li3414728242299228316_set_a > set_li3414728242299228316_set_a ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    insert4789241225314331020od_c_a: list_P6327159017948738492od_c_a > set_li1159382662694783132od_c_a > set_li1159382662694783132od_c_a ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mtf__c_J_J,type,
    insert6227932334100060350od_b_c: list_P903359562653991662od_b_c > set_li6436108459499378894od_b_c > set_li6436108459499378894od_b_c ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Set__Oset_Itf__a_J_J_J_J,type,
    insert7572610720624169478_set_a: produc373761375140688950_set_a > set_Pr3565145081747743382_set_a > set_Pr3565145081747743382_set_a ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    insert8105206408023001606od_c_a: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a > set_Pr8357624050790843030od_c_a ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
    insert_set_a: set_a > set_set_a > set_set_a ).

thf(sy_c_Set_Oinsert_001tf__a,type,
    insert_a: a > set_a > set_a ).

thf(sy_c_Set_Oinsert_001tf__b,type,
    insert_b: b > set_b > set_b ).

thf(sy_c_Set_Oinsert_001tf__c,type,
    insert_c: c > set_c > set_c ).

thf(sy_c_Set_Othe__elem_001t__Code____Numeral__Ointeger,type,
    the_el4656377885804969768nteger: set_Code_integer > code_integer ).

thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: set_nat > nat ).

thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
    the_elem_set_a: set_set_a > set_a ).

thf(sy_c_Set_Othe__elem_001tf__a,type,
    the_elem_a: set_a > a ).

thf(sy_c_Util_Oassign__indices_001t__Code____Numeral__Ointeger,type,
    assign469752435954302728nteger: set_Code_integer > code_integer > nat ).

thf(sy_c_Util_Oassign__indices_001t__Nat__Onat,type,
    assign_indices_nat: set_nat > nat > nat ).

thf(sy_c_Util_Oassign__indices_001tf__a,type,
    assign_indices_a: set_a > a > nat ).

thf(sy_c_Wellfounded_Oaccp_001t__FSM__Ofsm_It__Code____Numeral__Ointeger_Mtf__b_Mtf__c_J,type,
    accp_f9084625243946205319er_b_c: ( fsm_Code_integer_b_c > fsm_Code_integer_b_c > $o ) > fsm_Code_integer_b_c > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__FSM__Ofsm_It__Nat__Onat_Mtf__b_Mtf__c_J,type,
    accp_fsm_nat_b_c: ( fsm_nat_b_c > fsm_nat_b_c > $o ) > fsm_nat_b_c > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__FSM__Ofsm_It__Set__Oset_Itf__a_J_Mtf__b_Mtf__c_J,type,
    accp_fsm_set_a_b_c: ( fsm_set_a_b_c > fsm_set_a_b_c > $o ) > fsm_set_a_b_c > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J,type,
    accp_fsm_a_b_c: ( fsm_a_b_c > fsm_a_b_c > $o ) > fsm_a_b_c > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__FSM__Ofsm_It__Set__Oset_Itf__a_J_Mtf__b_Mtf__c_J_Mt__FSM__Ofsm_It__Set__Oset_Itf__a_J_Mtf__b_Mtf__c_J_J,type,
    accp_P4877812173494536944_a_b_c: ( produc7144419203860858599_a_b_c > produc7144419203860858599_a_b_c > $o ) > produc7144419203860858599_a_b_c > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__FSM__Ofsm_It__Set__Oset_Itf__a_J_Mtf__b_Mtf__c_J_Mt__Set__Oset_Itf__a_J_J,type,
    accp_P2020493475874802763_set_a: ( produc7304979456409760962_set_a > produc7304979456409760962_set_a > $o ) > produc7304979456409760962_set_a > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J_Mt__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J_J,type,
    accp_P4943880149483332336_a_b_c: ( produc5795920974755469031_a_b_c > produc5795920974755469031_a_b_c > $o ) > produc5795920974755469031_a_b_c > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__FSM__Ofsm_Itf__a_Mtf__b_Mtf__c_J_Mtf__a_J,type,
    accp_P8775296850406592587_b_c_a: ( produc7386240147096006850_b_c_a > produc7386240147096006850_b_c_a > $o ) > produc7386240147096006850_b_c_a > $o ).

thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > set_Code_integer > $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__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Code____Numeral__Ointeger_J_J_J,type,
    member5112046945434310049nteger: produc558599814252099832nteger > set_Pr7509222024173591768nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6766774627807740288at_nat: produc8642769642335960151at_nat > set_Pr58315729134567223at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J,type,
    member3717341653178413575_c_nat: produc8586872805302537566_c_nat > set_Pr7601921274955925054_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mt__Set__Oset_Itf__a_J_J_J_J,type,
    member6517745037573369951_set_a: produc373761375140688950_set_a > set_Pr3565145081747743382_set_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_Itf__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__Code____Numeral__Ointeger_J,type,
    member7093370293991928355nteger: produc3993223824263494796nteger > set_Pr3404113227065285442nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J,type,
    member8195077246299207702_c_nat: product_prod_c_nat > set_Pr6903500605879609269_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Set__Oset_Itf__a_J_J,type,
    member7460900956140502318_set_a: product_prod_c_set_a > set_Pr146186216032501733_set_a > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    member1834298987299537450nteger: set_Code_integer > set_set_Code_integer > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
    member5679158588528110291_a_a_a: sum_su5154756782089200444_a_a_a > set_Su8591296761904255986_a_a_a > $o ).

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

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

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

thf(sy_v_M,type,
    m: fsm_a_b_c ).

% Relevant facts (1275)
thf(fact_0_assms,axiom,
    observable_a_b_c @ m ).

% assms
thf(fact_1_index__states__observable,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( observable_nat_b_c @ ( index_states_nat_b_c @ M ) ) ) ).

% index_states_observable
thf(fact_2_index__states__observable,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( observable_nat_b_c @ ( index_5743659002960661343er_b_c @ M ) ) ) ).

% index_states_observable
thf(fact_3_index__states__observable,axiom,
    ! [M: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( observable_nat_b_c @ ( index_states_a_b_c @ M ) ) ) ).

% index_states_observable
thf(fact_4_index__states__integer__observable,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( observ7524490533585752726er_b_c @ ( index_5652677021022579920at_b_c @ M ) ) ) ).

% index_states_integer_observable
thf(fact_5_index__states__integer__observable,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( observ7524490533585752726er_b_c @ ( index_3398277517725508317er_b_c @ M ) ) ) ).

% index_states_integer_observable
thf(fact_6_index__states__integer__observable,axiom,
    ! [M: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( observ7524490533585752726er_b_c @ ( index_2500966223721395136_a_b_c @ M ) ) ) ).

% index_states_integer_observable
thf(fact_7_minimise__props_I4_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c] :
      ( ( observ6810049560328217077_a_b_c @ M )
     => ( ( output5387264885956845650_a_b_c @ ( minimi5636679632172786532_a_b_c @ M ) )
        = ( output2259889051082456732_a_b_c @ M ) ) ) ).

% minimise_props(4)
thf(fact_8_minimise__props_I4_J,axiom,
    ! [M: fsm_nat_nat_nat] :
      ( ( observ3518789870599552490at_nat @ M )
     => ( ( output4772581906129750727at_nat @ ( minimise_nat_nat_nat @ M ) )
        = ( outputs_nat_nat_nat @ M ) ) ) ).

% minimise_props(4)
thf(fact_9_minimise__props_I4_J,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( outputs_set_nat_b_c @ ( minimise_nat_b_c @ M ) )
        = ( outputs_nat_b_c @ M ) ) ) ).

% minimise_props(4)
thf(fact_10_minimise__props_I4_J,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( output3616175663950214771er_b_c @ ( minimi9001410862329526533er_b_c @ M ) )
        = ( output924834633477831741er_b_c @ M ) ) ) ).

% minimise_props(4)
thf(fact_11_minimise__props_I4_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( outputs_set_a_b_c @ ( minimise_a_b_c @ M ) )
        = ( outputs_a_b_c @ M ) ) ) ).

% minimise_props(4)
thf(fact_12_minimise__props_I3_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c] :
      ( ( observ6810049560328217077_a_b_c @ M )
     => ( ( inputs6338826817688532945_a_b_c @ ( minimi5636679632172786532_a_b_c @ M ) )
        = ( inputs3694848887974842907_a_b_c @ M ) ) ) ).

% minimise_props(3)
thf(fact_13_minimise__props_I3_J,axiom,
    ! [M: fsm_nat_nat_nat] :
      ( ( observ3518789870599552490at_nat @ M )
     => ( ( inputs110340292907730630at_nat @ ( minimise_nat_nat_nat @ M ) )
        = ( inputs_nat_nat_nat @ M ) ) ) ).

% minimise_props(3)
thf(fact_14_minimise__props_I3_J,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( inputs_set_nat_b_c @ ( minimise_nat_b_c @ M ) )
        = ( inputs_nat_b_c @ M ) ) ) ).

% minimise_props(3)
thf(fact_15_minimise__props_I3_J,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( inputs1631762778656711410er_b_c @ ( minimi9001410862329526533er_b_c @ M ) )
        = ( inputs4521787250163688636er_b_c @ M ) ) ) ).

% minimise_props(3)
thf(fact_16_minimise__props_I3_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( inputs_set_a_b_c @ ( minimise_a_b_c @ M ) )
        = ( inputs_a_b_c @ M ) ) ) ).

% minimise_props(3)
thf(fact_17_restrict__to__reachable__states__observable,axiom,
    ! [M: fsm_se7249436556146598534er_b_c] :
      ( ( observ8366386160172968140er_b_c @ M )
     => ( observ8366386160172968140er_b_c @ ( restri6051775343782813657er_b_c @ M ) ) ) ).

% restrict_to_reachable_states_observable
thf(fact_18_restrict__to__reachable__states__observable,axiom,
    ! [M: fsm_set_nat_b_c] :
      ( ( observ2138676753445337407at_b_c @ M )
     => ( observ2138676753445337407at_b_c @ ( restri7930285009982923852at_b_c @ M ) ) ) ).

% restrict_to_reachable_states_observable
thf(fact_19_restrict__to__reachable__states__observable,axiom,
    ! [M: fsm_set_set_a_b_c] :
      ( ( observ3077046029229394695_a_b_c @ M )
     => ( observ3077046029229394695_a_b_c @ ( restri4473554248985132346_a_b_c @ M ) ) ) ).

% restrict_to_reachable_states_observable
thf(fact_20_restrict__to__reachable__states__observable,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( observable_nat_b_c @ ( restri2801621895680909206at_b_c @ M ) ) ) ).

% restrict_to_reachable_states_observable
thf(fact_21_restrict__to__reachable__states__observable,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( observ7524490533585752726er_b_c @ ( restri4092726753268563363er_b_c @ M ) ) ) ).

% restrict_to_reachable_states_observable
thf(fact_22_restrict__to__reachable__states__observable,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( observable_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) ) ) ).

% restrict_to_reachable_states_observable
thf(fact_23_restrict__to__reachable__states__observable,axiom,
    ! [M: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( observable_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) ) ) ).

% restrict_to_reachable_states_observable
thf(fact_24_submachine__observable,axiom,
    ! [S: fsm_se7249436556146598534er_b_c,M: fsm_se7249436556146598534er_b_c] :
      ( ( is_sub1762352666996382522er_b_c @ S @ M )
     => ( ( observ8366386160172968140er_b_c @ M )
       => ( observ8366386160172968140er_b_c @ S ) ) ) ).

% submachine_observable
thf(fact_25_submachine__observable,axiom,
    ! [S: fsm_set_nat_b_c,M: fsm_set_nat_b_c] :
      ( ( is_sub96770212967015341at_b_c @ S @ M )
     => ( ( observ2138676753445337407at_b_c @ M )
       => ( observ2138676753445337407at_b_c @ S ) ) ) ).

% submachine_observable
thf(fact_26_submachine__observable,axiom,
    ! [S: fsm_set_set_a_b_c,M: fsm_set_set_a_b_c] :
      ( ( is_sub4633271590407526425_a_b_c @ S @ M )
     => ( ( observ3077046029229394695_a_b_c @ M )
       => ( observ3077046029229394695_a_b_c @ S ) ) ) ).

% submachine_observable
thf(fact_27_submachine__observable,axiom,
    ! [S: fsm_nat_b_c,M: fsm_nat_b_c] :
      ( ( is_sub6065310311083418359at_b_c @ S @ M )
     => ( ( observable_nat_b_c @ M )
       => ( observable_nat_b_c @ S ) ) ) ).

% submachine_observable
thf(fact_28_submachine__observable,axiom,
    ! [S: fsm_Code_integer_b_c,M: fsm_Code_integer_b_c] :
      ( ( is_sub5482583993107430660er_b_c @ S @ M )
     => ( ( observ7524490533585752726er_b_c @ M )
       => ( observ7524490533585752726er_b_c @ S ) ) ) ).

% submachine_observable
thf(fact_29_submachine__observable,axiom,
    ! [S: fsm_set_a_b_c,M: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ S @ M )
     => ( ( observable_set_a_b_c @ M )
       => ( observable_set_a_b_c @ S ) ) ) ).

% submachine_observable
thf(fact_30_submachine__observable,axiom,
    ! [S: fsm_a_b_c,M: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ S @ M )
     => ( ( observable_a_b_c @ M )
       => ( observable_a_b_c @ S ) ) ) ).

% submachine_observable
thf(fact_31_from__FSM__observable,axiom,
    ! [M: fsm_se7249436556146598534er_b_c,Q: set_Code_integer] :
      ( ( observ8366386160172968140er_b_c @ M )
     => ( observ8366386160172968140er_b_c @ ( from_F1896587134240577335er_b_c @ M @ Q ) ) ) ).

% from_FSM_observable
thf(fact_32_from__FSM__observable,axiom,
    ! [M: fsm_set_nat_b_c,Q: set_nat] :
      ( ( observ2138676753445337407at_b_c @ M )
     => ( observ2138676753445337407at_b_c @ ( from_FSM_set_nat_b_c @ M @ Q ) ) ) ).

% from_FSM_observable
thf(fact_33_from__FSM__observable,axiom,
    ! [M: fsm_set_set_a_b_c,Q: set_set_a] :
      ( ( observ3077046029229394695_a_b_c @ M )
     => ( observ3077046029229394695_a_b_c @ ( from_F6314652777202278748_a_b_c @ M @ Q ) ) ) ).

% from_FSM_observable
thf(fact_34_from__FSM__observable,axiom,
    ! [M: fsm_nat_b_c,Q: nat] :
      ( ( observable_nat_b_c @ M )
     => ( observable_nat_b_c @ ( from_FSM_nat_b_c @ M @ Q ) ) ) ).

% from_FSM_observable
thf(fact_35_from__FSM__observable,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( observ7524490533585752726er_b_c @ ( from_F2639493813852433665er_b_c @ M @ Q ) ) ) ).

% from_FSM_observable
thf(fact_36_from__FSM__observable,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( observable_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) ) ) ).

% from_FSM_observable
thf(fact_37_from__FSM__observable,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( observable_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) ) ).

% from_FSM_observable
thf(fact_38_restrict__to__reachable__states__simps_I3_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c] :
      ( ( inputs3694848887974842907_a_b_c @ ( restri4463730293876628994_a_b_c @ M ) )
      = ( inputs3694848887974842907_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(3)
thf(fact_39_restrict__to__reachable__states__simps_I3_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c] :
      ( ( inputs1631762778656711410er_b_c @ ( restri6051775343782813657er_b_c @ M ) )
      = ( inputs1631762778656711410er_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(3)
thf(fact_40_restrict__to__reachable__states__simps_I3_J,axiom,
    ! [M: fsm_set_nat_b_c] :
      ( ( inputs_set_nat_b_c @ ( restri7930285009982923852at_b_c @ M ) )
      = ( inputs_set_nat_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(3)
thf(fact_41_restrict__to__reachable__states__simps_I3_J,axiom,
    ! [M: fsm_nat_nat_nat] :
      ( ( inputs_nat_nat_nat @ ( restri2246964829936590711at_nat @ M ) )
      = ( inputs_nat_nat_nat @ M ) ) ).

% restrict_to_reachable_states_simps(3)
thf(fact_42_restrict__to__reachable__states__simps_I3_J,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( inputs4521787250163688636er_b_c @ ( restri4092726753268563363er_b_c @ M ) )
      = ( inputs4521787250163688636er_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(3)
thf(fact_43_restrict__to__reachable__states__simps_I3_J,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( inputs_nat_b_c @ ( restri2801621895680909206at_b_c @ M ) )
      = ( inputs_nat_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(3)
thf(fact_44_restrict__to__reachable__states__simps_I3_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( inputs_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) )
      = ( inputs_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(3)
thf(fact_45_restrict__to__reachable__states__simps_I3_J,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( inputs_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) )
      = ( inputs_set_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(3)
thf(fact_46_restrict__to__reachable__states__simps_I4_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c] :
      ( ( output2259889051082456732_a_b_c @ ( restri4463730293876628994_a_b_c @ M ) )
      = ( output2259889051082456732_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(4)
thf(fact_47_restrict__to__reachable__states__simps_I4_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c] :
      ( ( output3616175663950214771er_b_c @ ( restri6051775343782813657er_b_c @ M ) )
      = ( output3616175663950214771er_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(4)
thf(fact_48_restrict__to__reachable__states__simps_I4_J,axiom,
    ! [M: fsm_set_nat_b_c] :
      ( ( outputs_set_nat_b_c @ ( restri7930285009982923852at_b_c @ M ) )
      = ( outputs_set_nat_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(4)
thf(fact_49_restrict__to__reachable__states__simps_I4_J,axiom,
    ! [M: fsm_nat_nat_nat] :
      ( ( outputs_nat_nat_nat @ ( restri2246964829936590711at_nat @ M ) )
      = ( outputs_nat_nat_nat @ M ) ) ).

% restrict_to_reachable_states_simps(4)
thf(fact_50_restrict__to__reachable__states__simps_I4_J,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( output924834633477831741er_b_c @ ( restri4092726753268563363er_b_c @ M ) )
      = ( output924834633477831741er_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(4)
thf(fact_51_restrict__to__reachable__states__simps_I4_J,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( outputs_nat_b_c @ ( restri2801621895680909206at_b_c @ M ) )
      = ( outputs_nat_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(4)
thf(fact_52_restrict__to__reachable__states__simps_I4_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( outputs_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) )
      = ( outputs_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(4)
thf(fact_53_restrict__to__reachable__states__simps_I4_J,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( outputs_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) )
      = ( outputs_set_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(4)
thf(fact_54_submachine__simps_I4_J,axiom,
    ! [A: fsm_Su3173892059090327343_a_b_c,B: fsm_Su3173892059090327343_a_b_c] :
      ( ( is_sub8484252162375176803_a_b_c @ A @ B )
     => ( ( output2259889051082456732_a_b_c @ A )
        = ( output2259889051082456732_a_b_c @ B ) ) ) ).

% submachine_simps(4)
thf(fact_55_submachine__simps_I4_J,axiom,
    ! [A: fsm_se7249436556146598534er_b_c,B: fsm_se7249436556146598534er_b_c] :
      ( ( is_sub1762352666996382522er_b_c @ A @ B )
     => ( ( output3616175663950214771er_b_c @ A )
        = ( output3616175663950214771er_b_c @ B ) ) ) ).

% submachine_simps(4)
thf(fact_56_submachine__simps_I4_J,axiom,
    ! [A: fsm_set_nat_b_c,B: fsm_set_nat_b_c] :
      ( ( is_sub96770212967015341at_b_c @ A @ B )
     => ( ( outputs_set_nat_b_c @ A )
        = ( outputs_set_nat_b_c @ B ) ) ) ).

% submachine_simps(4)
thf(fact_57_submachine__simps_I4_J,axiom,
    ! [A: fsm_nat_nat_nat,B: fsm_nat_nat_nat] :
      ( ( is_sub7029232139394105176at_nat @ A @ B )
     => ( ( outputs_nat_nat_nat @ A )
        = ( outputs_nat_nat_nat @ B ) ) ) ).

% submachine_simps(4)
thf(fact_58_submachine__simps_I4_J,axiom,
    ! [A: fsm_Code_integer_b_c,B: fsm_Code_integer_b_c] :
      ( ( is_sub5482583993107430660er_b_c @ A @ B )
     => ( ( output924834633477831741er_b_c @ A )
        = ( output924834633477831741er_b_c @ B ) ) ) ).

% submachine_simps(4)
thf(fact_59_submachine__simps_I4_J,axiom,
    ! [A: fsm_nat_b_c,B: fsm_nat_b_c] :
      ( ( is_sub6065310311083418359at_b_c @ A @ B )
     => ( ( outputs_nat_b_c @ A )
        = ( outputs_nat_b_c @ B ) ) ) ).

% submachine_simps(4)
thf(fact_60_submachine__simps_I4_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( outputs_a_b_c @ A )
        = ( outputs_a_b_c @ B ) ) ) ).

% submachine_simps(4)
thf(fact_61_submachine__simps_I4_J,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ( outputs_set_a_b_c @ A )
        = ( outputs_set_a_b_c @ B ) ) ) ).

% submachine_simps(4)
thf(fact_62_submachine__simps_I3_J,axiom,
    ! [A: fsm_Su3173892059090327343_a_b_c,B: fsm_Su3173892059090327343_a_b_c] :
      ( ( is_sub8484252162375176803_a_b_c @ A @ B )
     => ( ( inputs3694848887974842907_a_b_c @ A )
        = ( inputs3694848887974842907_a_b_c @ B ) ) ) ).

% submachine_simps(3)
thf(fact_63_submachine__simps_I3_J,axiom,
    ! [A: fsm_se7249436556146598534er_b_c,B: fsm_se7249436556146598534er_b_c] :
      ( ( is_sub1762352666996382522er_b_c @ A @ B )
     => ( ( inputs1631762778656711410er_b_c @ A )
        = ( inputs1631762778656711410er_b_c @ B ) ) ) ).

% submachine_simps(3)
thf(fact_64_submachine__simps_I3_J,axiom,
    ! [A: fsm_set_nat_b_c,B: fsm_set_nat_b_c] :
      ( ( is_sub96770212967015341at_b_c @ A @ B )
     => ( ( inputs_set_nat_b_c @ A )
        = ( inputs_set_nat_b_c @ B ) ) ) ).

% submachine_simps(3)
thf(fact_65_submachine__simps_I3_J,axiom,
    ! [A: fsm_nat_nat_nat,B: fsm_nat_nat_nat] :
      ( ( is_sub7029232139394105176at_nat @ A @ B )
     => ( ( inputs_nat_nat_nat @ A )
        = ( inputs_nat_nat_nat @ B ) ) ) ).

% submachine_simps(3)
thf(fact_66_submachine__simps_I3_J,axiom,
    ! [A: fsm_Code_integer_b_c,B: fsm_Code_integer_b_c] :
      ( ( is_sub5482583993107430660er_b_c @ A @ B )
     => ( ( inputs4521787250163688636er_b_c @ A )
        = ( inputs4521787250163688636er_b_c @ B ) ) ) ).

% submachine_simps(3)
thf(fact_67_submachine__simps_I3_J,axiom,
    ! [A: fsm_nat_b_c,B: fsm_nat_b_c] :
      ( ( is_sub6065310311083418359at_b_c @ A @ B )
     => ( ( inputs_nat_b_c @ A )
        = ( inputs_nat_b_c @ B ) ) ) ).

% submachine_simps(3)
thf(fact_68_submachine__simps_I3_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( inputs_a_b_c @ A )
        = ( inputs_a_b_c @ B ) ) ) ).

% submachine_simps(3)
thf(fact_69_submachine__simps_I3_J,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ( inputs_set_a_b_c @ A )
        = ( inputs_set_a_b_c @ B ) ) ) ).

% submachine_simps(3)
thf(fact_70_submachine__transitive,axiom,
    ! [S: fsm_Code_integer_b_c,M: fsm_Code_integer_b_c,S2: fsm_Code_integer_b_c] :
      ( ( is_sub5482583993107430660er_b_c @ S @ M )
     => ( ( is_sub5482583993107430660er_b_c @ S2 @ S )
       => ( is_sub5482583993107430660er_b_c @ S2 @ M ) ) ) ).

% submachine_transitive
thf(fact_71_submachine__transitive,axiom,
    ! [S: fsm_nat_b_c,M: fsm_nat_b_c,S2: fsm_nat_b_c] :
      ( ( is_sub6065310311083418359at_b_c @ S @ M )
     => ( ( is_sub6065310311083418359at_b_c @ S2 @ S )
       => ( is_sub6065310311083418359at_b_c @ S2 @ M ) ) ) ).

% submachine_transitive
thf(fact_72_submachine__transitive,axiom,
    ! [S: fsm_a_b_c,M: fsm_a_b_c,S2: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ S @ M )
     => ( ( is_submachine_a_b_c @ S2 @ S )
       => ( is_submachine_a_b_c @ S2 @ M ) ) ) ).

% submachine_transitive
thf(fact_73_submachine__transitive,axiom,
    ! [S: fsm_set_a_b_c,M: fsm_set_a_b_c,S2: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ S @ M )
     => ( ( is_sub1377638108373535417_a_b_c @ S2 @ S )
       => ( is_sub1377638108373535417_a_b_c @ S2 @ M ) ) ) ).

% submachine_transitive
thf(fact_74_add__input__simps_I3_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,X: b] :
      ( ( output2259889051082456732_a_b_c @ ( add_in7389665105193716379_a_b_c @ M @ X ) )
      = ( output2259889051082456732_a_b_c @ M ) ) ).

% add_input_simps(3)
thf(fact_75_add__input__simps_I3_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c,X: b] :
      ( ( output3616175663950214771er_b_c @ ( add_in5283742766163303282er_b_c @ M @ X ) )
      = ( output3616175663950214771er_b_c @ M ) ) ).

% add_input_simps(3)
thf(fact_76_add__input__simps_I3_J,axiom,
    ! [M: fsm_set_nat_b_c,X: b] :
      ( ( outputs_set_nat_b_c @ ( add_in7503527852466619365at_b_c @ M @ X ) )
      = ( outputs_set_nat_b_c @ M ) ) ).

% add_input_simps(3)
thf(fact_77_add__input__simps_I3_J,axiom,
    ! [M: fsm_nat_nat_nat,X: nat] :
      ( ( outputs_nat_nat_nat @ ( add_in6123057044259584912at_nat @ M @ X ) )
      = ( outputs_nat_nat_nat @ M ) ) ).

% add_input_simps(3)
thf(fact_78_add__input__simps_I3_J,axiom,
    ! [M: fsm_nat_b_c,X: b] :
      ( ( outputs_nat_b_c @ ( add_input_nat_b_c @ M @ X ) )
      = ( outputs_nat_b_c @ M ) ) ).

% add_input_simps(3)
thf(fact_79_add__input__simps_I3_J,axiom,
    ! [M: fsm_set_a_b_c,X: b] :
      ( ( outputs_set_a_b_c @ ( add_input_set_a_b_c @ M @ X ) )
      = ( outputs_set_a_b_c @ M ) ) ).

% add_input_simps(3)
thf(fact_80_add__input__simps_I3_J,axiom,
    ! [M: fsm_a_b_c,X: b] :
      ( ( outputs_a_b_c @ ( add_input_a_b_c @ M @ X ) )
      = ( outputs_a_b_c @ M ) ) ).

% add_input_simps(3)
thf(fact_81_add__state__simps_I3_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a] :
      ( ( output2259889051082456732_a_b_c @ ( add_st4776560808360540098_a_b_c @ M @ Q ) )
      = ( output2259889051082456732_a_b_c @ M ) ) ).

% add_state_simps(3)
thf(fact_82_add__state__simps_I3_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c,Q: set_Code_integer] :
      ( ( output3616175663950214771er_b_c @ ( add_st798595002496711065er_b_c @ M @ Q ) )
      = ( output3616175663950214771er_b_c @ M ) ) ).

% add_state_simps(3)
thf(fact_83_add__state__simps_I3_J,axiom,
    ! [M: fsm_set_nat_b_c,Q: set_nat] :
      ( ( outputs_set_nat_b_c @ ( add_st8360318785478835724at_b_c @ M @ Q ) )
      = ( outputs_set_nat_b_c @ M ) ) ).

% add_state_simps(3)
thf(fact_84_add__state__simps_I3_J,axiom,
    ! [M: fsm_nat_nat_nat,Q: nat] :
      ( ( outputs_nat_nat_nat @ ( add_st626164701413644599at_nat @ M @ Q ) )
      = ( outputs_nat_nat_nat @ M ) ) ).

% add_state_simps(3)
thf(fact_85_add__state__simps_I3_J,axiom,
    ! [M: fsm_nat_b_c,Q: nat] :
      ( ( outputs_nat_b_c @ ( add_state_nat_b_c @ M @ Q ) )
      = ( outputs_nat_b_c @ M ) ) ).

% add_state_simps(3)
thf(fact_86_add__state__simps_I3_J,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a] :
      ( ( outputs_set_a_b_c @ ( add_state_set_a_b_c @ M @ Q ) )
      = ( outputs_set_a_b_c @ M ) ) ).

% add_state_simps(3)
thf(fact_87_add__state__simps_I3_J,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( outputs_a_b_c @ ( add_state_a_b_c @ M @ Q ) )
      = ( outputs_a_b_c @ M ) ) ).

% add_state_simps(3)
thf(fact_88_add__state__simps_I2_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Q: sum_su5154756782089200444_a_a_a] :
      ( ( inputs3694848887974842907_a_b_c @ ( add_st4776560808360540098_a_b_c @ M @ Q ) )
      = ( inputs3694848887974842907_a_b_c @ M ) ) ).

% add_state_simps(2)
thf(fact_89_add__state__simps_I2_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c,Q: set_Code_integer] :
      ( ( inputs1631762778656711410er_b_c @ ( add_st798595002496711065er_b_c @ M @ Q ) )
      = ( inputs1631762778656711410er_b_c @ M ) ) ).

% add_state_simps(2)
thf(fact_90_add__state__simps_I2_J,axiom,
    ! [M: fsm_set_nat_b_c,Q: set_nat] :
      ( ( inputs_set_nat_b_c @ ( add_st8360318785478835724at_b_c @ M @ Q ) )
      = ( inputs_set_nat_b_c @ M ) ) ).

% add_state_simps(2)
thf(fact_91_add__state__simps_I2_J,axiom,
    ! [M: fsm_nat_nat_nat,Q: nat] :
      ( ( inputs_nat_nat_nat @ ( add_st626164701413644599at_nat @ M @ Q ) )
      = ( inputs_nat_nat_nat @ M ) ) ).

% add_state_simps(2)
thf(fact_92_add__state__simps_I2_J,axiom,
    ! [M: fsm_nat_b_c,Q: nat] :
      ( ( inputs_nat_b_c @ ( add_state_nat_b_c @ M @ Q ) )
      = ( inputs_nat_b_c @ M ) ) ).

% add_state_simps(2)
thf(fact_93_add__state__simps_I2_J,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a] :
      ( ( inputs_set_a_b_c @ ( add_state_set_a_b_c @ M @ Q ) )
      = ( inputs_set_a_b_c @ M ) ) ).

% add_state_simps(2)
thf(fact_94_add__state__simps_I2_J,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( inputs_a_b_c @ ( add_state_a_b_c @ M @ Q ) )
      = ( inputs_a_b_c @ M ) ) ).

% add_state_simps(2)
thf(fact_95_add__output__simps_I2_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Y: c] :
      ( ( inputs3694848887974842907_a_b_c @ ( add_ou235458353079876748_a_b_c @ M @ Y ) )
      = ( inputs3694848887974842907_a_b_c @ M ) ) ).

% add_output_simps(2)
thf(fact_96_add__output__simps_I2_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c,Y: c] :
      ( ( inputs1631762778656711410er_b_c @ ( add_ou7817685156754277475er_b_c @ M @ Y ) )
      = ( inputs1631762778656711410er_b_c @ M ) ) ).

% add_output_simps(2)
thf(fact_97_add__output__simps_I2_J,axiom,
    ! [M: fsm_set_nat_b_c,Y: c] :
      ( ( inputs_set_nat_b_c @ ( add_ou8618801446620803286at_b_c @ M @ Y ) )
      = ( inputs_set_nat_b_c @ M ) ) ).

% add_output_simps(2)
thf(fact_98_add__output__simps_I2_J,axiom,
    ! [M: fsm_nat_nat_nat,Y: nat] :
      ( ( inputs_nat_nat_nat @ ( add_ou6503059941141532929at_nat @ M @ Y ) )
      = ( inputs_nat_nat_nat @ M ) ) ).

% add_output_simps(2)
thf(fact_99_add__output__simps_I2_J,axiom,
    ! [M: fsm_nat_b_c,Y: c] :
      ( ( inputs_nat_b_c @ ( add_output_nat_b_c @ M @ Y ) )
      = ( inputs_nat_b_c @ M ) ) ).

% add_output_simps(2)
thf(fact_100_add__output__simps_I2_J,axiom,
    ! [M: fsm_set_a_b_c,Y: c] :
      ( ( inputs_set_a_b_c @ ( add_output_set_a_b_c @ M @ Y ) )
      = ( inputs_set_a_b_c @ M ) ) ).

% add_output_simps(2)
thf(fact_101_add__output__simps_I2_J,axiom,
    ! [M: fsm_a_b_c,Y: c] :
      ( ( inputs_a_b_c @ ( add_output_a_b_c @ M @ Y ) )
      = ( inputs_a_b_c @ M ) ) ).

% add_output_simps(2)
thf(fact_102_filter__transitions__simps_I4_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,P: produc884577481120224310_a_a_a > $o] :
      ( ( output2259889051082456732_a_b_c @ ( filter7058864375813322832_a_b_c @ M @ P ) )
      = ( output2259889051082456732_a_b_c @ M ) ) ).

% filter_transitions_simps(4)
thf(fact_103_filter__transitions__simps_I4_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c,P: produc9083820074870880868nteger > $o] :
      ( ( output3616175663950214771er_b_c @ ( filter8485183139320716327er_b_c @ M @ P ) )
      = ( output3616175663950214771er_b_c @ M ) ) ).

% filter_transitions_simps(4)
thf(fact_104_filter__transitions__simps_I4_J,axiom,
    ! [M: fsm_set_nat_b_c,P: produc4100985854500036554et_nat > $o] :
      ( ( outputs_set_nat_b_c @ ( filter6002735969965398170at_b_c @ M @ P ) )
      = ( outputs_set_nat_b_c @ M ) ) ).

% filter_transitions_simps(4)
thf(fact_105_filter__transitions__simps_I4_J,axiom,
    ! [M: fsm_nat_nat_nat,P: produc8642769642335960151at_nat > $o] :
      ( ( outputs_nat_nat_nat @ ( filter5128389180105923269at_nat @ M @ P ) )
      = ( outputs_nat_nat_nat @ M ) ) ).

% filter_transitions_simps(4)
thf(fact_106_filter__transitions__simps_I4_J,axiom,
    ! [M: fsm_nat_b_c,P: produc8586872805302537566_c_nat > $o] :
      ( ( outputs_nat_b_c @ ( filter4406304812386798564at_b_c @ M @ P ) )
      = ( outputs_nat_b_c @ M ) ) ).

% filter_transitions_simps(4)
thf(fact_107_filter__transitions__simps_I4_J,axiom,
    ! [M: fsm_set_a_b_c,P: produc373761375140688950_set_a > $o] :
      ( ( outputs_set_a_b_c @ ( filter824366976161576716_a_b_c @ M @ P ) )
      = ( outputs_set_a_b_c @ M ) ) ).

% filter_transitions_simps(4)
thf(fact_108_filter__transitions__simps_I4_J,axiom,
    ! [M: fsm_a_b_c,P: produc2669932201312201782od_c_a > $o] :
      ( ( outputs_a_b_c @ ( filter4876964422427095468_a_b_c @ M @ P ) )
      = ( outputs_a_b_c @ M ) ) ).

% filter_transitions_simps(4)
thf(fact_109_from__FSM__simps_I3_J,axiom,
    ! [Q: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c] :
      ( ( member5679158588528110291_a_a_a @ Q @ ( states3035385874452377268_a_b_c @ M ) )
     => ( ( output2259889051082456732_a_b_c @ ( from_F8539169374725194592_a_b_c @ M @ Q ) )
        = ( output2259889051082456732_a_b_c @ M ) ) ) ).

% from_FSM_simps(3)
thf(fact_110_from__FSM__simps_I3_J,axiom,
    ! [Q: set_Code_integer,M: fsm_se7249436556146598534er_b_c] :
      ( ( member1834298987299537450nteger @ Q @ ( states5991589681222143627er_b_c @ M ) )
     => ( ( output3616175663950214771er_b_c @ ( from_F1896587134240577335er_b_c @ M @ Q ) )
        = ( output3616175663950214771er_b_c @ M ) ) ) ).

% from_FSM_simps(3)
thf(fact_111_from__FSM__simps_I3_J,axiom,
    ! [Q: set_nat,M: fsm_set_nat_b_c] :
      ( ( member_set_nat @ Q @ ( states_set_nat_b_c @ M ) )
     => ( ( outputs_set_nat_b_c @ ( from_FSM_set_nat_b_c @ M @ Q ) )
        = ( outputs_set_nat_b_c @ M ) ) ) ).

% from_FSM_simps(3)
thf(fact_112_from__FSM__simps_I3_J,axiom,
    ! [Q: nat,M: fsm_nat_nat_nat] :
      ( ( member_nat @ Q @ ( states_nat_nat_nat @ M ) )
     => ( ( outputs_nat_nat_nat @ ( from_FSM_nat_nat_nat @ M @ Q ) )
        = ( outputs_nat_nat_nat @ M ) ) ) ).

% from_FSM_simps(3)
thf(fact_113_from__FSM__simps_I3_J,axiom,
    ! [Q: code_integer,M: fsm_Code_integer_b_c] :
      ( ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ M ) )
     => ( ( output924834633477831741er_b_c @ ( from_F2639493813852433665er_b_c @ M @ Q ) )
        = ( output924834633477831741er_b_c @ M ) ) ) ).

% from_FSM_simps(3)
thf(fact_114_from__FSM__simps_I3_J,axiom,
    ! [Q: nat,M: fsm_nat_b_c] :
      ( ( member_nat @ Q @ ( states_nat_b_c @ M ) )
     => ( ( outputs_nat_b_c @ ( from_FSM_nat_b_c @ M @ Q ) )
        = ( outputs_nat_b_c @ M ) ) ) ).

% from_FSM_simps(3)
thf(fact_115_from__FSM__simps_I3_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( outputs_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = ( outputs_a_b_c @ M ) ) ) ).

% from_FSM_simps(3)
thf(fact_116_from__FSM__simps_I3_J,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
     => ( ( outputs_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) )
        = ( outputs_set_a_b_c @ M ) ) ) ).

% from_FSM_simps(3)
thf(fact_117_from__FSM__simps_I2_J,axiom,
    ! [Q: sum_su5154756782089200444_a_a_a,M: fsm_Su3173892059090327343_a_b_c] :
      ( ( member5679158588528110291_a_a_a @ Q @ ( states3035385874452377268_a_b_c @ M ) )
     => ( ( inputs3694848887974842907_a_b_c @ ( from_F8539169374725194592_a_b_c @ M @ Q ) )
        = ( inputs3694848887974842907_a_b_c @ M ) ) ) ).

% from_FSM_simps(2)
thf(fact_118_from__FSM__simps_I2_J,axiom,
    ! [Q: set_Code_integer,M: fsm_se7249436556146598534er_b_c] :
      ( ( member1834298987299537450nteger @ Q @ ( states5991589681222143627er_b_c @ M ) )
     => ( ( inputs1631762778656711410er_b_c @ ( from_F1896587134240577335er_b_c @ M @ Q ) )
        = ( inputs1631762778656711410er_b_c @ M ) ) ) ).

% from_FSM_simps(2)
thf(fact_119_from__FSM__simps_I2_J,axiom,
    ! [Q: set_nat,M: fsm_set_nat_b_c] :
      ( ( member_set_nat @ Q @ ( states_set_nat_b_c @ M ) )
     => ( ( inputs_set_nat_b_c @ ( from_FSM_set_nat_b_c @ M @ Q ) )
        = ( inputs_set_nat_b_c @ M ) ) ) ).

% from_FSM_simps(2)
thf(fact_120_from__FSM__simps_I2_J,axiom,
    ! [Q: nat,M: fsm_nat_nat_nat] :
      ( ( member_nat @ Q @ ( states_nat_nat_nat @ M ) )
     => ( ( inputs_nat_nat_nat @ ( from_FSM_nat_nat_nat @ M @ Q ) )
        = ( inputs_nat_nat_nat @ M ) ) ) ).

% from_FSM_simps(2)
thf(fact_121_from__FSM__simps_I2_J,axiom,
    ! [Q: code_integer,M: fsm_Code_integer_b_c] :
      ( ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ M ) )
     => ( ( inputs4521787250163688636er_b_c @ ( from_F2639493813852433665er_b_c @ M @ Q ) )
        = ( inputs4521787250163688636er_b_c @ M ) ) ) ).

% from_FSM_simps(2)
thf(fact_122_from__FSM__simps_I2_J,axiom,
    ! [Q: nat,M: fsm_nat_b_c] :
      ( ( member_nat @ Q @ ( states_nat_b_c @ M ) )
     => ( ( inputs_nat_b_c @ ( from_FSM_nat_b_c @ M @ Q ) )
        = ( inputs_nat_b_c @ M ) ) ) ).

% from_FSM_simps(2)
thf(fact_123_from__FSM__simps_I2_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( inputs_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = ( inputs_a_b_c @ M ) ) ) ).

% from_FSM_simps(2)
thf(fact_124_from__FSM__simps_I2_J,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
     => ( ( inputs_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) )
        = ( inputs_set_a_b_c @ M ) ) ) ).

% from_FSM_simps(2)
thf(fact_125_filter__transitions__simps_I3_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,P: produc884577481120224310_a_a_a > $o] :
      ( ( inputs3694848887974842907_a_b_c @ ( filter7058864375813322832_a_b_c @ M @ P ) )
      = ( inputs3694848887974842907_a_b_c @ M ) ) ).

% filter_transitions_simps(3)
thf(fact_126_filter__transitions__simps_I3_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c,P: produc9083820074870880868nteger > $o] :
      ( ( inputs1631762778656711410er_b_c @ ( filter8485183139320716327er_b_c @ M @ P ) )
      = ( inputs1631762778656711410er_b_c @ M ) ) ).

% filter_transitions_simps(3)
thf(fact_127_filter__transitions__simps_I3_J,axiom,
    ! [M: fsm_set_nat_b_c,P: produc4100985854500036554et_nat > $o] :
      ( ( inputs_set_nat_b_c @ ( filter6002735969965398170at_b_c @ M @ P ) )
      = ( inputs_set_nat_b_c @ M ) ) ).

% filter_transitions_simps(3)
thf(fact_128_filter__transitions__simps_I3_J,axiom,
    ! [M: fsm_nat_nat_nat,P: produc8642769642335960151at_nat > $o] :
      ( ( inputs_nat_nat_nat @ ( filter5128389180105923269at_nat @ M @ P ) )
      = ( inputs_nat_nat_nat @ M ) ) ).

% filter_transitions_simps(3)
thf(fact_129_filter__transitions__simps_I3_J,axiom,
    ! [M: fsm_nat_b_c,P: produc8586872805302537566_c_nat > $o] :
      ( ( inputs_nat_b_c @ ( filter4406304812386798564at_b_c @ M @ P ) )
      = ( inputs_nat_b_c @ M ) ) ).

% filter_transitions_simps(3)
thf(fact_130_filter__transitions__simps_I3_J,axiom,
    ! [M: fsm_set_a_b_c,P: produc373761375140688950_set_a > $o] :
      ( ( inputs_set_a_b_c @ ( filter824366976161576716_a_b_c @ M @ P ) )
      = ( inputs_set_a_b_c @ M ) ) ).

% filter_transitions_simps(3)
thf(fact_131_filter__transitions__simps_I3_J,axiom,
    ! [M: fsm_a_b_c,P: produc2669932201312201782od_c_a > $o] :
      ( ( inputs_a_b_c @ ( filter4876964422427095468_a_b_c @ M @ P ) )
      = ( inputs_a_b_c @ M ) ) ).

% filter_transitions_simps(3)
thf(fact_132_rename__states__simps_I4_J,axiom,
    ! [M: fsm_a_b_c,F: a > a] :
      ( ( outputs_a_b_c @ ( rename4556187690751568563_b_c_a @ M @ F ) )
      = ( outputs_a_b_c @ M ) ) ).

% rename_states_simps(4)
thf(fact_133_rename__states__simps_I4_J,axiom,
    ! [M: fsm_a_b_c,F: a > set_a] :
      ( ( outputs_set_a_b_c @ ( rename6977240347653668883_set_a @ M @ F ) )
      = ( outputs_a_b_c @ M ) ) ).

% rename_states_simps(4)
thf(fact_134_rename__states__simps_I4_J,axiom,
    ! [M: fsm_set_a_b_c,F: set_a > a] :
      ( ( outputs_a_b_c @ ( rename6603781358153168211_b_c_a @ M @ F ) )
      = ( outputs_set_a_b_c @ M ) ) ).

% rename_states_simps(4)
thf(fact_135_rename__states__simps_I4_J,axiom,
    ! [M: fsm_set_a_b_c,F: set_a > set_a] :
      ( ( outputs_set_a_b_c @ ( rename4970260902906481331_set_a @ M @ F ) )
      = ( outputs_set_a_b_c @ M ) ) ).

% rename_states_simps(4)
thf(fact_136_rename__states__simps_I4_J,axiom,
    ! [M: fsm_nat_nat_nat,F: nat > nat] :
      ( ( outputs_nat_nat_nat @ ( rename3390199371875448312at_nat @ M @ F ) )
      = ( outputs_nat_nat_nat @ M ) ) ).

% rename_states_simps(4)
thf(fact_137_rename__states__simps_I4_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > nat] :
      ( ( outputs_nat_b_c @ ( rename4185420143961928215_c_nat @ M @ F ) )
      = ( outputs_nat_b_c @ M ) ) ).

% rename_states_simps(4)
thf(fact_138_rename__states__simps_I4_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > a] :
      ( ( outputs_a_b_c @ ( rename7200612048645949559_b_c_a @ M @ F ) )
      = ( outputs_nat_b_c @ M ) ) ).

% rename_states_simps(4)
thf(fact_139_rename__states__simps_I4_J,axiom,
    ! [M: fsm_a_b_c,F: a > code_integer] :
      ( ( output924834633477831741er_b_c @ ( rename9064899991920257384nteger @ M @ F ) )
      = ( outputs_a_b_c @ M ) ) ).

% rename_states_simps(4)
thf(fact_140_rename__states__simps_I4_J,axiom,
    ! [M: fsm_a_b_c,F: a > nat] :
      ( ( outputs_nat_b_c @ ( rename7348797690608482523_c_nat @ M @ F ) )
      = ( outputs_a_b_c @ M ) ) ).

% rename_states_simps(4)
thf(fact_141_rename__states__simps_I4_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > set_a] :
      ( ( outputs_set_a_b_c @ ( rename7954113337160224215_set_a @ M @ F ) )
      = ( outputs_nat_b_c @ M ) ) ).

% rename_states_simps(4)
thf(fact_142_rename__states__simps_I3_J,axiom,
    ! [M: fsm_a_b_c,F: a > a] :
      ( ( inputs_a_b_c @ ( rename4556187690751568563_b_c_a @ M @ F ) )
      = ( inputs_a_b_c @ M ) ) ).

% rename_states_simps(3)
thf(fact_143_rename__states__simps_I3_J,axiom,
    ! [M: fsm_a_b_c,F: a > set_a] :
      ( ( inputs_set_a_b_c @ ( rename6977240347653668883_set_a @ M @ F ) )
      = ( inputs_a_b_c @ M ) ) ).

% rename_states_simps(3)
thf(fact_144_rename__states__simps_I3_J,axiom,
    ! [M: fsm_set_a_b_c,F: set_a > a] :
      ( ( inputs_a_b_c @ ( rename6603781358153168211_b_c_a @ M @ F ) )
      = ( inputs_set_a_b_c @ M ) ) ).

% rename_states_simps(3)
thf(fact_145_rename__states__simps_I3_J,axiom,
    ! [M: fsm_set_a_b_c,F: set_a > set_a] :
      ( ( inputs_set_a_b_c @ ( rename4970260902906481331_set_a @ M @ F ) )
      = ( inputs_set_a_b_c @ M ) ) ).

% rename_states_simps(3)
thf(fact_146_rename__states__simps_I3_J,axiom,
    ! [M: fsm_nat_nat_nat,F: nat > nat] :
      ( ( inputs_nat_nat_nat @ ( rename3390199371875448312at_nat @ M @ F ) )
      = ( inputs_nat_nat_nat @ M ) ) ).

% rename_states_simps(3)
thf(fact_147_rename__states__simps_I3_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > nat] :
      ( ( inputs_nat_b_c @ ( rename4185420143961928215_c_nat @ M @ F ) )
      = ( inputs_nat_b_c @ M ) ) ).

% rename_states_simps(3)
thf(fact_148_rename__states__simps_I3_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > a] :
      ( ( inputs_a_b_c @ ( rename7200612048645949559_b_c_a @ M @ F ) )
      = ( inputs_nat_b_c @ M ) ) ).

% rename_states_simps(3)
thf(fact_149_rename__states__simps_I3_J,axiom,
    ! [M: fsm_a_b_c,F: a > code_integer] :
      ( ( inputs4521787250163688636er_b_c @ ( rename9064899991920257384nteger @ M @ F ) )
      = ( inputs_a_b_c @ M ) ) ).

% rename_states_simps(3)
thf(fact_150_rename__states__simps_I3_J,axiom,
    ! [M: fsm_a_b_c,F: a > nat] :
      ( ( inputs_nat_b_c @ ( rename7348797690608482523_c_nat @ M @ F ) )
      = ( inputs_a_b_c @ M ) ) ).

% rename_states_simps(3)
thf(fact_151_rename__states__simps_I3_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > set_a] :
      ( ( inputs_set_a_b_c @ ( rename7954113337160224215_set_a @ M @ F ) )
      = ( inputs_nat_b_c @ M ) ) ).

% rename_states_simps(3)
thf(fact_152_submachine__from,axiom,
    ! [S: fsm_set_nat_b_c,M: fsm_set_nat_b_c,Q: set_nat] :
      ( ( is_sub96770212967015341at_b_c @ S @ M )
     => ( ( member_set_nat @ Q @ ( states_set_nat_b_c @ S ) )
       => ( is_sub96770212967015341at_b_c @ ( from_FSM_set_nat_b_c @ S @ Q ) @ ( from_FSM_set_nat_b_c @ M @ Q ) ) ) ) ).

% submachine_from
thf(fact_153_submachine__from,axiom,
    ! [S: fsm_nat_nat_nat,M: fsm_nat_nat_nat,Q: nat] :
      ( ( is_sub7029232139394105176at_nat @ S @ M )
     => ( ( member_nat @ Q @ ( states_nat_nat_nat @ S ) )
       => ( is_sub7029232139394105176at_nat @ ( from_FSM_nat_nat_nat @ S @ Q ) @ ( from_FSM_nat_nat_nat @ M @ Q ) ) ) ) ).

% submachine_from
thf(fact_154_submachine__from,axiom,
    ! [S: fsm_Code_integer_b_c,M: fsm_Code_integer_b_c,Q: code_integer] :
      ( ( is_sub5482583993107430660er_b_c @ S @ M )
     => ( ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ S ) )
       => ( is_sub5482583993107430660er_b_c @ ( from_F2639493813852433665er_b_c @ S @ Q ) @ ( from_F2639493813852433665er_b_c @ M @ Q ) ) ) ) ).

% submachine_from
thf(fact_155_submachine__from,axiom,
    ! [S: fsm_nat_b_c,M: fsm_nat_b_c,Q: nat] :
      ( ( is_sub6065310311083418359at_b_c @ S @ M )
     => ( ( member_nat @ Q @ ( states_nat_b_c @ S ) )
       => ( is_sub6065310311083418359at_b_c @ ( from_FSM_nat_b_c @ S @ Q ) @ ( from_FSM_nat_b_c @ M @ Q ) ) ) ) ).

% submachine_from
thf(fact_156_submachine__from,axiom,
    ! [S: fsm_a_b_c,M: fsm_a_b_c,Q: a] :
      ( ( is_submachine_a_b_c @ S @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ S ) )
       => ( is_submachine_a_b_c @ ( from_FSM_a_b_c @ S @ Q ) @ ( from_FSM_a_b_c @ M @ Q ) ) ) ) ).

% submachine_from
thf(fact_157_submachine__from,axiom,
    ! [S: fsm_set_a_b_c,M: fsm_set_a_b_c,Q: set_a] :
      ( ( is_sub1377638108373535417_a_b_c @ S @ M )
     => ( ( member_set_a @ Q @ ( states_set_a_b_c @ S ) )
       => ( is_sub1377638108373535417_a_b_c @ ( from_FSM_set_a_b_c @ S @ Q ) @ ( from_FSM_set_a_b_c @ M @ Q ) ) ) ) ).

% submachine_from
thf(fact_158_from__FSM__single__input,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer] :
      ( ( single2384623283392041037er_b_c @ M )
     => ( single2384623283392041037er_b_c @ ( from_F2639493813852433665er_b_c @ M @ Q ) ) ) ).

% from_FSM_single_input
thf(fact_159_from__FSM__single__input,axiom,
    ! [M: fsm_nat_b_c,Q: nat] :
      ( ( single_input_nat_b_c @ M )
     => ( single_input_nat_b_c @ ( from_FSM_nat_b_c @ M @ Q ) ) ) ).

% from_FSM_single_input
thf(fact_160_from__FSM__single__input,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( single_input_a_b_c @ M )
     => ( single_input_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) ) ).

% from_FSM_single_input
thf(fact_161_from__FSM__single__input,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a] :
      ( ( single847711263812794288_a_b_c @ M )
     => ( single847711263812794288_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) ) ) ).

% from_FSM_single_input
thf(fact_162_from__from,axiom,
    ! [Q1: set_nat,M: fsm_set_nat_b_c,Q12: set_nat] :
      ( ( member_set_nat @ Q1 @ ( states_set_nat_b_c @ M ) )
     => ( ( member_set_nat @ Q12 @ ( states_set_nat_b_c @ M ) )
       => ( ( from_FSM_set_nat_b_c @ ( from_FSM_set_nat_b_c @ M @ Q1 ) @ Q12 )
          = ( from_FSM_set_nat_b_c @ M @ Q12 ) ) ) ) ).

% from_from
thf(fact_163_from__from,axiom,
    ! [Q1: nat,M: fsm_nat_nat_nat,Q12: nat] :
      ( ( member_nat @ Q1 @ ( states_nat_nat_nat @ M ) )
     => ( ( member_nat @ Q12 @ ( states_nat_nat_nat @ M ) )
       => ( ( from_FSM_nat_nat_nat @ ( from_FSM_nat_nat_nat @ M @ Q1 ) @ Q12 )
          = ( from_FSM_nat_nat_nat @ M @ Q12 ) ) ) ) ).

% from_from
thf(fact_164_from__from,axiom,
    ! [Q1: code_integer,M: fsm_Code_integer_b_c,Q12: code_integer] :
      ( ( member_Code_integer @ Q1 @ ( states1921009325540848725er_b_c @ M ) )
     => ( ( member_Code_integer @ Q12 @ ( states1921009325540848725er_b_c @ M ) )
       => ( ( from_F2639493813852433665er_b_c @ ( from_F2639493813852433665er_b_c @ M @ Q1 ) @ Q12 )
          = ( from_F2639493813852433665er_b_c @ M @ Q12 ) ) ) ) ).

% from_from
thf(fact_165_from__from,axiom,
    ! [Q1: nat,M: fsm_nat_b_c,Q12: nat] :
      ( ( member_nat @ Q1 @ ( states_nat_b_c @ M ) )
     => ( ( member_nat @ Q12 @ ( states_nat_b_c @ M ) )
       => ( ( from_FSM_nat_b_c @ ( from_FSM_nat_b_c @ M @ Q1 ) @ Q12 )
          = ( from_FSM_nat_b_c @ M @ Q12 ) ) ) ) ).

% from_from
thf(fact_166_from__from,axiom,
    ! [Q1: a,M: fsm_a_b_c,Q12: a] :
      ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
     => ( ( member_a @ Q12 @ ( states_a_b_c @ M ) )
       => ( ( from_FSM_a_b_c @ ( from_FSM_a_b_c @ M @ Q1 ) @ Q12 )
          = ( from_FSM_a_b_c @ M @ Q12 ) ) ) ) ).

% from_from
thf(fact_167_from__from,axiom,
    ! [Q1: set_a,M: fsm_set_a_b_c,Q12: set_a] :
      ( ( member_set_a @ Q1 @ ( states_set_a_b_c @ M ) )
     => ( ( member_set_a @ Q12 @ ( states_set_a_b_c @ M ) )
       => ( ( from_FSM_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q1 ) @ Q12 )
          = ( from_FSM_set_a_b_c @ M @ Q12 ) ) ) ) ).

% from_from
thf(fact_168_from__FSM__simps_I5_J,axiom,
    ! [Q: set_nat,M: fsm_set_nat_b_c] :
      ( ( member_set_nat @ Q @ ( states_set_nat_b_c @ M ) )
     => ( ( states_set_nat_b_c @ ( from_FSM_set_nat_b_c @ M @ Q ) )
        = ( states_set_nat_b_c @ M ) ) ) ).

% from_FSM_simps(5)
thf(fact_169_from__FSM__simps_I5_J,axiom,
    ! [Q: nat,M: fsm_nat_nat_nat] :
      ( ( member_nat @ Q @ ( states_nat_nat_nat @ M ) )
     => ( ( states_nat_nat_nat @ ( from_FSM_nat_nat_nat @ M @ Q ) )
        = ( states_nat_nat_nat @ M ) ) ) ).

% from_FSM_simps(5)
thf(fact_170_from__FSM__simps_I5_J,axiom,
    ! [Q: code_integer,M: fsm_Code_integer_b_c] :
      ( ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ M ) )
     => ( ( states1921009325540848725er_b_c @ ( from_F2639493813852433665er_b_c @ M @ Q ) )
        = ( states1921009325540848725er_b_c @ M ) ) ) ).

% from_FSM_simps(5)
thf(fact_171_from__FSM__simps_I5_J,axiom,
    ! [Q: nat,M: fsm_nat_b_c] :
      ( ( member_nat @ Q @ ( states_nat_b_c @ M ) )
     => ( ( states_nat_b_c @ ( from_FSM_nat_b_c @ M @ Q ) )
        = ( states_nat_b_c @ M ) ) ) ).

% from_FSM_simps(5)
thf(fact_172_from__FSM__simps_I5_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( states_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = ( states_a_b_c @ M ) ) ) ).

% from_FSM_simps(5)
thf(fact_173_from__FSM__simps_I5_J,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
     => ( ( states_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) )
        = ( states_set_a_b_c @ M ) ) ) ).

% from_FSM_simps(5)
thf(fact_174_filter__transitions__simps_I2_J,axiom,
    ! [M: fsm_set_nat_b_c,P: produc4100985854500036554et_nat > $o] :
      ( ( states_set_nat_b_c @ ( filter6002735969965398170at_b_c @ M @ P ) )
      = ( states_set_nat_b_c @ M ) ) ).

% filter_transitions_simps(2)
thf(fact_175_filter__transitions__simps_I2_J,axiom,
    ! [M: fsm_nat_nat_nat,P: produc8642769642335960151at_nat > $o] :
      ( ( states_nat_nat_nat @ ( filter5128389180105923269at_nat @ M @ P ) )
      = ( states_nat_nat_nat @ M ) ) ).

% filter_transitions_simps(2)
thf(fact_176_filter__transitions__simps_I2_J,axiom,
    ! [M: fsm_nat_b_c,P: produc8586872805302537566_c_nat > $o] :
      ( ( states_nat_b_c @ ( filter4406304812386798564at_b_c @ M @ P ) )
      = ( states_nat_b_c @ M ) ) ).

% filter_transitions_simps(2)
thf(fact_177_filter__transitions__simps_I2_J,axiom,
    ! [M: fsm_a_b_c,P: produc2669932201312201782od_c_a > $o] :
      ( ( states_a_b_c @ ( filter4876964422427095468_a_b_c @ M @ P ) )
      = ( states_a_b_c @ M ) ) ).

% filter_transitions_simps(2)
thf(fact_178_filter__transitions__simps_I2_J,axiom,
    ! [M: fsm_set_a_b_c,P: produc373761375140688950_set_a > $o] :
      ( ( states_set_a_b_c @ ( filter824366976161576716_a_b_c @ M @ P ) )
      = ( states_set_a_b_c @ M ) ) ).

% filter_transitions_simps(2)
thf(fact_179_add__input__simps_I5_J,axiom,
    ! [M: fsm_set_nat_b_c,X: b] :
      ( ( states_set_nat_b_c @ ( add_in7503527852466619365at_b_c @ M @ X ) )
      = ( states_set_nat_b_c @ M ) ) ).

% add_input_simps(5)
thf(fact_180_add__input__simps_I5_J,axiom,
    ! [M: fsm_nat_nat_nat,X: nat] :
      ( ( states_nat_nat_nat @ ( add_in6123057044259584912at_nat @ M @ X ) )
      = ( states_nat_nat_nat @ M ) ) ).

% add_input_simps(5)
thf(fact_181_add__input__simps_I5_J,axiom,
    ! [M: fsm_nat_b_c,X: b] :
      ( ( states_nat_b_c @ ( add_input_nat_b_c @ M @ X ) )
      = ( states_nat_b_c @ M ) ) ).

% add_input_simps(5)
thf(fact_182_add__input__simps_I5_J,axiom,
    ! [M: fsm_a_b_c,X: b] :
      ( ( states_a_b_c @ ( add_input_a_b_c @ M @ X ) )
      = ( states_a_b_c @ M ) ) ).

% add_input_simps(5)
thf(fact_183_add__input__simps_I5_J,axiom,
    ! [M: fsm_set_a_b_c,X: b] :
      ( ( states_set_a_b_c @ ( add_input_set_a_b_c @ M @ X ) )
      = ( states_set_a_b_c @ M ) ) ).

% add_input_simps(5)
thf(fact_184_add__output__simps_I5_J,axiom,
    ! [M: fsm_set_nat_b_c,Y: c] :
      ( ( states_set_nat_b_c @ ( add_ou8618801446620803286at_b_c @ M @ Y ) )
      = ( states_set_nat_b_c @ M ) ) ).

% add_output_simps(5)
thf(fact_185_add__output__simps_I5_J,axiom,
    ! [M: fsm_nat_nat_nat,Y: nat] :
      ( ( states_nat_nat_nat @ ( add_ou6503059941141532929at_nat @ M @ Y ) )
      = ( states_nat_nat_nat @ M ) ) ).

% add_output_simps(5)
thf(fact_186_add__output__simps_I5_J,axiom,
    ! [M: fsm_nat_b_c,Y: c] :
      ( ( states_nat_b_c @ ( add_output_nat_b_c @ M @ Y ) )
      = ( states_nat_b_c @ M ) ) ).

% add_output_simps(5)
thf(fact_187_add__output__simps_I5_J,axiom,
    ! [M: fsm_a_b_c,Y: c] :
      ( ( states_a_b_c @ ( add_output_a_b_c @ M @ Y ) )
      = ( states_a_b_c @ M ) ) ).

% add_output_simps(5)
thf(fact_188_add__output__simps_I5_J,axiom,
    ! [M: fsm_set_a_b_c,Y: c] :
      ( ( states_set_a_b_c @ ( add_output_set_a_b_c @ M @ Y ) )
      = ( states_set_a_b_c @ M ) ) ).

% add_output_simps(5)
thf(fact_189_filter__transitions__submachine,axiom,
    ! [M: fsm_Code_integer_b_c,P: produc558599814252099832nteger > $o] : ( is_sub5482583993107430660er_b_c @ ( filter2165177713251037681er_b_c @ M @ P ) @ M ) ).

% filter_transitions_submachine
thf(fact_190_filter__transitions__submachine,axiom,
    ! [M: fsm_nat_b_c,P: produc8586872805302537566_c_nat > $o] : ( is_sub6065310311083418359at_b_c @ ( filter4406304812386798564at_b_c @ M @ P ) @ M ) ).

% filter_transitions_submachine
thf(fact_191_filter__transitions__submachine,axiom,
    ! [M: fsm_a_b_c,P: produc2669932201312201782od_c_a > $o] : ( is_submachine_a_b_c @ ( filter4876964422427095468_a_b_c @ M @ P ) @ M ) ).

% filter_transitions_submachine
thf(fact_192_filter__transitions__submachine,axiom,
    ! [M: fsm_set_a_b_c,P: produc373761375140688950_set_a > $o] : ( is_sub1377638108373535417_a_b_c @ ( filter824366976161576716_a_b_c @ M @ P ) @ M ) ).

% filter_transitions_submachine
thf(fact_193_index__states_Osimps,axiom,
    ( index_492992408057744179at_nat
    = ( ^ [M2: fsm_nat_nat_nat] : ( rename3390199371875448312at_nat @ M2 @ ( assign_indices_nat @ ( states_nat_nat_nat @ M2 ) ) ) ) ) ).

% index_states.simps
thf(fact_194_index__states_Osimps,axiom,
    ( index_5743659002960661343er_b_c
    = ( ^ [M2: fsm_Code_integer_b_c] : ( rename5224757547639086116_c_nat @ M2 @ ( assign469752435954302728nteger @ ( states1921009325540848725er_b_c @ M2 ) ) ) ) ) ).

% index_states.simps
thf(fact_195_index__states_Osimps,axiom,
    ( index_states_nat_b_c
    = ( ^ [M2: fsm_nat_b_c] : ( rename4185420143961928215_c_nat @ M2 @ ( assign_indices_nat @ ( states_nat_b_c @ M2 ) ) ) ) ) ).

% index_states.simps
thf(fact_196_index__states_Osimps,axiom,
    ( index_states_a_b_c
    = ( ^ [M2: fsm_a_b_c] : ( rename7348797690608482523_c_nat @ M2 @ ( assign_indices_a @ ( states_a_b_c @ M2 ) ) ) ) ) ).

% index_states.simps
thf(fact_197_index__states_Oelims,axiom,
    ! [X: fsm_nat_nat_nat,Y: fsm_nat_nat_nat] :
      ( ( ( index_492992408057744179at_nat @ X )
        = Y )
     => ( Y
        = ( rename3390199371875448312at_nat @ X @ ( assign_indices_nat @ ( states_nat_nat_nat @ X ) ) ) ) ) ).

% index_states.elims
thf(fact_198_index__states_Oelims,axiom,
    ! [X: fsm_Code_integer_b_c,Y: fsm_nat_b_c] :
      ( ( ( index_5743659002960661343er_b_c @ X )
        = Y )
     => ( Y
        = ( rename5224757547639086116_c_nat @ X @ ( assign469752435954302728nteger @ ( states1921009325540848725er_b_c @ X ) ) ) ) ) ).

% index_states.elims
thf(fact_199_index__states_Oelims,axiom,
    ! [X: fsm_nat_b_c,Y: fsm_nat_b_c] :
      ( ( ( index_states_nat_b_c @ X )
        = Y )
     => ( Y
        = ( rename4185420143961928215_c_nat @ X @ ( assign_indices_nat @ ( states_nat_b_c @ X ) ) ) ) ) ).

% index_states.elims
thf(fact_200_index__states_Oelims,axiom,
    ! [X: fsm_a_b_c,Y: fsm_nat_b_c] :
      ( ( ( index_states_a_b_c @ X )
        = Y )
     => ( Y
        = ( rename7348797690608482523_c_nat @ X @ ( assign_indices_a @ ( states_a_b_c @ X ) ) ) ) ) ).

% index_states.elims
thf(fact_201_add__output__simps_I3_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,Y: c] :
      ( ( output2259889051082456732_a_b_c @ ( add_ou235458353079876748_a_b_c @ M @ Y ) )
      = ( insert_c @ Y @ ( output2259889051082456732_a_b_c @ M ) ) ) ).

% add_output_simps(3)
thf(fact_202_add__output__simps_I3_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c,Y: c] :
      ( ( output3616175663950214771er_b_c @ ( add_ou7817685156754277475er_b_c @ M @ Y ) )
      = ( insert_c @ Y @ ( output3616175663950214771er_b_c @ M ) ) ) ).

% add_output_simps(3)
thf(fact_203_add__output__simps_I3_J,axiom,
    ! [M: fsm_set_nat_b_c,Y: c] :
      ( ( outputs_set_nat_b_c @ ( add_ou8618801446620803286at_b_c @ M @ Y ) )
      = ( insert_c @ Y @ ( outputs_set_nat_b_c @ M ) ) ) ).

% add_output_simps(3)
thf(fact_204_add__output__simps_I3_J,axiom,
    ! [M: fsm_nat_nat_nat,Y: nat] :
      ( ( outputs_nat_nat_nat @ ( add_ou6503059941141532929at_nat @ M @ Y ) )
      = ( insert_nat @ Y @ ( outputs_nat_nat_nat @ M ) ) ) ).

% add_output_simps(3)
thf(fact_205_add__output__simps_I3_J,axiom,
    ! [M: fsm_nat_b_c,Y: c] :
      ( ( outputs_nat_b_c @ ( add_output_nat_b_c @ M @ Y ) )
      = ( insert_c @ Y @ ( outputs_nat_b_c @ M ) ) ) ).

% add_output_simps(3)
thf(fact_206_add__output__simps_I3_J,axiom,
    ! [M: fsm_set_a_b_c,Y: c] :
      ( ( outputs_set_a_b_c @ ( add_output_set_a_b_c @ M @ Y ) )
      = ( insert_c @ Y @ ( outputs_set_a_b_c @ M ) ) ) ).

% add_output_simps(3)
thf(fact_207_add__output__simps_I3_J,axiom,
    ! [M: fsm_a_b_c,Y: c] :
      ( ( outputs_a_b_c @ ( add_output_a_b_c @ M @ Y ) )
      = ( insert_c @ Y @ ( outputs_a_b_c @ M ) ) ) ).

% add_output_simps(3)
thf(fact_208_add__input__simps_I2_J,axiom,
    ! [M: fsm_nat_nat_nat,X: nat] :
      ( ( inputs_nat_nat_nat @ ( add_in6123057044259584912at_nat @ M @ X ) )
      = ( insert_nat @ X @ ( inputs_nat_nat_nat @ M ) ) ) ).

% add_input_simps(2)
thf(fact_209_add__input__simps_I2_J,axiom,
    ! [M: fsm_Su3173892059090327343_a_b_c,X: b] :
      ( ( inputs3694848887974842907_a_b_c @ ( add_in7389665105193716379_a_b_c @ M @ X ) )
      = ( insert_b @ X @ ( inputs3694848887974842907_a_b_c @ M ) ) ) ).

% add_input_simps(2)
thf(fact_210_add__input__simps_I2_J,axiom,
    ! [M: fsm_se7249436556146598534er_b_c,X: b] :
      ( ( inputs1631762778656711410er_b_c @ ( add_in5283742766163303282er_b_c @ M @ X ) )
      = ( insert_b @ X @ ( inputs1631762778656711410er_b_c @ M ) ) ) ).

% add_input_simps(2)
thf(fact_211_add__input__simps_I2_J,axiom,
    ! [M: fsm_set_nat_b_c,X: b] :
      ( ( inputs_set_nat_b_c @ ( add_in7503527852466619365at_b_c @ M @ X ) )
      = ( insert_b @ X @ ( inputs_set_nat_b_c @ M ) ) ) ).

% add_input_simps(2)
thf(fact_212_add__input__simps_I2_J,axiom,
    ! [M: fsm_nat_b_c,X: b] :
      ( ( inputs_nat_b_c @ ( add_input_nat_b_c @ M @ X ) )
      = ( insert_b @ X @ ( inputs_nat_b_c @ M ) ) ) ).

% add_input_simps(2)
thf(fact_213_add__input__simps_I2_J,axiom,
    ! [M: fsm_set_a_b_c,X: b] :
      ( ( inputs_set_a_b_c @ ( add_input_set_a_b_c @ M @ X ) )
      = ( insert_b @ X @ ( inputs_set_a_b_c @ M ) ) ) ).

% add_input_simps(2)
thf(fact_214_add__input__simps_I2_J,axiom,
    ! [M: fsm_a_b_c,X: b] :
      ( ( inputs_a_b_c @ ( add_input_a_b_c @ M @ X ) )
      = ( insert_b @ X @ ( inputs_a_b_c @ M ) ) ) ).

% add_input_simps(2)
thf(fact_215_add__state__simps_I5_J,axiom,
    ! [M: fsm_set_nat_b_c,Q: set_nat] :
      ( ( states_set_nat_b_c @ ( add_st8360318785478835724at_b_c @ M @ Q ) )
      = ( insert_set_nat @ Q @ ( states_set_nat_b_c @ M ) ) ) ).

% add_state_simps(5)
thf(fact_216_add__state__simps_I5_J,axiom,
    ! [M: fsm_nat_nat_nat,Q: nat] :
      ( ( states_nat_nat_nat @ ( add_st626164701413644599at_nat @ M @ Q ) )
      = ( insert_nat @ Q @ ( states_nat_nat_nat @ M ) ) ) ).

% add_state_simps(5)
thf(fact_217_add__state__simps_I5_J,axiom,
    ! [M: fsm_nat_b_c,Q: nat] :
      ( ( states_nat_b_c @ ( add_state_nat_b_c @ M @ Q ) )
      = ( insert_nat @ Q @ ( states_nat_b_c @ M ) ) ) ).

% add_state_simps(5)
thf(fact_218_add__state__simps_I5_J,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( states_a_b_c @ ( add_state_a_b_c @ M @ Q ) )
      = ( insert_a @ Q @ ( states_a_b_c @ M ) ) ) ).

% add_state_simps(5)
thf(fact_219_add__state__simps_I5_J,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a] :
      ( ( states_set_a_b_c @ ( add_state_set_a_b_c @ M @ Q ) )
      = ( insert_set_a @ Q @ ( states_set_a_b_c @ M ) ) ) ).

% add_state_simps(5)
thf(fact_220_restrict__to__reachable__states__simps_I2_J,axiom,
    ! [M: fsm_set_nat_b_c] :
      ( ( states_set_nat_b_c @ ( restri7930285009982923852at_b_c @ M ) )
      = ( reacha6474151152035807107at_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(2)
thf(fact_221_restrict__to__reachable__states__simps_I2_J,axiom,
    ! [M: fsm_nat_nat_nat] :
      ( ( states_nat_nat_nat @ ( restri2246964829936590711at_nat @ M ) )
      = ( reacha3101573547803616814at_nat @ M ) ) ).

% restrict_to_reachable_states_simps(2)
thf(fact_222_restrict__to__reachable__states__simps_I2_J,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( states1921009325540848725er_b_c @ ( restri4092726753268563363er_b_c @ M ) )
      = ( reacha2636592895321446618er_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(2)
thf(fact_223_restrict__to__reachable__states__simps_I2_J,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( states_nat_b_c @ ( restri2801621895680909206at_b_c @ M ) )
      = ( reacha6243244737040045773at_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(2)
thf(fact_224_restrict__to__reachable__states__simps_I2_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( states_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) )
      = ( reacha1620305530751930115_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(2)
thf(fact_225_restrict__to__reachable__states__simps_I2_J,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( states_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) )
      = ( reacha7078015818774689891_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(2)
thf(fact_226_rename__states__simps_I2_J,axiom,
    ! [M: fsm_nat_nat_nat,F: nat > a] :
      ( ( states_a_nat_nat @ ( rename2576486212611275478_nat_a @ M @ F ) )
      = ( image_nat_a @ F @ ( states_nat_nat_nat @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_227_rename__states__simps_I2_J,axiom,
    ! [M: fsm_a_nat_nat,F: a > nat] :
      ( ( states_nat_nat_nat @ ( rename3121724683823550140at_nat @ M @ F ) )
      = ( image_a_nat @ F @ ( states_a_nat_nat @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_228_rename__states__simps_I2_J,axiom,
    ! [M: fsm_nat_nat_nat,F: nat > nat] :
      ( ( states_nat_nat_nat @ ( rename3390199371875448312at_nat @ M @ F ) )
      = ( image_nat_nat @ F @ ( states_nat_nat_nat @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_229_rename__states__simps_I2_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > nat] :
      ( ( states_nat_b_c @ ( rename4185420143961928215_c_nat @ M @ F ) )
      = ( image_nat_nat @ F @ ( states_nat_b_c @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_230_rename__states__simps_I2_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > a] :
      ( ( states_a_b_c @ ( rename7200612048645949559_b_c_a @ M @ F ) )
      = ( image_nat_a @ F @ ( states_nat_b_c @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_231_rename__states__simps_I2_J,axiom,
    ! [M: fsm_a_b_c,F: a > code_integer] :
      ( ( states1921009325540848725er_b_c @ ( rename9064899991920257384nteger @ M @ F ) )
      = ( image_a_Code_integer @ F @ ( states_a_b_c @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_232_rename__states__simps_I2_J,axiom,
    ! [M: fsm_a_b_c,F: a > nat] :
      ( ( states_nat_b_c @ ( rename7348797690608482523_c_nat @ M @ F ) )
      = ( image_a_nat @ F @ ( states_a_b_c @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_233_rename__states__simps_I2_J,axiom,
    ! [M: fsm_a_b_c,F: a > a] :
      ( ( states_a_b_c @ ( rename4556187690751568563_b_c_a @ M @ F ) )
      = ( image_a_a @ F @ ( states_a_b_c @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_234_rename__states__simps_I2_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > set_nat] :
      ( ( states_set_nat_b_c @ ( rename8160155668314554829et_nat @ M @ F ) )
      = ( image_nat_set_nat @ F @ ( states_nat_b_c @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_235_rename__states__simps_I2_J,axiom,
    ! [M: fsm_a_b_c,F: a > set_nat] :
      ( ( states_set_nat_b_c @ ( rename4200259836811344017et_nat @ M @ F ) )
      = ( image_a_set_nat @ F @ ( states_a_b_c @ M ) ) ) ).

% rename_states_simps(2)
thf(fact_236_from__FSM__simps_I1_J,axiom,
    ! [Q: set_nat,M: fsm_set_nat_b_c] :
      ( ( member_set_nat @ Q @ ( states_set_nat_b_c @ M ) )
     => ( ( initial_set_nat_b_c @ ( from_FSM_set_nat_b_c @ M @ Q ) )
        = Q ) ) ).

% from_FSM_simps(1)
thf(fact_237_from__FSM__simps_I1_J,axiom,
    ! [Q: nat,M: fsm_nat_nat_nat] :
      ( ( member_nat @ Q @ ( states_nat_nat_nat @ M ) )
     => ( ( initial_nat_nat_nat @ ( from_FSM_nat_nat_nat @ M @ Q ) )
        = Q ) ) ).

% from_FSM_simps(1)
thf(fact_238_from__FSM__simps_I1_J,axiom,
    ! [Q: code_integer,M: fsm_Code_integer_b_c] :
      ( ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ M ) )
     => ( ( initia4873860033875475119er_b_c @ ( from_F2639493813852433665er_b_c @ M @ Q ) )
        = Q ) ) ).

% from_FSM_simps(1)
thf(fact_239_from__FSM__simps_I1_J,axiom,
    ! [Q: nat,M: fsm_nat_b_c] :
      ( ( member_nat @ Q @ ( states_nat_b_c @ M ) )
     => ( ( initial_nat_b_c @ ( from_FSM_nat_b_c @ M @ Q ) )
        = Q ) ) ).

% from_FSM_simps(1)
thf(fact_240_from__FSM__simps_I1_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = Q ) ) ).

% from_FSM_simps(1)
thf(fact_241_from__FSM__simps_I1_J,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
     => ( ( initial_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) )
        = Q ) ) ).

% from_FSM_simps(1)
thf(fact_242_from__FSM__simps_I4_J,axiom,
    ! [Q: code_integer,M: fsm_Code_integer_b_c] :
      ( ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ M ) )
     => ( ( transi6707074847254859401er_b_c @ ( from_F2639493813852433665er_b_c @ M @ Q ) )
        = ( transi6707074847254859401er_b_c @ M ) ) ) ).

% from_FSM_simps(4)
thf(fact_243_from__FSM__simps_I4_J,axiom,
    ! [Q: set_nat,M: fsm_set_nat_b_c] :
      ( ( member_set_nat @ Q @ ( states_set_nat_b_c @ M ) )
     => ( ( transi1321261067114444082at_b_c @ ( from_FSM_set_nat_b_c @ M @ Q ) )
        = ( transi1321261067114444082at_b_c @ M ) ) ) ).

% from_FSM_simps(4)
thf(fact_244_from__FSM__simps_I4_J,axiom,
    ! [Q: nat,M: fsm_nat_nat_nat] :
      ( ( member_nat @ Q @ ( states_nat_nat_nat @ M ) )
     => ( ( transi2178791137084441437at_nat @ ( from_FSM_nat_nat_nat @ M @ Q ) )
        = ( transi2178791137084441437at_nat @ M ) ) ) ).

% from_FSM_simps(4)
thf(fact_245_from__FSM__simps_I4_J,axiom,
    ! [Q: nat,M: fsm_nat_b_c] :
      ( ( member_nat @ Q @ ( states_nat_b_c @ M ) )
     => ( ( transitions_nat_b_c @ ( from_FSM_nat_b_c @ M @ Q ) )
        = ( transitions_nat_b_c @ M ) ) ) ).

% from_FSM_simps(4)
thf(fact_246_from__FSM__simps_I4_J,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( transitions_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) )
        = ( transitions_a_b_c @ M ) ) ) ).

% from_FSM_simps(4)
thf(fact_247_from__FSM__simps_I4_J,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
     => ( ( transi8205445899535437940_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) )
        = ( transi8205445899535437940_a_b_c @ M ) ) ) ).

% from_FSM_simps(4)
thf(fact_248_restrict__to__reachable__states__reachable__states,axiom,
    ! [M: fsm_set_nat_b_c] :
      ( ( reacha6474151152035807107at_b_c @ ( restri7930285009982923852at_b_c @ M ) )
      = ( states_set_nat_b_c @ ( restri7930285009982923852at_b_c @ M ) ) ) ).

% restrict_to_reachable_states_reachable_states
thf(fact_249_restrict__to__reachable__states__reachable__states,axiom,
    ! [M: fsm_nat_nat_nat] :
      ( ( reacha3101573547803616814at_nat @ ( restri2246964829936590711at_nat @ M ) )
      = ( states_nat_nat_nat @ ( restri2246964829936590711at_nat @ M ) ) ) ).

% restrict_to_reachable_states_reachable_states
thf(fact_250_restrict__to__reachable__states__reachable__states,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( reacha2636592895321446618er_b_c @ ( restri4092726753268563363er_b_c @ M ) )
      = ( states1921009325540848725er_b_c @ ( restri4092726753268563363er_b_c @ M ) ) ) ).

% restrict_to_reachable_states_reachable_states
thf(fact_251_restrict__to__reachable__states__reachable__states,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( reacha6243244737040045773at_b_c @ ( restri2801621895680909206at_b_c @ M ) )
      = ( states_nat_b_c @ ( restri2801621895680909206at_b_c @ M ) ) ) ).

% restrict_to_reachable_states_reachable_states
thf(fact_252_restrict__to__reachable__states__reachable__states,axiom,
    ! [M: fsm_a_b_c] :
      ( ( reacha1620305530751930115_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) )
      = ( states_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) ) ) ).

% restrict_to_reachable_states_reachable_states
thf(fact_253_restrict__to__reachable__states__reachable__states,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( reacha7078015818774689891_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) )
      = ( states_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) ) ) ).

% restrict_to_reachable_states_reachable_states
thf(fact_254_submachine__simps_I2_J,axiom,
    ! [A: fsm_Code_integer_b_c,B: fsm_Code_integer_b_c] :
      ( ( is_sub5482583993107430660er_b_c @ A @ B )
     => ( ord_le7084787975880047091nteger @ ( states1921009325540848725er_b_c @ A ) @ ( states1921009325540848725er_b_c @ B ) ) ) ).

% submachine_simps(2)
thf(fact_255_submachine__simps_I2_J,axiom,
    ! [A: fsm_set_nat_b_c,B: fsm_set_nat_b_c] :
      ( ( is_sub96770212967015341at_b_c @ A @ B )
     => ( ord_le6893508408891458716et_nat @ ( states_set_nat_b_c @ A ) @ ( states_set_nat_b_c @ B ) ) ) ).

% submachine_simps(2)
thf(fact_256_submachine__simps_I2_J,axiom,
    ! [A: fsm_nat_nat_nat,B: fsm_nat_nat_nat] :
      ( ( is_sub7029232139394105176at_nat @ A @ B )
     => ( ord_less_eq_set_nat @ ( states_nat_nat_nat @ A ) @ ( states_nat_nat_nat @ B ) ) ) ).

% submachine_simps(2)
thf(fact_257_submachine__simps_I2_J,axiom,
    ! [A: fsm_nat_b_c,B: fsm_nat_b_c] :
      ( ( is_sub6065310311083418359at_b_c @ A @ B )
     => ( ord_less_eq_set_nat @ ( states_nat_b_c @ A ) @ ( states_nat_b_c @ B ) ) ) ).

% submachine_simps(2)
thf(fact_258_submachine__simps_I2_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ord_less_eq_set_a @ ( states_a_b_c @ A ) @ ( states_a_b_c @ B ) ) ) ).

% submachine_simps(2)
thf(fact_259_submachine__simps_I2_J,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ord_le3724670747650509150_set_a @ ( states_set_a_b_c @ A ) @ ( states_set_a_b_c @ B ) ) ) ).

% submachine_simps(2)
thf(fact_260_fsm__initial,axiom,
    ! [M: fsm_set_nat_b_c] : ( member_set_nat @ ( initial_set_nat_b_c @ M ) @ ( states_set_nat_b_c @ M ) ) ).

% fsm_initial
thf(fact_261_fsm__initial,axiom,
    ! [M: fsm_set_a_b_c] : ( member_set_a @ ( initial_set_a_b_c @ M ) @ ( states_set_a_b_c @ M ) ) ).

% fsm_initial
thf(fact_262_fsm__initial,axiom,
    ! [M: fsm_nat_nat_nat] : ( member_nat @ ( initial_nat_nat_nat @ M ) @ ( states_nat_nat_nat @ M ) ) ).

% fsm_initial
thf(fact_263_fsm__initial,axiom,
    ! [M: fsm_nat_b_c] : ( member_nat @ ( initial_nat_b_c @ M ) @ ( states_nat_b_c @ M ) ) ).

% fsm_initial
thf(fact_264_fsm__initial,axiom,
    ! [M: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M ) @ ( states_a_b_c @ M ) ) ).

% fsm_initial
thf(fact_265_rename__states__simps_I1_J,axiom,
    ! [M: fsm_nat_nat_nat,F: nat > nat] :
      ( ( initial_nat_nat_nat @ ( rename3390199371875448312at_nat @ M @ F ) )
      = ( F @ ( initial_nat_nat_nat @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_266_rename__states__simps_I1_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > nat] :
      ( ( initial_nat_b_c @ ( rename4185420143961928215_c_nat @ M @ F ) )
      = ( F @ ( initial_nat_b_c @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_267_rename__states__simps_I1_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > a] :
      ( ( initial_a_b_c @ ( rename7200612048645949559_b_c_a @ M @ F ) )
      = ( F @ ( initial_nat_b_c @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_268_rename__states__simps_I1_J,axiom,
    ! [M: fsm_a_b_c,F: a > code_integer] :
      ( ( initia4873860033875475119er_b_c @ ( rename9064899991920257384nteger @ M @ F ) )
      = ( F @ ( initial_a_b_c @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_269_rename__states__simps_I1_J,axiom,
    ! [M: fsm_a_b_c,F: a > nat] :
      ( ( initial_nat_b_c @ ( rename7348797690608482523_c_nat @ M @ F ) )
      = ( F @ ( initial_a_b_c @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_270_rename__states__simps_I1_J,axiom,
    ! [M: fsm_a_b_c,F: a > a] :
      ( ( initial_a_b_c @ ( rename4556187690751568563_b_c_a @ M @ F ) )
      = ( F @ ( initial_a_b_c @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_271_rename__states__simps_I1_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > set_nat] :
      ( ( initial_set_nat_b_c @ ( rename8160155668314554829et_nat @ M @ F ) )
      = ( F @ ( initial_nat_b_c @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_272_rename__states__simps_I1_J,axiom,
    ! [M: fsm_a_b_c,F: a > set_nat] :
      ( ( initial_set_nat_b_c @ ( rename4200259836811344017et_nat @ M @ F ) )
      = ( F @ ( initial_a_b_c @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_273_rename__states__simps_I1_J,axiom,
    ! [M: fsm_nat_b_c,F: nat > set_a] :
      ( ( initial_set_a_b_c @ ( rename7954113337160224215_set_a @ M @ F ) )
      = ( F @ ( initial_nat_b_c @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_274_rename__states__simps_I1_J,axiom,
    ! [M: fsm_set_nat_b_c,F: set_nat > nat] :
      ( ( initial_nat_b_c @ ( rename7907741259076083405_c_nat @ M @ F ) )
      = ( F @ ( initial_set_nat_b_c @ M ) ) ) ).

% rename_states_simps(1)
thf(fact_275_restrict__to__reachable__states__simps_I1_J,axiom,
    ! [M: fsm_set_nat_b_c] :
      ( ( initial_set_nat_b_c @ ( restri7930285009982923852at_b_c @ M ) )
      = ( initial_set_nat_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(1)
thf(fact_276_restrict__to__reachable__states__simps_I1_J,axiom,
    ! [M: fsm_nat_nat_nat] :
      ( ( initial_nat_nat_nat @ ( restri2246964829936590711at_nat @ M ) )
      = ( initial_nat_nat_nat @ M ) ) ).

% restrict_to_reachable_states_simps(1)
thf(fact_277_restrict__to__reachable__states__simps_I1_J,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( initia4873860033875475119er_b_c @ ( restri4092726753268563363er_b_c @ M ) )
      = ( initia4873860033875475119er_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(1)
thf(fact_278_restrict__to__reachable__states__simps_I1_J,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( initial_nat_b_c @ ( restri2801621895680909206at_b_c @ M ) )
      = ( initial_nat_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(1)
thf(fact_279_restrict__to__reachable__states__simps_I1_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( initial_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) )
      = ( initial_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(1)
thf(fact_280_restrict__to__reachable__states__simps_I1_J,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( initial_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) )
      = ( initial_set_a_b_c @ M ) ) ).

% restrict_to_reachable_states_simps(1)
thf(fact_281_mem__Collect__eq,axiom,
    ! [A2: c,P: c > $o] :
      ( ( member_c @ A2 @ ( collect_c @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_282_mem__Collect__eq,axiom,
    ! [A2: produc2669932201312201782od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ A2 @ ( collec8405073525749625505od_c_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_283_mem__Collect__eq,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_284_mem__Collect__eq,axiom,
    ! [A2: set_a,P: set_a > $o] :
      ( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_285_mem__Collect__eq,axiom,
    ! [A2: b,P: b > $o] :
      ( ( member_b @ A2 @ ( collect_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_286_mem__Collect__eq,axiom,
    ! [A2: a,P: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_287_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_288_Collect__mem__eq,axiom,
    ! [A: set_c] :
      ( ( collect_c
        @ ^ [X2: c] : ( member_c @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_289_Collect__mem__eq,axiom,
    ! [A: set_Pr8357624050790843030od_c_a] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X2: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_290_Collect__mem__eq,axiom,
    ! [A: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_291_Collect__mem__eq,axiom,
    ! [A: set_set_a] :
      ( ( collect_set_a
        @ ^ [X2: set_a] : ( member_set_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_292_Collect__mem__eq,axiom,
    ! [A: set_b] :
      ( ( collect_b
        @ ^ [X2: b] : ( member_b @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_293_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X2: a] : ( member_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_294_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_295_Collect__cong,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,Q2: produc2669932201312201782od_c_a > $o] :
      ( ! [X3: produc2669932201312201782od_c_a] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec8405073525749625505od_c_a @ P )
        = ( collec8405073525749625505od_c_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_296_Collect__cong,axiom,
    ! [P: set_nat > $o,Q2: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_297_Collect__cong,axiom,
    ! [P: set_a > $o,Q2: set_a > $o] :
      ( ! [X3: set_a] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_set_a @ P )
        = ( collect_set_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_298_Collect__cong,axiom,
    ! [P: b > $o,Q2: b > $o] :
      ( ! [X3: b] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_b @ P )
        = ( collect_b @ Q2 ) ) ) ).

% Collect_cong
thf(fact_299_Collect__cong,axiom,
    ! [P: a > $o,Q2: a > $o] :
      ( ! [X3: a] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_a @ P )
        = ( collect_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_300_Collect__cong,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_301_filter__transitions__simps_I1_J,axiom,
    ! [M: fsm_set_nat_b_c,P: produc4100985854500036554et_nat > $o] :
      ( ( initial_set_nat_b_c @ ( filter6002735969965398170at_b_c @ M @ P ) )
      = ( initial_set_nat_b_c @ M ) ) ).

% filter_transitions_simps(1)
thf(fact_302_filter__transitions__simps_I1_J,axiom,
    ! [M: fsm_nat_nat_nat,P: produc8642769642335960151at_nat > $o] :
      ( ( initial_nat_nat_nat @ ( filter5128389180105923269at_nat @ M @ P ) )
      = ( initial_nat_nat_nat @ M ) ) ).

% filter_transitions_simps(1)
thf(fact_303_filter__transitions__simps_I1_J,axiom,
    ! [M: fsm_nat_b_c,P: produc8586872805302537566_c_nat > $o] :
      ( ( initial_nat_b_c @ ( filter4406304812386798564at_b_c @ M @ P ) )
      = ( initial_nat_b_c @ M ) ) ).

% filter_transitions_simps(1)
thf(fact_304_filter__transitions__simps_I1_J,axiom,
    ! [M: fsm_a_b_c,P: produc2669932201312201782od_c_a > $o] :
      ( ( initial_a_b_c @ ( filter4876964422427095468_a_b_c @ M @ P ) )
      = ( initial_a_b_c @ M ) ) ).

% filter_transitions_simps(1)
thf(fact_305_filter__transitions__simps_I1_J,axiom,
    ! [M: fsm_set_a_b_c,P: produc373761375140688950_set_a > $o] :
      ( ( initial_set_a_b_c @ ( filter824366976161576716_a_b_c @ M @ P ) )
      = ( initial_set_a_b_c @ M ) ) ).

% filter_transitions_simps(1)
thf(fact_306_add__output__simps_I1_J,axiom,
    ! [M: fsm_set_nat_b_c,Y: c] :
      ( ( initial_set_nat_b_c @ ( add_ou8618801446620803286at_b_c @ M @ Y ) )
      = ( initial_set_nat_b_c @ M ) ) ).

% add_output_simps(1)
thf(fact_307_add__output__simps_I1_J,axiom,
    ! [M: fsm_nat_nat_nat,Y: nat] :
      ( ( initial_nat_nat_nat @ ( add_ou6503059941141532929at_nat @ M @ Y ) )
      = ( initial_nat_nat_nat @ M ) ) ).

% add_output_simps(1)
thf(fact_308_add__output__simps_I1_J,axiom,
    ! [M: fsm_nat_b_c,Y: c] :
      ( ( initial_nat_b_c @ ( add_output_nat_b_c @ M @ Y ) )
      = ( initial_nat_b_c @ M ) ) ).

% add_output_simps(1)
thf(fact_309_add__output__simps_I1_J,axiom,
    ! [M: fsm_a_b_c,Y: c] :
      ( ( initial_a_b_c @ ( add_output_a_b_c @ M @ Y ) )
      = ( initial_a_b_c @ M ) ) ).

% add_output_simps(1)
thf(fact_310_add__output__simps_I1_J,axiom,
    ! [M: fsm_set_a_b_c,Y: c] :
      ( ( initial_set_a_b_c @ ( add_output_set_a_b_c @ M @ Y ) )
      = ( initial_set_a_b_c @ M ) ) ).

% add_output_simps(1)
thf(fact_311_add__state__simps_I1_J,axiom,
    ! [M: fsm_set_nat_b_c,Q: set_nat] :
      ( ( initial_set_nat_b_c @ ( add_st8360318785478835724at_b_c @ M @ Q ) )
      = ( initial_set_nat_b_c @ M ) ) ).

% add_state_simps(1)
thf(fact_312_add__state__simps_I1_J,axiom,
    ! [M: fsm_nat_nat_nat,Q: nat] :
      ( ( initial_nat_nat_nat @ ( add_st626164701413644599at_nat @ M @ Q ) )
      = ( initial_nat_nat_nat @ M ) ) ).

% add_state_simps(1)
thf(fact_313_add__state__simps_I1_J,axiom,
    ! [M: fsm_nat_b_c,Q: nat] :
      ( ( initial_nat_b_c @ ( add_state_nat_b_c @ M @ Q ) )
      = ( initial_nat_b_c @ M ) ) ).

% add_state_simps(1)
thf(fact_314_add__state__simps_I1_J,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( initial_a_b_c @ ( add_state_a_b_c @ M @ Q ) )
      = ( initial_a_b_c @ M ) ) ).

% add_state_simps(1)
thf(fact_315_add__state__simps_I1_J,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a] :
      ( ( initial_set_a_b_c @ ( add_state_set_a_b_c @ M @ Q ) )
      = ( initial_set_a_b_c @ M ) ) ).

% add_state_simps(1)
thf(fact_316_add__input__simps_I1_J,axiom,
    ! [M: fsm_set_nat_b_c,X: b] :
      ( ( initial_set_nat_b_c @ ( add_in7503527852466619365at_b_c @ M @ X ) )
      = ( initial_set_nat_b_c @ M ) ) ).

% add_input_simps(1)
thf(fact_317_add__input__simps_I1_J,axiom,
    ! [M: fsm_nat_nat_nat,X: nat] :
      ( ( initial_nat_nat_nat @ ( add_in6123057044259584912at_nat @ M @ X ) )
      = ( initial_nat_nat_nat @ M ) ) ).

% add_input_simps(1)
thf(fact_318_add__input__simps_I1_J,axiom,
    ! [M: fsm_nat_b_c,X: b] :
      ( ( initial_nat_b_c @ ( add_input_nat_b_c @ M @ X ) )
      = ( initial_nat_b_c @ M ) ) ).

% add_input_simps(1)
thf(fact_319_add__input__simps_I1_J,axiom,
    ! [M: fsm_a_b_c,X: b] :
      ( ( initial_a_b_c @ ( add_input_a_b_c @ M @ X ) )
      = ( initial_a_b_c @ M ) ) ).

% add_input_simps(1)
thf(fact_320_add__input__simps_I1_J,axiom,
    ! [M: fsm_set_a_b_c,X: b] :
      ( ( initial_set_a_b_c @ ( add_input_set_a_b_c @ M @ X ) )
      = ( initial_set_a_b_c @ M ) ) ).

% add_input_simps(1)
thf(fact_321_add__output__simps_I4_J,axiom,
    ! [M: fsm_set_nat_b_c,Y: c] :
      ( ( transi1321261067114444082at_b_c @ ( add_ou8618801446620803286at_b_c @ M @ Y ) )
      = ( transi1321261067114444082at_b_c @ M ) ) ).

% add_output_simps(4)
thf(fact_322_add__output__simps_I4_J,axiom,
    ! [M: fsm_set_a_b_c,Y: c] :
      ( ( transi8205445899535437940_a_b_c @ ( add_output_set_a_b_c @ M @ Y ) )
      = ( transi8205445899535437940_a_b_c @ M ) ) ).

% add_output_simps(4)
thf(fact_323_add__output__simps_I4_J,axiom,
    ! [M: fsm_nat_nat_nat,Y: nat] :
      ( ( transi2178791137084441437at_nat @ ( add_ou6503059941141532929at_nat @ M @ Y ) )
      = ( transi2178791137084441437at_nat @ M ) ) ).

% add_output_simps(4)
thf(fact_324_add__output__simps_I4_J,axiom,
    ! [M: fsm_nat_b_c,Y: c] :
      ( ( transitions_nat_b_c @ ( add_output_nat_b_c @ M @ Y ) )
      = ( transitions_nat_b_c @ M ) ) ).

% add_output_simps(4)
thf(fact_325_add__output__simps_I4_J,axiom,
    ! [M: fsm_a_b_c,Y: c] :
      ( ( transitions_a_b_c @ ( add_output_a_b_c @ M @ Y ) )
      = ( transitions_a_b_c @ M ) ) ).

% add_output_simps(4)
thf(fact_326_add__state__simps_I4_J,axiom,
    ! [M: fsm_set_nat_b_c,Q: set_nat] :
      ( ( transi1321261067114444082at_b_c @ ( add_st8360318785478835724at_b_c @ M @ Q ) )
      = ( transi1321261067114444082at_b_c @ M ) ) ).

% add_state_simps(4)
thf(fact_327_add__state__simps_I4_J,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a] :
      ( ( transi8205445899535437940_a_b_c @ ( add_state_set_a_b_c @ M @ Q ) )
      = ( transi8205445899535437940_a_b_c @ M ) ) ).

% add_state_simps(4)
thf(fact_328_add__state__simps_I4_J,axiom,
    ! [M: fsm_nat_nat_nat,Q: nat] :
      ( ( transi2178791137084441437at_nat @ ( add_st626164701413644599at_nat @ M @ Q ) )
      = ( transi2178791137084441437at_nat @ M ) ) ).

% add_state_simps(4)
thf(fact_329_add__state__simps_I4_J,axiom,
    ! [M: fsm_nat_b_c,Q: nat] :
      ( ( transitions_nat_b_c @ ( add_state_nat_b_c @ M @ Q ) )
      = ( transitions_nat_b_c @ M ) ) ).

% add_state_simps(4)
thf(fact_330_add__state__simps_I4_J,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( transitions_a_b_c @ ( add_state_a_b_c @ M @ Q ) )
      = ( transitions_a_b_c @ M ) ) ).

% add_state_simps(4)
thf(fact_331_add__input__simps_I4_J,axiom,
    ! [M: fsm_set_nat_b_c,X: b] :
      ( ( transi1321261067114444082at_b_c @ ( add_in7503527852466619365at_b_c @ M @ X ) )
      = ( transi1321261067114444082at_b_c @ M ) ) ).

% add_input_simps(4)
thf(fact_332_add__input__simps_I4_J,axiom,
    ! [M: fsm_set_a_b_c,X: b] :
      ( ( transi8205445899535437940_a_b_c @ ( add_input_set_a_b_c @ M @ X ) )
      = ( transi8205445899535437940_a_b_c @ M ) ) ).

% add_input_simps(4)
thf(fact_333_add__input__simps_I4_J,axiom,
    ! [M: fsm_nat_nat_nat,X: nat] :
      ( ( transi2178791137084441437at_nat @ ( add_in6123057044259584912at_nat @ M @ X ) )
      = ( transi2178791137084441437at_nat @ M ) ) ).

% add_input_simps(4)
thf(fact_334_add__input__simps_I4_J,axiom,
    ! [M: fsm_nat_b_c,X: b] :
      ( ( transitions_nat_b_c @ ( add_input_nat_b_c @ M @ X ) )
      = ( transitions_nat_b_c @ M ) ) ).

% add_input_simps(4)
thf(fact_335_add__input__simps_I4_J,axiom,
    ! [M: fsm_a_b_c,X: b] :
      ( ( transitions_a_b_c @ ( add_input_a_b_c @ M @ X ) )
      = ( transitions_a_b_c @ M ) ) ).

% add_input_simps(4)
thf(fact_336_reachable__states__initial,axiom,
    ! [M: fsm_set_nat_b_c] : ( member_set_nat @ ( initial_set_nat_b_c @ M ) @ ( reacha6474151152035807107at_b_c @ M ) ) ).

% reachable_states_initial
thf(fact_337_reachable__states__initial,axiom,
    ! [M: fsm_nat_nat_nat] : ( member_nat @ ( initial_nat_nat_nat @ M ) @ ( reacha3101573547803616814at_nat @ M ) ) ).

% reachable_states_initial
thf(fact_338_reachable__states__initial,axiom,
    ! [M: fsm_set_a_b_c] : ( member_set_a @ ( initial_set_a_b_c @ M ) @ ( reacha7078015818774689891_a_b_c @ M ) ) ).

% reachable_states_initial
thf(fact_339_reachable__states__initial,axiom,
    ! [M: fsm_a_b_c] : ( member_a @ ( initial_a_b_c @ M ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ).

% reachable_states_initial
thf(fact_340_reachable__states__initial,axiom,
    ! [M: fsm_nat_b_c] : ( member_nat @ ( initial_nat_b_c @ M ) @ ( reacha6243244737040045773at_b_c @ M ) ) ).

% reachable_states_initial
thf(fact_341_reachable__states__initial,axiom,
    ! [M: fsm_Code_integer_b_c] : ( member_Code_integer @ ( initia4873860033875475119er_b_c @ M ) @ ( reacha2636592895321446618er_b_c @ M ) ) ).

% reachable_states_initial
thf(fact_342_submachine__simps_I5_J,axiom,
    ! [A: fsm_set_nat_b_c,B: fsm_set_nat_b_c] :
      ( ( is_sub96770212967015341at_b_c @ A @ B )
     => ( ord_le724355047373210186et_nat @ ( transi1321261067114444082at_b_c @ A ) @ ( transi1321261067114444082at_b_c @ B ) ) ) ).

% submachine_simps(5)
thf(fact_343_submachine__simps_I5_J,axiom,
    ! [A: fsm_nat_nat_nat,B: fsm_nat_nat_nat] :
      ( ( is_sub7029232139394105176at_nat @ A @ B )
     => ( ord_le3588339395105511127at_nat @ ( transi2178791137084441437at_nat @ A ) @ ( transi2178791137084441437at_nat @ B ) ) ) ).

% submachine_simps(5)
thf(fact_344_submachine__simps_I5_J,axiom,
    ! [A: fsm_Code_integer_b_c,B: fsm_Code_integer_b_c] :
      ( ( is_sub5482583993107430660er_b_c @ A @ B )
     => ( ord_le690246243407062136nteger @ ( transi6707074847254859401er_b_c @ A ) @ ( transi6707074847254859401er_b_c @ B ) ) ) ).

% submachine_simps(5)
thf(fact_345_submachine__simps_I5_J,axiom,
    ! [A: fsm_nat_b_c,B: fsm_nat_b_c] :
      ( ( is_sub6065310311083418359at_b_c @ A @ B )
     => ( ord_le6809018855743278558_c_nat @ ( transitions_nat_b_c @ A ) @ ( transitions_nat_b_c @ B ) ) ) ).

% submachine_simps(5)
thf(fact_346_submachine__simps_I5_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ A ) @ ( transitions_a_b_c @ B ) ) ) ).

% submachine_simps(5)
thf(fact_347_submachine__simps_I5_J,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ord_le4557092201414138422_set_a @ ( transi8205445899535437940_a_b_c @ A ) @ ( transi8205445899535437940_a_b_c @ B ) ) ) ).

% submachine_simps(5)
thf(fact_348_from__FSM__reachable__states,axiom,
    ! [Q: code_integer,M: fsm_Code_integer_b_c] :
      ( ( member_Code_integer @ Q @ ( reacha2636592895321446618er_b_c @ M ) )
     => ( ord_le7084787975880047091nteger @ ( reacha2636592895321446618er_b_c @ ( from_F2639493813852433665er_b_c @ M @ Q ) ) @ ( reacha2636592895321446618er_b_c @ M ) ) ) ).

% from_FSM_reachable_states
thf(fact_349_from__FSM__reachable__states,axiom,
    ! [Q: nat,M: fsm_nat_b_c] :
      ( ( member_nat @ Q @ ( reacha6243244737040045773at_b_c @ M ) )
     => ( ord_less_eq_set_nat @ ( reacha6243244737040045773at_b_c @ ( from_FSM_nat_b_c @ M @ Q ) ) @ ( reacha6243244737040045773at_b_c @ M ) ) ) ).

% from_FSM_reachable_states
thf(fact_350_from__FSM__reachable__states,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) )
     => ( ord_less_eq_set_a @ ( reacha1620305530751930115_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ) ).

% from_FSM_reachable_states
thf(fact_351_from__FSM__reachable__states,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ Q @ ( reacha7078015818774689891_a_b_c @ M ) )
     => ( ord_le3724670747650509150_set_a @ ( reacha7078015818774689891_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) ) @ ( reacha7078015818774689891_a_b_c @ M ) ) ) ).

% from_FSM_reachable_states
thf(fact_352_submachine__reachable__subset,axiom,
    ! [A: fsm_Code_integer_b_c,B: fsm_Code_integer_b_c] :
      ( ( is_sub5482583993107430660er_b_c @ A @ B )
     => ( ord_le7084787975880047091nteger @ ( reacha2636592895321446618er_b_c @ A ) @ ( reacha2636592895321446618er_b_c @ B ) ) ) ).

% submachine_reachable_subset
thf(fact_353_submachine__reachable__subset,axiom,
    ! [A: fsm_nat_b_c,B: fsm_nat_b_c] :
      ( ( is_sub6065310311083418359at_b_c @ A @ B )
     => ( ord_less_eq_set_nat @ ( reacha6243244737040045773at_b_c @ A ) @ ( reacha6243244737040045773at_b_c @ B ) ) ) ).

% submachine_reachable_subset
thf(fact_354_submachine__reachable__subset,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ord_less_eq_set_a @ ( reacha1620305530751930115_a_b_c @ A ) @ ( reacha1620305530751930115_a_b_c @ B ) ) ) ).

% submachine_reachable_subset
thf(fact_355_submachine__reachable__subset,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ord_le3724670747650509150_set_a @ ( reacha7078015818774689891_a_b_c @ A ) @ ( reacha7078015818774689891_a_b_c @ B ) ) ) ).

% submachine_reachable_subset
thf(fact_356_is__submachine_Oelims_I3_J,axiom,
    ! [X: fsm_Su3173892059090327343_a_b_c,Xa: fsm_Su3173892059090327343_a_b_c] :
      ( ~ ( is_sub8484252162375176803_a_b_c @ X @ Xa )
     => ~ ( ( ( initia839594848065427214_a_b_c @ X )
            = ( initia839594848065427214_a_b_c @ Xa ) )
          & ( ord_le8158965334049305270_a_a_a @ ( transi749694810082594024_a_b_c @ X ) @ ( transi749694810082594024_a_b_c @ Xa ) )
          & ( ( inputs3694848887974842907_a_b_c @ X )
            = ( inputs3694848887974842907_a_b_c @ Xa ) )
          & ( ( output2259889051082456732_a_b_c @ X )
            = ( output2259889051082456732_a_b_c @ Xa ) )
          & ( ord_le4476889455585342546_a_a_a @ ( states3035385874452377268_a_b_c @ X ) @ ( states3035385874452377268_a_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(3)
thf(fact_357_is__submachine_Oelims_I3_J,axiom,
    ! [X: fsm_se7249436556146598534er_b_c,Xa: fsm_se7249436556146598534er_b_c] :
      ( ~ ( is_sub1762352666996382522er_b_c @ X @ Xa )
     => ~ ( ( ( initia8790449882354265317er_b_c @ X )
            = ( initia8790449882354265317er_b_c @ Xa ) )
          & ( ord_le3280107110435850212nteger @ ( transi4510574700103178431er_b_c @ X ) @ ( transi4510574700103178431er_b_c @ Xa ) )
          & ( ( inputs1631762778656711410er_b_c @ X )
            = ( inputs1631762778656711410er_b_c @ Xa ) )
          & ( ( output3616175663950214771er_b_c @ X )
            = ( output3616175663950214771er_b_c @ Xa ) )
          & ( ord_le1914454125413604393nteger @ ( states5991589681222143627er_b_c @ X ) @ ( states5991589681222143627er_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(3)
thf(fact_358_is__submachine_Oelims_I3_J,axiom,
    ! [X: fsm_Code_integer_b_c,Xa: fsm_Code_integer_b_c] :
      ( ~ ( is_sub5482583993107430660er_b_c @ X @ Xa )
     => ~ ( ( ( initia4873860033875475119er_b_c @ X )
            = ( initia4873860033875475119er_b_c @ Xa ) )
          & ( ord_le690246243407062136nteger @ ( transi6707074847254859401er_b_c @ X ) @ ( transi6707074847254859401er_b_c @ Xa ) )
          & ( ( inputs4521787250163688636er_b_c @ X )
            = ( inputs4521787250163688636er_b_c @ Xa ) )
          & ( ( output924834633477831741er_b_c @ X )
            = ( output924834633477831741er_b_c @ Xa ) )
          & ( ord_le7084787975880047091nteger @ ( states1921009325540848725er_b_c @ X ) @ ( states1921009325540848725er_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(3)
thf(fact_359_is__submachine_Oelims_I3_J,axiom,
    ! [X: fsm_set_nat_b_c,Xa: fsm_set_nat_b_c] :
      ( ~ ( is_sub96770212967015341at_b_c @ X @ Xa )
     => ~ ( ( ( initial_set_nat_b_c @ X )
            = ( initial_set_nat_b_c @ Xa ) )
          & ( ord_le724355047373210186et_nat @ ( transi1321261067114444082at_b_c @ X ) @ ( transi1321261067114444082at_b_c @ Xa ) )
          & ( ( inputs_set_nat_b_c @ X )
            = ( inputs_set_nat_b_c @ Xa ) )
          & ( ( outputs_set_nat_b_c @ X )
            = ( outputs_set_nat_b_c @ Xa ) )
          & ( ord_le6893508408891458716et_nat @ ( states_set_nat_b_c @ X ) @ ( states_set_nat_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(3)
thf(fact_360_is__submachine_Oelims_I3_J,axiom,
    ! [X: fsm_nat_nat_nat,Xa: fsm_nat_nat_nat] :
      ( ~ ( is_sub7029232139394105176at_nat @ X @ Xa )
     => ~ ( ( ( initial_nat_nat_nat @ X )
            = ( initial_nat_nat_nat @ Xa ) )
          & ( ord_le3588339395105511127at_nat @ ( transi2178791137084441437at_nat @ X ) @ ( transi2178791137084441437at_nat @ Xa ) )
          & ( ( inputs_nat_nat_nat @ X )
            = ( inputs_nat_nat_nat @ Xa ) )
          & ( ( outputs_nat_nat_nat @ X )
            = ( outputs_nat_nat_nat @ Xa ) )
          & ( ord_less_eq_set_nat @ ( states_nat_nat_nat @ X ) @ ( states_nat_nat_nat @ Xa ) ) ) ) ).

% is_submachine.elims(3)
thf(fact_361_is__submachine_Oelims_I3_J,axiom,
    ! [X: fsm_nat_b_c,Xa: fsm_nat_b_c] :
      ( ~ ( is_sub6065310311083418359at_b_c @ X @ Xa )
     => ~ ( ( ( initial_nat_b_c @ X )
            = ( initial_nat_b_c @ Xa ) )
          & ( ord_le6809018855743278558_c_nat @ ( transitions_nat_b_c @ X ) @ ( transitions_nat_b_c @ Xa ) )
          & ( ( inputs_nat_b_c @ X )
            = ( inputs_nat_b_c @ Xa ) )
          & ( ( outputs_nat_b_c @ X )
            = ( outputs_nat_b_c @ Xa ) )
          & ( ord_less_eq_set_nat @ ( states_nat_b_c @ X ) @ ( states_nat_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(3)
thf(fact_362_is__submachine_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c,Xa: fsm_a_b_c] :
      ( ~ ( is_submachine_a_b_c @ X @ Xa )
     => ~ ( ( ( initial_a_b_c @ X )
            = ( initial_a_b_c @ Xa ) )
          & ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ X ) @ ( transitions_a_b_c @ Xa ) )
          & ( ( inputs_a_b_c @ X )
            = ( inputs_a_b_c @ Xa ) )
          & ( ( outputs_a_b_c @ X )
            = ( outputs_a_b_c @ Xa ) )
          & ( ord_less_eq_set_a @ ( states_a_b_c @ X ) @ ( states_a_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(3)
thf(fact_363_is__submachine_Oelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: fsm_set_a_b_c] :
      ( ~ ( is_sub1377638108373535417_a_b_c @ X @ Xa )
     => ~ ( ( ( initial_set_a_b_c @ X )
            = ( initial_set_a_b_c @ Xa ) )
          & ( ord_le4557092201414138422_set_a @ ( transi8205445899535437940_a_b_c @ X ) @ ( transi8205445899535437940_a_b_c @ Xa ) )
          & ( ( inputs_set_a_b_c @ X )
            = ( inputs_set_a_b_c @ Xa ) )
          & ( ( outputs_set_a_b_c @ X )
            = ( outputs_set_a_b_c @ Xa ) )
          & ( ord_le3724670747650509150_set_a @ ( states_set_a_b_c @ X ) @ ( states_set_a_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(3)
thf(fact_364_is__submachine_Oelims_I2_J,axiom,
    ! [X: fsm_Su3173892059090327343_a_b_c,Xa: fsm_Su3173892059090327343_a_b_c] :
      ( ( is_sub8484252162375176803_a_b_c @ X @ Xa )
     => ( ( ( initia839594848065427214_a_b_c @ X )
          = ( initia839594848065427214_a_b_c @ Xa ) )
        & ( ord_le8158965334049305270_a_a_a @ ( transi749694810082594024_a_b_c @ X ) @ ( transi749694810082594024_a_b_c @ Xa ) )
        & ( ( inputs3694848887974842907_a_b_c @ X )
          = ( inputs3694848887974842907_a_b_c @ Xa ) )
        & ( ( output2259889051082456732_a_b_c @ X )
          = ( output2259889051082456732_a_b_c @ Xa ) )
        & ( ord_le4476889455585342546_a_a_a @ ( states3035385874452377268_a_b_c @ X ) @ ( states3035385874452377268_a_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(2)
thf(fact_365_is__submachine_Oelims_I2_J,axiom,
    ! [X: fsm_se7249436556146598534er_b_c,Xa: fsm_se7249436556146598534er_b_c] :
      ( ( is_sub1762352666996382522er_b_c @ X @ Xa )
     => ( ( ( initia8790449882354265317er_b_c @ X )
          = ( initia8790449882354265317er_b_c @ Xa ) )
        & ( ord_le3280107110435850212nteger @ ( transi4510574700103178431er_b_c @ X ) @ ( transi4510574700103178431er_b_c @ Xa ) )
        & ( ( inputs1631762778656711410er_b_c @ X )
          = ( inputs1631762778656711410er_b_c @ Xa ) )
        & ( ( output3616175663950214771er_b_c @ X )
          = ( output3616175663950214771er_b_c @ Xa ) )
        & ( ord_le1914454125413604393nteger @ ( states5991589681222143627er_b_c @ X ) @ ( states5991589681222143627er_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(2)
thf(fact_366_is__submachine_Oelims_I2_J,axiom,
    ! [X: fsm_Code_integer_b_c,Xa: fsm_Code_integer_b_c] :
      ( ( is_sub5482583993107430660er_b_c @ X @ Xa )
     => ( ( ( initia4873860033875475119er_b_c @ X )
          = ( initia4873860033875475119er_b_c @ Xa ) )
        & ( ord_le690246243407062136nteger @ ( transi6707074847254859401er_b_c @ X ) @ ( transi6707074847254859401er_b_c @ Xa ) )
        & ( ( inputs4521787250163688636er_b_c @ X )
          = ( inputs4521787250163688636er_b_c @ Xa ) )
        & ( ( output924834633477831741er_b_c @ X )
          = ( output924834633477831741er_b_c @ Xa ) )
        & ( ord_le7084787975880047091nteger @ ( states1921009325540848725er_b_c @ X ) @ ( states1921009325540848725er_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(2)
thf(fact_367_is__submachine_Oelims_I2_J,axiom,
    ! [X: fsm_set_nat_b_c,Xa: fsm_set_nat_b_c] :
      ( ( is_sub96770212967015341at_b_c @ X @ Xa )
     => ( ( ( initial_set_nat_b_c @ X )
          = ( initial_set_nat_b_c @ Xa ) )
        & ( ord_le724355047373210186et_nat @ ( transi1321261067114444082at_b_c @ X ) @ ( transi1321261067114444082at_b_c @ Xa ) )
        & ( ( inputs_set_nat_b_c @ X )
          = ( inputs_set_nat_b_c @ Xa ) )
        & ( ( outputs_set_nat_b_c @ X )
          = ( outputs_set_nat_b_c @ Xa ) )
        & ( ord_le6893508408891458716et_nat @ ( states_set_nat_b_c @ X ) @ ( states_set_nat_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(2)
thf(fact_368_is__submachine_Oelims_I2_J,axiom,
    ! [X: fsm_nat_nat_nat,Xa: fsm_nat_nat_nat] :
      ( ( is_sub7029232139394105176at_nat @ X @ Xa )
     => ( ( ( initial_nat_nat_nat @ X )
          = ( initial_nat_nat_nat @ Xa ) )
        & ( ord_le3588339395105511127at_nat @ ( transi2178791137084441437at_nat @ X ) @ ( transi2178791137084441437at_nat @ Xa ) )
        & ( ( inputs_nat_nat_nat @ X )
          = ( inputs_nat_nat_nat @ Xa ) )
        & ( ( outputs_nat_nat_nat @ X )
          = ( outputs_nat_nat_nat @ Xa ) )
        & ( ord_less_eq_set_nat @ ( states_nat_nat_nat @ X ) @ ( states_nat_nat_nat @ Xa ) ) ) ) ).

% is_submachine.elims(2)
thf(fact_369_is__submachine_Oelims_I2_J,axiom,
    ! [X: fsm_nat_b_c,Xa: fsm_nat_b_c] :
      ( ( is_sub6065310311083418359at_b_c @ X @ Xa )
     => ( ( ( initial_nat_b_c @ X )
          = ( initial_nat_b_c @ Xa ) )
        & ( ord_le6809018855743278558_c_nat @ ( transitions_nat_b_c @ X ) @ ( transitions_nat_b_c @ Xa ) )
        & ( ( inputs_nat_b_c @ X )
          = ( inputs_nat_b_c @ Xa ) )
        & ( ( outputs_nat_b_c @ X )
          = ( outputs_nat_b_c @ Xa ) )
        & ( ord_less_eq_set_nat @ ( states_nat_b_c @ X ) @ ( states_nat_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(2)
thf(fact_370_is__submachine_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c,Xa: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ X @ Xa )
     => ( ( ( initial_a_b_c @ X )
          = ( initial_a_b_c @ Xa ) )
        & ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ X ) @ ( transitions_a_b_c @ Xa ) )
        & ( ( inputs_a_b_c @ X )
          = ( inputs_a_b_c @ Xa ) )
        & ( ( outputs_a_b_c @ X )
          = ( outputs_a_b_c @ Xa ) )
        & ( ord_less_eq_set_a @ ( states_a_b_c @ X ) @ ( states_a_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(2)
thf(fact_371_is__submachine_Oelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ X @ Xa )
     => ( ( ( initial_set_a_b_c @ X )
          = ( initial_set_a_b_c @ Xa ) )
        & ( ord_le4557092201414138422_set_a @ ( transi8205445899535437940_a_b_c @ X ) @ ( transi8205445899535437940_a_b_c @ Xa ) )
        & ( ( inputs_set_a_b_c @ X )
          = ( inputs_set_a_b_c @ Xa ) )
        & ( ( outputs_set_a_b_c @ X )
          = ( outputs_set_a_b_c @ Xa ) )
        & ( ord_le3724670747650509150_set_a @ ( states_set_a_b_c @ X ) @ ( states_set_a_b_c @ Xa ) ) ) ) ).

% is_submachine.elims(2)
thf(fact_372_is__submachine_Oelims_I1_J,axiom,
    ! [X: fsm_Su3173892059090327343_a_b_c,Xa: fsm_Su3173892059090327343_a_b_c,Y: $o] :
      ( ( ( is_sub8484252162375176803_a_b_c @ X @ Xa )
        = Y )
     => ( Y
        = ( ( ( initia839594848065427214_a_b_c @ X )
            = ( initia839594848065427214_a_b_c @ Xa ) )
          & ( ord_le8158965334049305270_a_a_a @ ( transi749694810082594024_a_b_c @ X ) @ ( transi749694810082594024_a_b_c @ Xa ) )
          & ( ( inputs3694848887974842907_a_b_c @ X )
            = ( inputs3694848887974842907_a_b_c @ Xa ) )
          & ( ( output2259889051082456732_a_b_c @ X )
            = ( output2259889051082456732_a_b_c @ Xa ) )
          & ( ord_le4476889455585342546_a_a_a @ ( states3035385874452377268_a_b_c @ X ) @ ( states3035385874452377268_a_b_c @ Xa ) ) ) ) ) ).

% is_submachine.elims(1)
thf(fact_373_is__submachine_Oelims_I1_J,axiom,
    ! [X: fsm_se7249436556146598534er_b_c,Xa: fsm_se7249436556146598534er_b_c,Y: $o] :
      ( ( ( is_sub1762352666996382522er_b_c @ X @ Xa )
        = Y )
     => ( Y
        = ( ( ( initia8790449882354265317er_b_c @ X )
            = ( initia8790449882354265317er_b_c @ Xa ) )
          & ( ord_le3280107110435850212nteger @ ( transi4510574700103178431er_b_c @ X ) @ ( transi4510574700103178431er_b_c @ Xa ) )
          & ( ( inputs1631762778656711410er_b_c @ X )
            = ( inputs1631762778656711410er_b_c @ Xa ) )
          & ( ( output3616175663950214771er_b_c @ X )
            = ( output3616175663950214771er_b_c @ Xa ) )
          & ( ord_le1914454125413604393nteger @ ( states5991589681222143627er_b_c @ X ) @ ( states5991589681222143627er_b_c @ Xa ) ) ) ) ) ).

% is_submachine.elims(1)
thf(fact_374_is__submachine_Oelims_I1_J,axiom,
    ! [X: fsm_Code_integer_b_c,Xa: fsm_Code_integer_b_c,Y: $o] :
      ( ( ( is_sub5482583993107430660er_b_c @ X @ Xa )
        = Y )
     => ( Y
        = ( ( ( initia4873860033875475119er_b_c @ X )
            = ( initia4873860033875475119er_b_c @ Xa ) )
          & ( ord_le690246243407062136nteger @ ( transi6707074847254859401er_b_c @ X ) @ ( transi6707074847254859401er_b_c @ Xa ) )
          & ( ( inputs4521787250163688636er_b_c @ X )
            = ( inputs4521787250163688636er_b_c @ Xa ) )
          & ( ( output924834633477831741er_b_c @ X )
            = ( output924834633477831741er_b_c @ Xa ) )
          & ( ord_le7084787975880047091nteger @ ( states1921009325540848725er_b_c @ X ) @ ( states1921009325540848725er_b_c @ Xa ) ) ) ) ) ).

% is_submachine.elims(1)
thf(fact_375_is__submachine_Oelims_I1_J,axiom,
    ! [X: fsm_set_nat_b_c,Xa: fsm_set_nat_b_c,Y: $o] :
      ( ( ( is_sub96770212967015341at_b_c @ X @ Xa )
        = Y )
     => ( Y
        = ( ( ( initial_set_nat_b_c @ X )
            = ( initial_set_nat_b_c @ Xa ) )
          & ( ord_le724355047373210186et_nat @ ( transi1321261067114444082at_b_c @ X ) @ ( transi1321261067114444082at_b_c @ Xa ) )
          & ( ( inputs_set_nat_b_c @ X )
            = ( inputs_set_nat_b_c @ Xa ) )
          & ( ( outputs_set_nat_b_c @ X )
            = ( outputs_set_nat_b_c @ Xa ) )
          & ( ord_le6893508408891458716et_nat @ ( states_set_nat_b_c @ X ) @ ( states_set_nat_b_c @ Xa ) ) ) ) ) ).

% is_submachine.elims(1)
thf(fact_376_is__submachine_Oelims_I1_J,axiom,
    ! [X: fsm_nat_nat_nat,Xa: fsm_nat_nat_nat,Y: $o] :
      ( ( ( is_sub7029232139394105176at_nat @ X @ Xa )
        = Y )
     => ( Y
        = ( ( ( initial_nat_nat_nat @ X )
            = ( initial_nat_nat_nat @ Xa ) )
          & ( ord_le3588339395105511127at_nat @ ( transi2178791137084441437at_nat @ X ) @ ( transi2178791137084441437at_nat @ Xa ) )
          & ( ( inputs_nat_nat_nat @ X )
            = ( inputs_nat_nat_nat @ Xa ) )
          & ( ( outputs_nat_nat_nat @ X )
            = ( outputs_nat_nat_nat @ Xa ) )
          & ( ord_less_eq_set_nat @ ( states_nat_nat_nat @ X ) @ ( states_nat_nat_nat @ Xa ) ) ) ) ) ).

% is_submachine.elims(1)
thf(fact_377_is__submachine_Oelims_I1_J,axiom,
    ! [X: fsm_nat_b_c,Xa: fsm_nat_b_c,Y: $o] :
      ( ( ( is_sub6065310311083418359at_b_c @ X @ Xa )
        = Y )
     => ( Y
        = ( ( ( initial_nat_b_c @ X )
            = ( initial_nat_b_c @ Xa ) )
          & ( ord_le6809018855743278558_c_nat @ ( transitions_nat_b_c @ X ) @ ( transitions_nat_b_c @ Xa ) )
          & ( ( inputs_nat_b_c @ X )
            = ( inputs_nat_b_c @ Xa ) )
          & ( ( outputs_nat_b_c @ X )
            = ( outputs_nat_b_c @ Xa ) )
          & ( ord_less_eq_set_nat @ ( states_nat_b_c @ X ) @ ( states_nat_b_c @ Xa ) ) ) ) ) ).

% is_submachine.elims(1)
thf(fact_378_is__submachine_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Xa: fsm_a_b_c,Y: $o] :
      ( ( ( is_submachine_a_b_c @ X @ Xa )
        = Y )
     => ( Y
        = ( ( ( initial_a_b_c @ X )
            = ( initial_a_b_c @ Xa ) )
          & ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ X ) @ ( transitions_a_b_c @ Xa ) )
          & ( ( inputs_a_b_c @ X )
            = ( inputs_a_b_c @ Xa ) )
          & ( ( outputs_a_b_c @ X )
            = ( outputs_a_b_c @ Xa ) )
          & ( ord_less_eq_set_a @ ( states_a_b_c @ X ) @ ( states_a_b_c @ Xa ) ) ) ) ) ).

% is_submachine.elims(1)
thf(fact_379_is__submachine_Oelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: fsm_set_a_b_c,Y: $o] :
      ( ( ( is_sub1377638108373535417_a_b_c @ X @ Xa )
        = Y )
     => ( Y
        = ( ( ( initial_set_a_b_c @ X )
            = ( initial_set_a_b_c @ Xa ) )
          & ( ord_le4557092201414138422_set_a @ ( transi8205445899535437940_a_b_c @ X ) @ ( transi8205445899535437940_a_b_c @ Xa ) )
          & ( ( inputs_set_a_b_c @ X )
            = ( inputs_set_a_b_c @ Xa ) )
          & ( ( outputs_set_a_b_c @ X )
            = ( outputs_set_a_b_c @ Xa ) )
          & ( ord_le3724670747650509150_set_a @ ( states_set_a_b_c @ X ) @ ( states_set_a_b_c @ Xa ) ) ) ) ) ).

% is_submachine.elims(1)
thf(fact_380_is__submachine_Osimps,axiom,
    ( is_sub8484252162375176803_a_b_c
    = ( ^ [A3: fsm_Su3173892059090327343_a_b_c,B2: fsm_Su3173892059090327343_a_b_c] :
          ( ( ( initia839594848065427214_a_b_c @ A3 )
            = ( initia839594848065427214_a_b_c @ B2 ) )
          & ( ord_le8158965334049305270_a_a_a @ ( transi749694810082594024_a_b_c @ A3 ) @ ( transi749694810082594024_a_b_c @ B2 ) )
          & ( ( inputs3694848887974842907_a_b_c @ A3 )
            = ( inputs3694848887974842907_a_b_c @ B2 ) )
          & ( ( output2259889051082456732_a_b_c @ A3 )
            = ( output2259889051082456732_a_b_c @ B2 ) )
          & ( ord_le4476889455585342546_a_a_a @ ( states3035385874452377268_a_b_c @ A3 ) @ ( states3035385874452377268_a_b_c @ B2 ) ) ) ) ) ).

% is_submachine.simps
thf(fact_381_is__submachine_Osimps,axiom,
    ( is_sub1762352666996382522er_b_c
    = ( ^ [A3: fsm_se7249436556146598534er_b_c,B2: fsm_se7249436556146598534er_b_c] :
          ( ( ( initia8790449882354265317er_b_c @ A3 )
            = ( initia8790449882354265317er_b_c @ B2 ) )
          & ( ord_le3280107110435850212nteger @ ( transi4510574700103178431er_b_c @ A3 ) @ ( transi4510574700103178431er_b_c @ B2 ) )
          & ( ( inputs1631762778656711410er_b_c @ A3 )
            = ( inputs1631762778656711410er_b_c @ B2 ) )
          & ( ( output3616175663950214771er_b_c @ A3 )
            = ( output3616175663950214771er_b_c @ B2 ) )
          & ( ord_le1914454125413604393nteger @ ( states5991589681222143627er_b_c @ A3 ) @ ( states5991589681222143627er_b_c @ B2 ) ) ) ) ) ).

% is_submachine.simps
thf(fact_382_is__submachine_Osimps,axiom,
    ( is_sub5482583993107430660er_b_c
    = ( ^ [A3: fsm_Code_integer_b_c,B2: fsm_Code_integer_b_c] :
          ( ( ( initia4873860033875475119er_b_c @ A3 )
            = ( initia4873860033875475119er_b_c @ B2 ) )
          & ( ord_le690246243407062136nteger @ ( transi6707074847254859401er_b_c @ A3 ) @ ( transi6707074847254859401er_b_c @ B2 ) )
          & ( ( inputs4521787250163688636er_b_c @ A3 )
            = ( inputs4521787250163688636er_b_c @ B2 ) )
          & ( ( output924834633477831741er_b_c @ A3 )
            = ( output924834633477831741er_b_c @ B2 ) )
          & ( ord_le7084787975880047091nteger @ ( states1921009325540848725er_b_c @ A3 ) @ ( states1921009325540848725er_b_c @ B2 ) ) ) ) ) ).

% is_submachine.simps
thf(fact_383_is__submachine_Osimps,axiom,
    ( is_sub96770212967015341at_b_c
    = ( ^ [A3: fsm_set_nat_b_c,B2: fsm_set_nat_b_c] :
          ( ( ( initial_set_nat_b_c @ A3 )
            = ( initial_set_nat_b_c @ B2 ) )
          & ( ord_le724355047373210186et_nat @ ( transi1321261067114444082at_b_c @ A3 ) @ ( transi1321261067114444082at_b_c @ B2 ) )
          & ( ( inputs_set_nat_b_c @ A3 )
            = ( inputs_set_nat_b_c @ B2 ) )
          & ( ( outputs_set_nat_b_c @ A3 )
            = ( outputs_set_nat_b_c @ B2 ) )
          & ( ord_le6893508408891458716et_nat @ ( states_set_nat_b_c @ A3 ) @ ( states_set_nat_b_c @ B2 ) ) ) ) ) ).

% is_submachine.simps
thf(fact_384_is__submachine_Osimps,axiom,
    ( is_sub7029232139394105176at_nat
    = ( ^ [A3: fsm_nat_nat_nat,B2: fsm_nat_nat_nat] :
          ( ( ( initial_nat_nat_nat @ A3 )
            = ( initial_nat_nat_nat @ B2 ) )
          & ( ord_le3588339395105511127at_nat @ ( transi2178791137084441437at_nat @ A3 ) @ ( transi2178791137084441437at_nat @ B2 ) )
          & ( ( inputs_nat_nat_nat @ A3 )
            = ( inputs_nat_nat_nat @ B2 ) )
          & ( ( outputs_nat_nat_nat @ A3 )
            = ( outputs_nat_nat_nat @ B2 ) )
          & ( ord_less_eq_set_nat @ ( states_nat_nat_nat @ A3 ) @ ( states_nat_nat_nat @ B2 ) ) ) ) ) ).

% is_submachine.simps
thf(fact_385_is__submachine_Osimps,axiom,
    ( is_sub6065310311083418359at_b_c
    = ( ^ [A3: fsm_nat_b_c,B2: fsm_nat_b_c] :
          ( ( ( initial_nat_b_c @ A3 )
            = ( initial_nat_b_c @ B2 ) )
          & ( ord_le6809018855743278558_c_nat @ ( transitions_nat_b_c @ A3 ) @ ( transitions_nat_b_c @ B2 ) )
          & ( ( inputs_nat_b_c @ A3 )
            = ( inputs_nat_b_c @ B2 ) )
          & ( ( outputs_nat_b_c @ A3 )
            = ( outputs_nat_b_c @ B2 ) )
          & ( ord_less_eq_set_nat @ ( states_nat_b_c @ A3 ) @ ( states_nat_b_c @ B2 ) ) ) ) ) ).

% is_submachine.simps
thf(fact_386_is__submachine_Osimps,axiom,
    ( is_submachine_a_b_c
    = ( ^ [A3: fsm_a_b_c,B2: fsm_a_b_c] :
          ( ( ( initial_a_b_c @ A3 )
            = ( initial_a_b_c @ B2 ) )
          & ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ A3 ) @ ( transitions_a_b_c @ B2 ) )
          & ( ( inputs_a_b_c @ A3 )
            = ( inputs_a_b_c @ B2 ) )
          & ( ( outputs_a_b_c @ A3 )
            = ( outputs_a_b_c @ B2 ) )
          & ( ord_less_eq_set_a @ ( states_a_b_c @ A3 ) @ ( states_a_b_c @ B2 ) ) ) ) ) ).

% is_submachine.simps
thf(fact_387_is__submachine_Osimps,axiom,
    ( is_sub1377638108373535417_a_b_c
    = ( ^ [A3: fsm_set_a_b_c,B2: fsm_set_a_b_c] :
          ( ( ( initial_set_a_b_c @ A3 )
            = ( initial_set_a_b_c @ B2 ) )
          & ( ord_le4557092201414138422_set_a @ ( transi8205445899535437940_a_b_c @ A3 ) @ ( transi8205445899535437940_a_b_c @ B2 ) )
          & ( ( inputs_set_a_b_c @ A3 )
            = ( inputs_set_a_b_c @ B2 ) )
          & ( ( outputs_set_a_b_c @ A3 )
            = ( outputs_set_a_b_c @ B2 ) )
          & ( ord_le3724670747650509150_set_a @ ( states_set_a_b_c @ A3 ) @ ( states_set_a_b_c @ B2 ) ) ) ) ) ).

% is_submachine.simps
thf(fact_388_reachable__state__is__state,axiom,
    ! [Q: set_nat,M: fsm_set_nat_b_c] :
      ( ( member_set_nat @ Q @ ( reacha6474151152035807107at_b_c @ M ) )
     => ( member_set_nat @ Q @ ( states_set_nat_b_c @ M ) ) ) ).

% reachable_state_is_state
thf(fact_389_reachable__state__is__state,axiom,
    ! [Q: nat,M: fsm_nat_nat_nat] :
      ( ( member_nat @ Q @ ( reacha3101573547803616814at_nat @ M ) )
     => ( member_nat @ Q @ ( states_nat_nat_nat @ M ) ) ) ).

% reachable_state_is_state
thf(fact_390_reachable__state__is__state,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ Q @ ( reacha7078015818774689891_a_b_c @ M ) )
     => ( member_set_a @ Q @ ( states_set_a_b_c @ M ) ) ) ).

% reachable_state_is_state
thf(fact_391_reachable__state__is__state,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) )
     => ( member_a @ Q @ ( states_a_b_c @ M ) ) ) ).

% reachable_state_is_state
thf(fact_392_reachable__state__is__state,axiom,
    ! [Q: nat,M: fsm_nat_b_c] :
      ( ( member_nat @ Q @ ( reacha6243244737040045773at_b_c @ M ) )
     => ( member_nat @ Q @ ( states_nat_b_c @ M ) ) ) ).

% reachable_state_is_state
thf(fact_393_reachable__state__is__state,axiom,
    ! [Q: code_integer,M: fsm_Code_integer_b_c] :
      ( ( member_Code_integer @ Q @ ( reacha2636592895321446618er_b_c @ M ) )
     => ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ M ) ) ) ).

% reachable_state_is_state
thf(fact_394_submachine__simps_I1_J,axiom,
    ! [A: fsm_set_nat_b_c,B: fsm_set_nat_b_c] :
      ( ( is_sub96770212967015341at_b_c @ A @ B )
     => ( ( initial_set_nat_b_c @ A )
        = ( initial_set_nat_b_c @ B ) ) ) ).

% submachine_simps(1)
thf(fact_395_submachine__simps_I1_J,axiom,
    ! [A: fsm_nat_nat_nat,B: fsm_nat_nat_nat] :
      ( ( is_sub7029232139394105176at_nat @ A @ B )
     => ( ( initial_nat_nat_nat @ A )
        = ( initial_nat_nat_nat @ B ) ) ) ).

% submachine_simps(1)
thf(fact_396_submachine__simps_I1_J,axiom,
    ! [A: fsm_Code_integer_b_c,B: fsm_Code_integer_b_c] :
      ( ( is_sub5482583993107430660er_b_c @ A @ B )
     => ( ( initia4873860033875475119er_b_c @ A )
        = ( initia4873860033875475119er_b_c @ B ) ) ) ).

% submachine_simps(1)
thf(fact_397_submachine__simps_I1_J,axiom,
    ! [A: fsm_nat_b_c,B: fsm_nat_b_c] :
      ( ( is_sub6065310311083418359at_b_c @ A @ B )
     => ( ( initial_nat_b_c @ A )
        = ( initial_nat_b_c @ B ) ) ) ).

% submachine_simps(1)
thf(fact_398_submachine__simps_I1_J,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( initial_a_b_c @ A )
        = ( initial_a_b_c @ B ) ) ) ).

% submachine_simps(1)
thf(fact_399_submachine__simps_I1_J,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ( initial_set_a_b_c @ A )
        = ( initial_set_a_b_c @ B ) ) ) ).

% submachine_simps(1)
thf(fact_400_filter__transitions__reachable__states,axiom,
    ! [Q: nat,M: fsm_nat_b_c,P: produc8586872805302537566_c_nat > $o] :
      ( ( member_nat @ Q @ ( reacha6243244737040045773at_b_c @ ( filter4406304812386798564at_b_c @ M @ P ) ) )
     => ( member_nat @ Q @ ( reacha6243244737040045773at_b_c @ M ) ) ) ).

% filter_transitions_reachable_states
thf(fact_401_filter__transitions__reachable__states,axiom,
    ! [Q: code_integer,M: fsm_Code_integer_b_c,P: produc558599814252099832nteger > $o] :
      ( ( member_Code_integer @ Q @ ( reacha2636592895321446618er_b_c @ ( filter2165177713251037681er_b_c @ M @ P ) ) )
     => ( member_Code_integer @ Q @ ( reacha2636592895321446618er_b_c @ M ) ) ) ).

% filter_transitions_reachable_states
thf(fact_402_filter__transitions__reachable__states,axiom,
    ! [Q: a,M: fsm_a_b_c,P: produc2669932201312201782od_c_a > $o] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ ( filter4876964422427095468_a_b_c @ M @ P ) ) )
     => ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) ) ) ).

% filter_transitions_reachable_states
thf(fact_403_filter__transitions__reachable__states,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c,P: produc373761375140688950_set_a > $o] :
      ( ( member_set_a @ Q @ ( reacha7078015818774689891_a_b_c @ ( filter824366976161576716_a_b_c @ M @ P ) ) )
     => ( member_set_a @ Q @ ( reacha7078015818774689891_a_b_c @ M ) ) ) ).

% filter_transitions_reachable_states
thf(fact_404_insert__subset,axiom,
    ! [X: c,A: set_c,B: set_c] :
      ( ( ord_less_eq_set_c @ ( insert_c @ X @ A ) @ B )
      = ( ( member_c @ X @ B )
        & ( ord_less_eq_set_c @ A @ B ) ) ) ).

% insert_subset
thf(fact_405_insert__subset,axiom,
    ! [X: b,A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ ( insert_b @ X @ A ) @ B )
      = ( ( member_b @ X @ B )
        & ( ord_less_eq_set_b @ A @ B ) ) ) ).

% insert_subset
thf(fact_406_insert__subset,axiom,
    ! [X: list_P903359562653991662od_b_c,A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ ( insert6227932334100060350od_b_c @ X @ A ) @ B )
      = ( ( member6330420149250801815od_b_c @ X @ B )
        & ( ord_le282488521294790766od_b_c @ A @ B ) ) ) ).

% insert_subset
thf(fact_407_insert__subset,axiom,
    ! [X: set_nat,A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A ) @ B )
      = ( ( member_set_nat @ X @ B )
        & ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_408_insert__subset,axiom,
    ! [X: set_a,A: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A ) @ B )
      = ( ( member_set_a @ X @ B )
        & ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).

% insert_subset
thf(fact_409_insert__subset,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
      = ( ( member_nat @ X @ B )
        & ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_410_insert__subset,axiom,
    ! [X: a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
      = ( ( member_a @ X @ B )
        & ( ord_less_eq_set_a @ A @ B ) ) ) ).

% insert_subset
thf(fact_411_image__insert,axiom,
    ! [F: nat > nat,A2: nat,B: set_nat] :
      ( ( image_nat_nat @ F @ ( insert_nat @ A2 @ B ) )
      = ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).

% image_insert
thf(fact_412_image__insert,axiom,
    ! [F: nat > c,A2: nat,B: set_nat] :
      ( ( image_nat_c @ F @ ( insert_nat @ A2 @ B ) )
      = ( insert_c @ ( F @ A2 ) @ ( image_nat_c @ F @ B ) ) ) ).

% image_insert
thf(fact_413_image__insert,axiom,
    ! [F: nat > b,A2: nat,B: set_nat] :
      ( ( image_nat_b @ F @ ( insert_nat @ A2 @ B ) )
      = ( insert_b @ ( F @ A2 ) @ ( image_nat_b @ F @ B ) ) ) ).

% image_insert
thf(fact_414_image__insert,axiom,
    ! [F: nat > a,A2: nat,B: set_nat] :
      ( ( image_nat_a @ F @ ( insert_nat @ A2 @ B ) )
      = ( insert_a @ ( F @ A2 ) @ ( image_nat_a @ F @ B ) ) ) ).

% image_insert
thf(fact_415_image__insert,axiom,
    ! [F: c > nat,A2: c,B: set_c] :
      ( ( image_c_nat @ F @ ( insert_c @ A2 @ B ) )
      = ( insert_nat @ ( F @ A2 ) @ ( image_c_nat @ F @ B ) ) ) ).

% image_insert
thf(fact_416_image__insert,axiom,
    ! [F: c > c,A2: c,B: set_c] :
      ( ( image_c_c @ F @ ( insert_c @ A2 @ B ) )
      = ( insert_c @ ( F @ A2 ) @ ( image_c_c @ F @ B ) ) ) ).

% image_insert
thf(fact_417_image__insert,axiom,
    ! [F: c > b,A2: c,B: set_c] :
      ( ( image_c_b @ F @ ( insert_c @ A2 @ B ) )
      = ( insert_b @ ( F @ A2 ) @ ( image_c_b @ F @ B ) ) ) ).

% image_insert
thf(fact_418_image__insert,axiom,
    ! [F: c > a,A2: c,B: set_c] :
      ( ( image_c_a @ F @ ( insert_c @ A2 @ B ) )
      = ( insert_a @ ( F @ A2 ) @ ( image_c_a @ F @ B ) ) ) ).

% image_insert
thf(fact_419_image__insert,axiom,
    ! [F: b > nat,A2: b,B: set_b] :
      ( ( image_b_nat @ F @ ( insert_b @ A2 @ B ) )
      = ( insert_nat @ ( F @ A2 ) @ ( image_b_nat @ F @ B ) ) ) ).

% image_insert
thf(fact_420_image__insert,axiom,
    ! [F: b > c,A2: b,B: set_b] :
      ( ( image_b_c @ F @ ( insert_b @ A2 @ B ) )
      = ( insert_c @ ( F @ A2 ) @ ( image_b_c @ F @ B ) ) ) ).

% image_insert
thf(fact_421_insert__image,axiom,
    ! [X: nat,A: set_nat,F: nat > nat] :
      ( ( member_nat @ X @ A )
     => ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) )
        = ( image_nat_nat @ F @ A ) ) ) ).

% insert_image
thf(fact_422_insert__image,axiom,
    ! [X: nat,A: set_nat,F: nat > c] :
      ( ( member_nat @ X @ A )
     => ( ( insert_c @ ( F @ X ) @ ( image_nat_c @ F @ A ) )
        = ( image_nat_c @ F @ A ) ) ) ).

% insert_image
thf(fact_423_insert__image,axiom,
    ! [X: nat,A: set_nat,F: nat > b] :
      ( ( member_nat @ X @ A )
     => ( ( insert_b @ ( F @ X ) @ ( image_nat_b @ F @ A ) )
        = ( image_nat_b @ F @ A ) ) ) ).

% insert_image
thf(fact_424_insert__image,axiom,
    ! [X: nat,A: set_nat,F: nat > a] :
      ( ( member_nat @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_nat_a @ F @ A ) )
        = ( image_nat_a @ F @ A ) ) ) ).

% insert_image
thf(fact_425_insert__image,axiom,
    ! [X: c,A: set_c,F: c > nat] :
      ( ( member_c @ X @ A )
     => ( ( insert_nat @ ( F @ X ) @ ( image_c_nat @ F @ A ) )
        = ( image_c_nat @ F @ A ) ) ) ).

% insert_image
thf(fact_426_insert__image,axiom,
    ! [X: c,A: set_c,F: c > c] :
      ( ( member_c @ X @ A )
     => ( ( insert_c @ ( F @ X ) @ ( image_c_c @ F @ A ) )
        = ( image_c_c @ F @ A ) ) ) ).

% insert_image
thf(fact_427_insert__image,axiom,
    ! [X: c,A: set_c,F: c > b] :
      ( ( member_c @ X @ A )
     => ( ( insert_b @ ( F @ X ) @ ( image_c_b @ F @ A ) )
        = ( image_c_b @ F @ A ) ) ) ).

% insert_image
thf(fact_428_insert__image,axiom,
    ! [X: c,A: set_c,F: c > a] :
      ( ( member_c @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_c_a @ F @ A ) )
        = ( image_c_a @ F @ A ) ) ) ).

% insert_image
thf(fact_429_insert__image,axiom,
    ! [X: b,A: set_b,F: b > nat] :
      ( ( member_b @ X @ A )
     => ( ( insert_nat @ ( F @ X ) @ ( image_b_nat @ F @ A ) )
        = ( image_b_nat @ F @ A ) ) ) ).

% insert_image
thf(fact_430_insert__image,axiom,
    ! [X: b,A: set_b,F: b > c] :
      ( ( member_b @ X @ A )
     => ( ( insert_c @ ( F @ X ) @ ( image_b_c @ F @ A ) )
        = ( image_b_c @ F @ A ) ) ) ).

% insert_image
thf(fact_431_create__fsm__from__self,axiom,
    ! [M3: fsm_Su3173892059090327343_a_b_c] :
      ( M3
      = ( create2774977162279808585_a_b_c @ ( initia839594848065427214_a_b_c @ M3 ) @ ( states3035385874452377268_a_b_c @ M3 ) @ ( inputs3694848887974842907_a_b_c @ M3 ) @ ( output2259889051082456732_a_b_c @ M3 ) @ ( transi749694810082594024_a_b_c @ M3 ) ) ) ).

% create_fsm_from_self
thf(fact_432_create__fsm__from__self,axiom,
    ! [M3: fsm_se7249436556146598534er_b_c] :
      ( M3
      = ( create1325583021688733984er_b_c @ ( initia8790449882354265317er_b_c @ M3 ) @ ( states5991589681222143627er_b_c @ M3 ) @ ( inputs1631762778656711410er_b_c @ M3 ) @ ( output3616175663950214771er_b_c @ M3 ) @ ( transi4510574700103178431er_b_c @ M3 ) ) ) ).

% create_fsm_from_self
thf(fact_433_create__fsm__from__self,axiom,
    ! [M3: fsm_set_nat_b_c] :
      ( M3
      = ( create4007149607805723027at_b_c @ ( initial_set_nat_b_c @ M3 ) @ ( states_set_nat_b_c @ M3 ) @ ( inputs_set_nat_b_c @ M3 ) @ ( outputs_set_nat_b_c @ M3 ) @ ( transi1321261067114444082at_b_c @ M3 ) ) ) ).

% create_fsm_from_self
thf(fact_434_create__fsm__from__self,axiom,
    ! [M3: fsm_nat_b_c] :
      ( M3
      = ( create3407544177645123805at_b_c @ ( initial_nat_b_c @ M3 ) @ ( states_nat_b_c @ M3 ) @ ( inputs_nat_b_c @ M3 ) @ ( outputs_nat_b_c @ M3 ) @ ( transitions_nat_b_c @ M3 ) ) ) ).

% create_fsm_from_self
thf(fact_435_create__fsm__from__self,axiom,
    ! [M3: fsm_nat_nat_nat] :
      ( M3
      = ( create4800940341628538942at_nat @ ( initial_nat_nat_nat @ M3 ) @ ( states_nat_nat_nat @ M3 ) @ ( inputs_nat_nat_nat @ M3 ) @ ( outputs_nat_nat_nat @ M3 ) @ ( transi2178791137084441437at_nat @ M3 ) ) ) ).

% create_fsm_from_self
thf(fact_436_create__fsm__from__self,axiom,
    ! [M3: fsm_set_a_b_c] :
      ( M3
      = ( create7113092633863273043_a_b_c @ ( initial_set_a_b_c @ M3 ) @ ( states_set_a_b_c @ M3 ) @ ( inputs_set_a_b_c @ M3 ) @ ( outputs_set_a_b_c @ M3 ) @ ( transi8205445899535437940_a_b_c @ M3 ) ) ) ).

% create_fsm_from_self
thf(fact_437_create__fsm__from__self,axiom,
    ! [M3: fsm_a_b_c] :
      ( M3
      = ( create3933655864275538163_a_b_c @ ( initial_a_b_c @ M3 ) @ ( states_a_b_c @ M3 ) @ ( inputs_a_b_c @ M3 ) @ ( outputs_a_b_c @ M3 ) @ ( transitions_a_b_c @ M3 ) ) ) ).

% create_fsm_from_self
thf(fact_438_insertCI,axiom,
    ! [A2: set_a,B: set_set_a,B3: set_a] :
      ( ( ~ ( member_set_a @ A2 @ B )
       => ( A2 = B3 ) )
     => ( member_set_a @ A2 @ ( insert_set_a @ B3 @ B ) ) ) ).

% insertCI
thf(fact_439_insertCI,axiom,
    ! [A2: nat,B: set_nat,B3: nat] :
      ( ( ~ ( member_nat @ A2 @ B )
       => ( A2 = B3 ) )
     => ( member_nat @ A2 @ ( insert_nat @ B3 @ B ) ) ) ).

% insertCI
thf(fact_440_insertCI,axiom,
    ! [A2: c,B: set_c,B3: c] :
      ( ( ~ ( member_c @ A2 @ B )
       => ( A2 = B3 ) )
     => ( member_c @ A2 @ ( insert_c @ B3 @ B ) ) ) ).

% insertCI
thf(fact_441_insertCI,axiom,
    ! [A2: b,B: set_b,B3: b] :
      ( ( ~ ( member_b @ A2 @ B )
       => ( A2 = B3 ) )
     => ( member_b @ A2 @ ( insert_b @ B3 @ B ) ) ) ).

% insertCI
thf(fact_442_insertCI,axiom,
    ! [A2: a,B: set_a,B3: a] :
      ( ( ~ ( member_a @ A2 @ B )
       => ( A2 = B3 ) )
     => ( member_a @ A2 @ ( insert_a @ B3 @ B ) ) ) ).

% insertCI
thf(fact_443_insert__iff,axiom,
    ! [A2: set_a,B3: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ ( insert_set_a @ B3 @ A ) )
      = ( ( A2 = B3 )
        | ( member_set_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_444_insert__iff,axiom,
    ! [A2: nat,B3: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A ) )
      = ( ( A2 = B3 )
        | ( member_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_445_insert__iff,axiom,
    ! [A2: c,B3: c,A: set_c] :
      ( ( member_c @ A2 @ ( insert_c @ B3 @ A ) )
      = ( ( A2 = B3 )
        | ( member_c @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_446_insert__iff,axiom,
    ! [A2: b,B3: b,A: set_b] :
      ( ( member_b @ A2 @ ( insert_b @ B3 @ A ) )
      = ( ( A2 = B3 )
        | ( member_b @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_447_insert__iff,axiom,
    ! [A2: a,B3: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a @ B3 @ A ) )
      = ( ( A2 = B3 )
        | ( member_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_448_insert__absorb2,axiom,
    ! [X: set_a,A: set_set_a] :
      ( ( insert_set_a @ X @ ( insert_set_a @ X @ A ) )
      = ( insert_set_a @ X @ A ) ) ).

% insert_absorb2
thf(fact_449_insert__absorb2,axiom,
    ! [X: nat,A: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
      = ( insert_nat @ X @ A ) ) ).

% insert_absorb2
thf(fact_450_insert__absorb2,axiom,
    ! [X: c,A: set_c] :
      ( ( insert_c @ X @ ( insert_c @ X @ A ) )
      = ( insert_c @ X @ A ) ) ).

% insert_absorb2
thf(fact_451_insert__absorb2,axiom,
    ! [X: b,A: set_b] :
      ( ( insert_b @ X @ ( insert_b @ X @ A ) )
      = ( insert_b @ X @ A ) ) ).

% insert_absorb2
thf(fact_452_insert__absorb2,axiom,
    ! [X: a,A: set_a] :
      ( ( insert_a @ X @ ( insert_a @ X @ A ) )
      = ( insert_a @ X @ A ) ) ).

% insert_absorb2
thf(fact_453_subsetI,axiom,
    ! [A: set_c,B: set_c] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_c @ X3 @ B ) )
     => ( ord_less_eq_set_c @ A @ B ) ) ).

% subsetI
thf(fact_454_subsetI,axiom,
    ! [A: set_b,B: set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_b @ X3 @ B ) )
     => ( ord_less_eq_set_b @ A @ B ) ) ).

% subsetI
thf(fact_455_subsetI,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ! [X3: list_P903359562653991662od_b_c] :
          ( ( member6330420149250801815od_b_c @ X3 @ A )
         => ( member6330420149250801815od_b_c @ X3 @ B ) )
     => ( ord_le282488521294790766od_b_c @ A @ B ) ) ).

% subsetI
thf(fact_456_subsetI,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ( member_set_nat @ X3 @ B ) )
     => ( ord_le6893508408891458716et_nat @ A @ B ) ) ).

% subsetI
thf(fact_457_subsetI,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ! [X3: set_a] :
          ( ( member_set_a @ X3 @ A )
         => ( member_set_a @ X3 @ B ) )
     => ( ord_le3724670747650509150_set_a @ A @ B ) ) ).

% subsetI
thf(fact_458_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ X3 @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% subsetI
thf(fact_459_subsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a @ X3 @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% subsetI
thf(fact_460_subset__antisym,axiom,
    ! [A: set_li6436108459499378894od_b_c,B: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A @ B )
     => ( ( ord_le282488521294790766od_b_c @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_461_subset__antisym,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( ord_le6893508408891458716et_nat @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_462_subset__antisym,axiom,
    ! [A: set_set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_463_subset__antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_464_subset__antisym,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_465_image__eqI,axiom,
    ! [B3: nat,F: nat > nat,X: nat,A: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_466_image__eqI,axiom,
    ! [B3: c,F: nat > c,X: nat,A: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A )
       => ( member_c @ B3 @ ( image_nat_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_467_image__eqI,axiom,
    ! [B3: b,F: nat > b,X: nat,A: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A )
       => ( member_b @ B3 @ ( image_nat_b @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_468_image__eqI,axiom,
    ! [B3: a,F: nat > a,X: nat,A: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A )
       => ( member_a @ B3 @ ( image_nat_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_469_image__eqI,axiom,
    ! [B3: nat,F: c > nat,X: c,A: set_c] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_c @ X @ A )
       => ( member_nat @ B3 @ ( image_c_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_470_image__eqI,axiom,
    ! [B3: c,F: c > c,X: c,A: set_c] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_c @ X @ A )
       => ( member_c @ B3 @ ( image_c_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_471_image__eqI,axiom,
    ! [B3: b,F: c > b,X: c,A: set_c] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_c @ X @ A )
       => ( member_b @ B3 @ ( image_c_b @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_472_image__eqI,axiom,
    ! [B3: a,F: c > a,X: c,A: set_c] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_c @ X @ A )
       => ( member_a @ B3 @ ( image_c_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_473_image__eqI,axiom,
    ! [B3: nat,F: b > nat,X: b,A: set_b] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_b @ X @ A )
       => ( member_nat @ B3 @ ( image_b_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_474_image__eqI,axiom,
    ! [B3: c,F: b > c,X: b,A: set_b] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_b @ X @ A )
       => ( member_c @ B3 @ ( image_b_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_475_order__refl,axiom,
    ! [X: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ X @ X ) ).

% order_refl
thf(fact_476_order__refl,axiom,
    ! [X: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ X ) ).

% order_refl
thf(fact_477_order__refl,axiom,
    ! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).

% order_refl
thf(fact_478_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_479_order__refl,axiom,
    ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).

% order_refl
thf(fact_480_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_481_dual__order_Orefl,axiom,
    ! [A2: set_li6436108459499378894od_b_c] : ( ord_le282488521294790766od_b_c @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_482_dual__order_Orefl,axiom,
    ! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_483_dual__order_Orefl,axiom,
    ! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_484_dual__order_Orefl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_485_dual__order_Orefl,axiom,
    ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_486_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_487_order__antisym__conv,axiom,
    ! [Y: set_li6436108459499378894od_b_c,X: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ Y @ X )
     => ( ( ord_le282488521294790766od_b_c @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_488_order__antisym__conv,axiom,
    ! [Y: set_set_nat,X: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ Y @ X )
     => ( ( ord_le6893508408891458716et_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_489_order__antisym__conv,axiom,
    ! [Y: set_set_a,X: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ Y @ X )
     => ( ( ord_le3724670747650509150_set_a @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_490_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_491_order__antisym__conv,axiom,
    ! [Y: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X )
     => ( ( ord_less_eq_set_a @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_492_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_493_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_494_ord__le__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_495_ord__le__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_496_ord__le__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_497_ord__le__eq__subst,axiom,
    ! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_498_ord__le__eq__subst,axiom,
    ! [A2: set_a,B3: set_a,F: set_a > nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_499_ord__le__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: nat > set_set_nat,C: set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_500_ord__le__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: nat > set_set_a,C: set_set_a] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_501_ord__le__eq__subst,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,F: set_set_nat > nat,C: nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: set_set_nat,Y2: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_502_ord__le__eq__subst,axiom,
    ! [A2: set_set_a,B3: set_set_a,F: set_set_a > nat,C: nat] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_503_ord__le__eq__subst,axiom,
    ! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_504_ord__eq__le__subst,axiom,
    ! [A2: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_505_ord__eq__le__subst,axiom,
    ! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_506_ord__eq__le__subst,axiom,
    ! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_507_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_nat @ B3 @ C )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_508_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_a > nat,B3: set_a,C: set_a] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X3: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_509_ord__eq__le__subst,axiom,
    ! [A2: set_set_nat,F: nat > set_set_nat,B3: nat,C: nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_510_ord__eq__le__subst,axiom,
    ! [A2: set_set_a,F: nat > set_set_a,B3: nat,C: nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_511_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_set_nat > nat,B3: set_set_nat,C: set_set_nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_le6893508408891458716et_nat @ B3 @ C )
       => ( ! [X3: set_set_nat,Y2: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_512_ord__eq__le__subst,axiom,
    ! [A2: nat,F: set_set_a > nat,B3: set_set_a,C: set_set_a] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ C )
       => ( ! [X3: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_513_ord__eq__le__subst,axiom,
    ! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_nat @ B3 @ C )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_514_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_515_order__eq__refl,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( X = Y )
     => ( ord_le282488521294790766od_b_c @ X @ Y ) ) ).

% order_eq_refl
thf(fact_516_order__eq__refl,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( X = Y )
     => ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_517_order__eq__refl,axiom,
    ! [X: set_set_a,Y: set_set_a] :
      ( ( X = Y )
     => ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).

% order_eq_refl
thf(fact_518_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_519_order__eq__refl,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( X = Y )
     => ( ord_less_eq_set_a @ X @ Y ) ) ).

% order_eq_refl
thf(fact_520_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_521_order__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_522_order__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_523_order__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_524_order__subst2,axiom,
    ! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_525_order__subst2,axiom,
    ! [A2: set_a,B3: set_a,F: set_a > nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_526_order__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > set_set_nat,C: set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_527_order__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > set_set_a,C: set_set_a] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_528_order__subst2,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,F: set_set_nat > nat,C: nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: set_set_nat,Y2: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_529_order__subst2,axiom,
    ! [A2: set_set_a,B3: set_set_a,F: set_set_a > nat,C: nat] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_530_order__subst2,axiom,
    ! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_531_order__subst1,axiom,
    ! [A2: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_532_order__subst1,axiom,
    ! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_nat @ B3 @ C )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_533_order__subst1,axiom,
    ! [A2: nat,F: set_a > nat,B3: set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ! [X3: set_a,Y2: set_a] :
              ( ( ord_less_eq_set_a @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_534_order__subst1,axiom,
    ! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_535_order__subst1,axiom,
    ! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
      ( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_536_order__subst1,axiom,
    ! [A2: nat,F: set_set_nat > nat,B3: set_set_nat,C: set_set_nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_le6893508408891458716et_nat @ B3 @ C )
       => ( ! [X3: set_set_nat,Y2: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_537_order__subst1,axiom,
    ! [A2: nat,F: set_set_a > nat,B3: set_set_a,C: set_set_a] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ C )
       => ( ! [X3: set_set_a,Y2: set_set_a] :
              ( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_538_order__subst1,axiom,
    ! [A2: set_set_nat,F: nat > set_set_nat,B3: nat,C: nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_539_order__subst1,axiom,
    ! [A2: set_set_a,F: nat > set_set_a,B3: nat,C: nat] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_540_order__subst1,axiom,
    ! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_nat @ B3 @ C )
       => ( ! [X3: set_nat,Y2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_541_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] : ( Y3 = Z ) )
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ A4 @ B4 )
          & ( ord_le282488521294790766od_b_c @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_542_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_set_nat,Z: set_set_nat] : ( Y3 = Z ) )
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
          & ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_543_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_set_a,B4: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
          & ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_544_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B4 )
          & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_545_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ B4 )
          & ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_546_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_547_antisym,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B3 )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_548_antisym,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_549_antisym,axiom,
    ! [A2: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_550_antisym,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_551_antisym,axiom,
    ! [A2: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_552_antisym,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_553_dual__order_Otrans,axiom,
    ! [B3: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B3 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ C @ B3 )
       => ( ord_le282488521294790766od_b_c @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_554_dual__order_Otrans,axiom,
    ! [B3: set_set_nat,A2: set_set_nat,C: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ C @ B3 )
       => ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_555_dual__order_Otrans,axiom,
    ! [B3: set_set_a,A2: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
     => ( ( ord_le3724670747650509150_set_a @ C @ B3 )
       => ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_556_dual__order_Otrans,axiom,
    ! [B3: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( ord_less_eq_set_nat @ C @ B3 )
       => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_557_dual__order_Otrans,axiom,
    ! [B3: set_a,A2: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A2 )
     => ( ( ord_less_eq_set_a @ C @ B3 )
       => ( ord_less_eq_set_a @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_558_dual__order_Otrans,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B3 )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_559_dual__order_Oantisym,axiom,
    ! [B3: set_li6436108459499378894od_b_c,A2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ B3 @ A2 )
     => ( ( ord_le282488521294790766od_b_c @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_560_dual__order_Oantisym,axiom,
    ! [B3: set_set_nat,A2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_561_dual__order_Oantisym,axiom,
    ! [B3: set_set_a,A2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
     => ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_562_dual__order_Oantisym,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_563_dual__order_Oantisym,axiom,
    ! [B3: set_a,A2: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A2 )
     => ( ( ord_less_eq_set_a @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_564_dual__order_Oantisym,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_565_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_li6436108459499378894od_b_c,Z: set_li6436108459499378894od_b_c] : ( Y3 = Z ) )
    = ( ^ [A4: set_li6436108459499378894od_b_c,B4: set_li6436108459499378894od_b_c] :
          ( ( ord_le282488521294790766od_b_c @ B4 @ A4 )
          & ( ord_le282488521294790766od_b_c @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_566_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_set_nat,Z: set_set_nat] : ( Y3 = Z ) )
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
          & ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_567_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_set_a,B4: set_set_a] :
          ( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
          & ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_568_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A4 )
          & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_569_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
    = ( ^ [A4: set_a,B4: set_a] :
          ( ( ord_less_eq_set_a @ B4 @ A4 )
          & ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_570_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_571_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B3: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat,B5: nat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A2 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_572_order__trans,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c,Z2: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( ord_le282488521294790766od_b_c @ Y @ Z2 )
       => ( ord_le282488521294790766od_b_c @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_573_order__trans,axiom,
    ! [X: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X @ Y )
     => ( ( ord_le6893508408891458716et_nat @ Y @ Z2 )
       => ( ord_le6893508408891458716et_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_574_order__trans,axiom,
    ! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X @ Y )
     => ( ( ord_le3724670747650509150_set_a @ Y @ Z2 )
       => ( ord_le3724670747650509150_set_a @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_575_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z2 )
       => ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_576_order__trans,axiom,
    ! [X: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ Z2 )
       => ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_577_order__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_578_order_Otrans,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B3 )
     => ( ( ord_le282488521294790766od_b_c @ B3 @ C )
       => ( ord_le282488521294790766od_b_c @ A2 @ C ) ) ) ).

% order.trans
thf(fact_579_order_Otrans,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ B3 @ C )
       => ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_580_order_Otrans,axiom,
    ! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ B3 @ C )
       => ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).

% order.trans
thf(fact_581_order_Otrans,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_582_order_Otrans,axiom,
    ! [A2: set_a,B3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% order.trans
thf(fact_583_order_Otrans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_584_order__antisym,axiom,
    ! [X: set_li6436108459499378894od_b_c,Y: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ X @ Y )
     => ( ( ord_le282488521294790766od_b_c @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_585_order__antisym,axiom,
    ! [X: set_set_nat,Y: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X @ Y )
     => ( ( ord_le6893508408891458716et_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_586_order__antisym,axiom,
    ! [X: set_set_a,Y: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ X @ Y )
     => ( ( ord_le3724670747650509150_set_a @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_587_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_588_order__antisym,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_589_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_590_ord__le__eq__trans,axiom,
    ! [A2: set_li6436108459499378894od_b_c,B3: set_li6436108459499378894od_b_c,C: set_li6436108459499378894od_b_c] :
      ( ( ord_le282488521294790766od_b_c @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_le282488521294790766od_b_c @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_591_ord__le__eq__trans,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_592_ord__le__eq__trans,axiom,
    ! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_593_ord__le__eq__trans,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_594_ord__le__eq__trans,axiom,
    ! [A2: set_a,B3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_set_a @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_595_ord__le__eq__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_596_ord__eq__le__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( A2 = B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_597_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_598_le__cases3,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_599_nle__le,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
      = ( ( ord_less_eq_nat @ B3 @ A2 )
        & ( B3 != A2 ) ) ) ).

% nle_le
thf(fact_600_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) )
      = ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( Q2 @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_601_Collect__mono,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q2 @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) ) ) ).

% Collect_mono
thf(fact_602_transition__subset__paths,axiom,
    ! [S: fsm_set_a_b_c,M: fsm_set_a_b_c,P2: list_P6923848007992943548_set_a] :
      ( ( ord_le4557092201414138422_set_a @ ( transi8205445899535437940_a_b_c @ S ) @ ( transi8205445899535437940_a_b_c @ M ) )
     => ( ( member_set_a @ ( initial_set_a_b_c @ S ) @ ( states_set_a_b_c @ M ) )
       => ( ( ( inputs_set_a_b_c @ S )
            = ( inputs_set_a_b_c @ M ) )
         => ( ( ( outputs_set_a_b_c @ S )
              = ( outputs_set_a_b_c @ M ) )
           => ( ( path_set_a_b_c @ S @ ( initial_set_a_b_c @ S ) @ P2 )
             => ( path_set_a_b_c @ M @ ( initial_set_a_b_c @ S ) @ P2 ) ) ) ) ) ) ).

% transition_subset_paths
thf(fact_603_transition__subset__paths,axiom,
    ! [S: fsm_a_b_c,M: fsm_a_b_c,P2: list_P6327159017948738492od_c_a] :
      ( ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ S ) @ ( transitions_a_b_c @ M ) )
     => ( ( member_a @ ( initial_a_b_c @ S ) @ ( states_a_b_c @ M ) )
       => ( ( ( inputs_a_b_c @ S )
            = ( inputs_a_b_c @ M ) )
         => ( ( ( outputs_a_b_c @ S )
              = ( outputs_a_b_c @ M ) )
           => ( ( path_a_b_c @ S @ ( initial_a_b_c @ S ) @ P2 )
             => ( path_a_b_c @ M @ ( initial_a_b_c @ S ) @ P2 ) ) ) ) ) ) ).

% transition_subset_paths
thf(fact_604_filter__states__simps_I4_J,axiom,
    ! [P: set_a > $o,M: fsm_set_a_b_c] :
      ( ( P @ ( initial_set_a_b_c @ M ) )
     => ( ( outputs_set_a_b_c @ ( filter2961209833878020368_a_b_c @ M @ P ) )
        = ( outputs_set_a_b_c @ M ) ) ) ).

% filter_states_simps(4)
thf(fact_605_filter__states__simps_I4_J,axiom,
    ! [P: a > $o,M: fsm_a_b_c] :
      ( ( P @ ( initial_a_b_c @ M ) )
     => ( ( outputs_a_b_c @ ( filter_states_a_b_c @ M @ P ) )
        = ( outputs_a_b_c @ M ) ) ) ).

% filter_states_simps(4)
thf(fact_606_filter__states__simps_I3_J,axiom,
    ! [P: set_a > $o,M: fsm_set_a_b_c] :
      ( ( P @ ( initial_set_a_b_c @ M ) )
     => ( ( inputs_set_a_b_c @ ( filter2961209833878020368_a_b_c @ M @ P ) )
        = ( inputs_set_a_b_c @ M ) ) ) ).

% filter_states_simps(3)
thf(fact_607_filter__states__simps_I3_J,axiom,
    ! [P: a > $o,M: fsm_a_b_c] :
      ( ( P @ ( initial_a_b_c @ M ) )
     => ( ( inputs_a_b_c @ ( filter_states_a_b_c @ M @ P ) )
        = ( inputs_a_b_c @ M ) ) ) ).

% filter_states_simps(3)
thf(fact_608_submachine__reduction,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( is_io_1347772631898979393_b_c_a @ A @ ( initial_a_b_c @ A ) @ B @ ( initial_a_b_c @ B ) ) ) ).

% submachine_reduction
thf(fact_609_submachine__reduction,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( is_io_8703620250697608257_set_a @ A @ ( initial_set_a_b_c @ A ) @ B @ ( initial_set_a_b_c @ B ) ) ) ).

% submachine_reduction
thf(fact_610_index__states_Opelims,axiom,
    ! [X: fsm_a_b_c,Y: fsm_nat_b_c] :
      ( ( ( index_states_a_b_c @ X )
        = Y )
     => ( ( accp_fsm_a_b_c @ index_4146448100405152581_a_b_c @ X )
       => ~ ( ( Y
              = ( rename7348797690608482523_c_nat @ X @ ( assign_indices_a @ ( states_a_b_c @ X ) ) ) )
           => ~ ( accp_fsm_a_b_c @ index_4146448100405152581_a_b_c @ X ) ) ) ) ).

% index_states.pelims
thf(fact_611_rename__states__observable,axiom,
    ! [F: a > a,M: fsm_a_b_c] :
      ( ( bij_betw_a_a @ F @ ( states_a_b_c @ M ) @ ( image_a_a @ F @ ( states_a_b_c @ M ) ) )
     => ( ( observable_a_b_c @ M )
       => ( observable_a_b_c @ ( rename4556187690751568563_b_c_a @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_612_rename__states__observable,axiom,
    ! [F: a > nat,M: fsm_a_b_c] :
      ( ( bij_betw_a_nat @ F @ ( states_a_b_c @ M ) @ ( image_a_nat @ F @ ( states_a_b_c @ M ) ) )
     => ( ( observable_a_b_c @ M )
       => ( observable_nat_b_c @ ( rename7348797690608482523_c_nat @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_613_rename__states__observable,axiom,
    ! [F: a > code_integer,M: fsm_a_b_c] :
      ( ( bij_be3023389015827269733nteger @ F @ ( states_a_b_c @ M ) @ ( image_a_Code_integer @ F @ ( states_a_b_c @ M ) ) )
     => ( ( observable_a_b_c @ M )
       => ( observ7524490533585752726er_b_c @ ( rename9064899991920257384nteger @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_614_rename__states__observable,axiom,
    ! [F: nat > a,M: fsm_nat_b_c] :
      ( ( bij_betw_nat_a @ F @ ( states_nat_b_c @ M ) @ ( image_nat_a @ F @ ( states_nat_b_c @ M ) ) )
     => ( ( observable_nat_b_c @ M )
       => ( observable_a_b_c @ ( rename7200612048645949559_b_c_a @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_615_rename__states__observable,axiom,
    ! [F: nat > nat,M: fsm_nat_b_c] :
      ( ( bij_betw_nat_nat @ F @ ( states_nat_b_c @ M ) @ ( image_nat_nat @ F @ ( states_nat_b_c @ M ) ) )
     => ( ( observable_nat_b_c @ M )
       => ( observable_nat_b_c @ ( rename4185420143961928215_c_nat @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_616_rename__states__observable,axiom,
    ! [F: nat > code_integer,M: fsm_nat_b_c] :
      ( ( bij_be1603980594752706597nteger @ F @ ( states_nat_b_c @ M ) @ ( image_1215581382706833972nteger @ F @ ( states_nat_b_c @ M ) ) )
     => ( ( observable_nat_b_c @ M )
       => ( observ7524490533585752726er_b_c @ ( rename3801423786568692388nteger @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_617_rename__states__observable,axiom,
    ! [F: code_integer > a,M: fsm_Code_integer_b_c] :
      ( ( bij_be1274759345640864233eger_a @ F @ ( states1921009325540848725er_b_c @ M ) @ ( image_Code_integer_a @ F @ ( states1921009325540848725er_b_c @ M ) ) )
     => ( ( observ7524490533585752726er_b_c @ M )
       => ( observable_a_b_c @ ( rename8478575672702338986_b_c_a @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_618_rename__states__observable,axiom,
    ! [F: code_integer > nat,M: fsm_Code_integer_b_c] :
      ( ( bij_be1339425145973663781er_nat @ F @ ( states1921009325540848725er_b_c @ M ) @ ( image_951025933927791156er_nat @ F @ ( states1921009325540848725er_b_c @ M ) ) )
     => ( ( observ7524490533585752726er_b_c @ M )
       => ( observable_nat_b_c @ ( rename5224757547639086116_c_nat @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_619_rename__states__observable,axiom,
    ! [F: code_integer > code_integer,M: fsm_Code_integer_b_c] :
      ( ( bij_be9215044108114292914nteger @ F @ ( states1921009325540848725er_b_c @ M ) @ ( image_4470545334726330049nteger @ F @ ( states1921009325540848725er_b_c @ M ) ) )
     => ( ( observ7524490533585752726er_b_c @ M )
       => ( observ7524490533585752726er_b_c @ ( rename1031411498263718065nteger @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_620_rename__states__observable,axiom,
    ! [F: set_a > a,M: fsm_set_a_b_c] :
      ( ( bij_betw_set_a_a @ F @ ( states_set_a_b_c @ M ) @ ( image_set_a_a @ F @ ( states_set_a_b_c @ M ) ) )
     => ( ( observable_set_a_b_c @ M )
       => ( observable_a_b_c @ ( rename6603781358153168211_b_c_a @ M @ F ) ) ) ) ).

% rename_states_observable
thf(fact_621_filter__states__submachine,axiom,
    ! [P: a > $o,M: fsm_a_b_c] :
      ( ( P @ ( initial_a_b_c @ M ) )
     => ( is_submachine_a_b_c @ ( filter_states_a_b_c @ M @ P ) @ M ) ) ).

% filter_states_submachine
thf(fact_622_filter__states__submachine,axiom,
    ! [P: set_a > $o,M: fsm_set_a_b_c] :
      ( ( P @ ( initial_set_a_b_c @ M ) )
     => ( is_sub1377638108373535417_a_b_c @ ( filter2961209833878020368_a_b_c @ M @ P ) @ M ) ) ).

% filter_states_submachine
thf(fact_623_Greatest__equality,axiom,
    ! [P: nat > $o,X: nat] :
      ( ( P @ X )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ X ) )
       => ( ( order_Greatest_nat @ P )
          = X ) ) ) ).

% Greatest_equality
thf(fact_624_from__FSM__path__rev__initial,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( path_a_b_c @ M @ Q @ P2 )
     => ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ Q @ P2 ) ) ).

% from_FSM_path_rev_initial
thf(fact_625_from__FSM__path__rev__initial,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,P2: list_P6923848007992943548_set_a] :
      ( ( path_set_a_b_c @ M @ Q @ P2 )
     => ( path_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) @ Q @ P2 ) ) ).

% from_FSM_path_rev_initial
thf(fact_626_submachine__path,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( path_a_b_c @ A @ Q @ P2 )
       => ( path_a_b_c @ B @ Q @ P2 ) ) ) ).

% submachine_path
thf(fact_627_submachine__path,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c,Q: set_a,P2: list_P6923848007992943548_set_a] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ( path_set_a_b_c @ A @ Q @ P2 )
       => ( path_set_a_b_c @ B @ Q @ P2 ) ) ) ).

% submachine_path
thf(fact_628_from__FSM__path,axiom,
    ! [Q: a,M: fsm_a_b_c,Q3: a,P2: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ Q3 @ P2 )
       => ( path_a_b_c @ M @ Q3 @ P2 ) ) ) ).

% from_FSM_path
thf(fact_629_from__FSM__path,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c,Q3: set_a,P2: list_P6923848007992943548_set_a] :
      ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
     => ( ( path_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) @ Q3 @ P2 )
       => ( path_set_a_b_c @ M @ Q3 @ P2 ) ) ) ).

% from_FSM_path
thf(fact_630_submachine__path__initial,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c,P2: list_P6327159017948738492od_c_a] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( path_a_b_c @ A @ ( initial_a_b_c @ A ) @ P2 )
       => ( path_a_b_c @ B @ ( initial_a_b_c @ B ) @ P2 ) ) ) ).

% submachine_path_initial
thf(fact_631_submachine__path__initial,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c,P2: list_P6923848007992943548_set_a] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ( path_set_a_b_c @ A @ ( initial_set_a_b_c @ A ) @ P2 )
       => ( path_set_a_b_c @ B @ ( initial_set_a_b_c @ B ) @ P2 ) ) ) ).

% submachine_path_initial
thf(fact_632_restrict__to__reachable__states__path,axiom,
    ! [Q: a,M: fsm_a_b_c,P2: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) )
     => ( ( path_a_b_c @ M @ Q @ P2 )
        = ( path_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) @ Q @ P2 ) ) ) ).

% restrict_to_reachable_states_path
thf(fact_633_restrict__to__reachable__states__path,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c,P2: list_P6923848007992943548_set_a] :
      ( ( member_set_a @ Q @ ( reacha7078015818774689891_a_b_c @ M ) )
     => ( ( path_set_a_b_c @ M @ Q @ P2 )
        = ( path_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) @ Q @ P2 ) ) ) ).

% restrict_to_reachable_states_path
thf(fact_634_from__FSM__path__initial,axiom,
    ! [Q: a,M: fsm_a_b_c,P2: list_P6327159017948738492od_c_a] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( path_a_b_c @ M @ Q @ P2 )
        = ( path_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) @ P2 ) ) ) ).

% from_FSM_path_initial
thf(fact_635_from__FSM__path__initial,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c,P2: list_P6923848007992943548_set_a] :
      ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
     => ( ( path_set_a_b_c @ M @ Q @ P2 )
        = ( path_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) @ ( initial_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) ) @ P2 ) ) ) ).

% from_FSM_path_initial
thf(fact_636_GreatestI2__order,axiom,
    ! [P: nat > $o,X: nat,Q2: nat > $o] :
      ( ( P @ X )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ X ) )
       => ( ! [X3: nat] :
              ( ( P @ X3 )
             => ( ! [Y5: nat] :
                    ( ( P @ Y5 )
                   => ( ord_less_eq_nat @ Y5 @ X3 ) )
               => ( Q2 @ X3 ) ) )
         => ( Q2 @ ( order_Greatest_nat @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_637_index__states__integer__minimal,axiom,
    ! [M: fsm_a_b_c] :
      ( ( minimal_a_b_c @ M )
     => ( minima6064407838494594252er_b_c @ ( index_2500966223721395136_a_b_c @ M ) ) ) ).

% index_states_integer_minimal
thf(fact_638_index__states__minimal,axiom,
    ! [M: fsm_a_b_c] :
      ( ( minimal_a_b_c @ M )
     => ( minimal_nat_b_c @ ( index_states_a_b_c @ M ) ) ) ).

% index_states_minimal
thf(fact_639_index__states__language,axiom,
    ! [M: fsm_a_b_c] :
      ( ( lS_nat_b_c @ ( index_states_a_b_c @ M ) @ ( initial_nat_b_c @ ( index_states_a_b_c @ M ) ) )
      = ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ).

% index_states_language
thf(fact_640_index__states__integer__language,axiom,
    ! [M: fsm_a_b_c] :
      ( ( lS_Code_integer_b_c @ ( index_2500966223721395136_a_b_c @ M ) @ ( initia4873860033875475119er_b_c @ ( index_2500966223721395136_a_b_c @ M ) ) )
      = ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ).

% index_states_integer_language
thf(fact_641_restrict__to__reachable__states__minimal,axiom,
    ! [M: fsm_a_b_c] :
      ( ( minimal_a_b_c @ M )
     => ( minimal_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) ) ) ).

% restrict_to_reachable_states_minimal
thf(fact_642_restrict__to__reachable__states__minimal,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( minimal_set_a_b_c @ M )
     => ( minimal_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) ) ) ).

% restrict_to_reachable_states_minimal
thf(fact_643_submachine__language,axiom,
    ! [S: fsm_a_b_c,M: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ S @ M )
     => ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ S @ ( initial_a_b_c @ S ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ) ).

% submachine_language
thf(fact_644_submachine__language,axiom,
    ! [S: fsm_set_a_b_c,M: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ S @ M )
     => ( ord_le282488521294790766od_b_c @ ( lS_set_a_b_c @ S @ ( initial_set_a_b_c @ S ) ) @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) ) ) ).

% submachine_language
thf(fact_645_restrict__to__reachable__states__language,axiom,
    ! [M: fsm_a_b_c] :
      ( ( lS_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) @ ( initial_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) ) )
      = ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ).

% restrict_to_reachable_states_language
thf(fact_646_restrict__to__reachable__states__language,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( lS_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) @ ( initial_set_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) ) )
      = ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) ) ).

% restrict_to_reachable_states_language
thf(fact_647_from__FSM__language,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( states_a_b_c @ M ) )
     => ( ( lS_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) )
        = ( lS_a_b_c @ M @ Q ) ) ) ).

% from_FSM_language
thf(fact_648_from__FSM__language,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
     => ( ( lS_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) @ ( initial_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) ) )
        = ( lS_set_a_b_c @ M @ Q ) ) ) ).

% from_FSM_language
thf(fact_649_maximal__prefix__in__language__properties_I1_J,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
       => ( member6330420149250801815od_b_c @ ( maxima5703285411639229488_a_b_c @ M @ Q @ Io ) @ ( lS_set_a_b_c @ M @ Q ) ) ) ) ).

% maximal_prefix_in_language_properties(1)
thf(fact_650_maximal__prefix__in__language__properties_I1_J,axiom,
    ! [M: fsm_a_b_c,Q: a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( member6330420149250801815od_b_c @ ( maxima1559550560783484624_a_b_c @ M @ Q @ Io ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ).

% maximal_prefix_in_language_properties(1)
thf(fact_651_maximal__prefix__in__language__properties_I1_J,axiom,
    ! [M: fsm_nat_b_c,Q: nat,Io: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member_nat @ Q @ ( states_nat_b_c @ M ) )
       => ( member6330420149250801815od_b_c @ ( maxima124221583468524992at_b_c @ M @ Q @ Io ) @ ( lS_nat_b_c @ M @ Q ) ) ) ) ).

% maximal_prefix_in_language_properties(1)
thf(fact_652_maximal__prefix__in__language__properties_I1_J,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer,Io: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ M ) )
       => ( member6330420149250801815od_b_c @ ( maxima5001844549960690637er_b_c @ M @ Q @ Io ) @ ( lS_Code_integer_b_c @ M @ Q ) ) ) ) ).

% maximal_prefix_in_language_properties(1)
thf(fact_653_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_a_b_c,M22: fsm_a_b_c] :
      ( ( minimal_a_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observable_a_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) )
                = ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) )
             => ( ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M1 ) )
                = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_654_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_a_b_c,M22: fsm_Code_integer_b_c] :
      ( ( minimal_a_b_c @ M1 )
     => ( ( minima6064407838494594252er_b_c @ M22 )
       => ( ( observable_a_b_c @ M1 )
         => ( ( observ7524490533585752726er_b_c @ M22 )
           => ( ( ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) )
                = ( lS_Code_integer_b_c @ M22 @ ( initia4873860033875475119er_b_c @ M22 ) ) )
             => ( ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M1 ) )
                = ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_655_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_Code_integer_b_c,M22: fsm_a_b_c] :
      ( ( minima6064407838494594252er_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observ7524490533585752726er_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ( lS_Code_integer_b_c @ M1 @ ( initia4873860033875475119er_b_c @ M1 ) )
                = ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) )
             => ( ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M1 ) )
                = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_656_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_Code_integer_b_c,M22: fsm_Code_integer_b_c] :
      ( ( minima6064407838494594252er_b_c @ M1 )
     => ( ( minima6064407838494594252er_b_c @ M22 )
       => ( ( observ7524490533585752726er_b_c @ M1 )
         => ( ( observ7524490533585752726er_b_c @ M22 )
           => ( ( ( lS_Code_integer_b_c @ M1 @ ( initia4873860033875475119er_b_c @ M1 ) )
                = ( lS_Code_integer_b_c @ M22 @ ( initia4873860033875475119er_b_c @ M22 ) ) )
             => ( ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M1 ) )
                = ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_657_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_a_b_c,M22: fsm_nat_b_c] :
      ( ( minimal_a_b_c @ M1 )
     => ( ( minimal_nat_b_c @ M22 )
       => ( ( observable_a_b_c @ M1 )
         => ( ( observable_nat_b_c @ M22 )
           => ( ( ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) )
                = ( lS_nat_b_c @ M22 @ ( initial_nat_b_c @ M22 ) ) )
             => ( ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M1 ) )
                = ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_658_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_Code_integer_b_c,M22: fsm_nat_b_c] :
      ( ( minima6064407838494594252er_b_c @ M1 )
     => ( ( minimal_nat_b_c @ M22 )
       => ( ( observ7524490533585752726er_b_c @ M1 )
         => ( ( observable_nat_b_c @ M22 )
           => ( ( ( lS_Code_integer_b_c @ M1 @ ( initia4873860033875475119er_b_c @ M1 ) )
                = ( lS_nat_b_c @ M22 @ ( initial_nat_b_c @ M22 ) ) )
             => ( ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M1 ) )
                = ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_659_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_nat_b_c,M22: fsm_a_b_c] :
      ( ( minimal_nat_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observable_nat_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ( lS_nat_b_c @ M1 @ ( initial_nat_b_c @ M1 ) )
                = ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) )
             => ( ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M1 ) )
                = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_660_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_nat_b_c,M22: fsm_Code_integer_b_c] :
      ( ( minimal_nat_b_c @ M1 )
     => ( ( minima6064407838494594252er_b_c @ M22 )
       => ( ( observable_nat_b_c @ M1 )
         => ( ( observ7524490533585752726er_b_c @ M22 )
           => ( ( ( lS_nat_b_c @ M1 @ ( initial_nat_b_c @ M1 ) )
                = ( lS_Code_integer_b_c @ M22 @ ( initia4873860033875475119er_b_c @ M22 ) ) )
             => ( ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M1 ) )
                = ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_661_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_nat_b_c,M22: fsm_nat_b_c] :
      ( ( minimal_nat_b_c @ M1 )
     => ( ( minimal_nat_b_c @ M22 )
       => ( ( observable_nat_b_c @ M1 )
         => ( ( observable_nat_b_c @ M22 )
           => ( ( ( lS_nat_b_c @ M1 @ ( initial_nat_b_c @ M1 ) )
                = ( lS_nat_b_c @ M22 @ ( initial_nat_b_c @ M22 ) ) )
             => ( ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M1 ) )
                = ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_662_minimal__equivalence__size__r,axiom,
    ! [M1: fsm_set_a_b_c,M22: fsm_a_b_c] :
      ( ( minimal_set_a_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observable_set_a_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ( lS_set_a_b_c @ M1 @ ( initial_set_a_b_c @ M1 ) )
                = ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) )
             => ( ( finite_card_set_a @ ( reacha7078015818774689891_a_b_c @ M1 ) )
                = ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% minimal_equivalence_size_r
thf(fact_663_card__insert__le,axiom,
    ! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) ) ).

% card_insert_le
thf(fact_664_bij__betw__same__card,axiom,
    ! [F: nat > nat,A: set_nat,B: set_nat] :
      ( ( bij_betw_nat_nat @ F @ A @ B )
     => ( ( finite_card_nat @ A )
        = ( finite_card_nat @ B ) ) ) ).

% bij_betw_same_card
thf(fact_665_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_a_b_c,M22: fsm_a_b_c] :
      ( ( minimal_a_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observable_a_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M1 ) ) @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) )
             => ( ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) )
               != ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_666_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_a_b_c,M22: fsm_Code_integer_b_c] :
      ( ( minimal_a_b_c @ M1 )
     => ( ( minima6064407838494594252er_b_c @ M22 )
       => ( ( observable_a_b_c @ M1 )
         => ( ( observ7524490533585752726er_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M1 ) ) @ ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M22 ) ) )
             => ( ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) )
               != ( lS_Code_integer_b_c @ M22 @ ( initia4873860033875475119er_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_667_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_Code_integer_b_c,M22: fsm_a_b_c] :
      ( ( minima6064407838494594252er_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observ7524490533585752726er_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M1 ) ) @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) )
             => ( ( lS_Code_integer_b_c @ M1 @ ( initia4873860033875475119er_b_c @ M1 ) )
               != ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_668_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_Code_integer_b_c,M22: fsm_Code_integer_b_c] :
      ( ( minima6064407838494594252er_b_c @ M1 )
     => ( ( minima6064407838494594252er_b_c @ M22 )
       => ( ( observ7524490533585752726er_b_c @ M1 )
         => ( ( observ7524490533585752726er_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M1 ) ) @ ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M22 ) ) )
             => ( ( lS_Code_integer_b_c @ M1 @ ( initia4873860033875475119er_b_c @ M1 ) )
               != ( lS_Code_integer_b_c @ M22 @ ( initia4873860033875475119er_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_669_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_a_b_c,M22: fsm_nat_b_c] :
      ( ( minimal_a_b_c @ M1 )
     => ( ( minimal_nat_b_c @ M22 )
       => ( ( observable_a_b_c @ M1 )
         => ( ( observable_nat_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M1 ) ) @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M22 ) ) )
             => ( ( lS_a_b_c @ M1 @ ( initial_a_b_c @ M1 ) )
               != ( lS_nat_b_c @ M22 @ ( initial_nat_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_670_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_Code_integer_b_c,M22: fsm_nat_b_c] :
      ( ( minima6064407838494594252er_b_c @ M1 )
     => ( ( minimal_nat_b_c @ M22 )
       => ( ( observ7524490533585752726er_b_c @ M1 )
         => ( ( observable_nat_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M1 ) ) @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M22 ) ) )
             => ( ( lS_Code_integer_b_c @ M1 @ ( initia4873860033875475119er_b_c @ M1 ) )
               != ( lS_nat_b_c @ M22 @ ( initial_nat_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_671_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_nat_b_c,M22: fsm_a_b_c] :
      ( ( minimal_nat_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observable_nat_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M1 ) ) @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) )
             => ( ( lS_nat_b_c @ M1 @ ( initial_nat_b_c @ M1 ) )
               != ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_672_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_nat_b_c,M22: fsm_Code_integer_b_c] :
      ( ( minimal_nat_b_c @ M1 )
     => ( ( minima6064407838494594252er_b_c @ M22 )
       => ( ( observable_nat_b_c @ M1 )
         => ( ( observ7524490533585752726er_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M1 ) ) @ ( finite4902975817058060853nteger @ ( reacha2636592895321446618er_b_c @ M22 ) ) )
             => ( ( lS_nat_b_c @ M1 @ ( initial_nat_b_c @ M1 ) )
               != ( lS_Code_integer_b_c @ M22 @ ( initia4873860033875475119er_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_673_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_nat_b_c,M22: fsm_nat_b_c] :
      ( ( minimal_nat_b_c @ M1 )
     => ( ( minimal_nat_b_c @ M22 )
       => ( ( observable_nat_b_c @ M1 )
         => ( ( observable_nat_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M1 ) ) @ ( finite_card_nat @ ( reacha6243244737040045773at_b_c @ M22 ) ) )
             => ( ( lS_nat_b_c @ M1 @ ( initial_nat_b_c @ M1 ) )
               != ( lS_nat_b_c @ M22 @ ( initial_nat_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_674_observable__minimal__size__r__language__distinct,axiom,
    ! [M1: fsm_set_a_b_c,M22: fsm_a_b_c] :
      ( ( minimal_set_a_b_c @ M1 )
     => ( ( minimal_a_b_c @ M22 )
       => ( ( observable_set_a_b_c @ M1 )
         => ( ( observable_a_b_c @ M22 )
           => ( ( ord_less_nat @ ( finite_card_set_a @ ( reacha7078015818774689891_a_b_c @ M1 ) ) @ ( finite_card_a @ ( reacha1620305530751930115_a_b_c @ M22 ) ) )
             => ( ( lS_set_a_b_c @ M1 @ ( initial_set_a_b_c @ M1 ) )
               != ( lS_a_b_c @ M22 @ ( initial_a_b_c @ M22 ) ) ) ) ) ) ) ) ).

% observable_minimal_size_r_language_distinct
thf(fact_675_observable__io__targets,axiom,
    ! [M: fsm_set_a_b_c,Io: list_P903359562653991662od_b_c,Q: set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
       => ~ ! [Q4: set_a] :
              ( ( io_targets_set_a_b_c @ M @ Io @ Q )
             != ( insert_set_a @ Q4 @ bot_bot_set_set_a ) ) ) ) ).

% observable_io_targets
thf(fact_676_observable__io__targets,axiom,
    ! [M: fsm_a_b_c,Io: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
       => ~ ! [Q4: a] :
              ( ( io_targets_a_b_c @ M @ Io @ Q )
             != ( insert_a @ Q4 @ bot_bot_set_a ) ) ) ) ).

% observable_io_targets
thf(fact_677_observable__io__targets,axiom,
    ! [M: fsm_nat_b_c,Io: list_P903359562653991662od_b_c,Q: nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q ) )
       => ~ ! [Q4: nat] :
              ( ( io_targets_nat_b_c @ M @ Io @ Q )
             != ( insert_nat @ Q4 @ bot_bot_set_nat ) ) ) ) ).

% observable_io_targets
thf(fact_678_observable__io__targets,axiom,
    ! [M: fsm_Code_integer_b_c,Io: list_P903359562653991662od_b_c,Q: code_integer] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ~ ! [Q4: code_integer] :
              ( ( io_tar3403248182866227591er_b_c @ M @ Io @ Q )
             != ( insert_Code_integer @ Q4 @ bot_bo3990330152332043303nteger ) ) ) ) ).

% observable_io_targets
thf(fact_679_after__reachable__initial,axiom,
    ! [M: fsm_set_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
       => ( member_set_a @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ Io ) @ ( reacha7078015818774689891_a_b_c @ M ) ) ) ) ).

% after_reachable_initial
thf(fact_680_after__reachable__initial,axiom,
    ! [M: fsm_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
       => ( member_a @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ Io ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ) ) ).

% after_reachable_initial
thf(fact_681_after__reachable__initial,axiom,
    ! [M: fsm_nat_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
       => ( member_nat @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ Io ) @ ( reacha6243244737040045773at_b_c @ M ) ) ) ) ).

% after_reachable_initial
thf(fact_682_after__reachable__initial,axiom,
    ! [M: fsm_Code_integer_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
       => ( member_Code_integer @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ Io ) @ ( reacha2636592895321446618er_b_c @ M ) ) ) ) ).

% after_reachable_initial
thf(fact_683_is__in__language__iff,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
       => ( ( is_in_511448957490290147_a_b_c @ M @ Q @ Io )
          = ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) ) ) ) ) ).

% is_in_language_iff
thf(fact_684_is__in__language__iff,axiom,
    ! [M: fsm_a_b_c,Q: a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( ( is_in_language_a_b_c @ M @ Q @ Io )
          = ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% is_in_language_iff
thf(fact_685_is__in__language__iff,axiom,
    ! [M: fsm_nat_b_c,Q: nat,Io: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member_nat @ Q @ ( states_nat_b_c @ M ) )
       => ( ( is_in_7825730478733939021at_b_c @ M @ Q @ Io )
          = ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q ) ) ) ) ) ).

% is_in_language_iff
thf(fact_686_is__in__language__iff,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer,Io: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ M ) )
       => ( ( is_in_5902465164305961818er_b_c @ M @ Q @ Io )
          = ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q ) ) ) ) ) ).

% is_in_language_iff
thf(fact_687_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_688_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_689_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_690_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_691_order_Oasym,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ~ ( ord_less_nat @ B3 @ A2 ) ) ).

% order.asym
thf(fact_692_ord__eq__less__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( A2 = B3 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_693_ord__less__eq__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_694_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P @ Y5 ) )
         => ( P @ X3 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_695_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_696_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_697_dual__order_Oasym,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B3 ) ) ).

% dual_order.asym
thf(fact_698_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_699_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X4: nat] : ( P3 @ X4 ) )
    = ( ^ [P4: nat > $o] :
        ? [N: nat] :
          ( ( P4 @ N )
          & ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N )
             => ~ ( P4 @ M4 ) ) ) ) ) ).

% exists_least_iff
thf(fact_700_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B3: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A2 @ B3 ) ) ) ) ).

% linorder_less_wlog
thf(fact_701_order_Ostrict__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_702_bot_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_703_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_704_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_705_dual__order_Ostrict__trans,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ( ord_less_nat @ C @ B3 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_706_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
thf(fact_707_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_708_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_709_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_710_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_711_order__less__asym_H,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ~ ( ord_less_nat @ B3 @ A2 ) ) ).

% order_less_asym'
thf(fact_712_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_713_ord__eq__less__subst,axiom,
    ! [A2: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_714_ord__less__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_715_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_716_order__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_717_order__less__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_718_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_719_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_720_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_721_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_722_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_723_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_724_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_725_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_726_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_727_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_728_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_729_order__less__le__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_730_order__less__le__subst1,axiom,
    ! [A2: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_731_order__le__less__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_732_order__le__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ! [X3: nat,Y2: nat] :
              ( ( ord_less_nat @ X3 @ Y2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_733_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_734_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_735_order__neq__le__trans,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != B3 )
     => ( ( ord_less_eq_nat @ A2 @ B3 )
       => ( ord_less_nat @ A2 @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_736_order__le__neq__trans,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_nat @ A2 @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_737_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_738_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_739_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_740_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
          & ( X2 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_741_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_nat @ X2 @ Y4 )
          | ( X2 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_742_dual__order_Ostrict__implies__order,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ord_less_eq_nat @ B3 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_743_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ord_less_eq_nat @ A2 @ B3 ) ) ).

% order.strict_implies_order
thf(fact_744_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_745_dual__order_Ostrict__trans2,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B3 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_746_dual__order_Ostrict__trans1,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_less_nat @ C @ B3 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_747_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_748_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_749_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_750_order_Ostrict__trans2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_751_order_Ostrict__trans1,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_752_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_753_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_754_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_755_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_756_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_757_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_758_nless__le,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B3 )
        | ( A2 = B3 ) ) ) ).

% nless_le
thf(fact_759_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_760_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_761_observable__paths__for__io,axiom,
    ! [M: fsm_set_a_b_c,Io: list_P903359562653991662od_b_c,Q: set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
       => ~ ! [P5: list_P6923848007992943548_set_a] :
              ( ( paths_1493807728476235305_a_b_c @ M @ Q @ Io )
             != ( insert3727591976216729996_set_a @ P5 @ bot_bo5609605560744208648_set_a ) ) ) ) ).

% observable_paths_for_io
thf(fact_762_observable__paths__for__io,axiom,
    ! [M: fsm_a_b_c,Io: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
       => ~ ! [P5: list_P6327159017948738492od_c_a] :
              ( ( paths_for_io_a_b_c @ M @ Q @ Io )
             != ( insert4789241225314331020od_c_a @ P5 @ bot_bo6236370880139903240od_c_a ) ) ) ) ).

% observable_paths_for_io
thf(fact_763_observable__paths__for__io,axiom,
    ! [M: fsm_nat_b_c,Io: list_P903359562653991662od_b_c,Q: nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q ) )
       => ~ ! [P5: list_P1412026030649758308_c_nat] :
              ( ( paths_for_io_nat_b_c @ M @ Q @ Io )
             != ( insert8610875376133238836_c_nat @ P5 @ bot_bo519094879196661808_c_nat ) ) ) ) ).

% observable_paths_for_io
thf(fact_764_observable__paths__for__io,axiom,
    ! [M: fsm_Code_integer_b_c,Io: list_P903359562653991662od_b_c,Q: code_integer] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ~ ! [P5: list_P7609586222656207102nteger] :
              ( ( paths_6888318843778524564er_b_c @ M @ Q @ Io )
             != ( insert2637641413281142478nteger @ P5 @ bot_bo6259408951822219722nteger ) ) ) ) ).

% observable_paths_for_io
thf(fact_765_after__is__state,axiom,
    ! [M: fsm_set_a_b_c,Io: list_P903359562653991662od_b_c,Q: set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
       => ( member_set_a @ ( after_set_a_b_c @ M @ Q @ Io ) @ ( states_set_a_b_c @ M ) ) ) ) ).

% after_is_state
thf(fact_766_after__is__state,axiom,
    ! [M: fsm_a_b_c,Io: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
       => ( member_a @ ( after_a_b_c @ M @ Q @ Io ) @ ( states_a_b_c @ M ) ) ) ) ).

% after_is_state
thf(fact_767_after__is__state,axiom,
    ! [M: fsm_nat_b_c,Io: list_P903359562653991662od_b_c,Q: nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q ) )
       => ( member_nat @ ( after_nat_b_c @ M @ Q @ Io ) @ ( states_nat_b_c @ M ) ) ) ) ).

% after_is_state
thf(fact_768_after__is__state,axiom,
    ! [M: fsm_Code_integer_b_c,Io: list_P903359562653991662od_b_c,Q: code_integer] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ( member_Code_integer @ ( after_5090571588895392199er_b_c @ M @ Q @ Io ) @ ( states1921009325540848725er_b_c @ M ) ) ) ) ).

% after_is_state
thf(fact_769_after__reachable,axiom,
    ! [M: fsm_set_a_b_c,Io: list_P903359562653991662od_b_c,Q: set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
       => ( ( member_set_a @ Q @ ( reacha7078015818774689891_a_b_c @ M ) )
         => ( member_set_a @ ( after_set_a_b_c @ M @ Q @ Io ) @ ( reacha7078015818774689891_a_b_c @ M ) ) ) ) ) ).

% after_reachable
thf(fact_770_after__reachable,axiom,
    ! [M: fsm_a_b_c,Io: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
       => ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) )
         => ( member_a @ ( after_a_b_c @ M @ Q @ Io ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ) ) ) ).

% after_reachable
thf(fact_771_after__reachable,axiom,
    ! [M: fsm_nat_b_c,Io: list_P903359562653991662od_b_c,Q: nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q ) )
       => ( ( member_nat @ Q @ ( reacha6243244737040045773at_b_c @ M ) )
         => ( member_nat @ ( after_nat_b_c @ M @ Q @ Io ) @ ( reacha6243244737040045773at_b_c @ M ) ) ) ) ) ).

% after_reachable
thf(fact_772_after__reachable,axiom,
    ! [M: fsm_Code_integer_b_c,Io: list_P903359562653991662od_b_c,Q: code_integer] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ( ( member_Code_integer @ Q @ ( reacha2636592895321446618er_b_c @ M ) )
         => ( member_Code_integer @ ( after_5090571588895392199er_b_c @ M @ Q @ Io ) @ ( reacha2636592895321446618er_b_c @ M ) ) ) ) ) ).

% after_reachable
thf(fact_773_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N: nat] :
          ( ( ord_less_eq_nat @ M4 @ N )
          & ( M4 != N ) ) ) ) ).

% nat_less_le
thf(fact_774_less__imp__le__nat,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_nat @ M3 @ N2 )
     => ( ord_less_eq_nat @ M3 @ N2 ) ) ).

% less_imp_le_nat
thf(fact_775_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N: nat] :
          ( ( ord_less_nat @ M4 @ N )
          | ( M4 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_776_less__or__eq__imp__le,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ( ord_less_nat @ M3 @ N2 )
        | ( M3 = N2 ) )
     => ( ord_less_eq_nat @ M3 @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_777_le__neq__implies__less,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M3 @ N2 )
     => ( ( M3 != N2 )
       => ( ord_less_nat @ M3 @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_778_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_779_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_780_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B3 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_781_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B3: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B3 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_782_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B3 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_783_nat__le__linear,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M3 @ N2 )
      | ( ord_less_eq_nat @ N2 @ M3 ) ) ).

% nat_le_linear
thf(fact_784_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B3 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_785_le__antisym,axiom,
    ! [M3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M3 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M3 )
       => ( M3 = N2 ) ) ) ).

% le_antisym
thf(fact_786_eq__imp__le,axiom,
    ! [M3: nat,N2: nat] :
      ( ( M3 = N2 )
     => ( ord_less_eq_nat @ M3 @ N2 ) ) ).

% eq_imp_le
thf(fact_787_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_788_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_789_language__empty__io,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( ( ( inputs_set_a_b_c @ M )
          = bot_bot_set_b )
        | ( ( outputs_set_a_b_c @ M )
          = bot_bot_set_c ) )
     => ( ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) )
        = ( insert6227932334100060350od_b_c @ nil_Product_prod_b_c @ bot_bo4166481423041325370od_b_c ) ) ) ).

% language_empty_io
thf(fact_790_language__empty__io,axiom,
    ! [M: fsm_a_b_c] :
      ( ( ( ( inputs_a_b_c @ M )
          = bot_bot_set_b )
        | ( ( outputs_a_b_c @ M )
          = bot_bot_set_c ) )
     => ( ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) )
        = ( insert6227932334100060350od_b_c @ nil_Product_prod_b_c @ bot_bo4166481423041325370od_b_c ) ) ) ).

% language_empty_io
thf(fact_791_nat__descend__induct,axiom,
    ! [N2: nat,P: nat > $o,M3: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N2 @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K2 @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K2 ) ) )
       => ( P @ M3 ) ) ) ).

% nat_descend_induct
thf(fact_792_after__io__targets,axiom,
    ! [M: fsm_set_a_b_c,Io: list_P903359562653991662od_b_c,Q: set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
       => ( ( after_set_a_b_c @ M @ Q @ Io )
          = ( the_elem_set_a @ ( io_targets_set_a_b_c @ M @ Io @ Q ) ) ) ) ) ).

% after_io_targets
thf(fact_793_after__io__targets,axiom,
    ! [M: fsm_a_b_c,Io: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
       => ( ( after_a_b_c @ M @ Q @ Io )
          = ( the_elem_a @ ( io_targets_a_b_c @ M @ Io @ Q ) ) ) ) ) ).

% after_io_targets
thf(fact_794_after__io__targets,axiom,
    ! [M: fsm_nat_b_c,Io: list_P903359562653991662od_b_c,Q: nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q ) )
       => ( ( after_nat_b_c @ M @ Q @ Io )
          = ( the_elem_nat @ ( io_targets_nat_b_c @ M @ Io @ Q ) ) ) ) ) ).

% after_io_targets
thf(fact_795_after__io__targets,axiom,
    ! [M: fsm_Code_integer_b_c,Io: list_P903359562653991662od_b_c,Q: code_integer] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ( ( after_5090571588895392199er_b_c @ M @ Q @ Io )
          = ( the_el4656377885804969768nteger @ ( io_tar3403248182866227591er_b_c @ M @ Io @ Q ) ) ) ) ) ).

% after_io_targets
thf(fact_796_complete__interval,axiom,
    ! [A2: nat,B3: nat,P: nat > $o] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( P @ A2 )
       => ( ~ ( P @ B3 )
         => ? [C2: nat] :
              ( ( ord_less_eq_nat @ A2 @ C2 )
              & ( ord_less_eq_nat @ C2 @ B3 )
              & ! [X5: nat] :
                  ( ( ( ord_less_eq_nat @ A2 @ X5 )
                    & ( ord_less_nat @ X5 @ C2 ) )
                 => ( P @ X5 ) )
              & ! [D: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A2 @ X3 )
                        & ( ord_less_nat @ X3 @ D ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_797_verit__comp__simplify1_I3_J,axiom,
    ! [B6: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
      = ( ord_less_nat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_798_verit__la__disequality,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = B3 )
      | ~ ( ord_less_eq_nat @ A2 @ B3 )
      | ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).

% verit_la_disequality
thf(fact_799_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_800_minimally__distinguishes__after__append__initial,axiom,
    ! [M: fsm_set_a_b_c,U: list_P903359562653991662od_b_c,V: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( minimal_set_a_b_c @ M )
       => ( ( member6330420149250801815od_b_c @ U @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
         => ( ( member6330420149250801815od_b_c @ V @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
           => ( ( minima7468014887693519109_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ U ) @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ V ) @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima7468014887693519109_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ V @ W ) ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append_initial
thf(fact_801_minimally__distinguishes__after__append__initial,axiom,
    ! [M: fsm_a_b_c,U: list_P903359562653991662od_b_c,V: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( minimal_a_b_c @ M )
       => ( ( member6330420149250801815od_b_c @ U @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
         => ( ( member6330420149250801815od_b_c @ V @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
           => ( ( minima243535863231358885_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ U ) @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ V ) @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima243535863231358885_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ V @ W ) ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append_initial
thf(fact_802_minimally__distinguishes__after__append__initial,axiom,
    ! [M: fsm_nat_b_c,U: list_P903359562653991662od_b_c,V: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( minimal_nat_b_c @ M )
       => ( ( member6330420149250801815od_b_c @ U @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
         => ( ( member6330420149250801815od_b_c @ V @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
           => ( ( minima7825597830629587371at_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ U ) @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ V ) @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima7825597830629587371at_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ ( append2547753245680614915od_b_c @ V @ W ) ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append_initial
thf(fact_803_minimally__distinguishes__after__append__initial,axiom,
    ! [M: fsm_Code_integer_b_c,U: list_P903359562653991662od_b_c,V: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( minima6064407838494594252er_b_c @ M )
       => ( ( member6330420149250801815od_b_c @ U @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
         => ( ( member6330420149250801815od_b_c @ V @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
           => ( ( minima4576043283673695160er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ U ) @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ V ) @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima4576043283673695160er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ ( append2547753245680614915od_b_c @ V @ W ) ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append_initial
thf(fact_804_minimally__distinguishes__after__append,axiom,
    ! [M: fsm_set_a_b_c,Q1: set_a,Q22: set_a,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( minimal_set_a_b_c @ M )
       => ( ( member_set_a @ Q1 @ ( states_set_a_b_c @ M ) )
         => ( ( member_set_a @ Q22 @ ( states_set_a_b_c @ M ) )
           => ( ( minima7468014887693519109_a_b_c @ M @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima7468014887693519109_a_b_c @ M @ ( after_set_a_b_c @ M @ Q1 @ W ) @ ( after_set_a_b_c @ M @ Q22 @ W ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append
thf(fact_805_minimally__distinguishes__after__append,axiom,
    ! [M: fsm_a_b_c,Q1: a,Q22: a,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( minimal_a_b_c @ M )
       => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
         => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
           => ( ( minima243535863231358885_a_b_c @ M @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima243535863231358885_a_b_c @ M @ ( after_a_b_c @ M @ Q1 @ W ) @ ( after_a_b_c @ M @ Q22 @ W ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append
thf(fact_806_minimally__distinguishes__after__append,axiom,
    ! [M: fsm_nat_b_c,Q1: nat,Q22: nat,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( minimal_nat_b_c @ M )
       => ( ( member_nat @ Q1 @ ( states_nat_b_c @ M ) )
         => ( ( member_nat @ Q22 @ ( states_nat_b_c @ M ) )
           => ( ( minima7825597830629587371at_b_c @ M @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima7825597830629587371at_b_c @ M @ ( after_nat_b_c @ M @ Q1 @ W ) @ ( after_nat_b_c @ M @ Q22 @ W ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append
thf(fact_807_minimally__distinguishes__after__append,axiom,
    ! [M: fsm_Code_integer_b_c,Q1: code_integer,Q22: code_integer,W: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( minima6064407838494594252er_b_c @ M )
       => ( ( member_Code_integer @ Q1 @ ( states1921009325540848725er_b_c @ M ) )
         => ( ( member_Code_integer @ Q22 @ ( states1921009325540848725er_b_c @ M ) )
           => ( ( minima4576043283673695160er_b_c @ M @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ W @ W2 ) )
             => ( ( W2 != nil_Product_prod_b_c )
               => ( minima4576043283673695160er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ Q1 @ W ) @ ( after_5090571588895392199er_b_c @ M @ Q22 @ W ) @ W2 ) ) ) ) ) ) ) ).

% minimally_distinguishes_after_append
thf(fact_808_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_809_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z3 )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_810_submachine__deadlock,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c,Q: a] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( deadlock_state_a_b_c @ B @ Q )
       => ( deadlock_state_a_b_c @ A @ Q ) ) ) ).

% submachine_deadlock
thf(fact_811_submachine__deadlock,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c,Q: set_a] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ( deadlo5770673589738903088_a_b_c @ B @ Q )
       => ( deadlo5770673589738903088_a_b_c @ A @ Q ) ) ) ).

% submachine_deadlock
thf(fact_812_observable__after__language__append,axiom,
    ! [M: fsm_set_a_b_c,Io1: list_P903359562653991662od_b_c,Q: set_a,Io2: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_set_a_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_set_a_b_c @ M @ ( after_set_a_b_c @ M @ Q @ Io1 ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_set_a_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_append
thf(fact_813_observable__after__language__append,axiom,
    ! [M: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Q: a,Io2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Io1 ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_append
thf(fact_814_observable__after__language__append,axiom,
    ! [M: fsm_nat_b_c,Io1: list_P903359562653991662od_b_c,Q: nat,Io2: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_nat_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_nat_b_c @ M @ ( after_nat_b_c @ M @ Q @ Io1 ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_nat_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_append
thf(fact_815_observable__after__language__append,axiom,
    ! [M: fsm_Code_integer_b_c,Io1: list_P903359562653991662od_b_c,Q: code_integer,Io2: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_Code_integer_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ Q @ Io1 ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_Code_integer_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_append
thf(fact_816_observable__after__language__none,axiom,
    ! [M: fsm_set_a_b_c,Io1: list_P903359562653991662od_b_c,Q: set_a,Io2: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_set_a_b_c @ M @ Q ) )
       => ( ~ ( member6330420149250801815od_b_c @ Io2 @ ( lS_set_a_b_c @ M @ ( after_set_a_b_c @ M @ Q @ Io1 ) ) )
         => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_set_a_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_none
thf(fact_817_observable__after__language__none,axiom,
    ! [M: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Q: a,Io2: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M @ Q ) )
       => ( ~ ( member6330420149250801815od_b_c @ Io2 @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Io1 ) ) )
         => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_none
thf(fact_818_observable__after__language__none,axiom,
    ! [M: fsm_nat_b_c,Io1: list_P903359562653991662od_b_c,Q: nat,Io2: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_nat_b_c @ M @ Q ) )
       => ( ~ ( member6330420149250801815od_b_c @ Io2 @ ( lS_nat_b_c @ M @ ( after_nat_b_c @ M @ Q @ Io1 ) ) )
         => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_nat_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_none
thf(fact_819_observable__after__language__none,axiom,
    ! [M: fsm_Code_integer_b_c,Io1: list_P903359562653991662od_b_c,Q: code_integer,Io2: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ( ~ ( member6330420149250801815od_b_c @ Io2 @ ( lS_Code_integer_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ Q @ Io1 ) ) )
         => ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_Code_integer_b_c @ M @ Q ) ) ) ) ) ).

% observable_after_language_none
thf(fact_820_observable__after__eq,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( ( after_set_a_b_c @ M @ Q @ Io1 )
          = ( after_set_a_b_c @ M @ Q @ Io2 ) )
       => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_set_a_b_c @ M @ Q ) )
         => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_set_a_b_c @ M @ Q ) )
           => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_set_a_b_c @ M @ Q ) )
              = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_set_a_b_c @ M @ Q ) ) ) ) ) ) ) ).

% observable_after_eq
thf(fact_821_observable__after__eq,axiom,
    ! [M: fsm_a_b_c,Q: a,Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( ( after_a_b_c @ M @ Q @ Io1 )
          = ( after_a_b_c @ M @ Q @ Io2 ) )
       => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_a_b_c @ M @ Q ) )
         => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_a_b_c @ M @ Q ) )
           => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_a_b_c @ M @ Q ) )
              = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ) ) ).

% observable_after_eq
thf(fact_822_observable__after__eq,axiom,
    ! [M: fsm_nat_b_c,Q: nat,Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( ( after_nat_b_c @ M @ Q @ Io1 )
          = ( after_nat_b_c @ M @ Q @ Io2 ) )
       => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_nat_b_c @ M @ Q ) )
         => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_nat_b_c @ M @ Q ) )
           => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_nat_b_c @ M @ Q ) )
              = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_nat_b_c @ M @ Q ) ) ) ) ) ) ) ).

% observable_after_eq
thf(fact_823_observable__after__eq,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer,Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( ( after_5090571588895392199er_b_c @ M @ Q @ Io1 )
          = ( after_5090571588895392199er_b_c @ M @ Q @ Io2 ) )
       => ( ( member6330420149250801815od_b_c @ Io1 @ ( lS_Code_integer_b_c @ M @ Q ) )
         => ( ( member6330420149250801815od_b_c @ Io2 @ ( lS_Code_integer_b_c @ M @ Q ) )
           => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_Code_integer_b_c @ M @ Q ) )
              = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_Code_integer_b_c @ M @ Q ) ) ) ) ) ) ) ).

% observable_after_eq
thf(fact_824_after__language__iff,axiom,
    ! [M: fsm_set_a_b_c,Alpha: list_P903359562653991662od_b_c,Q: set_a,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_set_a_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Gamma @ ( lS_set_a_b_c @ M @ ( after_set_a_b_c @ M @ Q @ Alpha ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_set_a_b_c @ M @ Q ) ) ) ) ) ).

% after_language_iff
thf(fact_825_after__language__iff,axiom,
    ! [M: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Q: a,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_a_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Gamma @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Alpha ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% after_language_iff
thf(fact_826_after__language__iff,axiom,
    ! [M: fsm_nat_b_c,Alpha: list_P903359562653991662od_b_c,Q: nat,Gamma: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_nat_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Gamma @ ( lS_nat_b_c @ M @ ( after_nat_b_c @ M @ Q @ Alpha ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_nat_b_c @ M @ Q ) ) ) ) ) ).

% after_language_iff
thf(fact_827_after__language__iff,axiom,
    ! [M: fsm_Code_integer_b_c,Alpha: list_P903359562653991662od_b_c,Q: code_integer,Gamma: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Alpha @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ( ( member6330420149250801815od_b_c @ Gamma @ ( lS_Code_integer_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ Q @ Alpha ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_Code_integer_b_c @ M @ Q ) ) ) ) ) ).

% after_language_iff
thf(fact_828_after__split,axiom,
    ! [M: fsm_set_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Q: set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_set_a_b_c @ M @ Q ) )
       => ( ( after_set_a_b_c @ M @ ( after_set_a_b_c @ M @ Q @ Alpha ) @ Gamma )
          = ( after_set_a_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) ) ) ).

% after_split
thf(fact_829_after__split,axiom,
    ! [M: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M @ Q ) )
       => ( ( after_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Alpha ) @ Gamma )
          = ( after_a_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) ) ) ).

% after_split
thf(fact_830_after__split,axiom,
    ! [M: fsm_nat_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Q: nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_nat_b_c @ M @ Q ) )
       => ( ( after_nat_b_c @ M @ ( after_nat_b_c @ M @ Q @ Alpha ) @ Gamma )
          = ( after_nat_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) ) ) ).

% after_split
thf(fact_831_after__split,axiom,
    ! [M: fsm_Code_integer_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Q: code_integer] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ( ( after_5090571588895392199er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ Q @ Alpha ) @ Gamma )
          = ( after_5090571588895392199er_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) ) ) ).

% after_split
thf(fact_832_observable__io__targets__language,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M: fsm_set_a_b_c,Q1: set_a,Q22: set_a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_set_a_b_c @ M @ Q1 ) )
     => ( ( observable_set_a_b_c @ M )
       => ( ( member_set_a @ Q22 @ ( io_targets_set_a_b_c @ M @ Io1 @ Q1 ) )
         => ( member6330420149250801815od_b_c @ Io2 @ ( lS_set_a_b_c @ M @ Q22 ) ) ) ) ) ).

% observable_io_targets_language
thf(fact_833_observable__io__targets__language,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q1: a,Q22: a] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_a_b_c @ M @ Q1 ) )
     => ( ( observable_a_b_c @ M )
       => ( ( member_a @ Q22 @ ( io_targets_a_b_c @ M @ Io1 @ Q1 ) )
         => ( member6330420149250801815od_b_c @ Io2 @ ( lS_a_b_c @ M @ Q22 ) ) ) ) ) ).

% observable_io_targets_language
thf(fact_834_observable__io__targets__language,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M: fsm_nat_b_c,Q1: nat,Q22: nat] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_nat_b_c @ M @ Q1 ) )
     => ( ( observable_nat_b_c @ M )
       => ( ( member_nat @ Q22 @ ( io_targets_nat_b_c @ M @ Io1 @ Q1 ) )
         => ( member6330420149250801815od_b_c @ Io2 @ ( lS_nat_b_c @ M @ Q22 ) ) ) ) ) ).

% observable_io_targets_language
thf(fact_835_observable__io__targets__language,axiom,
    ! [Io1: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,M: fsm_Code_integer_b_c,Q1: code_integer,Q22: code_integer] :
      ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io2 ) @ ( lS_Code_integer_b_c @ M @ Q1 ) )
     => ( ( observ7524490533585752726er_b_c @ M )
       => ( ( member_Code_integer @ Q22 @ ( io_tar3403248182866227591er_b_c @ M @ Io1 @ Q1 ) )
         => ( member6330420149250801815od_b_c @ Io2 @ ( lS_Code_integer_b_c @ M @ Q22 ) ) ) ) ) ).

% observable_io_targets_language
thf(fact_836_after__language__subset,axiom,
    ! [M: fsm_set_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_set_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_set_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_subset
thf(fact_837_after__language__subset,axiom,
    ! [M: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_subset
thf(fact_838_after__language__subset,axiom,
    ! [M: fsm_nat_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_nat_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_nat_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_subset
thf(fact_839_after__language__subset,axiom,
    ! [M: fsm_Code_integer_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_Code_integer_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
         => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_Code_integer_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_subset
thf(fact_840_after__language__append__iff,axiom,
    ! [M: fsm_set_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_set_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_set_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_append_iff
thf(fact_841_after__language__append__iff,axiom,
    ! [M: fsm_a_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_append_iff
thf(fact_842_after__language__append__iff,axiom,
    ! [M: fsm_nat_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_nat_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_nat_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_append_iff
thf(fact_843_after__language__append__iff,axiom,
    ! [M: fsm_Code_integer_b_c,Alpha: list_P903359562653991662od_b_c,Gamma: list_P903359562653991662od_b_c,Beta: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ Beta @ ( lS_Code_integer_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ ( append2547753245680614915od_b_c @ Alpha @ Gamma ) ) ) )
          = ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Gamma @ Beta ) @ ( lS_Code_integer_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ Alpha ) ) ) ) ) ) ).

% after_language_append_iff
thf(fact_844_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z3 @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_845_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z3 @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_846_minimally__distinguishes__no__prefix,axiom,
    ! [M: fsm_set_a_b_c,U: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,V: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c,W3: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ U @ W ) @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ V @ W ) @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
         => ( ( minima7468014887693519109_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ U ) @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ V ) @ ( append2547753245680614915od_b_c @ W @ ( append2547753245680614915od_b_c @ W2 @ W3 ) ) )
           => ( ( W2 != nil_Product_prod_b_c )
             => ~ ( distin5364081577826273247_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ V @ W ) ) @ W3 ) ) ) ) ) ) ).

% minimally_distinguishes_no_prefix
thf(fact_847_minimally__distinguishes__no__prefix,axiom,
    ! [M: fsm_a_b_c,U: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,V: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c,W3: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ U @ W ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ V @ W ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
         => ( ( minima243535863231358885_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ U ) @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ V ) @ ( append2547753245680614915od_b_c @ W @ ( append2547753245680614915od_b_c @ W2 @ W3 ) ) )
           => ( ( W2 != nil_Product_prod_b_c )
             => ~ ( distinguishes_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ V @ W ) ) @ W3 ) ) ) ) ) ) ).

% minimally_distinguishes_no_prefix
thf(fact_848_minimally__distinguishes__no__prefix,axiom,
    ! [M: fsm_nat_b_c,U: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,V: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c,W3: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ U @ W ) @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ V @ W ) @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
         => ( ( minima7825597830629587371at_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ U ) @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ V ) @ ( append2547753245680614915od_b_c @ W @ ( append2547753245680614915od_b_c @ W2 @ W3 ) ) )
           => ( ( W2 != nil_Product_prod_b_c )
             => ~ ( distin957357775430201425at_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ ( append2547753245680614915od_b_c @ V @ W ) ) @ W3 ) ) ) ) ) ) ).

% minimally_distinguishes_no_prefix
thf(fact_849_minimally__distinguishes__no__prefix,axiom,
    ! [M: fsm_Code_integer_b_c,U: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c,V: list_P903359562653991662od_b_c,W2: list_P903359562653991662od_b_c,W3: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ U @ W ) @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ V @ W ) @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
         => ( ( minima4576043283673695160er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ U ) @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ V ) @ ( append2547753245680614915od_b_c @ W @ ( append2547753245680614915od_b_c @ W2 @ W3 ) ) )
           => ( ( W2 != nil_Product_prod_b_c )
             => ~ ( distin3396024279900119646er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ ( append2547753245680614915od_b_c @ U @ W ) ) @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ ( append2547753245680614915od_b_c @ V @ W ) ) @ W3 ) ) ) ) ) ) ).

% minimally_distinguishes_no_prefix
thf(fact_850_distinguish__prepend__initial,axiom,
    ! [M: fsm_set_a_b_c,Io1: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( distin5364081577826273247_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Io1 @ Io ) ) @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Io2 @ Io ) ) @ W )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
         => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
           => ( distin5364081577826273247_a_b_c @ M @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ Io1 ) @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) @ Io2 ) @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ).

% distinguish_prepend_initial
thf(fact_851_distinguish__prepend__initial,axiom,
    ! [M: fsm_a_b_c,Io1: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( distinguishes_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Io1 @ Io ) ) @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ ( append2547753245680614915od_b_c @ Io2 @ Io ) ) @ W )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
         => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
           => ( distinguishes_a_b_c @ M @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ Io1 ) @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) @ Io2 ) @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ).

% distinguish_prepend_initial
thf(fact_852_distinguish__prepend__initial,axiom,
    ! [M: fsm_nat_b_c,Io1: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( distin957357775430201425at_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ ( append2547753245680614915od_b_c @ Io1 @ Io ) ) @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ ( append2547753245680614915od_b_c @ Io2 @ Io ) ) @ W )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
         => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
           => ( distin957357775430201425at_b_c @ M @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ Io1 ) @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) @ Io2 ) @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ).

% distinguish_prepend_initial
thf(fact_853_distinguish__prepend__initial,axiom,
    ! [M: fsm_Code_integer_b_c,Io1: list_P903359562653991662od_b_c,Io: list_P903359562653991662od_b_c,Io2: list_P903359562653991662od_b_c,W: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( distin3396024279900119646er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ ( append2547753245680614915od_b_c @ Io1 @ Io ) ) @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ ( append2547753245680614915od_b_c @ Io2 @ Io ) ) @ W )
       => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io1 @ Io ) @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
         => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io2 @ Io ) @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
           => ( distin3396024279900119646er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ Io1 ) @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) @ Io2 ) @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ).

% distinguish_prepend_initial
thf(fact_854_distinguish__prepend,axiom,
    ! [M: fsm_set_a_b_c,Q1: set_a,Io: list_P903359562653991662od_b_c,Q22: set_a,W: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( distin5364081577826273247_a_b_c @ M @ ( after_set_a_b_c @ M @ Q1 @ Io ) @ ( after_set_a_b_c @ M @ Q22 @ Io ) @ W )
       => ( ( member_set_a @ Q1 @ ( states_set_a_b_c @ M ) )
         => ( ( member_set_a @ Q22 @ ( states_set_a_b_c @ M ) )
           => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q1 ) )
             => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q22 ) )
               => ( distin5364081577826273247_a_b_c @ M @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ) ) ).

% distinguish_prepend
thf(fact_855_distinguish__prepend,axiom,
    ! [M: fsm_a_b_c,Q1: a,Io: list_P903359562653991662od_b_c,Q22: a,W: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( distinguishes_a_b_c @ M @ ( after_a_b_c @ M @ Q1 @ Io ) @ ( after_a_b_c @ M @ Q22 @ Io ) @ W )
       => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
         => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
           => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q1 ) )
             => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q22 ) )
               => ( distinguishes_a_b_c @ M @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ) ) ).

% distinguish_prepend
thf(fact_856_distinguish__prepend,axiom,
    ! [M: fsm_nat_b_c,Q1: nat,Io: list_P903359562653991662od_b_c,Q22: nat,W: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( distin957357775430201425at_b_c @ M @ ( after_nat_b_c @ M @ Q1 @ Io ) @ ( after_nat_b_c @ M @ Q22 @ Io ) @ W )
       => ( ( member_nat @ Q1 @ ( states_nat_b_c @ M ) )
         => ( ( member_nat @ Q22 @ ( states_nat_b_c @ M ) )
           => ( ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q1 ) )
             => ( ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q22 ) )
               => ( distin957357775430201425at_b_c @ M @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ) ) ).

% distinguish_prepend
thf(fact_857_distinguish__prepend,axiom,
    ! [M: fsm_Code_integer_b_c,Q1: code_integer,Io: list_P903359562653991662od_b_c,Q22: code_integer,W: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( distin3396024279900119646er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ Q1 @ Io ) @ ( after_5090571588895392199er_b_c @ M @ Q22 @ Io ) @ W )
       => ( ( member_Code_integer @ Q1 @ ( states1921009325540848725er_b_c @ M ) )
         => ( ( member_Code_integer @ Q22 @ ( states1921009325540848725er_b_c @ M ) )
           => ( ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q1 ) )
             => ( ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q22 ) )
               => ( distin3396024279900119646er_b_c @ M @ Q1 @ Q22 @ ( append2547753245680614915od_b_c @ Io @ W ) ) ) ) ) ) ) ) ).

% distinguish_prepend
thf(fact_858_does__distinguish__correctness,axiom,
    ! [M: fsm_set_a_b_c,Q1: set_a,Q22: set_a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member_set_a @ Q1 @ ( states_set_a_b_c @ M ) )
       => ( ( member_set_a @ Q22 @ ( states_set_a_b_c @ M ) )
         => ( ( does_d4832897855063403206_a_b_c @ M @ Q1 @ Q22 @ Io )
            = ( distin5364081577826273247_a_b_c @ M @ Q1 @ Q22 @ Io ) ) ) ) ) ).

% does_distinguish_correctness
thf(fact_859_does__distinguish__correctness,axiom,
    ! [M: fsm_a_b_c,Q1: a,Q22: a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member_a @ Q1 @ ( states_a_b_c @ M ) )
       => ( ( member_a @ Q22 @ ( states_a_b_c @ M ) )
         => ( ( does_d3309555387068894566_a_b_c @ M @ Q1 @ Q22 @ Io )
            = ( distinguishes_a_b_c @ M @ Q1 @ Q22 @ Io ) ) ) ) ) ).

% does_distinguish_correctness
thf(fact_860_does__distinguish__correctness,axiom,
    ! [M: fsm_nat_b_c,Q1: nat,Q22: nat,Io: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member_nat @ Q1 @ ( states_nat_b_c @ M ) )
       => ( ( member_nat @ Q22 @ ( states_nat_b_c @ M ) )
         => ( ( does_d5425369375050208682at_b_c @ M @ Q1 @ Q22 @ Io )
            = ( distin957357775430201425at_b_c @ M @ Q1 @ Q22 @ Io ) ) ) ) ) ).

% does_distinguish_correctness
thf(fact_861_does__distinguish__correctness,axiom,
    ! [M: fsm_Code_integer_b_c,Q1: code_integer,Q22: code_integer,Io: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member_Code_integer @ Q1 @ ( states1921009325540848725er_b_c @ M ) )
       => ( ( member_Code_integer @ Q22 @ ( states1921009325540848725er_b_c @ M ) )
         => ( ( does_d7537200161847468983er_b_c @ M @ Q1 @ Q22 @ Io )
            = ( distin3396024279900119646er_b_c @ M @ Q1 @ Q22 @ Io ) ) ) ) ) ).

% does_distinguish_correctness
thf(fact_862_from__FSM__acyclic,axiom,
    ! [Q: a,M: fsm_a_b_c] :
      ( ( member_a @ Q @ ( reacha1620305530751930115_a_b_c @ M ) )
     => ( ( acyclic_a_b_c @ M )
       => ( acyclic_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) ) ) ).

% from_FSM_acyclic
thf(fact_863_from__FSM__acyclic,axiom,
    ! [Q: set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ Q @ ( reacha7078015818774689891_a_b_c @ M ) )
     => ( ( acyclic_set_a_b_c @ M )
       => ( acyclic_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) ) ) ) ).

% from_FSM_acyclic
thf(fact_864_deadlock__state__alt__def__h,axiom,
    ( deadlo5770673589738903088_a_b_c
    = ( ^ [M2: fsm_set_a_b_c,Q5: set_a] :
        ! [X2: b] :
          ( ( member_b @ X2 @ ( inputs_set_a_b_c @ M2 ) )
         => ( ( h_set_a_b_c @ M2 @ ( product_Pair_set_a_b @ Q5 @ X2 ) )
            = bot_bo4411629339732149841_set_a ) ) ) ) ).

% deadlock_state_alt_def_h
thf(fact_865_deadlock__state__alt__def__h,axiom,
    ( deadlock_state_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q5: a] :
        ! [X2: b] :
          ( ( member_b @ X2 @ ( inputs_a_b_c @ M2 ) )
         => ( ( h_a_b_c @ M2 @ ( product_Pair_a_b @ Q5 @ X2 ) )
            = bot_bo2086078286244720881od_c_a ) ) ) ) ).

% deadlock_state_alt_def_h
thf(fact_866_after__transition,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,X: b,Y: c,Q3: set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6517745037573369951_set_a @ ( produc3915556858435501990_set_a @ Q @ ( produc799526633598312797_set_a @ X @ ( product_Pair_c_set_a @ Y @ Q3 ) ) ) @ ( transi8205445899535437940_a_b_c @ M ) )
       => ( ( after_set_a_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
          = Q3 ) ) ) ).

% after_transition
thf(fact_867_after__transition,axiom,
    ! [M: fsm_a_b_c,Q: a,X: b,Y: c,Q3: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q @ ( produc3440865914000881661od_c_a @ X @ ( product_Pair_c_a @ Y @ Q3 ) ) ) @ ( transitions_a_b_c @ M ) )
       => ( ( after_a_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
          = Q3 ) ) ) ).

% after_transition
thf(fact_868_after__transition,axiom,
    ! [M: fsm_nat_b_c,Q: nat,X: b,Y: c,Q3: nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member3717341653178413575_c_nat @ ( produc2243381569475206934_c_nat @ Q @ ( produc2831854573163739687_c_nat @ X @ ( product_Pair_c_nat @ Y @ Q3 ) ) ) @ ( transitions_nat_b_c @ M ) )
       => ( ( after_nat_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
          = Q3 ) ) ) ).

% after_transition
thf(fact_869_after__transition,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer,X: b,Y: c,Q3: code_integer] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member5112046945434310049nteger @ ( produc1187772602920745264nteger @ Q @ ( produc1840861740276527156nteger @ X @ ( produc2131686081043252230nteger @ Y @ Q3 ) ) ) @ ( transi6707074847254859401er_b_c @ M ) )
       => ( ( after_5090571588895392199er_b_c @ M @ Q @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
          = Q3 ) ) ) ).

% after_transition
thf(fact_870_language__io_I1_J,axiom,
    ! [Io: list_P903359562653991662od_b_c,M: fsm_set_a_b_c,Q: set_a,X: b,Y: c] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ ( set_Product_prod_b_c2 @ Io ) )
       => ( member_b @ X @ ( inputs_set_a_b_c @ M ) ) ) ) ).

% language_io(1)
thf(fact_871_language__io_I1_J,axiom,
    ! [Io: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a,X: b,Y: c] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ ( set_Product_prod_b_c2 @ Io ) )
       => ( member_b @ X @ ( inputs_a_b_c @ M ) ) ) ) ).

% language_io(1)
thf(fact_872_language__io_I2_J,axiom,
    ! [Io: list_P903359562653991662od_b_c,M: fsm_set_a_b_c,Q: set_a,X: b,Y: c] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ ( set_Product_prod_b_c2 @ Io ) )
       => ( member_c @ Y @ ( outputs_set_a_b_c @ M ) ) ) ) ).

% language_io(2)
thf(fact_873_language__io_I2_J,axiom,
    ! [Io: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a,X: b,Y: c] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X @ Y ) @ ( set_Product_prod_b_c2 @ Io ) )
       => ( member_c @ Y @ ( outputs_a_b_c @ M ) ) ) ) ).

% language_io(2)
thf(fact_874_observable__alt__def,axiom,
    ( observable_set_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
        ! [Q13: set_a,X2: b,Y4: c,Q14: set_a,Q15: set_a] :
          ( ( ( member6517745037573369951_set_a @ ( produc3915556858435501990_set_a @ Q13 @ ( produc799526633598312797_set_a @ X2 @ ( product_Pair_c_set_a @ Y4 @ Q14 ) ) ) @ ( transi8205445899535437940_a_b_c @ M2 ) )
            & ( member6517745037573369951_set_a @ ( produc3915556858435501990_set_a @ Q13 @ ( produc799526633598312797_set_a @ X2 @ ( product_Pair_c_set_a @ Y4 @ Q15 ) ) ) @ ( transi8205445899535437940_a_b_c @ M2 ) ) )
         => ( Q14 = Q15 ) ) ) ) ).

% observable_alt_def
thf(fact_875_observable__alt__def,axiom,
    ( observable_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q13: a,X2: b,Y4: c,Q14: a,Q15: a] :
          ( ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q13 @ ( produc3440865914000881661od_c_a @ X2 @ ( product_Pair_c_a @ Y4 @ Q14 ) ) ) @ ( transitions_a_b_c @ M2 ) )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q13 @ ( produc3440865914000881661od_c_a @ X2 @ ( product_Pair_c_a @ Y4 @ Q15 ) ) ) @ ( transitions_a_b_c @ M2 ) ) )
         => ( Q14 = Q15 ) ) ) ) ).

% observable_alt_def
thf(fact_876_observable__alt__def,axiom,
    ( observable_nat_b_c
    = ( ^ [M2: fsm_nat_b_c] :
        ! [Q13: nat,X2: b,Y4: c,Q14: nat,Q15: nat] :
          ( ( ( member3717341653178413575_c_nat @ ( produc2243381569475206934_c_nat @ Q13 @ ( produc2831854573163739687_c_nat @ X2 @ ( product_Pair_c_nat @ Y4 @ Q14 ) ) ) @ ( transitions_nat_b_c @ M2 ) )
            & ( member3717341653178413575_c_nat @ ( produc2243381569475206934_c_nat @ Q13 @ ( produc2831854573163739687_c_nat @ X2 @ ( product_Pair_c_nat @ Y4 @ Q15 ) ) ) @ ( transitions_nat_b_c @ M2 ) ) )
         => ( Q14 = Q15 ) ) ) ) ).

% observable_alt_def
thf(fact_877_observable__alt__def,axiom,
    ( observ7524490533585752726er_b_c
    = ( ^ [M2: fsm_Code_integer_b_c] :
        ! [Q13: code_integer,X2: b,Y4: c,Q14: code_integer,Q15: code_integer] :
          ( ( ( member5112046945434310049nteger @ ( produc1187772602920745264nteger @ Q13 @ ( produc1840861740276527156nteger @ X2 @ ( produc2131686081043252230nteger @ Y4 @ Q14 ) ) ) @ ( transi6707074847254859401er_b_c @ M2 ) )
            & ( member5112046945434310049nteger @ ( produc1187772602920745264nteger @ Q13 @ ( produc1840861740276527156nteger @ X2 @ ( produc2131686081043252230nteger @ Y4 @ Q15 ) ) ) @ ( transi6707074847254859401er_b_c @ M2 ) ) )
         => ( Q14 = Q15 ) ) ) ) ).

% observable_alt_def
thf(fact_878_language__maximal__contained__prefix__ob,axiom,
    ! [Io: list_P903359562653991662od_b_c,M: fsm_set_a_b_c,Q: set_a] :
      ( ~ ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
     => ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
       => ( ( observable_set_a_b_c @ M )
         => ~ ! [Io3: list_P903359562653991662od_b_c,X3: b,Y2: c] :
                ( ? [Io4: list_P903359562653991662od_b_c] :
                    ( Io
                    = ( append2547753245680614915od_b_c @ Io3 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) @ nil_Product_prod_b_c ) @ Io4 ) ) )
               => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_set_a_b_c @ M @ Q ) )
                 => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) @ nil_Product_prod_b_c ) ) @ ( lS_set_a_b_c @ M @ Q ) ) ) ) ) ) ) ).

% language_maximal_contained_prefix_ob
thf(fact_879_language__maximal__contained__prefix__ob,axiom,
    ! [Io: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a] :
      ( ~ ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( ( observable_a_b_c @ M )
         => ~ ! [Io3: list_P903359562653991662od_b_c,X3: b,Y2: c] :
                ( ? [Io4: list_P903359562653991662od_b_c] :
                    ( Io
                    = ( append2547753245680614915od_b_c @ Io3 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) @ nil_Product_prod_b_c ) @ Io4 ) ) )
               => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_a_b_c @ M @ Q ) )
                 => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ) ) ).

% language_maximal_contained_prefix_ob
thf(fact_880_language__maximal__contained__prefix__ob,axiom,
    ! [Io: list_P903359562653991662od_b_c,M: fsm_nat_b_c,Q: nat] :
      ( ~ ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q ) )
     => ( ( member_nat @ Q @ ( states_nat_b_c @ M ) )
       => ( ( observable_nat_b_c @ M )
         => ~ ! [Io3: list_P903359562653991662od_b_c,X3: b,Y2: c] :
                ( ? [Io4: list_P903359562653991662od_b_c] :
                    ( Io
                    = ( append2547753245680614915od_b_c @ Io3 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) @ nil_Product_prod_b_c ) @ Io4 ) ) )
               => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_nat_b_c @ M @ Q ) )
                 => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) @ nil_Product_prod_b_c ) ) @ ( lS_nat_b_c @ M @ Q ) ) ) ) ) ) ) ).

% language_maximal_contained_prefix_ob
thf(fact_881_language__maximal__contained__prefix__ob,axiom,
    ! [Io: list_P903359562653991662od_b_c,M: fsm_Code_integer_b_c,Q: code_integer] :
      ( ~ ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q ) )
     => ( ( member_Code_integer @ Q @ ( states1921009325540848725er_b_c @ M ) )
       => ( ( observ7524490533585752726er_b_c @ M )
         => ~ ! [Io3: list_P903359562653991662od_b_c,X3: b,Y2: c] :
                ( ? [Io4: list_P903359562653991662od_b_c] :
                    ( Io
                    = ( append2547753245680614915od_b_c @ Io3 @ ( append2547753245680614915od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) @ nil_Product_prod_b_c ) @ Io4 ) ) )
               => ( ( member6330420149250801815od_b_c @ Io3 @ ( lS_Code_integer_b_c @ M @ Q ) )
                 => ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io3 @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X3 @ Y2 ) @ nil_Product_prod_b_c ) ) @ ( lS_Code_integer_b_c @ M @ Q ) ) ) ) ) ) ) ).

% language_maximal_contained_prefix_ob
thf(fact_882_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat @ X1 @ X22 )
          & ( ord_less_eq_nat @ Y1 @ Y22 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_883_less__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
      ( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X22 @ Y22 ) )
      = ( ( ord_less_nat @ X1 @ X22 )
        | ( ( ord_less_eq_nat @ X1 @ X22 )
          & ( ord_less_nat @ Y1 @ Y22 ) ) ) ) ).

% less_prod_simp
thf(fact_884_completely__specified__language__extension,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,Io: list_P903359562653991662od_b_c,X: b] :
      ( ( comple9199925562481299400_a_b_c @ M )
     => ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
       => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
         => ( ( member_b @ X @ ( inputs_set_a_b_c @ M ) )
           => ~ ! [Y2: c] :
                  ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ nil_Product_prod_b_c ) ) @ ( lS_set_a_b_c @ M @ Q ) ) ) ) ) ) ).

% completely_specified_language_extension
thf(fact_885_completely__specified__language__extension,axiom,
    ! [M: fsm_a_b_c,Q: a,Io: list_P903359562653991662od_b_c,X: b] :
      ( ( comple6900478119722646120_a_b_c @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
         => ( ( member_b @ X @ ( inputs_a_b_c @ M ) )
           => ~ ! [Y2: c] :
                  ~ ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y2 ) @ nil_Product_prod_b_c ) ) @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ) ).

% completely_specified_language_extension
thf(fact_886_output__complete__alt__def,axiom,
    ( output5228291214970145744_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
        ! [Q5: set_a,X2: b] :
          ( ? [Y4: c,Q6: set_a] : ( member6517745037573369951_set_a @ ( produc3915556858435501990_set_a @ Q5 @ ( produc799526633598312797_set_a @ X2 @ ( product_Pair_c_set_a @ Y4 @ Q6 ) ) ) @ ( transi8205445899535437940_a_b_c @ M2 ) )
         => ! [Y4: c] :
              ( ( member_c @ Y4 @ ( outputs_set_a_b_c @ M2 ) )
             => ? [Q6: set_a] : ( member6517745037573369951_set_a @ ( produc3915556858435501990_set_a @ Q5 @ ( produc799526633598312797_set_a @ X2 @ ( product_Pair_c_set_a @ Y4 @ Q6 ) ) ) @ ( transi8205445899535437940_a_b_c @ M2 ) ) ) ) ) ) ).

% output_complete_alt_def
thf(fact_887_output__complete__alt__def,axiom,
    ( output3834324022317553264_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q5: a,X2: b] :
          ( ? [Y4: c,Q6: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q5 @ ( produc3440865914000881661od_c_a @ X2 @ ( product_Pair_c_a @ Y4 @ Q6 ) ) ) @ ( transitions_a_b_c @ M2 ) )
         => ! [Y4: c] :
              ( ( member_c @ Y4 @ ( outputs_a_b_c @ M2 ) )
             => ? [Q6: a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Q5 @ ( produc3440865914000881661od_c_a @ X2 @ ( product_Pair_c_a @ Y4 @ Q6 ) ) ) @ ( transitions_a_b_c @ M2 ) ) ) ) ) ) ).

% output_complete_alt_def
thf(fact_888_from__FSM__completely__specified,axiom,
    ! [M: fsm_a_b_c,Q: a] :
      ( ( comple6900478119722646120_a_b_c @ M )
     => ( comple6900478119722646120_a_b_c @ ( from_FSM_a_b_c @ M @ Q ) ) ) ).

% from_FSM_completely_specified
thf(fact_889_from__FSM__completely__specified,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a] :
      ( ( comple9199925562481299400_a_b_c @ M )
     => ( comple9199925562481299400_a_b_c @ ( from_FSM_set_a_b_c @ M @ Q ) ) ) ).

% from_FSM_completely_specified
thf(fact_890_completely__specified__alt__def__h,axiom,
    ( comple9199925562481299400_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
        ! [X2: set_a] :
          ( ( member_set_a @ X2 @ ( states_set_a_b_c @ M2 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs_set_a_b_c @ M2 ) )
             => ( ( h_set_a_b_c @ M2 @ ( product_Pair_set_a_b @ X2 @ Y4 ) )
               != bot_bo4411629339732149841_set_a ) ) ) ) ) ).

% completely_specified_alt_def_h
thf(fact_891_completely__specified__alt__def__h,axiom,
    ( comple6900478119722646120_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X2: a] :
          ( ( member_a @ X2 @ ( states_a_b_c @ M2 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs_a_b_c @ M2 ) )
             => ( ( h_a_b_c @ M2 @ ( product_Pair_a_b @ X2 @ Y4 ) )
               != bot_bo2086078286244720881od_c_a ) ) ) ) ) ).

% completely_specified_alt_def_h
thf(fact_892_output__complete__alt__def__h,axiom,
    ( output5228291214970145744_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
        ! [Q5: set_a,X2: b] :
          ( ( ( h_set_a_b_c @ M2 @ ( product_Pair_set_a_b @ Q5 @ X2 ) )
           != bot_bo4411629339732149841_set_a )
         => ! [Y4: c] :
              ( ( member_c @ Y4 @ ( outputs_set_a_b_c @ M2 ) )
             => ? [Q6: set_a] : ( member7460900956140502318_set_a @ ( product_Pair_c_set_a @ Y4 @ Q6 ) @ ( h_set_a_b_c @ M2 @ ( product_Pair_set_a_b @ Q5 @ X2 ) ) ) ) ) ) ) ).

% output_complete_alt_def_h
thf(fact_893_output__complete__alt__def__h,axiom,
    ( output3834324022317553264_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q5: a,X2: b] :
          ( ( ( h_a_b_c @ M2 @ ( product_Pair_a_b @ Q5 @ X2 ) )
           != bot_bo2086078286244720881od_c_a )
         => ! [Y4: c] :
              ( ( member_c @ Y4 @ ( outputs_a_b_c @ M2 ) )
             => ? [Q6: a] : ( member5074992350434858958od_c_a @ ( product_Pair_c_a @ Y4 @ Q6 ) @ ( h_a_b_c @ M2 @ ( product_Pair_a_b @ Q5 @ X2 ) ) ) ) ) ) ) ).

% output_complete_alt_def_h
thf(fact_894_completely__specified__alt__def,axiom,
    ( comple9199925562481299400_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
        ! [X2: set_a] :
          ( ( member_set_a @ X2 @ ( states_set_a_b_c @ M2 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs_set_a_b_c @ M2 ) )
             => ? [Q6: set_a,Z4: c] : ( member6517745037573369951_set_a @ ( produc3915556858435501990_set_a @ X2 @ ( produc799526633598312797_set_a @ Y4 @ ( product_Pair_c_set_a @ Z4 @ Q6 ) ) ) @ ( transi8205445899535437940_a_b_c @ M2 ) ) ) ) ) ) ).

% completely_specified_alt_def
thf(fact_895_completely__specified__alt__def,axiom,
    ( comple6900478119722646120_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X2: a] :
          ( ( member_a @ X2 @ ( states_a_b_c @ M2 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs_a_b_c @ M2 ) )
             => ? [Q6: a,Z4: c] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X2 @ ( produc3440865914000881661od_c_a @ Y4 @ ( product_Pair_c_a @ Z4 @ Q6 ) ) ) @ ( transitions_a_b_c @ M2 ) ) ) ) ) ) ).

% completely_specified_alt_def
thf(fact_896_completely__specified__state__alt__def__h,axiom,
    ( comple6745895220685239651_a_b_c
    = ( ^ [M2: fsm_set_a_b_c,Q5: set_a] :
        ! [X2: b] :
          ( ( member_b @ X2 @ ( inputs_set_a_b_c @ M2 ) )
         => ( ( h_set_a_b_c @ M2 @ ( product_Pair_set_a_b @ Q5 @ X2 ) )
           != bot_bo4411629339732149841_set_a ) ) ) ) ).

% completely_specified_state_alt_def_h
thf(fact_897_completely__specified__state__alt__def__h,axiom,
    ( comple2039557650379672579_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q5: a] :
        ! [X2: b] :
          ( ( member_b @ X2 @ ( inputs_a_b_c @ M2 ) )
         => ( ( h_a_b_c @ M2 @ ( product_Pair_a_b @ Q5 @ X2 ) )
           != bot_bo2086078286244720881od_c_a ) ) ) ) ).

% completely_specified_state_alt_def_h
thf(fact_898_complete__submachine__initial,axiom,
    ! [A: fsm_a_b_c,B: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ A @ B )
     => ( ( comple6900478119722646120_a_b_c @ A )
       => ( comple2039557650379672579_a_b_c @ B @ ( initial_a_b_c @ B ) ) ) ) ).

% complete_submachine_initial
thf(fact_899_complete__submachine__initial,axiom,
    ! [A: fsm_set_a_b_c,B: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ A @ B )
     => ( ( comple9199925562481299400_a_b_c @ A )
       => ( comple6745895220685239651_a_b_c @ B @ ( initial_set_a_b_c @ B ) ) ) ) ).

% complete_submachine_initial
thf(fact_900_completely__specified__path__extension,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,P2: list_P6923848007992943548_set_a,X: b] :
      ( ( comple9199925562481299400_a_b_c @ M )
     => ( ( member_set_a @ Q @ ( states_set_a_b_c @ M ) )
       => ( ( path_set_a_b_c @ M @ Q @ P2 )
         => ( ( member_b @ X @ ( inputs_set_a_b_c @ M ) )
           => ~ ! [T2: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ T2 @ ( transi8205445899535437940_a_b_c @ M ) )
                 => ( ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T2 ) )
                      = X )
                   => ( ( produc7645221154340064658_set_a @ T2 )
                     != ( target_set_a_b_c @ Q @ P2 ) ) ) ) ) ) ) ) ).

% completely_specified_path_extension
thf(fact_901_completely__specified__path__extension,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a,X: b] :
      ( ( comple6900478119722646120_a_b_c @ M )
     => ( ( member_a @ Q @ ( states_a_b_c @ M ) )
       => ( ( path_a_b_c @ M @ Q @ P2 )
         => ( ( member_b @ X @ ( inputs_a_b_c @ M ) )
           => ~ ! [T2: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ T2 @ ( transitions_a_b_c @ M ) )
                 => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) )
                      = X )
                   => ( ( produc4225943634317630866od_c_a @ T2 )
                     != ( target_a_b_c @ Q @ P2 ) ) ) ) ) ) ) ) ).

% completely_specified_path_extension
thf(fact_902_completely__specified__io__targets,axiom,
    ! [M: fsm_set_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( comple9199925562481299400_a_b_c @ M )
     => ! [X5: set_a] :
          ( ( member_set_a @ X5 @ ( io_targets_set_a_b_c @ M @ Io @ ( initial_set_a_b_c @ M ) ) )
         => ! [Xa2: b] :
              ( ( member_b @ Xa2 @ ( inputs_set_a_b_c @ M ) )
             => ? [Xb: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ Xb @ ( transi8205445899535437940_a_b_c @ M ) )
                  & ( ( produc7645221154340064658_set_a @ Xb )
                    = X5 )
                  & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xb ) )
                    = Xa2 ) ) ) ) ) ).

% completely_specified_io_targets
thf(fact_903_completely__specified__io__targets,axiom,
    ! [M: fsm_a_b_c,Io: list_P903359562653991662od_b_c] :
      ( ( comple6900478119722646120_a_b_c @ M )
     => ! [X5: a] :
          ( ( member_a @ X5 @ ( io_targets_a_b_c @ M @ Io @ ( initial_a_b_c @ M ) ) )
         => ! [Xa2: b] :
              ( ( member_b @ Xa2 @ ( inputs_a_b_c @ M ) )
             => ? [Xb: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Xb @ ( transitions_a_b_c @ M ) )
                  & ( ( produc4225943634317630866od_c_a @ Xb )
                    = X5 )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb ) )
                    = Xa2 ) ) ) ) ) ).

% completely_specified_io_targets
thf(fact_904_completely__specified_Osimps,axiom,
    ( comple9199925562481299400_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
        ! [X2: set_a] :
          ( ( member_set_a @ X2 @ ( states_set_a_b_c @ M2 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs_set_a_b_c @ M2 ) )
             => ? [Z4: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ Z4 @ ( transi8205445899535437940_a_b_c @ M2 ) )
                  & ( ( produc7645221154340064658_set_a @ Z4 )
                    = X2 )
                  & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Z4 ) )
                    = Y4 ) ) ) ) ) ) ).

% completely_specified.simps
thf(fact_905_completely__specified_Osimps,axiom,
    ( comple6900478119722646120_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X2: a] :
          ( ( member_a @ X2 @ ( states_a_b_c @ M2 ) )
         => ! [Y4: b] :
              ( ( member_b @ Y4 @ ( inputs_a_b_c @ M2 ) )
             => ? [Z4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Z4 @ ( transitions_a_b_c @ M2 ) )
                  & ( ( produc4225943634317630866od_c_a @ Z4 )
                    = X2 )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) )
                    = Y4 ) ) ) ) ) ) ).

% completely_specified.simps
thf(fact_906_completely__specified_Oelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Y: $o] :
      ( ( ( comple9199925562481299400_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X2: set_a] :
              ( ( member_set_a @ X2 @ ( states_set_a_b_c @ X ) )
             => ! [Y4: b] :
                  ( ( member_b @ Y4 @ ( inputs_set_a_b_c @ X ) )
                 => ? [Z4: produc373761375140688950_set_a] :
                      ( ( member6517745037573369951_set_a @ Z4 @ ( transi8205445899535437940_a_b_c @ X ) )
                      & ( ( produc7645221154340064658_set_a @ Z4 )
                        = X2 )
                      & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Z4 ) )
                        = Y4 ) ) ) ) ) ) ) ).

% completely_specified.elims(1)
thf(fact_907_completely__specified_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( comple6900478119722646120_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X2: a] :
              ( ( member_a @ X2 @ ( states_a_b_c @ X ) )
             => ! [Y4: b] :
                  ( ( member_b @ Y4 @ ( inputs_a_b_c @ X ) )
                 => ? [Z4: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Z4 @ ( transitions_a_b_c @ X ) )
                      & ( ( produc4225943634317630866od_c_a @ Z4 )
                        = X2 )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) )
                        = Y4 ) ) ) ) ) ) ) ).

% completely_specified.elims(1)
thf(fact_908_fsm__transition__input,axiom,
    ! [T: produc373761375140688950_set_a,M: fsm_set_a_b_c] :
      ( ( member6517745037573369951_set_a @ T @ ( transi8205445899535437940_a_b_c @ M ) )
     => ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) @ ( inputs_set_a_b_c @ M ) ) ) ).

% fsm_transition_input
thf(fact_909_fsm__transition__input,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M ) ) ) ).

% fsm_transition_input
thf(fact_910_fsm__transition__output,axiom,
    ! [T: produc373761375140688950_set_a,M: fsm_set_a_b_c] :
      ( ( member6517745037573369951_set_a @ T @ ( transi8205445899535437940_a_b_c @ M ) )
     => ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( outputs_set_a_b_c @ M ) ) ) ).

% fsm_transition_output
thf(fact_911_fsm__transition__output,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
     => ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M ) ) ) ).

% fsm_transition_output
thf(fact_912_observable_Oelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ~ ( observable_set_a_b_c @ X )
     => ~ ! [X3: produc373761375140688950_set_a] :
            ( ( member6517745037573369951_set_a @ X3 @ ( transi8205445899535437940_a_b_c @ X ) )
           => ! [Xa3: produc373761375140688950_set_a] :
                ( ( member6517745037573369951_set_a @ Xa3 @ ( transi8205445899535437940_a_b_c @ X ) )
               => ( ( ( ( produc7645221154340064658_set_a @ X3 )
                      = ( produc7645221154340064658_set_a @ Xa3 ) )
                    & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X3 ) )
                      = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xa3 ) ) )
                    & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X3 ) ) )
                      = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xa3 ) ) ) ) )
                 => ( ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X3 ) ) )
                    = ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xa3 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_913_observable_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( observable_a_b_c @ X )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X ) )
           => ! [Xa3: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X ) )
               => ( ( ( ( produc4225943634317630866od_c_a @ X3 )
                      = ( produc4225943634317630866od_c_a @ Xa3 ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) )
                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                      = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) )
                 => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                    = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_914_observable_Oelims_I3_J,axiom,
    ! [X: fsm_nat_b_c] :
      ( ~ ( observable_nat_b_c @ X )
     => ~ ! [X3: produc8586872805302537566_c_nat] :
            ( ( member3717341653178413575_c_nat @ X3 @ ( transitions_nat_b_c @ X ) )
           => ! [Xa3: produc8586872805302537566_c_nat] :
                ( ( member3717341653178413575_c_nat @ Xa3 @ ( transitions_nat_b_c @ X ) )
               => ( ( ( ( produc532606554400396034_c_nat @ X3 )
                      = ( produc532606554400396034_c_nat @ Xa3 ) )
                    & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) )
                      = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ Xa3 ) ) )
                    & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) ) )
                      = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Xa3 ) ) ) ) )
                 => ( ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) ) )
                    = ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Xa3 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_915_observable_Oelims_I3_J,axiom,
    ! [X: fsm_Code_integer_b_c] :
      ( ~ ( observ7524490533585752726er_b_c @ X )
     => ~ ! [X3: produc558599814252099832nteger] :
            ( ( member5112046945434310049nteger @ X3 @ ( transi6707074847254859401er_b_c @ X ) )
           => ! [Xa3: produc558599814252099832nteger] :
                ( ( member5112046945434310049nteger @ Xa3 @ ( transi6707074847254859401er_b_c @ X ) )
               => ( ( ( ( produc7104377276419476764nteger @ X3 )
                      = ( produc7104377276419476764nteger @ Xa3 ) )
                    & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ X3 ) )
                      = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ Xa3 ) ) )
                    & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X3 ) ) )
                      = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Xa3 ) ) ) ) )
                 => ( ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X3 ) ) )
                    = ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Xa3 ) ) ) ) ) ) ) ) ).

% observable.elims(3)
thf(fact_916_observable_Oelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ( observable_set_a_b_c @ X )
     => ! [X5: produc373761375140688950_set_a] :
          ( ( member6517745037573369951_set_a @ X5 @ ( transi8205445899535437940_a_b_c @ X ) )
         => ! [Xa2: produc373761375140688950_set_a] :
              ( ( member6517745037573369951_set_a @ Xa2 @ ( transi8205445899535437940_a_b_c @ X ) )
             => ( ( ( ( produc7645221154340064658_set_a @ X5 )
                    = ( produc7645221154340064658_set_a @ Xa2 ) )
                  & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X5 ) )
                    = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xa2 ) ) )
                  & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X5 ) ) )
                    = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xa2 ) ) ) ) )
               => ( ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X5 ) ) )
                  = ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_917_observable_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( observable_a_b_c @ X )
     => ! [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X ) )
         => ! [Xa2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X5 )
                    = ( produc4225943634317630866od_c_a @ Xa2 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_918_observable_Oelims_I2_J,axiom,
    ! [X: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ X )
     => ! [X5: produc8586872805302537566_c_nat] :
          ( ( member3717341653178413575_c_nat @ X5 @ ( transitions_nat_b_c @ X ) )
         => ! [Xa2: produc8586872805302537566_c_nat] :
              ( ( member3717341653178413575_c_nat @ Xa2 @ ( transitions_nat_b_c @ X ) )
             => ( ( ( ( produc532606554400396034_c_nat @ X5 )
                    = ( produc532606554400396034_c_nat @ Xa2 ) )
                  & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ X5 ) )
                    = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ Xa2 ) ) )
                  & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X5 ) ) )
                    = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Xa2 ) ) ) ) )
               => ( ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X5 ) ) )
                  = ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_919_observable_Oelims_I2_J,axiom,
    ! [X: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ X )
     => ! [X5: produc558599814252099832nteger] :
          ( ( member5112046945434310049nteger @ X5 @ ( transi6707074847254859401er_b_c @ X ) )
         => ! [Xa2: produc558599814252099832nteger] :
              ( ( member5112046945434310049nteger @ Xa2 @ ( transi6707074847254859401er_b_c @ X ) )
             => ( ( ( ( produc7104377276419476764nteger @ X5 )
                    = ( produc7104377276419476764nteger @ Xa2 ) )
                  & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ X5 ) )
                    = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ Xa2 ) ) )
                  & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X5 ) ) )
                    = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Xa2 ) ) ) ) )
               => ( ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X5 ) ) )
                  = ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Xa2 ) ) ) ) ) ) ) ) ).

% observable.elims(2)
thf(fact_920_observable_Oelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Y: $o] :
      ( ( ( observable_set_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X2: produc373761375140688950_set_a] :
              ( ( member6517745037573369951_set_a @ X2 @ ( transi8205445899535437940_a_b_c @ X ) )
             => ! [Y4: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ Y4 @ ( transi8205445899535437940_a_b_c @ X ) )
                 => ( ( ( ( produc7645221154340064658_set_a @ X2 )
                        = ( produc7645221154340064658_set_a @ Y4 ) )
                      & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X2 ) )
                        = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Y4 ) ) )
                      & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X2 ) ) )
                        = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Y4 ) ) ) ) )
                   => ( ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X2 ) ) )
                      = ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_921_observable_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( observable_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X2 @ ( transitions_a_b_c @ X ) )
             => ! [Y4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X ) )
                 => ( ( ( ( produc4225943634317630866od_c_a @ X2 )
                        = ( produc4225943634317630866od_c_a @ Y4 ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) )
                      & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) ) )
                        = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) )
                   => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) ) )
                      = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_922_observable_Oelims_I1_J,axiom,
    ! [X: fsm_nat_b_c,Y: $o] :
      ( ( ( observable_nat_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X2: produc8586872805302537566_c_nat] :
              ( ( member3717341653178413575_c_nat @ X2 @ ( transitions_nat_b_c @ X ) )
             => ! [Y4: produc8586872805302537566_c_nat] :
                  ( ( member3717341653178413575_c_nat @ Y4 @ ( transitions_nat_b_c @ X ) )
                 => ( ( ( ( produc532606554400396034_c_nat @ X2 )
                        = ( produc532606554400396034_c_nat @ Y4 ) )
                      & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ X2 ) )
                        = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ Y4 ) ) )
                      & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X2 ) ) )
                        = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Y4 ) ) ) ) )
                   => ( ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X2 ) ) )
                      = ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_923_observable_Oelims_I1_J,axiom,
    ! [X: fsm_Code_integer_b_c,Y: $o] :
      ( ( ( observ7524490533585752726er_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X2: produc558599814252099832nteger] :
              ( ( member5112046945434310049nteger @ X2 @ ( transi6707074847254859401er_b_c @ X ) )
             => ! [Y4: produc558599814252099832nteger] :
                  ( ( member5112046945434310049nteger @ Y4 @ ( transi6707074847254859401er_b_c @ X ) )
                 => ( ( ( ( produc7104377276419476764nteger @ X2 )
                        = ( produc7104377276419476764nteger @ Y4 ) )
                      & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ X2 ) )
                        = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ Y4 ) ) )
                      & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X2 ) ) )
                        = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Y4 ) ) ) ) )
                   => ( ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X2 ) ) )
                      = ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Y4 ) ) ) ) ) ) ) ) ) ) ).

% observable.elims(1)
thf(fact_924_observable_Osimps,axiom,
    ( observable_set_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
        ! [X2: produc373761375140688950_set_a] :
          ( ( member6517745037573369951_set_a @ X2 @ ( transi8205445899535437940_a_b_c @ M2 ) )
         => ! [Y4: produc373761375140688950_set_a] :
              ( ( member6517745037573369951_set_a @ Y4 @ ( transi8205445899535437940_a_b_c @ M2 ) )
             => ( ( ( ( produc7645221154340064658_set_a @ X2 )
                    = ( produc7645221154340064658_set_a @ Y4 ) )
                  & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X2 ) )
                    = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Y4 ) ) )
                  & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X2 ) ) )
                    = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Y4 ) ) ) ) )
               => ( ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X2 ) ) )
                  = ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Y4 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_925_observable_Osimps,axiom,
    ( observable_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ ( transitions_a_b_c @ M2 ) )
         => ! [Y4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ M2 ) )
             => ( ( ( ( produc4225943634317630866od_c_a @ X2 )
                    = ( produc4225943634317630866od_c_a @ Y4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) )
               => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) ) )
                  = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_926_observable_Osimps,axiom,
    ( observable_nat_b_c
    = ( ^ [M2: fsm_nat_b_c] :
        ! [X2: produc8586872805302537566_c_nat] :
          ( ( member3717341653178413575_c_nat @ X2 @ ( transitions_nat_b_c @ M2 ) )
         => ! [Y4: produc8586872805302537566_c_nat] :
              ( ( member3717341653178413575_c_nat @ Y4 @ ( transitions_nat_b_c @ M2 ) )
             => ( ( ( ( produc532606554400396034_c_nat @ X2 )
                    = ( produc532606554400396034_c_nat @ Y4 ) )
                  & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ X2 ) )
                    = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ Y4 ) ) )
                  & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X2 ) ) )
                    = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Y4 ) ) ) ) )
               => ( ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X2 ) ) )
                  = ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Y4 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_927_observable_Osimps,axiom,
    ( observ7524490533585752726er_b_c
    = ( ^ [M2: fsm_Code_integer_b_c] :
        ! [X2: produc558599814252099832nteger] :
          ( ( member5112046945434310049nteger @ X2 @ ( transi6707074847254859401er_b_c @ M2 ) )
         => ! [Y4: produc558599814252099832nteger] :
              ( ( member5112046945434310049nteger @ Y4 @ ( transi6707074847254859401er_b_c @ M2 ) )
             => ( ( ( ( produc7104377276419476764nteger @ X2 )
                    = ( produc7104377276419476764nteger @ Y4 ) )
                  & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ X2 ) )
                    = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ Y4 ) ) )
                  & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X2 ) ) )
                    = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Y4 ) ) ) ) )
               => ( ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X2 ) ) )
                  = ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Y4 ) ) ) ) ) ) ) ) ) ).

% observable.simps
thf(fact_928_observable__transition__target__language__eq,axiom,
    ! [M: fsm_set_a_b_c,T1: produc373761375140688950_set_a,T22: produc373761375140688950_set_a] :
      ( ( ( lS_set_a_b_c @ M @ ( produc7645221154340064658_set_a @ T1 ) )
        = ( lS_set_a_b_c @ M @ ( produc7645221154340064658_set_a @ T22 ) ) )
     => ( ( member6517745037573369951_set_a @ T1 @ ( transi8205445899535437940_a_b_c @ M ) )
       => ( ( member6517745037573369951_set_a @ T22 @ ( transi8205445899535437940_a_b_c @ M ) )
         => ( ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T1 ) )
              = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T22 ) ) )
           => ( ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T1 ) ) )
                = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T22 ) ) ) )
             => ( ( observable_set_a_b_c @ M )
               => ( ( lS_set_a_b_c @ M @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T1 ) ) ) )
                  = ( lS_set_a_b_c @ M @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_929_observable__transition__target__language__eq,axiom,
    ! [M: fsm_a_b_c,T1: produc2669932201312201782od_c_a,T22: produc2669932201312201782od_c_a] :
      ( ( ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T1 ) )
        = ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T22 ) ) )
     => ( ( member5502040268208861279od_c_a @ T1 @ ( transitions_a_b_c @ M ) )
       => ( ( member5502040268208861279od_c_a @ T22 @ ( transitions_a_b_c @ M ) )
         => ( ( ( 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 @ M )
               => ( ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) ) )
                  = ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_930_observable__transition__target__language__eq,axiom,
    ! [M: fsm_nat_b_c,T1: produc8586872805302537566_c_nat,T22: produc8586872805302537566_c_nat] :
      ( ( ( lS_nat_b_c @ M @ ( produc532606554400396034_c_nat @ T1 ) )
        = ( lS_nat_b_c @ M @ ( produc532606554400396034_c_nat @ T22 ) ) )
     => ( ( member3717341653178413575_c_nat @ T1 @ ( transitions_nat_b_c @ M ) )
       => ( ( member3717341653178413575_c_nat @ T22 @ ( transitions_nat_b_c @ M ) )
         => ( ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T1 ) )
              = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T22 ) ) )
           => ( ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T1 ) ) )
                = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T22 ) ) ) )
             => ( ( observable_nat_b_c @ M )
               => ( ( lS_nat_b_c @ M @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T1 ) ) ) )
                  = ( lS_nat_b_c @ M @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_931_observable__transition__target__language__eq,axiom,
    ! [M: fsm_Code_integer_b_c,T1: produc558599814252099832nteger,T22: produc558599814252099832nteger] :
      ( ( ( lS_Code_integer_b_c @ M @ ( produc7104377276419476764nteger @ T1 ) )
        = ( lS_Code_integer_b_c @ M @ ( produc7104377276419476764nteger @ T22 ) ) )
     => ( ( member5112046945434310049nteger @ T1 @ ( transi6707074847254859401er_b_c @ M ) )
       => ( ( member5112046945434310049nteger @ T22 @ ( transi6707074847254859401er_b_c @ M ) )
         => ( ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T1 ) )
              = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T22 ) ) )
           => ( ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T1 ) ) )
                = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T22 ) ) ) )
             => ( ( observ7524490533585752726er_b_c @ M )
               => ( ( lS_Code_integer_b_c @ M @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T1 ) ) ) )
                  = ( lS_Code_integer_b_c @ M @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_eq
thf(fact_932_observable__transition__target__language__subset,axiom,
    ! [M: fsm_set_a_b_c,T1: produc373761375140688950_set_a,T22: produc373761375140688950_set_a] :
      ( ( ord_le282488521294790766od_b_c @ ( lS_set_a_b_c @ M @ ( produc7645221154340064658_set_a @ T1 ) ) @ ( lS_set_a_b_c @ M @ ( produc7645221154340064658_set_a @ T22 ) ) )
     => ( ( member6517745037573369951_set_a @ T1 @ ( transi8205445899535437940_a_b_c @ M ) )
       => ( ( member6517745037573369951_set_a @ T22 @ ( transi8205445899535437940_a_b_c @ M ) )
         => ( ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T1 ) )
              = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T22 ) ) )
           => ( ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T1 ) ) )
                = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T22 ) ) ) )
             => ( ( observable_set_a_b_c @ M )
               => ( ord_le282488521294790766od_b_c @ ( lS_set_a_b_c @ M @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T1 ) ) ) ) @ ( lS_set_a_b_c @ M @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_subset
thf(fact_933_observable__transition__target__language__subset,axiom,
    ! [M: fsm_a_b_c,T1: produc2669932201312201782od_c_a,T22: produc2669932201312201782od_c_a] :
      ( ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T1 ) ) @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T22 ) ) )
     => ( ( member5502040268208861279od_c_a @ T1 @ ( transitions_a_b_c @ M ) )
       => ( ( member5502040268208861279od_c_a @ T22 @ ( transitions_a_b_c @ M ) )
         => ( ( ( 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 @ M )
               => ( ord_le282488521294790766od_b_c @ ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T1 ) ) ) ) @ ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_subset
thf(fact_934_observable__transition__target__language__subset,axiom,
    ! [M: fsm_nat_b_c,T1: produc8586872805302537566_c_nat,T22: produc8586872805302537566_c_nat] :
      ( ( ord_le282488521294790766od_b_c @ ( lS_nat_b_c @ M @ ( produc532606554400396034_c_nat @ T1 ) ) @ ( lS_nat_b_c @ M @ ( produc532606554400396034_c_nat @ T22 ) ) )
     => ( ( member3717341653178413575_c_nat @ T1 @ ( transitions_nat_b_c @ M ) )
       => ( ( member3717341653178413575_c_nat @ T22 @ ( transitions_nat_b_c @ M ) )
         => ( ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T1 ) )
              = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T22 ) ) )
           => ( ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T1 ) ) )
                = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T22 ) ) ) )
             => ( ( observable_nat_b_c @ M )
               => ( ord_le282488521294790766od_b_c @ ( lS_nat_b_c @ M @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T1 ) ) ) ) @ ( lS_nat_b_c @ M @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_subset
thf(fact_935_observable__transition__target__language__subset,axiom,
    ! [M: fsm_Code_integer_b_c,T1: produc558599814252099832nteger,T22: produc558599814252099832nteger] :
      ( ( ord_le282488521294790766od_b_c @ ( lS_Code_integer_b_c @ M @ ( produc7104377276419476764nteger @ T1 ) ) @ ( lS_Code_integer_b_c @ M @ ( produc7104377276419476764nteger @ T22 ) ) )
     => ( ( member5112046945434310049nteger @ T1 @ ( transi6707074847254859401er_b_c @ M ) )
       => ( ( member5112046945434310049nteger @ T22 @ ( transi6707074847254859401er_b_c @ M ) )
         => ( ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T1 ) )
              = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T22 ) ) )
           => ( ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T1 ) ) )
                = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T22 ) ) ) )
             => ( ( observ7524490533585752726er_b_c @ M )
               => ( ord_le282488521294790766od_b_c @ ( lS_Code_integer_b_c @ M @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T1 ) ) ) ) @ ( lS_Code_integer_b_c @ M @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T22 ) ) ) ) ) ) ) ) ) ) ) ).

% observable_transition_target_language_subset
thf(fact_936_observable__transition__unique,axiom,
    ! [M: fsm_set_a_b_c,T: produc373761375140688950_set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6517745037573369951_set_a @ T @ ( transi8205445899535437940_a_b_c @ M ) )
       => ? [X3: produc373761375140688950_set_a] :
            ( ( member6517745037573369951_set_a @ X3 @ ( transi8205445899535437940_a_b_c @ M ) )
            & ( ( produc7645221154340064658_set_a @ X3 )
              = ( produc7645221154340064658_set_a @ T ) )
            & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X3 ) )
              = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) )
            & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X3 ) ) )
              = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) )
            & ! [Y5: produc373761375140688950_set_a] :
                ( ( ( member6517745037573369951_set_a @ Y5 @ ( transi8205445899535437940_a_b_c @ M ) )
                  & ( ( produc7645221154340064658_set_a @ Y5 )
                    = ( produc7645221154340064658_set_a @ T ) )
                  & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Y5 ) )
                    = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) )
                  & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Y5 ) ) )
                    = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_937_observable__transition__unique,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ M ) )
            & ( ( produc4225943634317630866od_c_a @ X3 )
              = ( produc4225943634317630866od_c_a @ T ) )
            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) )
            & ! [Y5: produc2669932201312201782od_c_a] :
                ( ( ( member5502040268208861279od_c_a @ Y5 @ ( transitions_a_b_c @ M ) )
                  & ( ( produc4225943634317630866od_c_a @ Y5 )
                    = ( produc4225943634317630866od_c_a @ T ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y5 ) ) )
                    = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_938_observable__transition__unique,axiom,
    ! [M: fsm_nat_b_c,T: produc8586872805302537566_c_nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member3717341653178413575_c_nat @ T @ ( transitions_nat_b_c @ M ) )
       => ? [X3: produc8586872805302537566_c_nat] :
            ( ( member3717341653178413575_c_nat @ X3 @ ( transitions_nat_b_c @ M ) )
            & ( ( produc532606554400396034_c_nat @ X3 )
              = ( produc532606554400396034_c_nat @ T ) )
            & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) )
              = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) )
            & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) ) )
              = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) ) )
            & ! [Y5: produc8586872805302537566_c_nat] :
                ( ( ( member3717341653178413575_c_nat @ Y5 @ ( transitions_nat_b_c @ M ) )
                  & ( ( produc532606554400396034_c_nat @ Y5 )
                    = ( produc532606554400396034_c_nat @ T ) )
                  & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ Y5 ) )
                    = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) )
                  & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Y5 ) ) )
                    = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_939_observable__transition__unique,axiom,
    ! [M: fsm_Code_integer_b_c,T: produc558599814252099832nteger] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member5112046945434310049nteger @ T @ ( transi6707074847254859401er_b_c @ M ) )
       => ? [X3: produc558599814252099832nteger] :
            ( ( member5112046945434310049nteger @ X3 @ ( transi6707074847254859401er_b_c @ M ) )
            & ( ( produc7104377276419476764nteger @ X3 )
              = ( produc7104377276419476764nteger @ T ) )
            & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ X3 ) )
              = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T ) ) )
            & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X3 ) ) )
              = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) ) )
            & ! [Y5: produc558599814252099832nteger] :
                ( ( ( member5112046945434310049nteger @ Y5 @ ( transi6707074847254859401er_b_c @ M ) )
                  & ( ( produc7104377276419476764nteger @ Y5 )
                    = ( produc7104377276419476764nteger @ T ) )
                  & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ Y5 ) )
                    = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T ) ) )
                  & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Y5 ) ) )
                    = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) ) ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% observable_transition_unique
thf(fact_940_observable__language__transition__target,axiom,
    ! [M: fsm_set_a_b_c,T: produc373761375140688950_set_a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6517745037573369951_set_a @ T @ ( transi8205445899535437940_a_b_c @ M ) )
       => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) ) @ Io ) @ ( lS_set_a_b_c @ M @ ( produc7645221154340064658_set_a @ T ) ) )
         => ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_941_observable__language__transition__target,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ Io ) @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) ) )
         => ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_942_observable__language__transition__target,axiom,
    ! [M: fsm_nat_b_c,T: produc8586872805302537566_c_nat,Io: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member3717341653178413575_c_nat @ T @ ( transitions_nat_b_c @ M ) )
       => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) ) ) @ Io ) @ ( lS_nat_b_c @ M @ ( produc532606554400396034_c_nat @ T ) ) )
         => ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_943_observable__language__transition__target,axiom,
    ! [M: fsm_Code_integer_b_c,T: produc558599814252099832nteger,Io: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member5112046945434310049nteger @ T @ ( transi6707074847254859401er_b_c @ M ) )
       => ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) ) ) @ Io ) @ ( lS_Code_integer_b_c @ M @ ( produc7104377276419476764nteger @ T ) ) )
         => ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) ) ) ) ) ) ) ).

% observable_language_transition_target
thf(fact_944_after__transition__exhaust,axiom,
    ! [M: fsm_set_a_b_c,T: produc373761375140688950_set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6517745037573369951_set_a @ T @ ( transi8205445899535437940_a_b_c @ M ) )
       => ( ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) )
          = ( after_set_a_b_c @ M @ ( produc7645221154340064658_set_a @ T ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_945_after__transition__exhaust,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
          = ( after_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_946_after__transition__exhaust,axiom,
    ! [M: fsm_nat_b_c,T: produc8586872805302537566_c_nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member3717341653178413575_c_nat @ T @ ( transitions_nat_b_c @ M ) )
       => ( ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) )
          = ( after_nat_b_c @ M @ ( produc532606554400396034_c_nat @ T ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_947_after__transition__exhaust,axiom,
    ! [M: fsm_Code_integer_b_c,T: produc558599814252099832nteger] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member5112046945434310049nteger @ T @ ( transi6707074847254859401er_b_c @ M ) )
       => ( ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) )
          = ( after_5090571588895392199er_b_c @ M @ ( produc7104377276419476764nteger @ T ) @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) ) ) @ nil_Product_prod_b_c ) ) ) ) ) ).

% after_transition_exhaust
thf(fact_948_observable__language__next,axiom,
    ! [Io: product_prod_b_c,Ios: list_P903359562653991662od_b_c,M: fsm_set_a_b_c,T: produc373761375140688950_set_a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ Io @ Ios ) @ ( lS_set_a_b_c @ M @ ( produc7645221154340064658_set_a @ T ) ) )
     => ( ( observable_set_a_b_c @ M )
       => ( ( member6517745037573369951_set_a @ T @ ( transi8205445899535437940_a_b_c @ M ) )
         => ( ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) )
              = ( product_fst_b_c @ Io ) )
           => ( ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) )
                = ( product_snd_b_c @ Io ) )
             => ( member6330420149250801815od_b_c @ Ios @ ( lS_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) ) @ ( initial_set_a_b_c @ ( from_FSM_set_a_b_c @ M @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_949_observable__language__next,axiom,
    ! [Io: product_prod_b_c,Ios: list_P903359562653991662od_b_c,M: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ Io @ Ios ) @ ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T ) ) )
     => ( ( observable_a_b_c @ M )
       => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
         => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) )
              = ( product_fst_b_c @ Io ) )
           => ( ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) )
                = ( product_snd_b_c @ Io ) )
             => ( member6330420149250801815od_b_c @ Ios @ ( lS_a_b_c @ ( from_FSM_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_950_observable__language__next,axiom,
    ! [Io: product_prod_b_c,Ios: list_P903359562653991662od_b_c,M: fsm_nat_b_c,T: produc8586872805302537566_c_nat] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ Io @ Ios ) @ ( lS_nat_b_c @ M @ ( produc532606554400396034_c_nat @ T ) ) )
     => ( ( observable_nat_b_c @ M )
       => ( ( member3717341653178413575_c_nat @ T @ ( transitions_nat_b_c @ M ) )
         => ( ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T ) )
              = ( product_fst_b_c @ Io ) )
           => ( ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) )
                = ( product_snd_b_c @ Io ) )
             => ( member6330420149250801815od_b_c @ Ios @ ( lS_nat_b_c @ ( from_FSM_nat_b_c @ M @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) ) ) @ ( initial_nat_b_c @ ( from_FSM_nat_b_c @ M @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_951_observable__language__next,axiom,
    ! [Io: product_prod_b_c,Ios: list_P903359562653991662od_b_c,M: fsm_Code_integer_b_c,T: produc558599814252099832nteger] :
      ( ( member6330420149250801815od_b_c @ ( cons_P4529483553340347422od_b_c @ Io @ Ios ) @ ( lS_Code_integer_b_c @ M @ ( produc7104377276419476764nteger @ T ) ) )
     => ( ( observ7524490533585752726er_b_c @ M )
       => ( ( member5112046945434310049nteger @ T @ ( transi6707074847254859401er_b_c @ M ) )
         => ( ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T ) )
              = ( product_fst_b_c @ Io ) )
           => ( ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) )
                = ( product_snd_b_c @ Io ) )
             => ( member6330420149250801815od_b_c @ Ios @ ( lS_Code_integer_b_c @ ( from_F2639493813852433665er_b_c @ M @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) ) ) @ ( initia4873860033875475119er_b_c @ ( from_F2639493813852433665er_b_c @ M @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) ) ) ) ) ) ) ) ) ) ) ).

% observable_language_next
thf(fact_952_output__complete_Osimps,axiom,
    ( output5228291214970145744_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
        ! [X2: produc373761375140688950_set_a] :
          ( ( member6517745037573369951_set_a @ X2 @ ( transi8205445899535437940_a_b_c @ M2 ) )
         => ! [Y4: c] :
              ( ( member_c @ Y4 @ ( outputs_set_a_b_c @ M2 ) )
             => ? [Z4: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ Z4 @ ( transi8205445899535437940_a_b_c @ M2 ) )
                  & ( ( produc7645221154340064658_set_a @ X2 )
                    = ( produc7645221154340064658_set_a @ Z4 ) )
                  & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X2 ) )
                    = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Z4 ) ) )
                  & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Z4 ) ) )
                    = Y4 ) ) ) ) ) ) ).

% output_complete.simps
thf(fact_953_output__complete_Osimps,axiom,
    ( output3834324022317553264_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ ( transitions_a_b_c @ M2 ) )
         => ! [Y4: c] :
              ( ( member_c @ Y4 @ ( outputs_a_b_c @ M2 ) )
             => ? [Z4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Z4 @ ( transitions_a_b_c @ M2 ) )
                  & ( ( produc4225943634317630866od_c_a @ X2 )
                    = ( produc4225943634317630866od_c_a @ Z4 ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                    = Y4 ) ) ) ) ) ) ).

% output_complete.simps
thf(fact_954_output__complete_Oelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Y: $o] :
      ( ( ( output5228291214970145744_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X2: produc373761375140688950_set_a] :
              ( ( member6517745037573369951_set_a @ X2 @ ( transi8205445899535437940_a_b_c @ X ) )
             => ! [Y4: c] :
                  ( ( member_c @ Y4 @ ( outputs_set_a_b_c @ X ) )
                 => ? [Z4: produc373761375140688950_set_a] :
                      ( ( member6517745037573369951_set_a @ Z4 @ ( transi8205445899535437940_a_b_c @ X ) )
                      & ( ( produc7645221154340064658_set_a @ X2 )
                        = ( produc7645221154340064658_set_a @ Z4 ) )
                      & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X2 ) )
                        = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Z4 ) ) )
                      & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Z4 ) ) )
                        = Y4 ) ) ) ) ) ) ) ).

% output_complete.elims(1)
thf(fact_955_output__complete_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( output3834324022317553264_a_b_c @ X )
        = Y )
     => ( Y
        = ( ! [X2: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X2 @ ( transitions_a_b_c @ X ) )
             => ! [Y4: c] :
                  ( ( member_c @ Y4 @ ( outputs_a_b_c @ X ) )
                 => ? [Z4: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Z4 @ ( transitions_a_b_c @ X ) )
                      & ( ( produc4225943634317630866od_c_a @ X2 )
                        = ( produc4225943634317630866od_c_a @ Z4 ) )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) )
                        = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                      & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                        = Y4 ) ) ) ) ) ) ) ).

% output_complete.elims(1)
thf(fact_956_output__complete_Oelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ( output5228291214970145744_a_b_c @ X )
     => ! [X5: produc373761375140688950_set_a] :
          ( ( member6517745037573369951_set_a @ X5 @ ( transi8205445899535437940_a_b_c @ X ) )
         => ! [Xa2: c] :
              ( ( member_c @ Xa2 @ ( outputs_set_a_b_c @ X ) )
             => ? [Xb: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ Xb @ ( transi8205445899535437940_a_b_c @ X ) )
                  & ( ( produc7645221154340064658_set_a @ X5 )
                    = ( produc7645221154340064658_set_a @ Xb ) )
                  & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X5 ) )
                    = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xb ) ) )
                  & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xb ) ) )
                    = Xa2 ) ) ) ) ) ).

% output_complete.elims(2)
thf(fact_957_output__complete_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( output3834324022317553264_a_b_c @ X )
     => ! [X5: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X ) )
         => ! [Xa2: c] :
              ( ( member_c @ Xa2 @ ( outputs_a_b_c @ X ) )
             => ? [Xb: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Xb @ ( transitions_a_b_c @ X ) )
                  & ( ( produc4225943634317630866od_c_a @ X5 )
                    = ( produc4225943634317630866od_c_a @ Xb ) )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                    = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb ) ) )
                  & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb ) ) )
                    = Xa2 ) ) ) ) ) ).

% output_complete.elims(2)
thf(fact_958_output__complete_Oelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ~ ( output5228291214970145744_a_b_c @ X )
     => ~ ! [X3: produc373761375140688950_set_a] :
            ( ( member6517745037573369951_set_a @ X3 @ ( transi8205445899535437940_a_b_c @ X ) )
           => ! [Xa3: c] :
                ( ( member_c @ Xa3 @ ( outputs_set_a_b_c @ X ) )
               => ? [Xb2: produc373761375140688950_set_a] :
                    ( ( member6517745037573369951_set_a @ Xb2 @ ( transi8205445899535437940_a_b_c @ X ) )
                    & ( ( produc7645221154340064658_set_a @ X3 )
                      = ( produc7645221154340064658_set_a @ Xb2 ) )
                    & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X3 ) )
                      = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xb2 ) ) )
                    & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xb2 ) ) )
                      = Xa3 ) ) ) ) ) ).

% output_complete.elims(3)
thf(fact_959_output__complete_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( output3834324022317553264_a_b_c @ X )
     => ~ ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X ) )
           => ! [Xa3: c] :
                ( ( member_c @ Xa3 @ ( outputs_a_b_c @ X ) )
               => ? [Xb2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xb2 @ ( transitions_a_b_c @ X ) )
                    & ( ( produc4225943634317630866od_c_a @ X3 )
                      = ( produc4225943634317630866od_c_a @ Xb2 ) )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                      = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                    & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                      = Xa3 ) ) ) ) ) ).

% output_complete.elims(3)
thf(fact_960_observable__alt__def__h,axiom,
    ( observable_set_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
        ! [Q13: set_a,X2: b,Yq: product_prod_c_set_a,Yq2: product_prod_c_set_a] :
          ( ( ( member7460900956140502318_set_a @ Yq @ ( h_set_a_b_c @ M2 @ ( product_Pair_set_a_b @ Q13 @ X2 ) ) )
            & ( member7460900956140502318_set_a @ Yq2 @ ( h_set_a_b_c @ M2 @ ( product_Pair_set_a_b @ Q13 @ X2 ) ) ) )
         => ( ( ( product_fst_c_set_a @ Yq )
              = ( product_fst_c_set_a @ Yq2 ) )
           => ( ( product_snd_c_set_a @ Yq )
              = ( product_snd_c_set_a @ Yq2 ) ) ) ) ) ) ).

% observable_alt_def_h
thf(fact_961_observable__alt__def__h,axiom,
    ( observable_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
        ! [Q13: a,X2: b,Yq: product_prod_c_a,Yq2: product_prod_c_a] :
          ( ( ( member5074992350434858958od_c_a @ Yq @ ( h_a_b_c @ M2 @ ( product_Pair_a_b @ Q13 @ X2 ) ) )
            & ( member5074992350434858958od_c_a @ Yq2 @ ( h_a_b_c @ M2 @ ( product_Pair_a_b @ Q13 @ X2 ) ) ) )
         => ( ( ( product_fst_c_a @ Yq )
              = ( product_fst_c_a @ Yq2 ) )
           => ( ( product_snd_c_a @ Yq )
              = ( product_snd_c_a @ Yq2 ) ) ) ) ) ) ).

% observable_alt_def_h
thf(fact_962_observable__alt__def__h,axiom,
    ( observable_nat_b_c
    = ( ^ [M2: fsm_nat_b_c] :
        ! [Q13: nat,X2: b,Yq: product_prod_c_nat,Yq2: product_prod_c_nat] :
          ( ( ( member8195077246299207702_c_nat @ Yq @ ( h_nat_b_c @ M2 @ ( product_Pair_nat_b @ Q13 @ X2 ) ) )
            & ( member8195077246299207702_c_nat @ Yq2 @ ( h_nat_b_c @ M2 @ ( product_Pair_nat_b @ Q13 @ X2 ) ) ) )
         => ( ( ( product_fst_c_nat @ Yq )
              = ( product_fst_c_nat @ Yq2 ) )
           => ( ( product_snd_c_nat @ Yq )
              = ( product_snd_c_nat @ Yq2 ) ) ) ) ) ) ).

% observable_alt_def_h
thf(fact_963_observable__alt__def__h,axiom,
    ( observ7524490533585752726er_b_c
    = ( ^ [M2: fsm_Code_integer_b_c] :
        ! [Q13: code_integer,X2: b,Yq: produc3993223824263494796nteger,Yq2: produc3993223824263494796nteger] :
          ( ( ( member7093370293991928355nteger @ Yq @ ( h_Code_integer_b_c @ M2 @ ( produc3961679347488720265eger_b @ Q13 @ X2 ) ) )
            & ( member7093370293991928355nteger @ Yq2 @ ( h_Code_integer_b_c @ M2 @ ( produc3961679347488720265eger_b @ Q13 @ X2 ) ) ) )
         => ( ( ( produc5481060579096546074nteger @ Yq )
              = ( produc5481060579096546074nteger @ Yq2 ) )
           => ( ( produc4666406780376491864nteger @ Yq )
              = ( produc4666406780376491864nteger @ Yq2 ) ) ) ) ) ) ).

% observable_alt_def_h
thf(fact_964_prod__less__def,axiom,
    ( ord_le1203424502768444845at_nat
    = ( ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X2 ) @ ( product_fst_nat_nat @ Y4 ) )
          | ( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X2 ) @ ( product_fst_nat_nat @ Y4 ) )
            & ( ord_less_nat @ ( product_snd_nat_nat @ X2 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ) ).

% prod_less_def
thf(fact_965_less__eq__prod__def,axiom,
    ( ord_le8460144461188290721at_nat
    = ( ^ [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X2 ) @ ( product_fst_nat_nat @ Y4 ) )
          | ( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X2 ) @ ( product_fst_nat_nat @ Y4 ) )
            & ( ord_less_eq_nat @ ( product_snd_nat_nat @ X2 ) @ ( product_snd_nat_nat @ Y4 ) ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_966_completely__specified__state_Oelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: set_a] :
      ( ~ ( comple6745895220685239651_a_b_c @ X @ Xa )
     => ~ ! [X3: b] :
            ( ( member_b @ X3 @ ( inputs_set_a_b_c @ X ) )
           => ? [Xa2: produc373761375140688950_set_a] :
                ( ( member6517745037573369951_set_a @ Xa2 @ ( transi8205445899535437940_a_b_c @ X ) )
                & ( ( produc7645221154340064658_set_a @ Xa2 )
                  = Xa )
                & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xa2 ) )
                  = X3 ) ) ) ) ).

% completely_specified_state.elims(3)
thf(fact_967_completely__specified__state_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c,Xa: a] :
      ( ~ ( comple2039557650379672579_a_b_c @ X @ Xa )
     => ~ ! [X3: b] :
            ( ( member_b @ X3 @ ( inputs_a_b_c @ X ) )
           => ? [Xa2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X ) )
                & ( ( produc4225943634317630866od_c_a @ Xa2 )
                  = Xa )
                & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) )
                  = X3 ) ) ) ) ).

% completely_specified_state.elims(3)
thf(fact_968_completely__specified__state_Oelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: set_a] :
      ( ( comple6745895220685239651_a_b_c @ X @ Xa )
     => ! [X5: b] :
          ( ( member_b @ X5 @ ( inputs_set_a_b_c @ X ) )
         => ? [Xa3: produc373761375140688950_set_a] :
              ( ( member6517745037573369951_set_a @ Xa3 @ ( transi8205445899535437940_a_b_c @ X ) )
              & ( ( produc7645221154340064658_set_a @ Xa3 )
                = Xa )
              & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xa3 ) )
                = X5 ) ) ) ) ).

% completely_specified_state.elims(2)
thf(fact_969_completely__specified__state_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c,Xa: a] :
      ( ( comple2039557650379672579_a_b_c @ X @ Xa )
     => ! [X5: b] :
          ( ( member_b @ X5 @ ( inputs_a_b_c @ X ) )
         => ? [Xa3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X ) )
              & ( ( produc4225943634317630866od_c_a @ Xa3 )
                = Xa )
              & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) )
                = X5 ) ) ) ) ).

% completely_specified_state.elims(2)
thf(fact_970_completely__specified__state_Oelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: set_a,Y: $o] :
      ( ( ( comple6745895220685239651_a_b_c @ X @ Xa )
        = Y )
     => ( Y
        = ( ! [X2: b] :
              ( ( member_b @ X2 @ ( inputs_set_a_b_c @ X ) )
             => ? [Y4: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ Y4 @ ( transi8205445899535437940_a_b_c @ X ) )
                  & ( ( produc7645221154340064658_set_a @ Y4 )
                    = Xa )
                  & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Y4 ) )
                    = X2 ) ) ) ) ) ) ).

% completely_specified_state.elims(1)
thf(fact_971_completely__specified__state_Oelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Xa: a,Y: $o] :
      ( ( ( comple2039557650379672579_a_b_c @ X @ Xa )
        = Y )
     => ( Y
        = ( ! [X2: b] :
              ( ( member_b @ X2 @ ( inputs_a_b_c @ X ) )
             => ? [Y4: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X ) )
                  & ( ( produc4225943634317630866od_c_a @ Y4 )
                    = Xa )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) )
                    = X2 ) ) ) ) ) ) ).

% completely_specified_state.elims(1)
thf(fact_972_completely__specified__state_Osimps,axiom,
    ( comple6745895220685239651_a_b_c
    = ( ^ [M2: fsm_set_a_b_c,Q5: set_a] :
        ! [X2: b] :
          ( ( member_b @ X2 @ ( inputs_set_a_b_c @ M2 ) )
         => ? [Y4: produc373761375140688950_set_a] :
              ( ( member6517745037573369951_set_a @ Y4 @ ( transi8205445899535437940_a_b_c @ M2 ) )
              & ( ( produc7645221154340064658_set_a @ Y4 )
                = Q5 )
              & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Y4 ) )
                = X2 ) ) ) ) ) ).

% completely_specified_state.simps
thf(fact_973_completely__specified__state_Osimps,axiom,
    ( comple2039557650379672579_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q5: a] :
        ! [X2: b] :
          ( ( member_b @ X2 @ ( inputs_a_b_c @ M2 ) )
         => ? [Y4: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ M2 ) )
              & ( ( produc4225943634317630866od_c_a @ Y4 )
                = Q5 )
              & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) )
                = X2 ) ) ) ) ) ).

% completely_specified_state.simps
thf(fact_974_completely__specified_Oelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ~ ( comple9199925562481299400_a_b_c @ X )
     => ~ ! [X3: set_a] :
            ( ( member_set_a @ X3 @ ( states_set_a_b_c @ X ) )
           => ! [Xa3: b] :
                ( ( member_b @ Xa3 @ ( inputs_set_a_b_c @ X ) )
               => ? [Xb2: produc373761375140688950_set_a] :
                    ( ( member6517745037573369951_set_a @ Xb2 @ ( transi8205445899535437940_a_b_c @ X ) )
                    & ( ( produc7645221154340064658_set_a @ Xb2 )
                      = X3 )
                    & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xb2 ) )
                      = Xa3 ) ) ) ) ) ).

% completely_specified.elims(3)
thf(fact_975_completely__specified_Oelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( comple6900478119722646120_a_b_c @ X )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ ( states_a_b_c @ X ) )
           => ! [Xa3: b] :
                ( ( member_b @ Xa3 @ ( inputs_a_b_c @ X ) )
               => ? [Xb2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xb2 @ ( transitions_a_b_c @ X ) )
                    & ( ( produc4225943634317630866od_c_a @ Xb2 )
                      = X3 )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) )
                      = Xa3 ) ) ) ) ) ).

% completely_specified.elims(3)
thf(fact_976_completely__specified_Oelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ( comple9199925562481299400_a_b_c @ X )
     => ! [X5: set_a] :
          ( ( member_set_a @ X5 @ ( states_set_a_b_c @ X ) )
         => ! [Xa2: b] :
              ( ( member_b @ Xa2 @ ( inputs_set_a_b_c @ X ) )
             => ? [Xb: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ Xb @ ( transi8205445899535437940_a_b_c @ X ) )
                  & ( ( produc7645221154340064658_set_a @ Xb )
                    = X5 )
                  & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xb ) )
                    = Xa2 ) ) ) ) ) ).

% completely_specified.elims(2)
thf(fact_977_completely__specified_Oelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( comple6900478119722646120_a_b_c @ X )
     => ! [X5: a] :
          ( ( member_a @ X5 @ ( states_a_b_c @ X ) )
         => ! [Xa2: b] :
              ( ( member_b @ Xa2 @ ( inputs_a_b_c @ X ) )
             => ? [Xb: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ Xb @ ( transitions_a_b_c @ X ) )
                  & ( ( produc4225943634317630866od_c_a @ Xb )
                    = X5 )
                  & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb ) )
                    = Xa2 ) ) ) ) ) ).

% completely_specified.elims(2)
thf(fact_978_add__transition__simps_I4_J,axiom,
    ! [T: produc373761375140688950_set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ ( produc7645221154340064658_set_a @ T ) @ ( states_set_a_b_c @ M ) )
     => ( ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) @ ( inputs_set_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( outputs_set_a_b_c @ M ) )
         => ( ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( states_set_a_b_c @ M ) )
           => ( ( transi8205445899535437940_a_b_c @ ( add_tr1235648555799902492_a_b_c @ M @ T ) )
              = ( insert7572610720624169478_set_a @ T @ ( transi8205445899535437940_a_b_c @ M ) ) ) ) ) ) ) ).

% add_transition_simps(4)
thf(fact_979_add__transition__simps_I4_J,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member_a @ ( produc4225943634317630866od_c_a @ T ) @ ( states_a_b_c @ M ) )
     => ( ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M ) )
         => ( ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( states_a_b_c @ M ) )
           => ( ( transitions_a_b_c @ ( add_transition_a_b_c @ M @ T ) )
              = ( insert8105206408023001606od_c_a @ T @ ( transitions_a_b_c @ M ) ) ) ) ) ) ) ).

% add_transition_simps(4)
thf(fact_980_add__transition__simps_I1_J,axiom,
    ! [T: produc373761375140688950_set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ ( produc7645221154340064658_set_a @ T ) @ ( states_set_a_b_c @ M ) )
     => ( ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) @ ( inputs_set_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( outputs_set_a_b_c @ M ) )
         => ( ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( states_set_a_b_c @ M ) )
           => ( ( initial_set_a_b_c @ ( add_tr1235648555799902492_a_b_c @ M @ T ) )
              = ( initial_set_a_b_c @ M ) ) ) ) ) ) ).

% add_transition_simps(1)
thf(fact_981_add__transition__simps_I1_J,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member_a @ ( produc4225943634317630866od_c_a @ T ) @ ( states_a_b_c @ M ) )
     => ( ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M ) )
         => ( ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( states_a_b_c @ M ) )
           => ( ( initial_a_b_c @ ( add_transition_a_b_c @ M @ T ) )
              = ( initial_a_b_c @ M ) ) ) ) ) ) ).

% add_transition_simps(1)
thf(fact_982_add__transition__simps_I5_J,axiom,
    ! [T: produc373761375140688950_set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ ( produc7645221154340064658_set_a @ T ) @ ( states_set_a_b_c @ M ) )
     => ( ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) @ ( inputs_set_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( outputs_set_a_b_c @ M ) )
         => ( ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( states_set_a_b_c @ M ) )
           => ( ( states_set_a_b_c @ ( add_tr1235648555799902492_a_b_c @ M @ T ) )
              = ( states_set_a_b_c @ M ) ) ) ) ) ) ).

% add_transition_simps(5)
thf(fact_983_add__transition__simps_I5_J,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member_a @ ( produc4225943634317630866od_c_a @ T ) @ ( states_a_b_c @ M ) )
     => ( ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M ) )
         => ( ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( states_a_b_c @ M ) )
           => ( ( states_a_b_c @ ( add_transition_a_b_c @ M @ T ) )
              = ( states_a_b_c @ M ) ) ) ) ) ) ).

% add_transition_simps(5)
thf(fact_984_add__transition__simps_I2_J,axiom,
    ! [T: produc373761375140688950_set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ ( produc7645221154340064658_set_a @ T ) @ ( states_set_a_b_c @ M ) )
     => ( ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) @ ( inputs_set_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( outputs_set_a_b_c @ M ) )
         => ( ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( states_set_a_b_c @ M ) )
           => ( ( inputs_set_a_b_c @ ( add_tr1235648555799902492_a_b_c @ M @ T ) )
              = ( inputs_set_a_b_c @ M ) ) ) ) ) ) ).

% add_transition_simps(2)
thf(fact_985_add__transition__simps_I2_J,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member_a @ ( produc4225943634317630866od_c_a @ T ) @ ( states_a_b_c @ M ) )
     => ( ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M ) )
         => ( ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( states_a_b_c @ M ) )
           => ( ( inputs_a_b_c @ ( add_transition_a_b_c @ M @ T ) )
              = ( inputs_a_b_c @ M ) ) ) ) ) ) ).

% add_transition_simps(2)
thf(fact_986_add__transition__simps_I3_J,axiom,
    ! [T: produc373761375140688950_set_a,M: fsm_set_a_b_c] :
      ( ( member_set_a @ ( produc7645221154340064658_set_a @ T ) @ ( states_set_a_b_c @ M ) )
     => ( ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) @ ( inputs_set_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( outputs_set_a_b_c @ M ) )
         => ( ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( states_set_a_b_c @ M ) )
           => ( ( outputs_set_a_b_c @ ( add_tr1235648555799902492_a_b_c @ M @ T ) )
              = ( outputs_set_a_b_c @ M ) ) ) ) ) ) ).

% add_transition_simps(3)
thf(fact_987_add__transition__simps_I3_J,axiom,
    ! [T: produc2669932201312201782od_c_a,M: fsm_a_b_c] :
      ( ( member_a @ ( produc4225943634317630866od_c_a @ T ) @ ( states_a_b_c @ M ) )
     => ( ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M ) )
       => ( ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M ) )
         => ( ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( states_a_b_c @ M ) )
           => ( ( outputs_a_b_c @ ( add_transition_a_b_c @ M @ T ) )
              = ( outputs_a_b_c @ M ) ) ) ) ) ) ).

% add_transition_simps(3)
thf(fact_988_output__complete_Opelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Y: $o] :
      ( ( ( output5228291214970145744_a_b_c @ X )
        = Y )
     => ( ( accp_fsm_set_a_b_c @ output2932112043462401139_a_b_c @ X )
       => ~ ( ( Y
              = ( ! [X2: produc373761375140688950_set_a] :
                    ( ( member6517745037573369951_set_a @ X2 @ ( transi8205445899535437940_a_b_c @ X ) )
                   => ! [Y4: c] :
                        ( ( member_c @ Y4 @ ( outputs_set_a_b_c @ X ) )
                       => ? [Z4: produc373761375140688950_set_a] :
                            ( ( member6517745037573369951_set_a @ Z4 @ ( transi8205445899535437940_a_b_c @ X ) )
                            & ( ( produc7645221154340064658_set_a @ X2 )
                              = ( produc7645221154340064658_set_a @ Z4 ) )
                            & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X2 ) )
                              = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Z4 ) ) )
                            & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Z4 ) ) )
                              = Y4 ) ) ) ) ) )
           => ~ ( accp_fsm_set_a_b_c @ output2932112043462401139_a_b_c @ X ) ) ) ) ).

% output_complete.pelims(1)
thf(fact_989_output__complete_Opelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( output3834324022317553264_a_b_c @ X )
        = Y )
     => ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X )
       => ~ ( ( Y
              = ( ! [X2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ X2 @ ( transitions_a_b_c @ X ) )
                   => ! [Y4: c] :
                        ( ( member_c @ Y4 @ ( outputs_a_b_c @ X ) )
                       => ? [Z4: produc2669932201312201782od_c_a] :
                            ( ( member5502040268208861279od_c_a @ Z4 @ ( transitions_a_b_c @ X ) )
                            & ( ( produc4225943634317630866od_c_a @ X2 )
                              = ( produc4225943634317630866od_c_a @ Z4 ) )
                            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) )
                              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) ) )
                              = Y4 ) ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X ) ) ) ) ).

% output_complete.pelims(1)
thf(fact_990_output__complete_Opelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ( output5228291214970145744_a_b_c @ X )
     => ( ( accp_fsm_set_a_b_c @ output2932112043462401139_a_b_c @ X )
       => ~ ( ( accp_fsm_set_a_b_c @ output2932112043462401139_a_b_c @ X )
           => ~ ! [X5: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ X5 @ ( transi8205445899535437940_a_b_c @ X ) )
                 => ! [Xa2: c] :
                      ( ( member_c @ Xa2 @ ( outputs_set_a_b_c @ X ) )
                     => ? [Xb: produc373761375140688950_set_a] :
                          ( ( member6517745037573369951_set_a @ Xb @ ( transi8205445899535437940_a_b_c @ X ) )
                          & ( ( produc7645221154340064658_set_a @ X5 )
                            = ( produc7645221154340064658_set_a @ Xb ) )
                          & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X5 ) )
                            = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xb ) ) )
                          & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xb ) ) )
                            = Xa2 ) ) ) ) ) ) ) ).

% output_complete.pelims(2)
thf(fact_991_output__complete_Opelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( output3834324022317553264_a_b_c @ X )
     => ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X )
       => ~ ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X )
           => ~ ! [X5: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X ) )
                 => ! [Xa2: c] :
                      ( ( member_c @ Xa2 @ ( outputs_a_b_c @ X ) )
                     => ? [Xb: produc2669932201312201782od_c_a] :
                          ( ( member5502040268208861279od_c_a @ Xb @ ( transitions_a_b_c @ X ) )
                          & ( ( produc4225943634317630866od_c_a @ X5 )
                            = ( produc4225943634317630866od_c_a @ Xb ) )
                          & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                            = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb ) ) )
                          & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb ) ) )
                            = Xa2 ) ) ) ) ) ) ) ).

% output_complete.pelims(2)
thf(fact_992_output__complete_Opelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ~ ( output5228291214970145744_a_b_c @ X )
     => ( ( accp_fsm_set_a_b_c @ output2932112043462401139_a_b_c @ X )
       => ~ ( ( accp_fsm_set_a_b_c @ output2932112043462401139_a_b_c @ X )
           => ! [X3: produc373761375140688950_set_a] :
                ( ( member6517745037573369951_set_a @ X3 @ ( transi8205445899535437940_a_b_c @ X ) )
               => ! [Xa3: c] :
                    ( ( member_c @ Xa3 @ ( outputs_set_a_b_c @ X ) )
                   => ? [Xb2: produc373761375140688950_set_a] :
                        ( ( member6517745037573369951_set_a @ Xb2 @ ( transi8205445899535437940_a_b_c @ X ) )
                        & ( ( produc7645221154340064658_set_a @ X3 )
                          = ( produc7645221154340064658_set_a @ Xb2 ) )
                        & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X3 ) )
                          = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xb2 ) ) )
                        & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xb2 ) ) )
                          = Xa3 ) ) ) ) ) ) ) ).

% output_complete.pelims(3)
thf(fact_993_output__complete_Opelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( output3834324022317553264_a_b_c @ X )
     => ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X )
       => ~ ( ( accp_fsm_a_b_c @ output8087333536391590675_a_b_c @ X )
           => ! [X3: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X ) )
               => ! [Xa3: c] :
                    ( ( member_c @ Xa3 @ ( outputs_a_b_c @ X ) )
                   => ? [Xb2: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Xb2 @ ( transitions_a_b_c @ X ) )
                        & ( ( produc4225943634317630866od_c_a @ X3 )
                          = ( produc4225943634317630866od_c_a @ Xb2 ) )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                          = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                        & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) ) )
                          = Xa3 ) ) ) ) ) ) ) ).

% output_complete.pelims(3)
thf(fact_994_observable_Opelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Y: $o] :
      ( ( ( observable_set_a_b_c @ X )
        = Y )
     => ( ( accp_fsm_set_a_b_c @ observ4119222391820296028_a_b_c @ X )
       => ~ ( ( Y
              = ( ! [X2: produc373761375140688950_set_a] :
                    ( ( member6517745037573369951_set_a @ X2 @ ( transi8205445899535437940_a_b_c @ X ) )
                   => ! [Y4: produc373761375140688950_set_a] :
                        ( ( member6517745037573369951_set_a @ Y4 @ ( transi8205445899535437940_a_b_c @ X ) )
                       => ( ( ( ( produc7645221154340064658_set_a @ X2 )
                              = ( produc7645221154340064658_set_a @ Y4 ) )
                            & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X2 ) )
                              = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Y4 ) ) )
                            & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X2 ) ) )
                              = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Y4 ) ) ) ) )
                         => ( ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X2 ) ) )
                            = ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Y4 ) ) ) ) ) ) ) ) )
           => ~ ( accp_fsm_set_a_b_c @ observ4119222391820296028_a_b_c @ X ) ) ) ) ).

% observable.pelims(1)
thf(fact_995_observable_Opelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( observable_a_b_c @ X )
        = Y )
     => ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X )
       => ~ ( ( Y
              = ( ! [X2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ X2 @ ( transitions_a_b_c @ X ) )
                   => ! [Y4: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X ) )
                       => ( ( ( ( produc4225943634317630866od_c_a @ X2 )
                              = ( produc4225943634317630866od_c_a @ Y4 ) )
                            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) )
                              = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) )
                            & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) ) )
                              = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) )
                         => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X2 ) ) )
                            = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) ) ) ) ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X ) ) ) ) ).

% observable.pelims(1)
thf(fact_996_observable_Opelims_I1_J,axiom,
    ! [X: fsm_nat_b_c,Y: $o] :
      ( ( ( observable_nat_b_c @ X )
        = Y )
     => ( ( accp_fsm_nat_b_c @ observ7905213042030342036at_b_c @ X )
       => ~ ( ( Y
              = ( ! [X2: produc8586872805302537566_c_nat] :
                    ( ( member3717341653178413575_c_nat @ X2 @ ( transitions_nat_b_c @ X ) )
                   => ! [Y4: produc8586872805302537566_c_nat] :
                        ( ( member3717341653178413575_c_nat @ Y4 @ ( transitions_nat_b_c @ X ) )
                       => ( ( ( ( produc532606554400396034_c_nat @ X2 )
                              = ( produc532606554400396034_c_nat @ Y4 ) )
                            & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ X2 ) )
                              = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ Y4 ) ) )
                            & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X2 ) ) )
                              = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Y4 ) ) ) ) )
                         => ( ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X2 ) ) )
                            = ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Y4 ) ) ) ) ) ) ) ) )
           => ~ ( accp_fsm_nat_b_c @ observ7905213042030342036at_b_c @ X ) ) ) ) ).

% observable.pelims(1)
thf(fact_997_observable_Opelims_I1_J,axiom,
    ! [X: fsm_Code_integer_b_c,Y: $o] :
      ( ( ( observ7524490533585752726er_b_c @ X )
        = Y )
     => ( ( accp_f9084625243946205319er_b_c @ observ6669331047229756833er_b_c @ X )
       => ~ ( ( Y
              = ( ! [X2: produc558599814252099832nteger] :
                    ( ( member5112046945434310049nteger @ X2 @ ( transi6707074847254859401er_b_c @ X ) )
                   => ! [Y4: produc558599814252099832nteger] :
                        ( ( member5112046945434310049nteger @ Y4 @ ( transi6707074847254859401er_b_c @ X ) )
                       => ( ( ( ( produc7104377276419476764nteger @ X2 )
                              = ( produc7104377276419476764nteger @ Y4 ) )
                            & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ X2 ) )
                              = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ Y4 ) ) )
                            & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X2 ) ) )
                              = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Y4 ) ) ) ) )
                         => ( ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X2 ) ) )
                            = ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Y4 ) ) ) ) ) ) ) ) )
           => ~ ( accp_f9084625243946205319er_b_c @ observ6669331047229756833er_b_c @ X ) ) ) ) ).

% observable.pelims(1)
thf(fact_998_observable_Opelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ( observable_set_a_b_c @ X )
     => ( ( accp_fsm_set_a_b_c @ observ4119222391820296028_a_b_c @ X )
       => ~ ( ( accp_fsm_set_a_b_c @ observ4119222391820296028_a_b_c @ X )
           => ~ ! [X5: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ X5 @ ( transi8205445899535437940_a_b_c @ X ) )
                 => ! [Xa2: produc373761375140688950_set_a] :
                      ( ( member6517745037573369951_set_a @ Xa2 @ ( transi8205445899535437940_a_b_c @ X ) )
                     => ( ( ( ( produc7645221154340064658_set_a @ X5 )
                            = ( produc7645221154340064658_set_a @ Xa2 ) )
                          & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X5 ) )
                            = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xa2 ) ) )
                          & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X5 ) ) )
                            = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xa2 ) ) ) ) )
                       => ( ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X5 ) ) )
                          = ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(2)
thf(fact_999_observable_Opelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( observable_a_b_c @ X )
     => ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X )
       => ~ ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X )
           => ~ ! [X5: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ X5 @ ( transitions_a_b_c @ X ) )
                 => ! [Xa2: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X ) )
                     => ( ( ( ( produc4225943634317630866od_c_a @ X5 )
                            = ( produc4225943634317630866od_c_a @ Xa2 ) )
                          & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) )
                            = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) )
                          & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                            = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) )
                       => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X5 ) ) )
                          = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(2)
thf(fact_1000_observable_Opelims_I2_J,axiom,
    ! [X: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ X )
     => ( ( accp_fsm_nat_b_c @ observ7905213042030342036at_b_c @ X )
       => ~ ( ( accp_fsm_nat_b_c @ observ7905213042030342036at_b_c @ X )
           => ~ ! [X5: produc8586872805302537566_c_nat] :
                  ( ( member3717341653178413575_c_nat @ X5 @ ( transitions_nat_b_c @ X ) )
                 => ! [Xa2: produc8586872805302537566_c_nat] :
                      ( ( member3717341653178413575_c_nat @ Xa2 @ ( transitions_nat_b_c @ X ) )
                     => ( ( ( ( produc532606554400396034_c_nat @ X5 )
                            = ( produc532606554400396034_c_nat @ Xa2 ) )
                          & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ X5 ) )
                            = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ Xa2 ) ) )
                          & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X5 ) ) )
                            = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Xa2 ) ) ) ) )
                       => ( ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X5 ) ) )
                          = ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(2)
thf(fact_1001_observable_Opelims_I2_J,axiom,
    ! [X: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ X )
     => ( ( accp_f9084625243946205319er_b_c @ observ6669331047229756833er_b_c @ X )
       => ~ ( ( accp_f9084625243946205319er_b_c @ observ6669331047229756833er_b_c @ X )
           => ~ ! [X5: produc558599814252099832nteger] :
                  ( ( member5112046945434310049nteger @ X5 @ ( transi6707074847254859401er_b_c @ X ) )
                 => ! [Xa2: produc558599814252099832nteger] :
                      ( ( member5112046945434310049nteger @ Xa2 @ ( transi6707074847254859401er_b_c @ X ) )
                     => ( ( ( ( produc7104377276419476764nteger @ X5 )
                            = ( produc7104377276419476764nteger @ Xa2 ) )
                          & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ X5 ) )
                            = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ Xa2 ) ) )
                          & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X5 ) ) )
                            = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Xa2 ) ) ) ) )
                       => ( ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X5 ) ) )
                          = ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(2)
thf(fact_1002_observable_Opelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ~ ( observable_set_a_b_c @ X )
     => ( ( accp_fsm_set_a_b_c @ observ4119222391820296028_a_b_c @ X )
       => ~ ( ( accp_fsm_set_a_b_c @ observ4119222391820296028_a_b_c @ X )
           => ! [X3: produc373761375140688950_set_a] :
                ( ( member6517745037573369951_set_a @ X3 @ ( transi8205445899535437940_a_b_c @ X ) )
               => ! [Xa3: produc373761375140688950_set_a] :
                    ( ( member6517745037573369951_set_a @ Xa3 @ ( transi8205445899535437940_a_b_c @ X ) )
                   => ( ( ( ( produc7645221154340064658_set_a @ X3 )
                          = ( produc7645221154340064658_set_a @ Xa3 ) )
                        & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X3 ) )
                          = ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xa3 ) ) )
                        & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X3 ) ) )
                          = ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xa3 ) ) ) ) )
                     => ( ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X3 ) ) )
                        = ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(3)
thf(fact_1003_observable_Opelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( observable_a_b_c @ X )
     => ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X )
       => ~ ( ( accp_fsm_a_b_c @ observable_rel_a_b_c @ X )
           => ! [X3: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ X ) )
               => ! [Xa3: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X ) )
                   => ( ( ( ( produc4225943634317630866od_c_a @ X3 )
                          = ( produc4225943634317630866od_c_a @ Xa3 ) )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                          = ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) )
                        & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                          = ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) )
                     => ( ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                        = ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(3)
thf(fact_1004_observable_Opelims_I3_J,axiom,
    ! [X: fsm_nat_b_c] :
      ( ~ ( observable_nat_b_c @ X )
     => ( ( accp_fsm_nat_b_c @ observ7905213042030342036at_b_c @ X )
       => ~ ( ( accp_fsm_nat_b_c @ observ7905213042030342036at_b_c @ X )
           => ! [X3: produc8586872805302537566_c_nat] :
                ( ( member3717341653178413575_c_nat @ X3 @ ( transitions_nat_b_c @ X ) )
               => ! [Xa3: produc8586872805302537566_c_nat] :
                    ( ( member3717341653178413575_c_nat @ Xa3 @ ( transitions_nat_b_c @ X ) )
                   => ( ( ( ( produc532606554400396034_c_nat @ X3 )
                          = ( produc532606554400396034_c_nat @ Xa3 ) )
                        & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) )
                          = ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ Xa3 ) ) )
                        & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) ) )
                          = ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Xa3 ) ) ) ) )
                     => ( ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) ) )
                        = ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(3)
thf(fact_1005_observable_Opelims_I3_J,axiom,
    ! [X: fsm_Code_integer_b_c] :
      ( ~ ( observ7524490533585752726er_b_c @ X )
     => ( ( accp_f9084625243946205319er_b_c @ observ6669331047229756833er_b_c @ X )
       => ~ ( ( accp_f9084625243946205319er_b_c @ observ6669331047229756833er_b_c @ X )
           => ! [X3: produc558599814252099832nteger] :
                ( ( member5112046945434310049nteger @ X3 @ ( transi6707074847254859401er_b_c @ X ) )
               => ! [Xa3: produc558599814252099832nteger] :
                    ( ( member5112046945434310049nteger @ Xa3 @ ( transi6707074847254859401er_b_c @ X ) )
                   => ( ( ( ( produc7104377276419476764nteger @ X3 )
                          = ( produc7104377276419476764nteger @ Xa3 ) )
                        & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ X3 ) )
                          = ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ Xa3 ) ) )
                        & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X3 ) ) )
                          = ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Xa3 ) ) ) ) )
                     => ( ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X3 ) ) )
                        = ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% observable.pelims(3)
thf(fact_1006_completely__specified_Opelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Y: $o] :
      ( ( ( comple9199925562481299400_a_b_c @ X )
        = Y )
     => ( ( accp_fsm_set_a_b_c @ comple1434486782884712059_a_b_c @ X )
       => ~ ( ( Y
              = ( ! [X2: set_a] :
                    ( ( member_set_a @ X2 @ ( states_set_a_b_c @ X ) )
                   => ! [Y4: b] :
                        ( ( member_b @ Y4 @ ( inputs_set_a_b_c @ X ) )
                       => ? [Z4: produc373761375140688950_set_a] :
                            ( ( member6517745037573369951_set_a @ Z4 @ ( transi8205445899535437940_a_b_c @ X ) )
                            & ( ( produc7645221154340064658_set_a @ Z4 )
                              = X2 )
                            & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Z4 ) )
                              = Y4 ) ) ) ) ) )
           => ~ ( accp_fsm_set_a_b_c @ comple1434486782884712059_a_b_c @ X ) ) ) ) ).

% completely_specified.pelims(1)
thf(fact_1007_completely__specified_Opelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Y: $o] :
      ( ( ( comple6900478119722646120_a_b_c @ X )
        = Y )
     => ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X )
       => ~ ( ( Y
              = ( ! [X2: a] :
                    ( ( member_a @ X2 @ ( states_a_b_c @ X ) )
                   => ! [Y4: b] :
                        ( ( member_b @ Y4 @ ( inputs_a_b_c @ X ) )
                       => ? [Z4: produc2669932201312201782od_c_a] :
                            ( ( member5502040268208861279od_c_a @ Z4 @ ( transitions_a_b_c @ X ) )
                            & ( ( produc4225943634317630866od_c_a @ Z4 )
                              = X2 )
                            & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Z4 ) )
                              = Y4 ) ) ) ) ) )
           => ~ ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X ) ) ) ) ).

% completely_specified.pelims(1)
thf(fact_1008_completely__specified_Opelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ( comple9199925562481299400_a_b_c @ X )
     => ( ( accp_fsm_set_a_b_c @ comple1434486782884712059_a_b_c @ X )
       => ~ ( ( accp_fsm_set_a_b_c @ comple1434486782884712059_a_b_c @ X )
           => ~ ! [X5: set_a] :
                  ( ( member_set_a @ X5 @ ( states_set_a_b_c @ X ) )
                 => ! [Xa2: b] :
                      ( ( member_b @ Xa2 @ ( inputs_set_a_b_c @ X ) )
                     => ? [Xb: produc373761375140688950_set_a] :
                          ( ( member6517745037573369951_set_a @ Xb @ ( transi8205445899535437940_a_b_c @ X ) )
                          & ( ( produc7645221154340064658_set_a @ Xb )
                            = X5 )
                          & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xb ) )
                            = Xa2 ) ) ) ) ) ) ) ).

% completely_specified.pelims(2)
thf(fact_1009_completely__specified_Opelims_I2_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ( comple6900478119722646120_a_b_c @ X )
     => ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X )
       => ~ ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X )
           => ~ ! [X5: a] :
                  ( ( member_a @ X5 @ ( states_a_b_c @ X ) )
                 => ! [Xa2: b] :
                      ( ( member_b @ Xa2 @ ( inputs_a_b_c @ X ) )
                     => ? [Xb: produc2669932201312201782od_c_a] :
                          ( ( member5502040268208861279od_c_a @ Xb @ ( transitions_a_b_c @ X ) )
                          & ( ( produc4225943634317630866od_c_a @ Xb )
                            = X5 )
                          & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb ) )
                            = Xa2 ) ) ) ) ) ) ) ).

% completely_specified.pelims(2)
thf(fact_1010_completely__specified_Opelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c] :
      ( ~ ( comple9199925562481299400_a_b_c @ X )
     => ( ( accp_fsm_set_a_b_c @ comple1434486782884712059_a_b_c @ X )
       => ~ ( ( accp_fsm_set_a_b_c @ comple1434486782884712059_a_b_c @ X )
           => ! [X3: set_a] :
                ( ( member_set_a @ X3 @ ( states_set_a_b_c @ X ) )
               => ! [Xa3: b] :
                    ( ( member_b @ Xa3 @ ( inputs_set_a_b_c @ X ) )
                   => ? [Xb2: produc373761375140688950_set_a] :
                        ( ( member6517745037573369951_set_a @ Xb2 @ ( transi8205445899535437940_a_b_c @ X ) )
                        & ( ( produc7645221154340064658_set_a @ Xb2 )
                          = X3 )
                        & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xb2 ) )
                          = Xa3 ) ) ) ) ) ) ) ).

% completely_specified.pelims(3)
thf(fact_1011_completely__specified_Opelims_I3_J,axiom,
    ! [X: fsm_a_b_c] :
      ( ~ ( comple6900478119722646120_a_b_c @ X )
     => ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X )
       => ~ ( ( accp_fsm_a_b_c @ comple1171262150424069403_a_b_c @ X )
           => ! [X3: a] :
                ( ( member_a @ X3 @ ( states_a_b_c @ X ) )
               => ! [Xa3: b] :
                    ( ( member_b @ Xa3 @ ( inputs_a_b_c @ X ) )
                   => ? [Xb2: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Xb2 @ ( transitions_a_b_c @ X ) )
                        & ( ( produc4225943634317630866od_c_a @ Xb2 )
                          = X3 )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xb2 ) )
                          = Xa3 ) ) ) ) ) ) ) ).

% completely_specified.pelims(3)
thf(fact_1012_add__transitions__simps_I1_J,axiom,
    ! [Ts: set_Pr3565145081747743382_set_a,M: fsm_set_a_b_c] :
      ( ! [T2: produc373761375140688950_set_a] :
          ( ( member6517745037573369951_set_a @ T2 @ Ts )
         => ( ( member_set_a @ ( produc7645221154340064658_set_a @ T2 ) @ ( states_set_a_b_c @ M ) )
            & ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) @ ( inputs_set_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( outputs_set_a_b_c @ M ) )
            & ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( states_set_a_b_c @ M ) ) ) )
     => ( ( initial_set_a_b_c @ ( add_tr3542012413116934189_a_b_c @ M @ Ts ) )
        = ( initial_set_a_b_c @ M ) ) ) ).

% add_transitions_simps(1)
thf(fact_1013_add__transitions__simps_I1_J,axiom,
    ! [Ts: set_Pr8357624050790843030od_c_a,M: fsm_a_b_c] :
      ( ! [T2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ T2 @ Ts )
         => ( ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ ( states_a_b_c @ M ) )
            & ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( inputs_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( outputs_a_b_c @ M ) )
            & ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( states_a_b_c @ M ) ) ) )
     => ( ( initial_a_b_c @ ( add_tr968201492682548941_a_b_c @ M @ Ts ) )
        = ( initial_a_b_c @ M ) ) ) ).

% add_transitions_simps(1)
thf(fact_1014_add__transitions__simps_I4_J,axiom,
    ! [Ts: set_Pr3565145081747743382_set_a,M: fsm_set_a_b_c] :
      ( ! [T2: produc373761375140688950_set_a] :
          ( ( member6517745037573369951_set_a @ T2 @ Ts )
         => ( ( member_set_a @ ( produc7645221154340064658_set_a @ T2 ) @ ( states_set_a_b_c @ M ) )
            & ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) @ ( inputs_set_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( outputs_set_a_b_c @ M ) )
            & ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( states_set_a_b_c @ M ) ) ) )
     => ( ( outputs_set_a_b_c @ ( add_tr3542012413116934189_a_b_c @ M @ Ts ) )
        = ( outputs_set_a_b_c @ M ) ) ) ).

% add_transitions_simps(4)
thf(fact_1015_add__transitions__simps_I4_J,axiom,
    ! [Ts: set_Pr8357624050790843030od_c_a,M: fsm_a_b_c] :
      ( ! [T2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ T2 @ Ts )
         => ( ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ ( states_a_b_c @ M ) )
            & ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( inputs_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( outputs_a_b_c @ M ) )
            & ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( states_a_b_c @ M ) ) ) )
     => ( ( outputs_a_b_c @ ( add_tr968201492682548941_a_b_c @ M @ Ts ) )
        = ( outputs_a_b_c @ M ) ) ) ).

% add_transitions_simps(4)
thf(fact_1016_add__transitions__simps_I3_J,axiom,
    ! [Ts: set_Pr3565145081747743382_set_a,M: fsm_set_a_b_c] :
      ( ! [T2: produc373761375140688950_set_a] :
          ( ( member6517745037573369951_set_a @ T2 @ Ts )
         => ( ( member_set_a @ ( produc7645221154340064658_set_a @ T2 ) @ ( states_set_a_b_c @ M ) )
            & ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) @ ( inputs_set_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( outputs_set_a_b_c @ M ) )
            & ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( states_set_a_b_c @ M ) ) ) )
     => ( ( inputs_set_a_b_c @ ( add_tr3542012413116934189_a_b_c @ M @ Ts ) )
        = ( inputs_set_a_b_c @ M ) ) ) ).

% add_transitions_simps(3)
thf(fact_1017_add__transitions__simps_I3_J,axiom,
    ! [Ts: set_Pr8357624050790843030od_c_a,M: fsm_a_b_c] :
      ( ! [T2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ T2 @ Ts )
         => ( ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ ( states_a_b_c @ M ) )
            & ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( inputs_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( outputs_a_b_c @ M ) )
            & ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( states_a_b_c @ M ) ) ) )
     => ( ( inputs_a_b_c @ ( add_tr968201492682548941_a_b_c @ M @ Ts ) )
        = ( inputs_a_b_c @ M ) ) ) ).

% add_transitions_simps(3)
thf(fact_1018_add__transitions__simps_I2_J,axiom,
    ! [Ts: set_Pr3565145081747743382_set_a,M: fsm_set_a_b_c] :
      ( ! [T2: produc373761375140688950_set_a] :
          ( ( member6517745037573369951_set_a @ T2 @ Ts )
         => ( ( member_set_a @ ( produc7645221154340064658_set_a @ T2 ) @ ( states_set_a_b_c @ M ) )
            & ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) @ ( inputs_set_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( outputs_set_a_b_c @ M ) )
            & ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( states_set_a_b_c @ M ) ) ) )
     => ( ( states_set_a_b_c @ ( add_tr3542012413116934189_a_b_c @ M @ Ts ) )
        = ( states_set_a_b_c @ M ) ) ) ).

% add_transitions_simps(2)
thf(fact_1019_add__transitions__simps_I2_J,axiom,
    ! [Ts: set_Pr8357624050790843030od_c_a,M: fsm_a_b_c] :
      ( ! [T2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ T2 @ Ts )
         => ( ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ ( states_a_b_c @ M ) )
            & ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( inputs_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( outputs_a_b_c @ M ) )
            & ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( states_a_b_c @ M ) ) ) )
     => ( ( states_a_b_c @ ( add_tr968201492682548941_a_b_c @ M @ Ts ) )
        = ( states_a_b_c @ M ) ) ) ).

% add_transitions_simps(2)
thf(fact_1020_add__transition__with__components__simps_I2_J,axiom,
    ! [M: fsm_set_a_b_c,T: produc373761375140688950_set_a] :
      ( ( inputs_set_a_b_c @ ( add_tr7888489378088620512_a_b_c @ M @ T ) )
      = ( insert_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T ) ) @ ( inputs_set_a_b_c @ M ) ) ) ).

% add_transition_with_components_simps(2)
thf(fact_1021_add__transition__with__components__simps_I2_J,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( inputs_a_b_c @ ( add_tr8224572408740125824_a_b_c @ M @ T ) )
      = ( insert_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) @ ( inputs_a_b_c @ M ) ) ) ).

% add_transition_with_components_simps(2)
thf(fact_1022_add__transition__with__components__simps_I3_J,axiom,
    ! [M: fsm_set_a_b_c,T: produc373761375140688950_set_a] :
      ( ( outputs_set_a_b_c @ ( add_tr7888489378088620512_a_b_c @ M @ T ) )
      = ( insert_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) @ ( outputs_set_a_b_c @ M ) ) ) ).

% add_transition_with_components_simps(3)
thf(fact_1023_add__transition__with__components__simps_I3_J,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( outputs_a_b_c @ ( add_tr8224572408740125824_a_b_c @ M @ T ) )
      = ( insert_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) @ ( outputs_a_b_c @ M ) ) ) ).

% add_transition_with_components_simps(3)
thf(fact_1024_add__transitions__simps_I5_J,axiom,
    ! [Ts: set_Pr3565145081747743382_set_a,M: fsm_set_a_b_c] :
      ( ! [T2: produc373761375140688950_set_a] :
          ( ( member6517745037573369951_set_a @ T2 @ Ts )
         => ( ( member_set_a @ ( produc7645221154340064658_set_a @ T2 ) @ ( states_set_a_b_c @ M ) )
            & ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) @ ( inputs_set_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( outputs_set_a_b_c @ M ) )
            & ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ ( states_set_a_b_c @ M ) ) ) )
     => ( ( transi8205445899535437940_a_b_c @ ( add_tr3542012413116934189_a_b_c @ M @ Ts ) )
        = ( sup_su4168421701146457194_set_a @ ( transi8205445899535437940_a_b_c @ M ) @ Ts ) ) ) ).

% add_transitions_simps(5)
thf(fact_1025_add__transitions__simps_I5_J,axiom,
    ! [Ts: set_Pr8357624050790843030od_c_a,M: fsm_a_b_c] :
      ( ! [T2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ T2 @ Ts )
         => ( ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ ( states_a_b_c @ M ) )
            & ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ ( inputs_a_b_c @ M ) )
            & ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( outputs_a_b_c @ M ) )
            & ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ ( states_a_b_c @ M ) ) ) )
     => ( ( transitions_a_b_c @ ( add_tr968201492682548941_a_b_c @ M @ Ts ) )
        = ( sup_su7950883165052597354od_c_a @ ( transitions_a_b_c @ M ) @ Ts ) ) ) ).

% add_transitions_simps(5)
thf(fact_1026_create__fsm__from__sets__simps_I4_J,axiom,
    ! [Q: set_a,Qs: set_set_a,Ins: set_b,Outs: set_c,Ts: set_Pr3565145081747743382_set_a] :
      ( ( member_set_a @ Q @ Qs )
     => ( ( finite_finite_set_a @ Qs )
       => ( ( finite_finite_b @ Ins )
         => ( ( finite_finite_c @ Outs )
           => ( ! [T2: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ T2 @ Ts )
                 => ( ( member_set_a @ ( produc7645221154340064658_set_a @ T2 ) @ Qs )
                    & ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) @ Ins )
                    & ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ Outs )
                    & ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ Qs ) ) )
             => ( ( outputs_set_a_b_c @ ( create7113092633863273043_a_b_c @ Q @ Qs @ Ins @ Outs @ Ts ) )
                = Outs ) ) ) ) ) ) ).

% create_fsm_from_sets_simps(4)
thf(fact_1027_create__fsm__from__sets__simps_I4_J,axiom,
    ! [Q: a,Qs: set_a,Ins: set_b,Outs: set_c,Ts: set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ Q @ Qs )
     => ( ( finite_finite_a @ Qs )
       => ( ( finite_finite_b @ Ins )
         => ( ( finite_finite_c @ Outs )
           => ( ! [T2: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ T2 @ Ts )
                 => ( ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ Qs )
                    & ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ Ins )
                    & ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ Outs )
                    & ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ Qs ) ) )
             => ( ( outputs_a_b_c @ ( create3933655864275538163_a_b_c @ Q @ Qs @ Ins @ Outs @ Ts ) )
                = Outs ) ) ) ) ) ) ).

% create_fsm_from_sets_simps(4)
thf(fact_1028_create__fsm__from__sets__simps_I4_J,axiom,
    ! [Q: nat,Qs: set_nat,Ins: set_nat,Outs: set_nat,Ts: set_Pr58315729134567223at_nat] :
      ( ( member_nat @ Q @ Qs )
     => ( ( finite_finite_nat @ Qs )
       => ( ( finite_finite_nat @ Ins )
         => ( ( finite_finite_nat @ Outs )
           => ( ! [T2: produc8642769642335960151at_nat] :
                  ( ( member6766774627807740288at_nat @ T2 @ Ts )
                 => ( ( member_nat @ ( produc3437397837235661307at_nat @ T2 ) @ Qs )
                    & ( member_nat @ ( produc7982242761980330092at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) @ Ins )
                    & ( member_nat @ ( product_fst_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Outs )
                    & ( member_nat @ ( product_snd_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Qs ) ) )
             => ( ( outputs_nat_nat_nat @ ( create4800940341628538942at_nat @ Q @ Qs @ Ins @ Outs @ Ts ) )
                = Outs ) ) ) ) ) ) ).

% create_fsm_from_sets_simps(4)
thf(fact_1029_create__fsm__from__sets__simps_I3_J,axiom,
    ! [Q: set_a,Qs: set_set_a,Ins: set_b,Outs: set_c,Ts: set_Pr3565145081747743382_set_a] :
      ( ( member_set_a @ Q @ Qs )
     => ( ( finite_finite_set_a @ Qs )
       => ( ( finite_finite_b @ Ins )
         => ( ( finite_finite_c @ Outs )
           => ( ! [T2: produc373761375140688950_set_a] :
                  ( ( member6517745037573369951_set_a @ T2 @ Ts )
                 => ( ( member_set_a @ ( produc7645221154340064658_set_a @ T2 ) @ Qs )
                    & ( member_b @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) @ Ins )
                    & ( member_c @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ Outs )
                    & ( member_set_a @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) ) @ Qs ) ) )
             => ( ( inputs_set_a_b_c @ ( create7113092633863273043_a_b_c @ Q @ Qs @ Ins @ Outs @ Ts ) )
                = Ins ) ) ) ) ) ) ).

% create_fsm_from_sets_simps(3)
thf(fact_1030_create__fsm__from__sets__simps_I3_J,axiom,
    ! [Q: a,Qs: set_a,Ins: set_b,Outs: set_c,Ts: set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ Q @ Qs )
     => ( ( finite_finite_a @ Qs )
       => ( ( finite_finite_b @ Ins )
         => ( ( finite_finite_c @ Outs )
           => ( ! [T2: produc2669932201312201782od_c_a] :
                  ( ( member5502040268208861279od_c_a @ T2 @ Ts )
                 => ( ( member_a @ ( produc4225943634317630866od_c_a @ T2 ) @ Qs )
                    & ( member_b @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) @ Ins )
                    & ( member_c @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ Outs )
                    & ( member_a @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) ) @ Qs ) ) )
             => ( ( inputs_a_b_c @ ( create3933655864275538163_a_b_c @ Q @ Qs @ Ins @ Outs @ Ts ) )
                = Ins ) ) ) ) ) ) ).

% create_fsm_from_sets_simps(3)
thf(fact_1031_create__fsm__from__sets__simps_I3_J,axiom,
    ! [Q: nat,Qs: set_nat,Ins: set_nat,Outs: set_nat,Ts: set_Pr58315729134567223at_nat] :
      ( ( member_nat @ Q @ Qs )
     => ( ( finite_finite_nat @ Qs )
       => ( ( finite_finite_nat @ Ins )
         => ( ( finite_finite_nat @ Outs )
           => ( ! [T2: produc8642769642335960151at_nat] :
                  ( ( member6766774627807740288at_nat @ T2 @ Ts )
                 => ( ( member_nat @ ( produc3437397837235661307at_nat @ T2 ) @ Qs )
                    & ( member_nat @ ( produc7982242761980330092at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) @ Ins )
                    & ( member_nat @ ( product_fst_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Outs )
                    & ( member_nat @ ( product_snd_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Qs ) ) )
             => ( ( inputs_nat_nat_nat @ ( create4800940341628538942at_nat @ Q @ Qs @ Ins @ Outs @ Ts ) )
                = Ins ) ) ) ) ) ) ).

% create_fsm_from_sets_simps(3)
thf(fact_1032_finite__imageI,axiom,
    ! [F2: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1033_finite__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
      = ( finite_finite_nat @ A ) ) ).

% finite_insert
thf(fact_1034_finite__Un,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
      = ( ( finite_finite_nat @ F2 )
        & ( finite_finite_nat @ G ) ) ) ).

% finite_Un
thf(fact_1035_finite__Pow__iff,axiom,
    ! [A: set_nat] :
      ( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
      = ( finite_finite_nat @ A ) ) ).

% finite_Pow_iff
thf(fact_1036_finite__set__min__param__ex,axiom,
    ! [XS: set_nat,P: nat > nat > $o] :
      ( ( finite_finite_nat @ XS )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ XS )
           => ? [K3: nat] :
              ! [K4: nat] :
                ( ( ord_less_eq_nat @ K3 @ K4 )
               => ( P @ X3 @ K4 ) ) )
       => ? [K2: nat] :
          ! [X5: nat] :
            ( ( member_nat @ X5 @ XS )
           => ( P @ X5 @ K2 ) ) ) ) ).

% finite_set_min_param_ex
thf(fact_1037_fsm__inputs__finite,axiom,
    ! [M: fsm_set_a_b_c] : ( finite_finite_b @ ( inputs_set_a_b_c @ M ) ) ).

% fsm_inputs_finite
thf(fact_1038_fsm__inputs__finite,axiom,
    ! [M: fsm_a_b_c] : ( finite_finite_b @ ( inputs_a_b_c @ M ) ) ).

% fsm_inputs_finite
thf(fact_1039_fsm__outputs__finite,axiom,
    ! [M: fsm_set_a_b_c] : ( finite_finite_c @ ( outputs_set_a_b_c @ M ) ) ).

% fsm_outputs_finite
thf(fact_1040_fsm__outputs__finite,axiom,
    ! [M: fsm_a_b_c] : ( finite_finite_c @ ( outputs_a_b_c @ M ) ) ).

% fsm_outputs_finite
thf(fact_1041_finite__UnI,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( finite_finite_nat @ G )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).

% finite_UnI
thf(fact_1042_Un__infinite,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T3 ) ) ) ).

% Un_infinite
thf(fact_1043_infinite__Un,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T3 ) ) )
      = ( ~ ( finite_finite_nat @ S )
        | ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_Un
thf(fact_1044_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_1045_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_1046_bij__betw__finite,axiom,
    ! [F: nat > nat,A: set_nat,B: set_nat] :
      ( ( bij_betw_nat_nat @ F @ A @ B )
     => ( ( finite_finite_nat @ A )
        = ( finite_finite_nat @ B ) ) ) ).

% bij_betw_finite
thf(fact_1047_finite__filter,axiom,
    ! [S: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( finite_finite_nat @ ( filter_nat @ P @ S ) ) ) ).

% finite_filter
thf(fact_1048_rev__finite__subset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_1049_infinite__super,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T3 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_1050_finite__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_1051_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ( ord_less_eq_nat @ X3 @ A2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1052_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ( ord_less_eq_nat @ A2 @ X3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1053_finite__psubset__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [A7: set_nat] :
            ( ( finite_finite_nat @ A7 )
           => ( ! [B7: set_nat] :
                  ( ( ord_less_set_nat @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_1054_finite_OinsertI,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_1055_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1056_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1057_all__finite__subset__image,axiom,
    ! [F: nat > nat,A: set_nat,P: set_nat > $o] :
      ( ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) ) )
           => ( P @ B2 ) ) )
      = ( ! [B2: set_nat] :
            ( ( ( finite_finite_nat @ B2 )
              & ( ord_less_eq_set_nat @ B2 @ A ) )
           => ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1058_ex__finite__subset__image,axiom,
    ! [F: nat > nat,A: set_nat,P: set_nat > $o] :
      ( ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
            & ( P @ B2 ) ) )
      = ( ? [B2: set_nat] :
            ( ( finite_finite_nat @ B2 )
            & ( ord_less_eq_set_nat @ B2 @ A )
            & ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1059_finite__subset__image,axiom,
    ! [B: set_nat,F: nat > nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
       => ? [C3: set_nat] :
            ( ( ord_less_eq_set_nat @ C3 @ A )
            & ( finite_finite_nat @ C3 )
            & ( B
              = ( image_nat_nat @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1060_finite__surj,axiom,
    ! [A: set_nat,B: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_surj
thf(fact_1061_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_1062_finite__ne__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
         => ( ! [X3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1063_finite__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_1064_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A4: set_nat] :
          ( ( A4 = bot_bot_set_nat )
          | ? [A3: set_nat,B4: nat] :
              ( ( A4
                = ( insert_nat @ B4 @ A3 ) )
              & ( finite_finite_nat @ A3 ) ) ) ) ) ).

% finite.simps
thf(fact_1065_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A5: nat] :
                  ( A2
                  = ( insert_nat @ A5 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1066_image__inj__card__helper,axiom,
    ! [X6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ X6 )
     => ( ! [A5: nat,B5: nat] :
            ( ( member_nat @ A5 @ X6 )
           => ( ( member_nat @ B5 @ X6 )
             => ( ( A5 != B5 )
               => ( ( F @ A5 )
                 != ( F @ B5 ) ) ) ) )
       => ( ( finite_card_nat @ ( image_nat_nat @ F @ X6 ) )
          = ( finite_card_nat @ X6 ) ) ) ) ).

% image_inj_card_helper
thf(fact_1067_infinite__arbitrarily__large,axiom,
    ! [A: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ A )
     => ? [B8: set_nat] :
          ( ( finite_finite_nat @ B8 )
          & ( ( finite_card_nat @ B8 )
            = N2 )
          & ( ord_less_eq_set_nat @ B8 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1068_card__subset__eq,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ( finite_card_nat @ A )
            = ( finite_card_nat @ B ) )
         => ( A = B ) ) ) ) ).

% card_subset_eq
thf(fact_1069_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S3: set_nat] :
              ( ( finite_finite_nat @ S3 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S3 )
                   => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_nat @ X3 @ S3 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_1070_finite__linorder__max__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A7 )
                   => ( ord_less_nat @ X5 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1071_finite__linorder__min__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A7 )
                   => ( ord_less_nat @ B5 @ X5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B5 @ A7 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_1072_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A5 @ A )
                 => ( ( ord_less_eq_set_nat @ F3 @ A )
                   => ( ~ ( member_nat @ A5 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1073_finite__subset__induct,axiom,
    ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A5 @ A )
                 => ( ~ ( member_nat @ A5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1074_finite__card__less__witnesses,axiom,
    ! [A: set_nat,G2: nat > nat,F: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ G2 @ A ) ) @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) )
       => ~ ! [A5: nat] :
              ( ( member_nat @ A5 @ A )
             => ! [B5: nat] :
                  ( ( member_nat @ B5 @ A )
                 => ( ( ( F @ A5 )
                     != ( F @ B5 ) )
                   => ( ( G2 @ A5 )
                     != ( G2 @ B5 ) ) ) ) ) ) ) ).

% finite_card_less_witnesses
thf(fact_1075_card__image__le,axiom,
    ! [A: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).

% card_image_le
thf(fact_1076_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_nat,C4: nat] :
      ( ! [G3: set_nat] :
          ( ( ord_less_eq_set_nat @ G3 @ F2 )
         => ( ( finite_finite_nat @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C4 ) ) )
     => ( ( finite_finite_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1077_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
     => ~ ! [T4: set_nat] :
            ( ( ord_less_eq_set_nat @ T4 @ S )
           => ( ( ( finite_card_nat @ T4 )
                = N2 )
             => ~ ( finite_finite_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1078_exists__subset__between,axiom,
    ! [A: set_nat,N2: nat,C4: set_nat] :
      ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C4 ) )
       => ( ( ord_less_eq_set_nat @ A @ C4 )
         => ( ( finite_finite_nat @ C4 )
           => ? [B8: set_nat] :
                ( ( ord_less_eq_set_nat @ A @ B8 )
                & ( ord_less_eq_set_nat @ B8 @ C4 )
                & ( ( finite_card_nat @ B8 )
                  = N2 ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_1079_card__seteq,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
         => ( A = B ) ) ) ) ).

% card_seteq
thf(fact_1080_card__mono,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).

% card_mono
thf(fact_1081_psubset__card__mono,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_set_nat @ A @ B )
       => ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).

% psubset_card_mono
thf(fact_1082_surj__card__le,axiom,
    ! [A: set_nat,B: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).

% surj_card_le
thf(fact_1083_card__psubset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
         => ( ord_less_set_nat @ A @ B ) ) ) ) ).

% card_psubset
thf(fact_1084_create__fsm__from__sets__simps_I2_J,axiom,
    ! [Q: nat,Qs: set_nat,Ins: set_nat,Outs: set_nat,Ts: set_Pr58315729134567223at_nat] :
      ( ( member_nat @ Q @ Qs )
     => ( ( finite_finite_nat @ Qs )
       => ( ( finite_finite_nat @ Ins )
         => ( ( finite_finite_nat @ Outs )
           => ( ! [T2: produc8642769642335960151at_nat] :
                  ( ( member6766774627807740288at_nat @ T2 @ Ts )
                 => ( ( member_nat @ ( produc3437397837235661307at_nat @ T2 ) @ Qs )
                    & ( member_nat @ ( produc7982242761980330092at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) @ Ins )
                    & ( member_nat @ ( product_fst_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Outs )
                    & ( member_nat @ ( product_snd_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Qs ) ) )
             => ( ( states_nat_nat_nat @ ( create4800940341628538942at_nat @ Q @ Qs @ Ins @ Outs @ Ts ) )
                = Qs ) ) ) ) ) ) ).

% create_fsm_from_sets_simps(2)
thf(fact_1085_create__fsm__from__sets__simps_I1_J,axiom,
    ! [Q: nat,Qs: set_nat,Ins: set_nat,Outs: set_nat,Ts: set_Pr58315729134567223at_nat] :
      ( ( member_nat @ Q @ Qs )
     => ( ( finite_finite_nat @ Qs )
       => ( ( finite_finite_nat @ Ins )
         => ( ( finite_finite_nat @ Outs )
           => ( ! [T2: produc8642769642335960151at_nat] :
                  ( ( member6766774627807740288at_nat @ T2 @ Ts )
                 => ( ( member_nat @ ( produc3437397837235661307at_nat @ T2 ) @ Qs )
                    & ( member_nat @ ( produc7982242761980330092at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) @ Ins )
                    & ( member_nat @ ( product_fst_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Outs )
                    & ( member_nat @ ( product_snd_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Qs ) ) )
             => ( ( initial_nat_nat_nat @ ( create4800940341628538942at_nat @ Q @ Qs @ Ins @ Outs @ Ts ) )
                = Q ) ) ) ) ) ) ).

% create_fsm_from_sets_simps(1)
thf(fact_1086_create__fsm__from__sets__simps_I5_J,axiom,
    ! [Q: nat,Qs: set_nat,Ins: set_nat,Outs: set_nat,Ts: set_Pr58315729134567223at_nat] :
      ( ( member_nat @ Q @ Qs )
     => ( ( finite_finite_nat @ Qs )
       => ( ( finite_finite_nat @ Ins )
         => ( ( finite_finite_nat @ Outs )
           => ( ! [T2: produc8642769642335960151at_nat] :
                  ( ( member6766774627807740288at_nat @ T2 @ Ts )
                 => ( ( member_nat @ ( produc3437397837235661307at_nat @ T2 ) @ Qs )
                    & ( member_nat @ ( produc7982242761980330092at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) @ Ins )
                    & ( member_nat @ ( product_fst_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Outs )
                    & ( member_nat @ ( product_snd_nat_nat @ ( produc6427046184213638570at_nat @ ( produc6631887067320006461at_nat @ T2 ) ) ) @ Qs ) ) )
             => ( ( transi2178791137084441437at_nat @ ( create4800940341628538942at_nat @ Q @ Qs @ Ins @ Outs @ Ts ) )
                = Ts ) ) ) ) ) ) ).

% create_fsm_from_sets_simps(5)
thf(fact_1087_infinite__imp__bij__betw2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ~ ( finite_finite_nat @ A )
     => ? [H2: nat > nat] : ( bij_betw_nat_nat @ H2 @ A @ ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% infinite_imp_bij_betw2
thf(fact_1088_sup_Obounded__iff,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_nat @ B3 @ A2 )
        & ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1089_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
      = ( ( ord_less_eq_nat @ X @ Z2 )
        & ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_1090_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1091_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1092_le__supE,axiom,
    ! [A2: nat,B3: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X )
     => ~ ( ( ord_less_eq_nat @ A2 @ X )
         => ~ ( ord_less_eq_nat @ B3 @ X ) ) ) ).

% le_supE
thf(fact_1093_le__supI,axiom,
    ! [A2: nat,X: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ( ord_less_eq_nat @ B3 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X ) ) ) ).

% le_supI
thf(fact_1094_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1095_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1096_le__supI1,axiom,
    ! [X: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% le_supI1
thf(fact_1097_le__supI2,axiom,
    ! [X: nat,B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ X @ B3 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% le_supI2
thf(fact_1098_sup_Omono,axiom,
    ! [C: nat,A2: nat,D2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ( ord_less_eq_nat @ D2 @ B3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B3 ) ) ) ) ).

% sup.mono
thf(fact_1099_sup__mono,axiom,
    ! [A2: nat,C: nat,B3: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B3 @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1100_sup__least,axiom,
    ! [Y: nat,X: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z2 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).

% sup_least
thf(fact_1101_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y4: nat] :
          ( ( sup_sup_nat @ X2 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_1102_sup_OorderE,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( A2
        = ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% sup.orderE
thf(fact_1103_sup_OorderI,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2
        = ( sup_sup_nat @ A2 @ B3 ) )
     => ( ord_less_eq_nat @ B3 @ A2 ) ) ).

% sup.orderI
thf(fact_1104_sup__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
     => ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
       => ( ! [X3: nat,Y2: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y2 @ X3 )
             => ( ( ord_less_eq_nat @ Z3 @ X3 )
               => ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_1105_sup_Oabsorb1,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_1106_sup_Oabsorb2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( sup_sup_nat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_1107_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1108_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1109_sup_OboundedE,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_nat @ B3 @ A2 )
         => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_1110_sup_OboundedI,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_1111_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_1112_sup_Ocobounded1,axiom,
    ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) ) ).

% sup.cobounded1
thf(fact_1113_sup_Ocobounded2,axiom,
    ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A2 @ B3 ) ) ).

% sup.cobounded2
thf(fact_1114_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_1115_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_1116_sup_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_1117_sup_OcoboundedI2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ B3 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_1118_finite__same__card__bij,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ( ( finite_card_nat @ A )
            = ( finite_card_nat @ B ) )
         => ? [H2: nat > nat] : ( bij_betw_nat_nat @ H2 @ A @ B ) ) ) ) ).

% finite_same_card_bij
thf(fact_1119_bij__betw__iff__card,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ( ? [F4: nat > nat] : ( bij_betw_nat_nat @ F4 @ A @ B ) )
          = ( ( finite_card_nat @ A )
            = ( finite_card_nat @ B ) ) ) ) ) ).

% bij_betw_iff_card
thf(fact_1120_card__le__if__inj__on__rel,axiom,
    ! [B: set_nat,A: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ B )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A )
           => ? [B9: nat] :
                ( ( member_nat @ B9 @ B )
                & ( R @ A5 @ B9 ) ) )
       => ( ! [A1: nat,A22: nat,B5: nat] :
              ( ( member_nat @ A1 @ A )
             => ( ( member_nat @ A22 @ A )
               => ( ( member_nat @ B5 @ B )
                 => ( ( R @ A1 @ B5 )
                   => ( ( R @ A22 @ B5 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1121_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N3: set_nat] :
        ? [M4: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N3 )
         => ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1122_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M: nat] :
      ( ( P @ X )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M ) )
       => ~ ! [M5: nat] :
              ( ( P @ M5 )
             => ~ ! [X5: nat] :
                    ( ( P @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1123_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1124_canonical__separator_H__simps__without__assm_I4_J,axiom,
    ! [P: fsm_Pr1307097241311862860_a_b_c,Q1: set_a,Q22: set_a,M: fsm_set_a_b_c] :
      ( ( ( ( initia9132321972455559959_a_b_c @ P )
          = ( produc9088192753505129239_set_a @ Q1 @ Q22 ) )
       => ( ( output4897735804923954172_a_b_c @ ( canoni669745312745679216_a_b_c @ M @ P @ Q1 @ Q22 ) )
          = ( sup_sup_set_c @ ( outputs_set_a_b_c @ M ) @ ( output4507429379629698825_a_b_c @ P ) ) ) )
      & ( ( ( initia9132321972455559959_a_b_c @ P )
         != ( produc9088192753505129239_set_a @ Q1 @ Q22 ) )
       => ( ( output4897735804923954172_a_b_c @ ( canoni669745312745679216_a_b_c @ M @ P @ Q1 @ Q22 ) )
          = bot_bot_set_c ) ) ) ).

% canonical_separator'_simps_without_assm(4)
thf(fact_1125_canonical__separator_H__simps__without__assm_I4_J,axiom,
    ! [P: fsm_Pr5633030146480774220_a_b_c,Q1: a,Q22: a,M: fsm_a_b_c] :
      ( ( ( ( initia4141967198046469911_a_b_c @ P )
          = ( product_Pair_a_a @ Q1 @ Q22 ) )
       => ( ( output2259889051082456732_a_b_c @ ( canoni1795827711669416976_a_b_c @ M @ P @ Q1 @ Q22 ) )
          = ( sup_sup_set_c @ ( outputs_a_b_c @ M ) @ ( output5831646040678877961_a_b_c @ P ) ) ) )
      & ( ( ( initia4141967198046469911_a_b_c @ P )
         != ( product_Pair_a_a @ Q1 @ Q22 ) )
       => ( ( output2259889051082456732_a_b_c @ ( canoni1795827711669416976_a_b_c @ M @ P @ Q1 @ Q22 ) )
          = bot_bot_set_c ) ) ) ).

% canonical_separator'_simps_without_assm(4)
thf(fact_1126_canonical__separator_H__simps__without__assm_I3_J,axiom,
    ! [P: fsm_Pr1307097241311862860_a_b_c,Q1: set_a,Q22: set_a,M: fsm_set_a_b_c] :
      ( ( ( ( initia9132321972455559959_a_b_c @ P )
          = ( produc9088192753505129239_set_a @ Q1 @ Q22 ) )
       => ( ( inputs6346031644807275387_a_b_c @ ( canoni669745312745679216_a_b_c @ M @ P @ Q1 @ Q22 ) )
          = ( sup_sup_set_b @ ( inputs_set_a_b_c @ M ) @ ( inputs2702761543428496202_a_b_c @ P ) ) ) )
      & ( ( ( initia9132321972455559959_a_b_c @ P )
         != ( produc9088192753505129239_set_a @ Q1 @ Q22 ) )
       => ( ( inputs6346031644807275387_a_b_c @ ( canoni669745312745679216_a_b_c @ M @ P @ Q1 @ Q22 ) )
          = bot_bot_set_b ) ) ) ).

% canonical_separator'_simps_without_assm(3)
thf(fact_1127_canonical__separator_H__simps__without__assm_I3_J,axiom,
    ! [P: fsm_Pr5633030146480774220_a_b_c,Q1: a,Q22: a,M: fsm_a_b_c] :
      ( ( ( ( initia4141967198046469911_a_b_c @ P )
          = ( product_Pair_a_a @ Q1 @ Q22 ) )
       => ( ( inputs3694848887974842907_a_b_c @ ( canoni1795827711669416976_a_b_c @ M @ P @ Q1 @ Q22 ) )
          = ( sup_sup_set_b @ ( inputs_a_b_c @ M ) @ ( inputs2873161410287034186_a_b_c @ P ) ) ) )
      & ( ( ( initia4141967198046469911_a_b_c @ P )
         != ( product_Pair_a_a @ Q1 @ Q22 ) )
       => ( ( inputs3694848887974842907_a_b_c @ ( canoni1795827711669416976_a_b_c @ M @ P @ Q1 @ Q22 ) )
          = bot_bot_set_b ) ) ) ).

% canonical_separator'_simps_without_assm(3)
thf(fact_1128_FSM_Ocanonical__separator_H__simps_I3_J,axiom,
    ! [P: fsm_Pr1307097241311862860_a_b_c,Q1: set_a,Q22: set_a,M: fsm_set_a_b_c] :
      ( ( ( initia9132321972455559959_a_b_c @ P )
        = ( produc9088192753505129239_set_a @ Q1 @ Q22 ) )
     => ( ( inputs6346031644807275387_a_b_c @ ( canoni669745312745679216_a_b_c @ M @ P @ Q1 @ Q22 ) )
        = ( sup_sup_set_b @ ( inputs_set_a_b_c @ M ) @ ( inputs2702761543428496202_a_b_c @ P ) ) ) ) ).

% FSM.canonical_separator'_simps(3)
thf(fact_1129_FSM_Ocanonical__separator_H__simps_I3_J,axiom,
    ! [P: fsm_Pr5633030146480774220_a_b_c,Q1: a,Q22: a,M: fsm_a_b_c] :
      ( ( ( initia4141967198046469911_a_b_c @ P )
        = ( product_Pair_a_a @ Q1 @ Q22 ) )
     => ( ( inputs3694848887974842907_a_b_c @ ( canoni1795827711669416976_a_b_c @ M @ P @ Q1 @ Q22 ) )
        = ( sup_sup_set_b @ ( inputs_a_b_c @ M ) @ ( inputs2873161410287034186_a_b_c @ P ) ) ) ) ).

% FSM.canonical_separator'_simps(3)
thf(fact_1130_FSM_Ocanonical__separator_H__simps_I4_J,axiom,
    ! [P: fsm_Pr1307097241311862860_a_b_c,Q1: set_a,Q22: set_a,M: fsm_set_a_b_c] :
      ( ( ( initia9132321972455559959_a_b_c @ P )
        = ( produc9088192753505129239_set_a @ Q1 @ Q22 ) )
     => ( ( output4897735804923954172_a_b_c @ ( canoni669745312745679216_a_b_c @ M @ P @ Q1 @ Q22 ) )
        = ( sup_sup_set_c @ ( outputs_set_a_b_c @ M ) @ ( output4507429379629698825_a_b_c @ P ) ) ) ) ).

% FSM.canonical_separator'_simps(4)
thf(fact_1131_FSM_Ocanonical__separator_H__simps_I4_J,axiom,
    ! [P: fsm_Pr5633030146480774220_a_b_c,Q1: a,Q22: a,M: fsm_a_b_c] :
      ( ( ( initia4141967198046469911_a_b_c @ P )
        = ( product_Pair_a_a @ Q1 @ Q22 ) )
     => ( ( output2259889051082456732_a_b_c @ ( canoni1795827711669416976_a_b_c @ M @ P @ Q1 @ Q22 ) )
        = ( sup_sup_set_c @ ( outputs_a_b_c @ M ) @ ( output5831646040678877961_a_b_c @ P ) ) ) ) ).

% FSM.canonical_separator'_simps(4)
thf(fact_1132_is__submachine_Opelims_I3_J,axiom,
    ! [X: fsm_a_b_c,Xa: fsm_a_b_c] :
      ( ~ ( is_submachine_a_b_c @ X @ Xa )
     => ( ( accp_P4943880149483332336_a_b_c @ is_sub2782940026310382442_a_b_c @ ( produc8329669653585234007_a_b_c @ X @ Xa ) )
       => ~ ( ( accp_P4943880149483332336_a_b_c @ is_sub2782940026310382442_a_b_c @ ( produc8329669653585234007_a_b_c @ X @ Xa ) )
           => ( ( ( initial_a_b_c @ X )
                = ( initial_a_b_c @ Xa ) )
              & ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ X ) @ ( transitions_a_b_c @ Xa ) )
              & ( ( inputs_a_b_c @ X )
                = ( inputs_a_b_c @ Xa ) )
              & ( ( outputs_a_b_c @ X )
                = ( outputs_a_b_c @ Xa ) )
              & ( ord_less_eq_set_a @ ( states_a_b_c @ X ) @ ( states_a_b_c @ Xa ) ) ) ) ) ) ).

% is_submachine.pelims(3)
thf(fact_1133_is__submachine_Opelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: fsm_set_a_b_c] :
      ( ~ ( is_sub1377638108373535417_a_b_c @ X @ Xa )
     => ( ( accp_P4877812173494536944_a_b_c @ is_sub7629664660614848714_a_b_c @ ( produc4665385101254049879_a_b_c @ X @ Xa ) )
       => ~ ( ( accp_P4877812173494536944_a_b_c @ is_sub7629664660614848714_a_b_c @ ( produc4665385101254049879_a_b_c @ X @ Xa ) )
           => ( ( ( initial_set_a_b_c @ X )
                = ( initial_set_a_b_c @ Xa ) )
              & ( ord_le4557092201414138422_set_a @ ( transi8205445899535437940_a_b_c @ X ) @ ( transi8205445899535437940_a_b_c @ Xa ) )
              & ( ( inputs_set_a_b_c @ X )
                = ( inputs_set_a_b_c @ Xa ) )
              & ( ( outputs_set_a_b_c @ X )
                = ( outputs_set_a_b_c @ Xa ) )
              & ( ord_le3724670747650509150_set_a @ ( states_set_a_b_c @ X ) @ ( states_set_a_b_c @ Xa ) ) ) ) ) ) ).

% is_submachine.pelims(3)
thf(fact_1134_is__submachine_Opelims_I2_J,axiom,
    ! [X: fsm_a_b_c,Xa: fsm_a_b_c] :
      ( ( is_submachine_a_b_c @ X @ Xa )
     => ( ( accp_P4943880149483332336_a_b_c @ is_sub2782940026310382442_a_b_c @ ( produc8329669653585234007_a_b_c @ X @ Xa ) )
       => ~ ( ( accp_P4943880149483332336_a_b_c @ is_sub2782940026310382442_a_b_c @ ( produc8329669653585234007_a_b_c @ X @ Xa ) )
           => ~ ( ( ( initial_a_b_c @ X )
                  = ( initial_a_b_c @ Xa ) )
                & ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ X ) @ ( transitions_a_b_c @ Xa ) )
                & ( ( inputs_a_b_c @ X )
                  = ( inputs_a_b_c @ Xa ) )
                & ( ( outputs_a_b_c @ X )
                  = ( outputs_a_b_c @ Xa ) )
                & ( ord_less_eq_set_a @ ( states_a_b_c @ X ) @ ( states_a_b_c @ Xa ) ) ) ) ) ) ).

% is_submachine.pelims(2)
thf(fact_1135_is__submachine_Opelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: fsm_set_a_b_c] :
      ( ( is_sub1377638108373535417_a_b_c @ X @ Xa )
     => ( ( accp_P4877812173494536944_a_b_c @ is_sub7629664660614848714_a_b_c @ ( produc4665385101254049879_a_b_c @ X @ Xa ) )
       => ~ ( ( accp_P4877812173494536944_a_b_c @ is_sub7629664660614848714_a_b_c @ ( produc4665385101254049879_a_b_c @ X @ Xa ) )
           => ~ ( ( ( initial_set_a_b_c @ X )
                  = ( initial_set_a_b_c @ Xa ) )
                & ( ord_le4557092201414138422_set_a @ ( transi8205445899535437940_a_b_c @ X ) @ ( transi8205445899535437940_a_b_c @ Xa ) )
                & ( ( inputs_set_a_b_c @ X )
                  = ( inputs_set_a_b_c @ Xa ) )
                & ( ( outputs_set_a_b_c @ X )
                  = ( outputs_set_a_b_c @ Xa ) )
                & ( ord_le3724670747650509150_set_a @ ( states_set_a_b_c @ X ) @ ( states_set_a_b_c @ Xa ) ) ) ) ) ) ).

% is_submachine.pelims(2)
thf(fact_1136_is__submachine_Opelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Xa: fsm_a_b_c,Y: $o] :
      ( ( ( is_submachine_a_b_c @ X @ Xa )
        = Y )
     => ( ( accp_P4943880149483332336_a_b_c @ is_sub2782940026310382442_a_b_c @ ( produc8329669653585234007_a_b_c @ X @ Xa ) )
       => ~ ( ( Y
              = ( ( ( initial_a_b_c @ X )
                  = ( initial_a_b_c @ Xa ) )
                & ( ord_le3226338165915859510od_c_a @ ( transitions_a_b_c @ X ) @ ( transitions_a_b_c @ Xa ) )
                & ( ( inputs_a_b_c @ X )
                  = ( inputs_a_b_c @ Xa ) )
                & ( ( outputs_a_b_c @ X )
                  = ( outputs_a_b_c @ Xa ) )
                & ( ord_less_eq_set_a @ ( states_a_b_c @ X ) @ ( states_a_b_c @ Xa ) ) ) )
           => ~ ( accp_P4943880149483332336_a_b_c @ is_sub2782940026310382442_a_b_c @ ( produc8329669653585234007_a_b_c @ X @ Xa ) ) ) ) ) ).

% is_submachine.pelims(1)
thf(fact_1137_is__submachine_Opelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: fsm_set_a_b_c,Y: $o] :
      ( ( ( is_sub1377638108373535417_a_b_c @ X @ Xa )
        = Y )
     => ( ( accp_P4877812173494536944_a_b_c @ is_sub7629664660614848714_a_b_c @ ( produc4665385101254049879_a_b_c @ X @ Xa ) )
       => ~ ( ( Y
              = ( ( ( initial_set_a_b_c @ X )
                  = ( initial_set_a_b_c @ Xa ) )
                & ( ord_le4557092201414138422_set_a @ ( transi8205445899535437940_a_b_c @ X ) @ ( transi8205445899535437940_a_b_c @ Xa ) )
                & ( ( inputs_set_a_b_c @ X )
                  = ( inputs_set_a_b_c @ Xa ) )
                & ( ( outputs_set_a_b_c @ X )
                  = ( outputs_set_a_b_c @ Xa ) )
                & ( ord_le3724670747650509150_set_a @ ( states_set_a_b_c @ X ) @ ( states_set_a_b_c @ Xa ) ) ) )
           => ~ ( accp_P4877812173494536944_a_b_c @ is_sub7629664660614848714_a_b_c @ ( produc4665385101254049879_a_b_c @ X @ Xa ) ) ) ) ) ).

% is_submachine.pelims(1)
thf(fact_1138_completely__specified__state_Opelims_I3_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: set_a] :
      ( ~ ( comple6745895220685239651_a_b_c @ X @ Xa )
     => ( ( accp_P2020493475874802763_set_a @ comple7599367055876033184_a_b_c @ ( produc4466573882972644914_set_a @ X @ Xa ) )
       => ~ ( ( accp_P2020493475874802763_set_a @ comple7599367055876033184_a_b_c @ ( produc4466573882972644914_set_a @ X @ Xa ) )
           => ! [X3: b] :
                ( ( member_b @ X3 @ ( inputs_set_a_b_c @ X ) )
               => ? [Xa2: produc373761375140688950_set_a] :
                    ( ( member6517745037573369951_set_a @ Xa2 @ ( transi8205445899535437940_a_b_c @ X ) )
                    & ( ( produc7645221154340064658_set_a @ Xa2 )
                      = Xa )
                    & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xa2 ) )
                      = X3 ) ) ) ) ) ) ).

% completely_specified_state.pelims(3)
thf(fact_1139_completely__specified__state_Opelims_I3_J,axiom,
    ! [X: fsm_a_b_c,Xa: a] :
      ( ~ ( comple2039557650379672579_a_b_c @ X @ Xa )
     => ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X @ Xa ) )
       => ~ ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X @ Xa ) )
           => ! [X3: b] :
                ( ( member_b @ X3 @ ( inputs_a_b_c @ X ) )
               => ? [Xa2: produc2669932201312201782od_c_a] :
                    ( ( member5502040268208861279od_c_a @ Xa2 @ ( transitions_a_b_c @ X ) )
                    & ( ( produc4225943634317630866od_c_a @ Xa2 )
                      = Xa )
                    & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa2 ) )
                      = X3 ) ) ) ) ) ) ).

% completely_specified_state.pelims(3)
thf(fact_1140_completely__specified__state_Opelims_I2_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: set_a] :
      ( ( comple6745895220685239651_a_b_c @ X @ Xa )
     => ( ( accp_P2020493475874802763_set_a @ comple7599367055876033184_a_b_c @ ( produc4466573882972644914_set_a @ X @ Xa ) )
       => ~ ( ( accp_P2020493475874802763_set_a @ comple7599367055876033184_a_b_c @ ( produc4466573882972644914_set_a @ X @ Xa ) )
           => ~ ! [X5: b] :
                  ( ( member_b @ X5 @ ( inputs_set_a_b_c @ X ) )
                 => ? [Xa3: produc373761375140688950_set_a] :
                      ( ( member6517745037573369951_set_a @ Xa3 @ ( transi8205445899535437940_a_b_c @ X ) )
                      & ( ( produc7645221154340064658_set_a @ Xa3 )
                        = Xa )
                      & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Xa3 ) )
                        = X5 ) ) ) ) ) ) ).

% completely_specified_state.pelims(2)
thf(fact_1141_completely__specified__state_Opelims_I2_J,axiom,
    ! [X: fsm_a_b_c,Xa: a] :
      ( ( comple2039557650379672579_a_b_c @ X @ Xa )
     => ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X @ Xa ) )
       => ~ ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X @ Xa ) )
           => ~ ! [X5: b] :
                  ( ( member_b @ X5 @ ( inputs_a_b_c @ X ) )
                 => ? [Xa3: produc2669932201312201782od_c_a] :
                      ( ( member5502040268208861279od_c_a @ Xa3 @ ( transitions_a_b_c @ X ) )
                      & ( ( produc4225943634317630866od_c_a @ Xa3 )
                        = Xa )
                      & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Xa3 ) )
                        = X5 ) ) ) ) ) ) ).

% completely_specified_state.pelims(2)
thf(fact_1142_completely__specified__state_Opelims_I1_J,axiom,
    ! [X: fsm_set_a_b_c,Xa: set_a,Y: $o] :
      ( ( ( comple6745895220685239651_a_b_c @ X @ Xa )
        = Y )
     => ( ( accp_P2020493475874802763_set_a @ comple7599367055876033184_a_b_c @ ( produc4466573882972644914_set_a @ X @ Xa ) )
       => ~ ( ( Y
              = ( ! [X2: b] :
                    ( ( member_b @ X2 @ ( inputs_set_a_b_c @ X ) )
                   => ? [Y4: produc373761375140688950_set_a] :
                        ( ( member6517745037573369951_set_a @ Y4 @ ( transi8205445899535437940_a_b_c @ X ) )
                        & ( ( produc7645221154340064658_set_a @ Y4 )
                          = Xa )
                        & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ Y4 ) )
                          = X2 ) ) ) ) )
           => ~ ( accp_P2020493475874802763_set_a @ comple7599367055876033184_a_b_c @ ( produc4466573882972644914_set_a @ X @ Xa ) ) ) ) ) ).

% completely_specified_state.pelims(1)
thf(fact_1143_completely__specified__state_Opelims_I1_J,axiom,
    ! [X: fsm_a_b_c,Xa: a,Y: $o] :
      ( ( ( comple2039557650379672579_a_b_c @ X @ Xa )
        = Y )
     => ( ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X @ Xa ) )
       => ~ ( ( Y
              = ( ! [X2: b] :
                    ( ( member_b @ X2 @ ( inputs_a_b_c @ X ) )
                   => ? [Y4: produc2669932201312201782od_c_a] :
                        ( ( member5502040268208861279od_c_a @ Y4 @ ( transitions_a_b_c @ X ) )
                        & ( ( produc4225943634317630866od_c_a @ Y4 )
                          = Xa )
                        & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ Y4 ) )
                          = X2 ) ) ) ) )
           => ~ ( accp_P8775296850406592587_b_c_a @ comple5150018596827829568_a_b_c @ ( produc1330280085856475698_b_c_a @ X @ Xa ) ) ) ) ) ).

% completely_specified_state.pelims(1)
thf(fact_1144_language__equivalence__classes__preserve__observability,axiom,
    ! [M6: fsm_set_set_a_b_c,M: fsm_set_a_b_c] :
      ( ( ( transi2712930047638073812_a_b_c @ M6 )
        = ( image_1275805176282611333_set_a
          @ ^ [T5: produc373761375140688950_set_a] :
              ( produc7351556691857002406_set_a
              @ ( collect_set_a
                @ ^ [Q5: set_a] :
                    ( ( member_set_a @ Q5 @ ( states_set_a_b_c @ M ) )
                    & ( ( lS_set_a_b_c @ M @ Q5 )
                      = ( lS_set_a_b_c @ M @ ( produc7645221154340064658_set_a @ T5 ) ) ) ) )
              @ ( produc8815262582889187005_set_a @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) )
                @ ( produc8721044967893965397_set_a @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) )
                  @ ( collect_set_a
                    @ ^ [Q5: set_a] :
                        ( ( member_set_a @ Q5 @ ( states_set_a_b_c @ M ) )
                        & ( ( lS_set_a_b_c @ M @ Q5 )
                          = ( lS_set_a_b_c @ M @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) ) ) ) ) ) ) )
          @ ( transi8205445899535437940_a_b_c @ M ) ) )
     => ( ( observable_set_a_b_c @ M )
       => ( observ3077046029229394695_a_b_c @ M6 ) ) ) ).

% language_equivalence_classes_preserve_observability
thf(fact_1145_language__equivalence__classes__preserve__observability,axiom,
    ! [M6: fsm_set_a_b_c,M: fsm_a_b_c] :
      ( ( ( transi8205445899535437940_a_b_c @ M6 )
        = ( image_5400094664181065349_set_a
          @ ^ [T5: produc2669932201312201782od_c_a] :
              ( produc3915556858435501990_set_a
              @ ( collect_a
                @ ^ [Q5: a] :
                    ( ( member_a @ Q5 @ ( states_a_b_c @ M ) )
                    & ( ( lS_a_b_c @ M @ Q5 )
                      = ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T5 ) ) ) ) )
              @ ( produc799526633598312797_set_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) )
                @ ( product_Pair_c_set_a @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) )
                  @ ( collect_a
                    @ ^ [Q5: a] :
                        ( ( member_a @ Q5 @ ( states_a_b_c @ M ) )
                        & ( ( lS_a_b_c @ M @ Q5 )
                          = ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) ) ) ) ) ) ) )
          @ ( transitions_a_b_c @ M ) ) )
     => ( ( observable_a_b_c @ M )
       => ( observable_set_a_b_c @ M6 ) ) ) ).

% language_equivalence_classes_preserve_observability
thf(fact_1146_language__equivalence__classes__preserve__observability,axiom,
    ! [M6: fsm_set_nat_b_c,M: fsm_nat_b_c] :
      ( ( ( transi1321261067114444082at_b_c @ M6 )
        = ( image_4951582970363610481et_nat
          @ ^ [T5: produc8586872805302537566_c_nat] :
              ( produc244806926284792962et_nat
              @ ( collect_nat
                @ ^ [Q5: nat] :
                    ( ( member_nat @ Q5 @ ( states_nat_b_c @ M ) )
                    & ( ( lS_nat_b_c @ M @ Q5 )
                      = ( lS_nat_b_c @ M @ ( produc532606554400396034_c_nat @ T5 ) ) ) ) )
              @ ( produc1649582173287938781et_nat @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) )
                @ ( produc6490417962789114671et_nat @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) )
                  @ ( collect_nat
                    @ ^ [Q5: nat] :
                        ( ( member_nat @ Q5 @ ( states_nat_b_c @ M ) )
                        & ( ( lS_nat_b_c @ M @ Q5 )
                          = ( lS_nat_b_c @ M @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) ) ) ) ) ) ) )
          @ ( transitions_nat_b_c @ M ) ) )
     => ( ( observable_nat_b_c @ M )
       => ( observ2138676753445337407at_b_c @ M6 ) ) ) ).

% language_equivalence_classes_preserve_observability
thf(fact_1147_language__equivalence__classes__preserve__observability,axiom,
    ! [M6: fsm_se7249436556146598534er_b_c,M: fsm_Code_integer_b_c] :
      ( ( ( transi4510574700103178431er_b_c @ M6 )
        = ( image_5626487323572879345nteger
          @ ^ [T5: produc558599814252099832nteger] :
              ( produc9186817357391146396nteger
              @ ( collect_Code_integer
                @ ^ [Q5: code_integer] :
                    ( ( member_Code_integer @ Q5 @ ( states1921009325540848725er_b_c @ M ) )
                    & ( ( lS_Code_integer_b_c @ M @ Q5 )
                      = ( lS_Code_integer_b_c @ M @ ( produc7104377276419476764nteger @ T5 ) ) ) ) )
              @ ( produc7495836630374010474nteger @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) )
                @ ( produc3812338474690849084nteger @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) )
                  @ ( collect_Code_integer
                    @ ^ [Q5: code_integer] :
                        ( ( member_Code_integer @ Q5 @ ( states1921009325540848725er_b_c @ M ) )
                        & ( ( lS_Code_integer_b_c @ M @ Q5 )
                          = ( lS_Code_integer_b_c @ M @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) ) ) ) ) ) ) )
          @ ( transi6707074847254859401er_b_c @ M ) ) )
     => ( ( observ7524490533585752726er_b_c @ M )
       => ( observ8366386160172968140er_b_c @ M6 ) ) ) ).

% language_equivalence_classes_preserve_observability
thf(fact_1148_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( P @ X2 )
              & ( Q2 @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1149_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( P @ X2 )
              | ( Q2 @ X2 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1150_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_1151_singleton__conv2,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ( ^ [Y3: nat,Z: nat] : ( Y3 = Z )
          @ A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_1152_singleton__conv,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( X2 = A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_1153_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_1154_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1155_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_1156_restrict__to__reachable__states__simps_I5_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( transitions_a_b_c @ ( restri9132545300209641082_a_b_c @ M ) )
      = ( collec8405073525749625505od_c_a
        @ ^ [T5: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ T5 @ ( transitions_a_b_c @ M ) )
            & ( member_a @ ( produc4225943634317630866od_c_a @ T5 ) @ ( reacha1620305530751930115_a_b_c @ M ) ) ) ) ) ).

% restrict_to_reachable_states_simps(5)
thf(fact_1157_restrict__to__reachable__states__simps_I5_J,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( transi8205445899535437940_a_b_c @ ( restri1885653365341205466_a_b_c @ M ) )
      = ( collec5473208570875536033_set_a
        @ ^ [T5: produc373761375140688950_set_a] :
            ( ( member6517745037573369951_set_a @ T5 @ ( transi8205445899535437940_a_b_c @ M ) )
            & ( member_set_a @ ( produc7645221154340064658_set_a @ T5 ) @ ( reacha7078015818774689891_a_b_c @ M ) ) ) ) ) ).

% restrict_to_reachable_states_simps(5)
thf(fact_1158_fsm__from__list__simps_I3_J,axiom,
    ! [Q: set_a,Ts: list_P6923848007992943548_set_a] :
      ( ( inputs_set_a_b_c @ ( fsm_fr8620171418138189770_a_b_c @ Q @ Ts ) )
      = ( image_5572330044970814030et_a_b
        @ ^ [A4: produc373761375140688950_set_a] : ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ A4 ) )
        @ ( set_Pr8219927823471507147_set_a @ Ts ) ) ) ).

% fsm_from_list_simps(3)
thf(fact_1159_fsm__from__list__simps_I3_J,axiom,
    ! [Q: a,Ts: list_P6327159017948738492od_c_a] :
      ( ( inputs_a_b_c @ ( fsm_from_list_a_b_c @ Q @ Ts ) )
      = ( image_9171994905208062542_c_a_b
        @ ^ [A4: produc2669932201312201782od_c_a] : ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) )
        @ ( set_Pr5664641634098845387od_c_a @ Ts ) ) ) ).

% fsm_from_list_simps(3)
thf(fact_1160_fsm__from__list__simps_I4_J,axiom,
    ! [Q: set_a,Ts: list_P6923848007992943548_set_a] :
      ( ( outputs_set_a_b_c @ ( fsm_fr8620171418138189770_a_b_c @ Q @ Ts ) )
      = ( image_5572330044970814031et_a_c
        @ ^ [A4: produc373761375140688950_set_a] : ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ A4 ) ) )
        @ ( set_Pr8219927823471507147_set_a @ Ts ) ) ) ).

% fsm_from_list_simps(4)
thf(fact_1161_fsm__from__list__simps_I4_J,axiom,
    ! [Q: a,Ts: list_P6327159017948738492od_c_a] :
      ( ( outputs_a_b_c @ ( fsm_from_list_a_b_c @ Q @ Ts ) )
      = ( image_9171994905208062543_c_a_c
        @ ^ [A4: produc2669932201312201782od_c_a] : ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ A4 ) ) )
        @ ( set_Pr5664641634098845387od_c_a @ Ts ) ) ) ).

% fsm_from_list_simps(4)
thf(fact_1162_insert__def,axiom,
    ( insert_nat
    = ( ^ [A4: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X2: nat] : ( X2 = A4 ) ) ) ) ) ).

% insert_def
thf(fact_1163_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A4: nat,B2: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ( ( X2 = A4 )
              | ( member_nat @ X2 @ B2 ) ) ) ) ) ).

% insert_compr
thf(fact_1164_insert__Collect,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( insert_nat @ A2 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_1165_Collect__conv__if2,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_1166_Collect__conv__if,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_1167_Collect__subset,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_1168_prop__restrict,axiom,
    ! [X: nat,Z5: set_nat,X6: set_nat,P: nat > $o] :
      ( ( member_nat @ X @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ X6 )
                & ( P @ X2 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_1169_Collect__restrict,axiom,
    ! [X6: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ X6 )
            & ( P @ X2 ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_1170_Compr__image__eq,axiom,
    ! [F: nat > nat,A: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_nat_nat @ F
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1171_Set_Ofilter__def,axiom,
    ( filter_nat
    = ( ^ [P4: nat > $o,A3: set_nat] :
          ( collect_nat
          @ ^ [A4: nat] :
              ( ( member_nat @ A4 @ A3 )
              & ( P4 @ A4 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1172_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X2: nat] : $false ) ) ).

% Set.empty_def
thf(fact_1173_restrict__to__reachable__states_Oelims,axiom,
    ! [X: fsm_a_b_c,Y: fsm_a_b_c] :
      ( ( ( restri9132545300209641082_a_b_c @ X )
        = Y )
     => ( Y
        = ( filter_states_a_b_c @ X
          @ ^ [Q5: a] : ( member_a @ Q5 @ ( reacha1620305530751930115_a_b_c @ X ) ) ) ) ) ).

% restrict_to_reachable_states.elims
thf(fact_1174_restrict__to__reachable__states_Oelims,axiom,
    ! [X: fsm_set_a_b_c,Y: fsm_set_a_b_c] :
      ( ( ( restri1885653365341205466_a_b_c @ X )
        = Y )
     => ( Y
        = ( filter2961209833878020368_a_b_c @ X
          @ ^ [Q5: set_a] : ( member_set_a @ Q5 @ ( reacha7078015818774689891_a_b_c @ X ) ) ) ) ) ).

% restrict_to_reachable_states.elims
thf(fact_1175_restrict__to__reachable__states_Osimps,axiom,
    ( restri9132545300209641082_a_b_c
    = ( ^ [M2: fsm_a_b_c] :
          ( filter_states_a_b_c @ M2
          @ ^ [Q5: a] : ( member_a @ Q5 @ ( reacha1620305530751930115_a_b_c @ M2 ) ) ) ) ) ).

% restrict_to_reachable_states.simps
thf(fact_1176_restrict__to__reachable__states_Osimps,axiom,
    ( restri1885653365341205466_a_b_c
    = ( ^ [M2: fsm_set_a_b_c] :
          ( filter2961209833878020368_a_b_c @ M2
          @ ^ [Q5: set_a] : ( member_set_a @ Q5 @ ( reacha7078015818774689891_a_b_c @ M2 ) ) ) ) ) ).

% restrict_to_reachable_states.simps
thf(fact_1177_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ( collect_nat
        @ ^ [X2: nat] :
            ( ( P @ X2 )
            | ( Q2 @ X2 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) ) ) ).

% Collect_disj_eq
thf(fact_1178_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B2: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
            @ ^ [X2: nat] : ( member_nat @ X2 @ B2 ) ) ) ) ) ).

% sup_set_def
thf(fact_1179_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B2: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A3 )
              | ( member_nat @ X2 @ B2 ) ) ) ) ) ).

% Un_def
thf(fact_1180_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1181_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_nat,R2: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1182_pigeonhole__infinite,axiom,
    ! [A: set_nat,F: nat > nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A4: nat] :
                      ( ( member_nat @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1183_nat__seg__image__imp__finite,axiom,
    ! [A: set_nat,F: nat > nat,N2: nat] :
      ( ( A
        = ( image_nat_nat @ F
          @ ( collect_nat
            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) )
     => ( finite_finite_nat @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1184_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A3: set_nat] :
        ? [N: nat,F4: nat > nat] :
          ( A3
          = ( image_nat_nat @ F4
            @ ( collect_nat
              @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1185_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A3: set_nat] :
          ( collect_set_nat
          @ ^ [X7: set_nat] :
              ( ( ord_less_eq_set_nat @ X7 @ A3 )
              & ( finite_finite_nat @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_1186_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_1187_minimise__props_I2_J,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( states_set_nat_b_c @ ( minimise_nat_b_c @ M ) )
        = ( image_nat_set_nat
          @ ^ [Q5: nat] :
              ( collect_nat
              @ ^ [Q6: nat] :
                  ( ( member_nat @ Q6 @ ( states_nat_b_c @ M ) )
                  & ( ( lS_nat_b_c @ M @ Q5 )
                    = ( lS_nat_b_c @ M @ Q6 ) ) ) )
          @ ( states_nat_b_c @ M ) ) ) ) ).

% minimise_props(2)
thf(fact_1188_minimise__props_I2_J,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( states5991589681222143627er_b_c @ ( minimi9001410862329526533er_b_c @ M ) )
        = ( image_2291390168754192375nteger
          @ ^ [Q5: code_integer] :
              ( collect_Code_integer
              @ ^ [Q6: code_integer] :
                  ( ( member_Code_integer @ Q6 @ ( states1921009325540848725er_b_c @ M ) )
                  & ( ( lS_Code_integer_b_c @ M @ Q5 )
                    = ( lS_Code_integer_b_c @ M @ Q6 ) ) ) )
          @ ( states1921009325540848725er_b_c @ M ) ) ) ) ).

% minimise_props(2)
thf(fact_1189_minimise__props_I2_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( states_set_a_b_c @ ( minimise_a_b_c @ M ) )
        = ( image_a_set_a
          @ ^ [Q5: a] :
              ( collect_a
              @ ^ [Q6: a] :
                  ( ( member_a @ Q6 @ ( states_a_b_c @ M ) )
                  & ( ( lS_a_b_c @ M @ Q5 )
                    = ( lS_a_b_c @ M @ Q6 ) ) ) )
          @ ( states_a_b_c @ M ) ) ) ) ).

% minimise_props(2)
thf(fact_1190_minimise__props_I1_J,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( initial_set_nat_b_c @ ( minimise_nat_b_c @ M ) )
        = ( collect_nat
          @ ^ [Q6: nat] :
              ( ( member_nat @ Q6 @ ( states_nat_b_c @ M ) )
              & ( ( lS_nat_b_c @ M @ Q6 )
                = ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) ) ) ) ) ) ).

% minimise_props(1)
thf(fact_1191_minimise__props_I1_J,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( initia8790449882354265317er_b_c @ ( minimi9001410862329526533er_b_c @ M ) )
        = ( collect_Code_integer
          @ ^ [Q6: code_integer] :
              ( ( member_Code_integer @ Q6 @ ( states1921009325540848725er_b_c @ M ) )
              & ( ( lS_Code_integer_b_c @ M @ Q6 )
                = ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) ) ) ) ) ) ).

% minimise_props(1)
thf(fact_1192_minimise__props_I1_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( initial_set_a_b_c @ ( minimise_a_b_c @ M ) )
        = ( collect_a
          @ ^ [Q6: a] :
              ( ( member_a @ Q6 @ ( states_a_b_c @ M ) )
              & ( ( lS_a_b_c @ M @ Q6 )
                = ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) ) ) ) ) ) ).

% minimise_props(1)
thf(fact_1193_observable__after__reachable__surj,axiom,
    ! [M: fsm_set_a_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( image_6303648431024068597_set_a @ ( after_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) @ ( lS_set_a_b_c @ M @ ( initial_set_a_b_c @ M ) ) )
        = ( reacha7078015818774689891_a_b_c @ M ) ) ) ).

% observable_after_reachable_surj
thf(fact_1194_observable__after__reachable__surj,axiom,
    ! [M: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( image_3173854338463926933_b_c_a @ ( after_a_b_c @ M @ ( initial_a_b_c @ M ) ) @ ( lS_a_b_c @ M @ ( initial_a_b_c @ M ) ) )
        = ( reacha1620305530751930115_a_b_c @ M ) ) ) ).

% observable_after_reachable_surj
thf(fact_1195_observable__after__reachable__surj,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( image_4545125072327328889_c_nat @ ( after_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) @ ( lS_nat_b_c @ M @ ( initial_nat_b_c @ M ) ) )
        = ( reacha6243244737040045773at_b_c @ M ) ) ) ).

% observable_after_reachable_surj
thf(fact_1196_observable__after__reachable__surj,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( image_3585316291785293062nteger @ ( after_5090571588895392199er_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) @ ( lS_Code_integer_b_c @ M @ ( initia4873860033875475119er_b_c @ M ) ) )
        = ( reacha2636592895321446618er_b_c @ M ) ) ) ).

% observable_after_reachable_surj
thf(fact_1197_minimise__props_I5_J,axiom,
    ! [M: fsm_nat_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( transi1321261067114444082at_b_c @ ( minimise_nat_b_c @ M ) )
        = ( image_4951582970363610481et_nat
          @ ^ [T5: produc8586872805302537566_c_nat] :
              ( produc244806926284792962et_nat
              @ ( collect_nat
                @ ^ [Q5: nat] :
                    ( ( member_nat @ Q5 @ ( states_nat_b_c @ M ) )
                    & ( ( lS_nat_b_c @ M @ Q5 )
                      = ( lS_nat_b_c @ M @ ( produc532606554400396034_c_nat @ T5 ) ) ) ) )
              @ ( produc1649582173287938781et_nat @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) )
                @ ( produc6490417962789114671et_nat @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) )
                  @ ( collect_nat
                    @ ^ [Q5: nat] :
                        ( ( member_nat @ Q5 @ ( states_nat_b_c @ M ) )
                        & ( ( lS_nat_b_c @ M @ Q5 )
                          = ( lS_nat_b_c @ M @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) ) ) ) ) ) ) )
          @ ( transitions_nat_b_c @ M ) ) ) ) ).

% minimise_props(5)
thf(fact_1198_minimise__props_I5_J,axiom,
    ! [M: fsm_Code_integer_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( transi4510574700103178431er_b_c @ ( minimi9001410862329526533er_b_c @ M ) )
        = ( image_5626487323572879345nteger
          @ ^ [T5: produc558599814252099832nteger] :
              ( produc9186817357391146396nteger
              @ ( collect_Code_integer
                @ ^ [Q5: code_integer] :
                    ( ( member_Code_integer @ Q5 @ ( states1921009325540848725er_b_c @ M ) )
                    & ( ( lS_Code_integer_b_c @ M @ Q5 )
                      = ( lS_Code_integer_b_c @ M @ ( produc7104377276419476764nteger @ T5 ) ) ) ) )
              @ ( produc7495836630374010474nteger @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) )
                @ ( produc3812338474690849084nteger @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) )
                  @ ( collect_Code_integer
                    @ ^ [Q5: code_integer] :
                        ( ( member_Code_integer @ Q5 @ ( states1921009325540848725er_b_c @ M ) )
                        & ( ( lS_Code_integer_b_c @ M @ Q5 )
                          = ( lS_Code_integer_b_c @ M @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) ) ) ) ) ) ) )
          @ ( transi6707074847254859401er_b_c @ M ) ) ) ) ).

% minimise_props(5)
thf(fact_1199_minimise__props_I5_J,axiom,
    ! [M: fsm_a_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( transi8205445899535437940_a_b_c @ ( minimise_a_b_c @ M ) )
        = ( image_5400094664181065349_set_a
          @ ^ [T5: produc2669932201312201782od_c_a] :
              ( produc3915556858435501990_set_a
              @ ( collect_a
                @ ^ [Q5: a] :
                    ( ( member_a @ Q5 @ ( states_a_b_c @ M ) )
                    & ( ( lS_a_b_c @ M @ Q5 )
                      = ( lS_a_b_c @ M @ ( produc4225943634317630866od_c_a @ T5 ) ) ) ) )
              @ ( produc799526633598312797_set_a @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) )
                @ ( product_Pair_c_set_a @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) )
                  @ ( collect_a
                    @ ^ [Q5: a] :
                        ( ( member_a @ Q5 @ ( states_a_b_c @ M ) )
                        & ( ( lS_a_b_c @ M @ Q5 )
                          = ( lS_a_b_c @ M @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) ) ) ) ) ) ) )
          @ ( transitions_a_b_c @ M ) ) ) ) ).

% minimise_props(5)
thf(fact_1200_restrict__to__reachable__states_Opelims,axiom,
    ! [X: fsm_a_b_c,Y: fsm_a_b_c] :
      ( ( ( restri9132545300209641082_a_b_c @ X )
        = Y )
     => ( ( accp_fsm_a_b_c @ restri5254034100565356425_a_b_c @ X )
       => ~ ( ( Y
              = ( filter_states_a_b_c @ X
                @ ^ [Q5: a] : ( member_a @ Q5 @ ( reacha1620305530751930115_a_b_c @ X ) ) ) )
           => ~ ( accp_fsm_a_b_c @ restri5254034100565356425_a_b_c @ X ) ) ) ) ).

% restrict_to_reachable_states.pelims
thf(fact_1201_restrict__to__reachable__states_Opelims,axiom,
    ! [X: fsm_set_a_b_c,Y: fsm_set_a_b_c] :
      ( ( ( restri1885653365341205466_a_b_c @ X )
        = Y )
     => ( ( accp_fsm_set_a_b_c @ restri2847397351157334249_a_b_c @ X )
       => ~ ( ( Y
              = ( filter2961209833878020368_a_b_c @ X
                @ ^ [Q5: set_a] : ( member_set_a @ Q5 @ ( reacha7078015818774689891_a_b_c @ X ) ) ) )
           => ~ ( accp_fsm_set_a_b_c @ restri2847397351157334249_a_b_c @ X ) ) ) ) ).

% restrict_to_reachable_states.pelims
thf(fact_1202_finite__UN,axiom,
    ! [A: set_nat,B: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
        = ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ( finite_finite_nat @ ( B @ X2 ) ) ) ) ) ) ).

% finite_UN
thf(fact_1203_finite__Union,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ! [M7: set_nat] :
            ( ( member_set_nat @ M7 @ A )
           => ( finite_finite_nat @ M7 ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).

% finite_Union
thf(fact_1204_finite__UN__I,axiom,
    ! [A: set_nat,B: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A )
           => ( finite_finite_nat @ ( B @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).

% finite_UN_I
thf(fact_1205_le__cSup__finite,axiom,
    ! [X6: set_nat,X: nat] :
      ( ( finite_finite_nat @ X6 )
     => ( ( member_nat @ X @ X6 )
       => ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ).

% le_cSup_finite
thf(fact_1206_cSup__eq__non__empty,axiom,
    ! [X6: set_nat,A2: nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
           => ( ord_less_eq_nat @ X3 @ A2 ) )
       => ( ! [Y2: nat] :
              ( ! [X5: nat] :
                  ( ( member_nat @ X5 @ X6 )
                 => ( ord_less_eq_nat @ X5 @ Y2 ) )
             => ( ord_less_eq_nat @ A2 @ Y2 ) )
         => ( ( complete_Sup_Sup_nat @ X6 )
            = A2 ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_1207_cSup__least,axiom,
    ! [X6: set_nat,Z2: nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
           => ( ord_less_eq_nat @ X3 @ Z2 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X6 ) @ Z2 ) ) ) ).

% cSup_least
thf(fact_1208_cSup__eq__maximum,axiom,
    ! [Z2: nat,X6: set_nat] :
      ( ( member_nat @ Z2 @ X6 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X6 )
           => ( ord_less_eq_nat @ X3 @ Z2 ) )
       => ( ( complete_Sup_Sup_nat @ X6 )
          = Z2 ) ) ) ).

% cSup_eq_maximum
thf(fact_1209_finite__UnionD,axiom,
    ! [A: set_set_nat] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) )
     => ( finite1152437895449049373et_nat @ A ) ) ).

% finite_UnionD
thf(fact_1210_card__union__of__singletons,axiom,
    ! [SS: set_set_nat] :
      ( ! [S3: set_nat] :
          ( ( member_set_nat @ S3 @ SS )
         => ? [T6: nat] :
              ( S3
              = ( insert_nat @ T6 @ bot_bot_set_nat ) ) )
     => ( ( finite_card_nat @ ( comple7399068483239264473et_nat @ SS ) )
        = ( finite_card_set_nat @ SS ) ) ) ).

% card_union_of_singletons
thf(fact_1211_finite__subset__Union,axiom,
    ! [A: set_nat,B10: set_set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B10 ) )
       => ~ ! [F5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F5 )
             => ( ( ord_le6893508408891458716et_nat @ F5 @ B10 )
               => ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F5 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_1212_language__path__append__transition__observable,axiom,
    ! [P2: list_P6923848007992943548_set_a,X: b,Y: c,M: fsm_set_a_b_c,Q: set_a] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr4209967418700163830od_b_c
            @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
            @ P2 )
          @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
        @ ( lS_set_a_b_c @ M @ Q ) )
     => ( ( path_set_a_b_c @ M @ Q @ P2 )
       => ( ( observable_set_a_b_c @ M )
         => ~ ! [T2: produc373761375140688950_set_a] :
                ( ( path_set_a_b_c @ M @ Q @ ( append4693096456853939665_set_a @ P2 @ ( cons_P163416695733681644_set_a @ T2 @ nil_Pr6810293327492551228_set_a ) ) )
               => ( ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T2 ) )
                    = X )
                 => ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T2 ) ) )
                   != Y ) ) ) ) ) ) ).

% language_path_append_transition_observable
thf(fact_1213_language__path__append__transition__observable,axiom,
    ! [P2: list_P6327159017948738492od_c_a,X: b,Y: c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr776894537227705078od_b_c
            @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
            @ P2 )
          @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
        @ ( lS_a_b_c @ M @ Q ) )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( observable_a_b_c @ M )
         => ~ ! [T2: produc2669932201312201782od_c_a] :
                ( ( path_a_b_c @ M @ Q @ ( append3884164603879732689od_c_a @ P2 @ ( cons_P8659537020926305772od_c_a @ T2 @ nil_Pr1342775757158464060od_c_a ) ) )
               => ( ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) )
                    = X )
                 => ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T2 ) ) )
                   != Y ) ) ) ) ) ) ).

% language_path_append_transition_observable
thf(fact_1214_language__path__append__transition__observable,axiom,
    ! [P2: list_P1412026030649758308_c_nat,X: b,Y: c,M: fsm_nat_b_c,Q: nat] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr6527000103508470350od_b_c
            @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
            @ P2 )
          @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
        @ ( lS_nat_b_c @ M @ Q ) )
     => ( ( path_nat_b_c @ M @ Q @ P2 )
       => ( ( observable_nat_b_c @ M )
         => ~ ! [T2: produc8586872805302537566_c_nat] :
                ( ( path_nat_b_c @ M @ Q @ ( append6901786371091433593_c_nat @ P2 @ ( cons_P2734614606773430804_c_nat @ T2 @ nil_Pr2729297026091755108_c_nat ) ) )
               => ( ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T2 ) )
                    = X )
                 => ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T2 ) ) )
                   != Y ) ) ) ) ) ) ).

% language_path_append_transition_observable
thf(fact_1215_language__path__append__transition__observable,axiom,
    ! [P2: list_P7609586222656207102nteger,X: b,Y: c,M: fsm_Code_integer_b_c,Q: code_integer] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr1075410694537583412od_b_c
            @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
            @ P2 )
          @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
        @ ( lS_Code_integer_b_c @ M @ Q ) )
     => ( ( path_C4304074165394969944er_b_c @ M @ Q @ P2 )
       => ( ( observ7524490533585752726er_b_c @ M )
         => ~ ! [T2: produc558599814252099832nteger] :
                ( ( path_C4304074165394969944er_b_c @ M @ Q @ ( append7583211363492240403nteger @ P2 @ ( cons_P4508506314314016174nteger @ T2 @ nil_Pr5017997283661177342nteger ) ) )
               => ( ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T2 ) )
                    = X )
                 => ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T2 ) ) )
                   != Y ) ) ) ) ) ) ).

% language_path_append_transition_observable
thf(fact_1216_conj__subset__def,axiom,
    ! [A: set_nat,P: nat > $o,Q2: nat > $o] :
      ( ( ord_less_eq_set_nat @ A
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( P @ X2 )
              & ( Q2 @ X2 ) ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q2 ) ) ) ) ).

% conj_subset_def
thf(fact_1217_observable__path__unique,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,P2: list_P6923848007992943548_set_a,P6: list_P6923848007992943548_set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( path_set_a_b_c @ M @ Q @ P2 )
       => ( ( path_set_a_b_c @ M @ Q @ P6 )
         => ( ( ( map_Pr4209967418700163830od_b_c
                @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
                @ P2 )
              = ( map_Pr4209967418700163830od_b_c
                @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
                @ P6 ) )
           => ( P2 = P6 ) ) ) ) ) ).

% observable_path_unique
thf(fact_1218_observable__path__unique,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a,P6: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( path_a_b_c @ M @ Q @ P6 )
         => ( ( ( map_Pr776894537227705078od_b_c
                @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
                @ P2 )
              = ( map_Pr776894537227705078od_b_c
                @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
                @ P6 ) )
           => ( P2 = P6 ) ) ) ) ) ).

% observable_path_unique
thf(fact_1219_observable__path__unique,axiom,
    ! [M: fsm_nat_b_c,Q: nat,P2: list_P1412026030649758308_c_nat,P6: list_P1412026030649758308_c_nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( path_nat_b_c @ M @ Q @ P2 )
       => ( ( path_nat_b_c @ M @ Q @ P6 )
         => ( ( ( map_Pr6527000103508470350od_b_c
                @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
                @ P2 )
              = ( map_Pr6527000103508470350od_b_c
                @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
                @ P6 ) )
           => ( P2 = P6 ) ) ) ) ) ).

% observable_path_unique
thf(fact_1220_observable__path__unique,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer,P2: list_P7609586222656207102nteger,P6: list_P7609586222656207102nteger] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( path_C4304074165394969944er_b_c @ M @ Q @ P2 )
       => ( ( path_C4304074165394969944er_b_c @ M @ Q @ P6 )
         => ( ( ( map_Pr1075410694537583412od_b_c
                @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
                @ P2 )
              = ( map_Pr1075410694537583412od_b_c
                @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
                @ P6 ) )
           => ( P2 = P6 ) ) ) ) ) ).

% observable_path_unique
thf(fact_1221_after__path,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,P2: list_P6923848007992943548_set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( path_set_a_b_c @ M @ Q @ P2 )
       => ( ( after_set_a_b_c @ M @ Q
            @ ( map_Pr4209967418700163830od_b_c
              @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
              @ P2 ) )
          = ( target_set_a_b_c @ Q @ P2 ) ) ) ) ).

% after_path
thf(fact_1222_after__path,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( after_a_b_c @ M @ Q
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
              @ P2 ) )
          = ( target_a_b_c @ Q @ P2 ) ) ) ) ).

% after_path
thf(fact_1223_after__path,axiom,
    ! [M: fsm_nat_b_c,Q: nat,P2: list_P1412026030649758308_c_nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( path_nat_b_c @ M @ Q @ P2 )
       => ( ( after_nat_b_c @ M @ Q
            @ ( map_Pr6527000103508470350od_b_c
              @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
              @ P2 ) )
          = ( target_nat_b_c @ Q @ P2 ) ) ) ) ).

% after_path
thf(fact_1224_after__path,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer,P2: list_P7609586222656207102nteger] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( path_C4304074165394969944er_b_c @ M @ Q @ P2 )
       => ( ( after_5090571588895392199er_b_c @ M @ Q
            @ ( map_Pr1075410694537583412od_b_c
              @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
              @ P2 ) )
          = ( target686291486632357924er_b_c @ Q @ P2 ) ) ) ) ).

% after_path
thf(fact_1225_observable__path__suffix,axiom,
    ! [P2: list_P6923848007992943548_set_a,Io: list_P903359562653991662od_b_c,M: fsm_set_a_b_c,Q: set_a] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr4209967418700163830od_b_c
            @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
            @ P2 )
          @ Io )
        @ ( lS_set_a_b_c @ M @ Q ) )
     => ( ( path_set_a_b_c @ M @ Q @ P2 )
       => ( ( observable_set_a_b_c @ M )
         => ~ ! [P7: list_P6923848007992943548_set_a] :
                ( ( path_set_a_b_c @ M @ ( target_set_a_b_c @ Q @ P2 ) @ P7 )
               => ( ( map_Pr4209967418700163830od_b_c
                    @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
                    @ P7 )
                 != Io ) ) ) ) ) ).

% observable_path_suffix
thf(fact_1226_observable__path__suffix,axiom,
    ! [P2: list_P6327159017948738492od_c_a,Io: list_P903359562653991662od_b_c,M: fsm_a_b_c,Q: a] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr776894537227705078od_b_c
            @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
            @ P2 )
          @ Io )
        @ ( lS_a_b_c @ M @ Q ) )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( observable_a_b_c @ M )
         => ~ ! [P7: list_P6327159017948738492od_c_a] :
                ( ( path_a_b_c @ M @ ( target_a_b_c @ Q @ P2 ) @ P7 )
               => ( ( map_Pr776894537227705078od_b_c
                    @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
                    @ P7 )
                 != Io ) ) ) ) ) ).

% observable_path_suffix
thf(fact_1227_observable__path__suffix,axiom,
    ! [P2: list_P1412026030649758308_c_nat,Io: list_P903359562653991662od_b_c,M: fsm_nat_b_c,Q: nat] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr6527000103508470350od_b_c
            @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
            @ P2 )
          @ Io )
        @ ( lS_nat_b_c @ M @ Q ) )
     => ( ( path_nat_b_c @ M @ Q @ P2 )
       => ( ( observable_nat_b_c @ M )
         => ~ ! [P7: list_P1412026030649758308_c_nat] :
                ( ( path_nat_b_c @ M @ ( target_nat_b_c @ Q @ P2 ) @ P7 )
               => ( ( map_Pr6527000103508470350od_b_c
                    @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
                    @ P7 )
                 != Io ) ) ) ) ) ).

% observable_path_suffix
thf(fact_1228_observable__path__suffix,axiom,
    ! [P2: list_P7609586222656207102nteger,Io: list_P903359562653991662od_b_c,M: fsm_Code_integer_b_c,Q: code_integer] :
      ( ( member6330420149250801815od_b_c
        @ ( append2547753245680614915od_b_c
          @ ( map_Pr1075410694537583412od_b_c
            @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
            @ P2 )
          @ Io )
        @ ( lS_Code_integer_b_c @ M @ Q ) )
     => ( ( path_C4304074165394969944er_b_c @ M @ Q @ P2 )
       => ( ( observ7524490533585752726er_b_c @ M )
         => ~ ! [P7: list_P7609586222656207102nteger] :
                ( ( path_C4304074165394969944er_b_c @ M @ ( target686291486632357924er_b_c @ Q @ P2 ) @ P7 )
               => ( ( map_Pr1075410694537583412od_b_c
                    @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
                    @ P7 )
                 != Io ) ) ) ) ) ).

% observable_path_suffix
thf(fact_1229_observable__after__path,axiom,
    ! [M: fsm_set_a_b_c,Io: list_P903359562653991662od_b_c,Q: set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ M @ Q ) )
       => ~ ! [P5: list_P6923848007992943548_set_a] :
              ( ( path_set_a_b_c @ M @ Q @ P5 )
             => ( ( ( map_Pr4209967418700163830od_b_c
                    @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
                    @ P5 )
                  = Io )
               => ( ( target_set_a_b_c @ Q @ P5 )
                 != ( after_set_a_b_c @ M @ Q @ Io ) ) ) ) ) ) ).

% observable_after_path
thf(fact_1230_observable__after__path,axiom,
    ! [M: fsm_a_b_c,Io: list_P903359562653991662od_b_c,Q: a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ M @ Q ) )
       => ~ ! [P5: list_P6327159017948738492od_c_a] :
              ( ( path_a_b_c @ M @ Q @ P5 )
             => ( ( ( map_Pr776894537227705078od_b_c
                    @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
                    @ P5 )
                  = Io )
               => ( ( target_a_b_c @ Q @ P5 )
                 != ( after_a_b_c @ M @ Q @ Io ) ) ) ) ) ) ).

% observable_after_path
thf(fact_1231_observable__after__path,axiom,
    ! [M: fsm_nat_b_c,Io: list_P903359562653991662od_b_c,Q: nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_nat_b_c @ M @ Q ) )
       => ~ ! [P5: list_P1412026030649758308_c_nat] :
              ( ( path_nat_b_c @ M @ Q @ P5 )
             => ( ( ( map_Pr6527000103508470350od_b_c
                    @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
                    @ P5 )
                  = Io )
               => ( ( target_nat_b_c @ Q @ P5 )
                 != ( after_nat_b_c @ M @ Q @ Io ) ) ) ) ) ) ).

% observable_after_path
thf(fact_1232_observable__after__path,axiom,
    ! [M: fsm_Code_integer_b_c,Io: list_P903359562653991662od_b_c,Q: code_integer] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ Io @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ~ ! [P5: list_P7609586222656207102nteger] :
              ( ( path_C4304074165394969944er_b_c @ M @ Q @ P5 )
             => ( ( ( map_Pr1075410694537583412od_b_c
                    @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
                    @ P5 )
                  = Io )
               => ( ( target686291486632357924er_b_c @ Q @ P5 )
                 != ( after_5090571588895392199er_b_c @ M @ Q @ Io ) ) ) ) ) ) ).

% observable_after_path
thf(fact_1233_observable__path__io__target,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,P2: list_P6923848007992943548_set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( path_set_a_b_c @ M @ Q @ P2 )
       => ( ( io_targets_set_a_b_c @ M
            @ ( map_Pr4209967418700163830od_b_c
              @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
              @ P2 )
            @ Q )
          = ( insert_set_a @ ( target_set_a_b_c @ Q @ P2 ) @ bot_bot_set_set_a ) ) ) ) ).

% observable_path_io_target
thf(fact_1234_observable__path__io__target,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ( io_targets_a_b_c @ M
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
              @ P2 )
            @ Q )
          = ( insert_a @ ( target_a_b_c @ Q @ P2 ) @ bot_bot_set_a ) ) ) ) ).

% observable_path_io_target
thf(fact_1235_observable__path__io__target,axiom,
    ! [M: fsm_nat_b_c,Q: nat,P2: list_P1412026030649758308_c_nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( path_nat_b_c @ M @ Q @ P2 )
       => ( ( io_targets_nat_b_c @ M
            @ ( map_Pr6527000103508470350od_b_c
              @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
              @ P2 )
            @ Q )
          = ( insert_nat @ ( target_nat_b_c @ Q @ P2 ) @ bot_bot_set_nat ) ) ) ) ).

% observable_path_io_target
thf(fact_1236_observable__path__io__target,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer,P2: list_P7609586222656207102nteger] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( path_C4304074165394969944er_b_c @ M @ Q @ P2 )
       => ( ( io_tar3403248182866227591er_b_c @ M
            @ ( map_Pr1075410694537583412od_b_c
              @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
              @ P2 )
            @ Q )
          = ( insert_Code_integer @ ( target686291486632357924er_b_c @ Q @ P2 ) @ bot_bo3990330152332043303nteger ) ) ) ) ).

% observable_path_io_target
thf(fact_1237_observable__after__target,axiom,
    ! [M: fsm_set_a_b_c,Io: list_P903359562653991662od_b_c,Io5: list_P903359562653991662od_b_c,Q: set_a,P2: list_P6923848007992943548_set_a] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ Io5 ) @ ( lS_set_a_b_c @ M @ Q ) )
       => ( ( path_set_a_b_c @ M @ ( after_set_a_b_c @ M @ Q @ Io ) @ P2 )
         => ( ( ( map_Pr4209967418700163830od_b_c
                @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
                @ P2 )
              = Io5 )
           => ( ( target_set_a_b_c @ ( after_set_a_b_c @ M @ Q @ Io ) @ P2 )
              = ( after_set_a_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Io @ Io5 ) ) ) ) ) ) ) ).

% observable_after_target
thf(fact_1238_observable__after__target,axiom,
    ! [M: fsm_a_b_c,Io: list_P903359562653991662od_b_c,Io5: list_P903359562653991662od_b_c,Q: a,P2: list_P6327159017948738492od_c_a] :
      ( ( observable_a_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ Io5 ) @ ( lS_a_b_c @ M @ Q ) )
       => ( ( path_a_b_c @ M @ ( after_a_b_c @ M @ Q @ Io ) @ P2 )
         => ( ( ( map_Pr776894537227705078od_b_c
                @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
                @ P2 )
              = Io5 )
           => ( ( target_a_b_c @ ( after_a_b_c @ M @ Q @ Io ) @ P2 )
              = ( after_a_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Io @ Io5 ) ) ) ) ) ) ) ).

% observable_after_target
thf(fact_1239_observable__after__target,axiom,
    ! [M: fsm_nat_b_c,Io: list_P903359562653991662od_b_c,Io5: list_P903359562653991662od_b_c,Q: nat,P2: list_P1412026030649758308_c_nat] :
      ( ( observable_nat_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ Io5 ) @ ( lS_nat_b_c @ M @ Q ) )
       => ( ( path_nat_b_c @ M @ ( after_nat_b_c @ M @ Q @ Io ) @ P2 )
         => ( ( ( map_Pr6527000103508470350od_b_c
                @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
                @ P2 )
              = Io5 )
           => ( ( target_nat_b_c @ ( after_nat_b_c @ M @ Q @ Io ) @ P2 )
              = ( after_nat_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Io @ Io5 ) ) ) ) ) ) ) ).

% observable_after_target
thf(fact_1240_observable__after__target,axiom,
    ! [M: fsm_Code_integer_b_c,Io: list_P903359562653991662od_b_c,Io5: list_P903359562653991662od_b_c,Q: code_integer,P2: list_P7609586222656207102nteger] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member6330420149250801815od_b_c @ ( append2547753245680614915od_b_c @ Io @ Io5 ) @ ( lS_Code_integer_b_c @ M @ Q ) )
       => ( ( path_C4304074165394969944er_b_c @ M @ ( after_5090571588895392199er_b_c @ M @ Q @ Io ) @ P2 )
         => ( ( ( map_Pr1075410694537583412od_b_c
                @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
                @ P2 )
              = Io5 )
           => ( ( target686291486632357924er_b_c @ ( after_5090571588895392199er_b_c @ M @ Q @ Io ) @ P2 )
              = ( after_5090571588895392199er_b_c @ M @ Q @ ( append2547753245680614915od_b_c @ Io @ Io5 ) ) ) ) ) ) ) ).

% observable_after_target
thf(fact_1241_observable__io__targets__next,axiom,
    ! [M: fsm_set_a_b_c,T: produc373761375140688950_set_a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( member6517745037573369951_set_a @ T @ ( transi8205445899535437940_a_b_c @ M ) )
       => ( ( io_targets_set_a_b_c @ M
            @ ( append2547753245680614915od_b_c
              @ ( map_Pr4209967418700163830od_b_c
                @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
                @ ( cons_P163416695733681644_set_a @ T @ nil_Pr6810293327492551228_set_a ) )
              @ Io )
            @ ( produc7645221154340064658_set_a @ T ) )
          = ( io_targets_set_a_b_c @ M @ Io @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) ) ) ) ) ).

% observable_io_targets_next
thf(fact_1242_observable__io__targets__next,axiom,
    ! [M: fsm_a_b_c,T: produc2669932201312201782od_c_a,Io: list_P903359562653991662od_b_c] :
      ( ( observable_a_b_c @ M )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ M ) )
       => ( ( io_targets_a_b_c @ M
            @ ( append2547753245680614915od_b_c
              @ ( map_Pr776894537227705078od_b_c
                @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
                @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) )
              @ Io )
            @ ( produc4225943634317630866od_c_a @ T ) )
          = ( io_targets_a_b_c @ M @ Io @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) ) ).

% observable_io_targets_next
thf(fact_1243_observable__io__targets__next,axiom,
    ! [M: fsm_nat_b_c,T: produc8586872805302537566_c_nat,Io: list_P903359562653991662od_b_c] :
      ( ( observable_nat_b_c @ M )
     => ( ( member3717341653178413575_c_nat @ T @ ( transitions_nat_b_c @ M ) )
       => ( ( io_targets_nat_b_c @ M
            @ ( append2547753245680614915od_b_c
              @ ( map_Pr6527000103508470350od_b_c
                @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
                @ ( cons_P2734614606773430804_c_nat @ T @ nil_Pr2729297026091755108_c_nat ) )
              @ Io )
            @ ( produc532606554400396034_c_nat @ T ) )
          = ( io_targets_nat_b_c @ M @ Io @ ( product_snd_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T ) ) ) ) ) ) ) ).

% observable_io_targets_next
thf(fact_1244_observable__io__targets__next,axiom,
    ! [M: fsm_Code_integer_b_c,T: produc558599814252099832nteger,Io: list_P903359562653991662od_b_c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( member5112046945434310049nteger @ T @ ( transi6707074847254859401er_b_c @ M ) )
       => ( ( io_tar3403248182866227591er_b_c @ M
            @ ( append2547753245680614915od_b_c
              @ ( map_Pr1075410694537583412od_b_c
                @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
                @ ( cons_P4508506314314016174nteger @ T @ nil_Pr5017997283661177342nteger ) )
              @ Io )
            @ ( produc7104377276419476764nteger @ T ) )
          = ( io_tar3403248182866227591er_b_c @ M @ Io @ ( produc4666406780376491864nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T ) ) ) ) ) ) ) ).

% observable_io_targets_next
thf(fact_1245_observable__path__language__step,axiom,
    ! [M: fsm_set_a_b_c,Q: set_a,P2: list_P6923848007992943548_set_a,X: b,Y: c] :
      ( ( observable_set_a_b_c @ M )
     => ( ( path_set_a_b_c @ M @ Q @ P2 )
       => ( ~ ? [X3: produc373761375140688950_set_a] :
                ( ( member6517745037573369951_set_a @ X3 @ ( transi8205445899535437940_a_b_c @ M ) )
                & ( ( produc7645221154340064658_set_a @ X3 )
                  = ( target_set_a_b_c @ Q @ P2 ) )
                & ( ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ X3 ) )
                  = X )
                & ( ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ X3 ) ) )
                  = Y ) )
         => ~ ( member6330420149250801815od_b_c
              @ ( append2547753245680614915od_b_c
                @ ( map_Pr4209967418700163830od_b_c
                  @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
                  @ P2 )
                @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
              @ ( lS_set_a_b_c @ M @ Q ) ) ) ) ) ).

% observable_path_language_step
thf(fact_1246_observable__path__language__step,axiom,
    ! [M: fsm_a_b_c,Q: a,P2: list_P6327159017948738492od_c_a,X: b,Y: c] :
      ( ( observable_a_b_c @ M )
     => ( ( path_a_b_c @ M @ Q @ P2 )
       => ( ~ ? [X3: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ X3 @ ( transitions_a_b_c @ M ) )
                & ( ( produc4225943634317630866od_c_a @ X3 )
                  = ( target_a_b_c @ Q @ P2 ) )
                & ( ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) )
                  = X )
                & ( ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ X3 ) ) )
                  = Y ) )
         => ~ ( member6330420149250801815od_b_c
              @ ( append2547753245680614915od_b_c
                @ ( map_Pr776894537227705078od_b_c
                  @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
                  @ P2 )
                @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
              @ ( lS_a_b_c @ M @ Q ) ) ) ) ) ).

% observable_path_language_step
thf(fact_1247_observable__path__language__step,axiom,
    ! [M: fsm_nat_b_c,Q: nat,P2: list_P1412026030649758308_c_nat,X: b,Y: c] :
      ( ( observable_nat_b_c @ M )
     => ( ( path_nat_b_c @ M @ Q @ P2 )
       => ( ~ ? [X3: produc8586872805302537566_c_nat] :
                ( ( member3717341653178413575_c_nat @ X3 @ ( transitions_nat_b_c @ M ) )
                & ( ( produc532606554400396034_c_nat @ X3 )
                  = ( target_nat_b_c @ Q @ P2 ) )
                & ( ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) )
                  = X )
                & ( ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ X3 ) ) )
                  = Y ) )
         => ~ ( member6330420149250801815od_b_c
              @ ( append2547753245680614915od_b_c
                @ ( map_Pr6527000103508470350od_b_c
                  @ ^ [T5: produc8586872805302537566_c_nat] : ( product_Pair_b_c @ ( produc1732631467370978619_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) @ ( product_fst_c_nat @ ( produc2285613626377700217_c_nat @ ( produc1114525031416538948_c_nat @ T5 ) ) ) )
                  @ P2 )
                @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
              @ ( lS_nat_b_c @ M @ Q ) ) ) ) ) ).

% observable_path_language_step
thf(fact_1248_observable__path__language__step,axiom,
    ! [M: fsm_Code_integer_b_c,Q: code_integer,P2: list_P7609586222656207102nteger,X: b,Y: c] :
      ( ( observ7524490533585752726er_b_c @ M )
     => ( ( path_C4304074165394969944er_b_c @ M @ Q @ P2 )
       => ( ~ ? [X3: produc558599814252099832nteger] :
                ( ( member5112046945434310049nteger @ X3 @ ( transi6707074847254859401er_b_c @ M ) )
                & ( ( produc7104377276419476764nteger @ X3 )
                  = ( target686291486632357924er_b_c @ Q @ P2 ) )
                & ( ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ X3 ) )
                  = X )
                & ( ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ X3 ) ) )
                  = Y ) )
         => ~ ( member6330420149250801815od_b_c
              @ ( append2547753245680614915od_b_c
                @ ( map_Pr1075410694537583412od_b_c
                  @ ^ [T5: produc558599814252099832nteger] : ( product_Pair_b_c @ ( produc4095665871784379208nteger @ ( produc1412116139327852894nteger @ T5 ) ) @ ( produc5481060579096546074nteger @ ( produc3851724270299738502nteger @ ( produc1412116139327852894nteger @ T5 ) ) ) )
                  @ P2 )
                @ ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X @ Y ) @ nil_Product_prod_b_c ) )
              @ ( lS_Code_integer_b_c @ M @ Q ) ) ) ) ) ).

% observable_path_language_step
thf(fact_1249_language__state__prepend__transition,axiom,
    ! [Io: list_P903359562653991662od_b_c,A: fsm_a_b_c,T: produc2669932201312201782od_c_a] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_a_b_c @ ( from_FSM_a_b_c @ A @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) @ ( initial_a_b_c @ ( from_FSM_a_b_c @ A @ ( product_snd_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T ) ) ) ) ) ) )
     => ( ( member5502040268208861279od_c_a @ T @ ( transitions_a_b_c @ A ) )
       => ( member6330420149250801815od_b_c
          @ ( append2547753245680614915od_b_c
            @ ( map_Pr776894537227705078od_b_c
              @ ^ [T5: produc2669932201312201782od_c_a] : ( product_Pair_b_c @ ( produc301949074746585577od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) @ ( product_fst_c_a @ ( produc5221446324160585515od_c_a @ ( produc8088130389382527188od_c_a @ T5 ) ) ) )
              @ ( cons_P8659537020926305772od_c_a @ T @ nil_Pr1342775757158464060od_c_a ) )
            @ Io )
          @ ( lS_a_b_c @ A @ ( produc4225943634317630866od_c_a @ T ) ) ) ) ) ).

% language_state_prepend_transition
thf(fact_1250_language__state__prepend__transition,axiom,
    ! [Io: list_P903359562653991662od_b_c,A: fsm_set_a_b_c,T: produc373761375140688950_set_a] :
      ( ( member6330420149250801815od_b_c @ Io @ ( lS_set_a_b_c @ ( from_FSM_set_a_b_c @ A @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) ) @ ( initial_set_a_b_c @ ( from_FSM_set_a_b_c @ A @ ( product_snd_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T ) ) ) ) ) ) )
     => ( ( member6517745037573369951_set_a @ T @ ( transi8205445899535437940_a_b_c @ A ) )
       => ( member6330420149250801815od_b_c
          @ ( append2547753245680614915od_b_c
            @ ( map_Pr4209967418700163830od_b_c
              @ ^ [T5: produc373761375140688950_set_a] : ( product_Pair_b_c @ ( produc4260161599997422409_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) @ ( product_fst_c_set_a @ ( produc2627626931654905995_set_a @ ( produc6575470948336425172_set_a @ T5 ) ) ) )
              @ ( cons_P163416695733681644_set_a @ T @ nil_Pr6810293327492551228_set_a ) )
            @ Io )
          @ ( lS_set_a_b_c @ A @ ( produc7645221154340064658_set_a @ T ) ) ) ) ) ).

% language_state_prepend_transition
thf(fact_1251_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q2: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X2: nat] :
              ? [Y4: nat] :
                ( ( P @ Y4 )
                & ( Q2 @ X2 @ Y4 ) ) ) )
        = ( ! [Y4: nat] :
              ( ( P @ Y4 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X2: nat] : ( Q2 @ X2 @ Y4 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1252_finite__image__set2,axiom,
    ! [P: nat > $o,Q2: nat > $o,F: nat > nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q2 ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X2: nat,Y4: nat] :
                ( ( Uu
                  = ( F @ X2 @ Y4 ) )
                & ( P @ X2 )
                & ( Q2 @ Y4 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1253_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X2: nat] :
              ( ( Uu
                = ( F @ X2 ) )
              & ( P @ X2 ) ) ) ) ) ).

% finite_image_set
thf(fact_1254_setcompr__eq__image,axiom,
    ! [F: nat > nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X2: nat] :
            ( ( Uu
              = ( F @ X2 ) )
            & ( P @ X2 ) ) )
      = ( image_nat_nat @ F @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1255_defined__inputs__set,axiom,
    ( define885092113723484480_a_b_c
    = ( ^ [M2: fsm_set_a_b_c,Q5: set_a] :
          ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( inputs_set_a_b_c @ M2 ) )
              & ( ( h_set_a_b_c @ M2 @ ( product_Pair_set_a_b @ Q5 @ X2 ) )
               != bot_bo4411629339732149841_set_a ) ) ) ) ) ).

% defined_inputs_set
thf(fact_1256_defined__inputs__set,axiom,
    ( defined_inputs_a_b_c
    = ( ^ [M2: fsm_a_b_c,Q5: a] :
          ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( inputs_a_b_c @ M2 ) )
              & ( ( h_a_b_c @ M2 @ ( product_Pair_a_b @ Q5 @ X2 ) )
               != bot_bo2086078286244720881od_c_a ) ) ) ) ) ).

% defined_inputs_set
thf(fact_1257_finite__Diff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).

% finite_Diff
thf(fact_1258_finite__Diff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_Diff2
thf(fact_1259_finite__Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_1260_card__le__sym__Diff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1261_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A3: set_nat,B2: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A3 )
              & ~ ( member_nat @ X2 @ B2 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1262_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S: set_nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1263_card__less__sym__Diff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1264_infinite__remove,axiom,
    ! [S: set_nat,A2: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_1265_infinite__coinduct,axiom,
    ! [X6: set_nat > $o,A: set_nat] :
      ( ( X6 @ A )
     => ( ! [A7: set_nat] :
            ( ( X6 @ A7 )
           => ? [X5: nat] :
                ( ( member_nat @ X5 @ A7 )
                & ( ( X6 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A ) ) ) ).

% infinite_coinduct
thf(fact_1266_finite__empty__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ A )
       => ( ! [A5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_1267_remove__induct,axiom,
    ! [P: set_nat > $o,B: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B )
         => ( P @ B ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_1268_finite__remove__induct,axiom,
    ! [B: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_1269_finite__induct__select,axiom,
    ! [S: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T4: set_nat] :
              ( ( ord_less_set_nat @ T4 @ S )
             => ( ( P @ T4 )
               => ? [X5: nat] :
                    ( ( member_nat @ X5 @ ( minus_minus_set_nat @ S @ T4 ) )
                    & ( P @ ( insert_nat @ X5 @ T4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_1270_card__Diff1__le,axiom,
    ! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ).

% card_Diff1_le
thf(fact_1271_infinite__imp__bij__betw,axiom,
    ! [A: set_nat,A2: nat] :
      ( ~ ( finite_finite_nat @ A )
     => ? [H2: nat > nat] : ( bij_betw_nat_nat @ H2 @ A @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% infinite_imp_bij_betw
thf(fact_1272_card__Diff1__less,axiom,
    ! [A: set_nat,X: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).

% card_Diff1_less
thf(fact_1273_card__Diff2__less,axiom,
    ! [A: set_nat,X: nat,Y: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X @ A )
       => ( ( member_nat @ Y @ A )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ) ).

% card_Diff2_less
thf(fact_1274_card__Diff1__less__iff,axiom,
    ! [A: set_nat,X: nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
      = ( ( finite_finite_nat @ A )
        & ( member_nat @ X @ A ) ) ) ).

% card_Diff1_less_iff

% Conjectures (1)
thf(conj_0,conjecture,
    observable_set_a_b_c @ ( minimise_a_b_c @ m ) ).

%------------------------------------------------------------------------------